Knockout driven reactions in complex molecules and their clusters

In figure 4 we show the setup used for studies of PAH⁺/${{\rm{C}}}_{60}^{+}$ + He/Ne/Ar/Xe collisions at the DESIREE facility in Stockholm [40, 42, 43]. Briefly, the molecular ions are produced in an electrospray ion source using the sample preparation method of Maziarz et al [44]. The ions are then collected by a radio-frequency ion funnel, mass selected by a quadrupole mass filter, accelerated, and guided through a 4 cm long collision cell containing the target gas (He/Ne/Ar/Xe). After the collision cell, the intact and fragment ions are analyzed by means of a cylindrical lens and two pairs of electrostatic deflector plates, and recorded by a position-sensitive microchannel plate (MCP) detector. This setup allows for measurements of fragment mass spectra and absolute total destruction cross sections through attenuations of the intact ion beams as functions of the target gas pressure in the cell [40, 42].

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The center-of-mass collision energies in these experiments are typically in the sub keV range, which is ideal for studies of nuclear stopping (knockout) driven reactions (see figure 2). In Aarhus, a similar setup was used in pioneering experiments demonstrating knockout processes in fullerenes [14] and more recently in collision induced dissociation studies of other complex molecular systems such as biomolecules [45].

The first studies of keV ion impact on isolated PAH molecules were carried out using the setup shown in figure 5 [46]. There, H⁺ and He²⁺ ions were produced in separate experiments from the electron cyclotron resonance (ECR) ion source located at the ZernikeLEIF facility (KVI Groningen). The ion beam was in the 4–40 keV range, and chopped with a typical repetition rate of 10 kHz and a pulse length of about 10 ns. The so formed ion beam pulse was guided to the collision chamber to interact with a neutral anthracene target, which was evaporated from a resistively heated oven at about 360 K. This temperature was kept as high as possible to yield sufficient target densities without dissociating the PAH molecules. The intact and fragment anthracene ions produced in the collisions were extracted through a diaphragm and a lens system and analyzed by means of a reflectron time-of-flight (TOF) spectrometer equipped with an MCP detector [46].

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Similar techniques have been used by other groups for studies of PAHs [47–53]. One particularly powerful technique is called Collision-Induced Dissociation under Energy Control (CIDEC) and was developed at the University of Lyon 1 [48]. This technique is based on anion formation by double electron capture in inelastic collisions between a singly charged atomic projectile and a neutral molecular (PAH) target, and relies on that there is only a single bound state in the anion. By measuring the kinetic energy loss of the anion and subtracting the energy defect of the collision, the internal energy of the molecular ion may be determined. This, together with multi-coincidence detection of ions and electrons allows for characterizing fragmentation processes in unprecedented detail [48].

In figure 6 we show the mass spectrometer setup at the ARIBE facility in Caen, which has been used for studies of PAHs, fullerenes and their clusters [47, 52, 55–58]. The latter are formed in a cluster aggregation source in which isolated molecules are evaporated from an oven and cooled by a He buffer gas at liquid nitrogen temperatures (see figure 6). This allows formation of weakly bound neutral clusters in a broad log-normal size distribution [54]. The neutral cluster targets interact with a pulsed ion beam in a similar fashion as in the Groningen experiments (see figure 5), but the collision product ions are here analyzed by means of a linear Wiley–McLaren type spectrometer terminated by a metal conversion plate. Secondary electrons from the conversion plate are guided by a weak magnetic field to a microchannel plate detector. This detection scheme gives close to unit efficiency which is crucial for coincidence measurements of all charged particles formed in a single collision event [59].

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In 1994, Walch et al reported the first experimental study of collisions between highly charged ions and C₆₀ fullerenes [60]. This work has been followed by a vast number of studies of keV ion impact on isolated or weakly bound clusters of fullerenes (see e.g. [31, 32] and references therein), and more recently on PAHs. It is well-established through these studies that electrons are captured at large distances such that multiply charged intact fullerene/PAH ions may be formed sufficiently cold to survive on the experimental microsecond timescales. The cross sections for the electron transfer processes may be rationalized by means of the classical over-the-barrier model. In this model an electron is allowed to be transferred from the target molecule to the (multiply) charged ion when the Stark-shifted binding energy of the electron equals the maximum of the potential energy barrier experienced by the electron as it moves between the target and the projectile. This simple approach has been successfully used for atomic ions interacting with atoms [61–63], metal and insulator surfaces [64, 65], isolated fullerenes or PAHs and clusters of fullerenes or PAHs [66–68], as well as for fullerene-fullerene [69] and fullerene-surface collisions [70, 71]. In the latter models, the individual fullerene and PAH molecules are described as metal spheres [69] and infinitely thin metal disks [68], respectively. This gives a good representation of the heights and the positions of the barriers as concluded from comparisons with Density Functional Theory (DFT) calculations [68, 72]. For PAHs, it was demonstrated that it is important to take orientation dependent polarizabilities into account, in particular for large systems [68].

To gain further insights into distant collisions dynamics, quantum chemical approaches are required. A full quantum chemical treatment of the collision is computationally too demanding for current computer technology, but quasi-molecular approaches have been used for He²⁺ + C₆₀ [73] and C⁶⁺ + C₆₀ collisions [74]. There, the electronic structure of C₆₀ was described using an extension of the spherical jellium model by Puska and Nieminen [75]. Non-spherically symmetric systems such as PAHs are considerably more challenging, and have to our knowledge not been treated in this fashion. This would lead to a more detailed understanding of the collision dynamics and is important for testing the validity of simple models for easy calculations of, e.g., charge transfer cross sections.

At small impact parameters the projectile atom or ion can interact directly with the electron cloud and nuclei of the target molecule. The knockout process itself is driven by nuclear scattering processes, however at high collision energies electronic stopping is often responsible for the main part of the energy transferred (see figure 2). It is thus important to bear both types of energy transfer in mind when modeling collisions. At high energies, the collisions typically take place on timescales of fs or less, and models for electronic stopping therefore usually assume a static molecular structure during the entire collision process.

Schlathölter et al presented a model for calculating the energy transferred through electronic stopping in collisions between He⁺ and C₆₀ with velocities ranging from 0.1 to 1 a.u. (center-of-mass energies in the range of 1 to 100 keV) [76]. Treating the large number of C₆₀ valence electrons as an electron gas, they noted that the inelastic energy loss through electronic scattering scaled linearly with the velocity v [77]. This allowed them to describe the stopping power as

$$\begin{eqnarray}&&S=\displaystyle \frac{{\rm{d}}{E}}{{\rm{d}}{R}}=\gamma ({r}_{s})v.\end{eqnarray} \tag{ 1 }$$

Here γ is a friction coefficient that depends on the density parameter ${r}_{s}={\left(\tfrac{4}{3}\pi {n}_{0}\right)}^{-1/3}$, which at any point is a function of the valence electron density n₀ [76]. Values for the friction coefficient as a function of electron density in an electron gas, $\gamma ({r}_{s})$, have been calculated by Puska and Nieminen for a range of projectile ions and energies [78]. Using these values together with a jellium shell model for the valence electron density of C₆₀ [75], Schlathölter et al calculated the total energy transferred in a given collision geometry by integrating the stopping power S along a straight line trajectory through the static electron cloud of C₆₀ [76].

This approach was expanded upon by Postma et al in their study of collisions between protons or alpha particles and anthracene (C₁₄H₁₀) at keV energies [46]. Following the scheme outlined by Schlathölter et al [76] for calculating the electronic stopping, Postma et al used Density Functional Theory to calculate the valence electron density of anthracene [46]. This approach of using quantum chemical methods to obtain the electronic density distribution may be applied on a wide range of molecular systems. Postma et al subsequently utilized a Monte Carlo approach to calculate the electronic stopping energy distribution for randomly selected collision trajectories between the ions and the anthracene target at different orientations [46]. The top panel of figure 7 shows the electronic stopping energy distributions for $5\times {10}^{6}$ randomly distributed straight-line trajectories of He ($v=0.2$ a.u., ${E}_{\mathrm{CM}}=3.9$ keV) colliding with anthracene face-on and edge-on (see insets in figure 7) [46]. The energy distribution from randomly oriented molecules is shown in the lower panel of figure 7.

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A similar method was used to study the interactions between 50–180 keV He⁺ or He²⁺ ions and naphthalene (C₁₀H₈) by Mishra et al [53]. Using DFT to calculate the valence electron density of the target molecules, they then determined the stopping power using the Local Density Approximation (LDA) model developed by Lindhard et al for swift low charge state ions interacting with an electron gas [79, 80].

At low center-of-mass collision energies (E_CM less than about 1 keV for H and He projectiles, see figure 2) nuclear stopping is the dominant mechanism for energy transfer. Nuclear scattering is the result of the projectile and target nuclei repelling each other through Coulomb interactions. This short range interaction can be described as binary collisions between a projectile ion and individual atoms in the target as

$$\begin{eqnarray}&&V(r)=\displaystyle \sum _{i}\displaystyle \frac{{Z}_{p}{Z}_{i}}{{r}_{{pi}}}f({x}_{{pi}}),\end{eqnarray} \tag{ 2 }$$

where Z_p and Z_i are the atomic numbers of the projectile and the i individual target atoms, and $f({x}_{{pi}})$ is a screening function for describing the combined shielding effect of the electrons surrounding the projectile and target nuclei.

The screened Bohr potential [81] and the Ziegler–Biersack-Littmark (ZBL) potential [7] are two potentials where such a screening function has been parameterized for a wide range of colliding atom/ion pairs. In both cases the screening function is a relatively simple function of the internuclear distance r_pi. In the screened Bohr potential the screening function is described as a power-law that depends on ${r}_{{pi}},{Z}_{p}$, and Z_i [81]. The ZBL potential uses a screening function of the form

$$\begin{eqnarray}{f}_{{\rm{ZBL}}}({x}_{{pi}}) & = & 0.1818{{\rm{e}}}^{-3.2{x}_{{pi}}}+0.5099{{\rm{e}}}^{-0.9423{x}_{{pi}}}\\ & & +0.2802{{\rm{e}}}^{-0.4029{x}_{{pi}}}+0.02817{{\rm{e}}}^{-0.2016{x}_{{pi}}}\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}&&{x}_{{pi}}=\displaystyle \frac{{r}_{{pi}}}{{a}_{{\rm{ZBL}}}}=\displaystyle \frac{{r}_{{pi}}({Z}_{p}^{0.23}+{Z}_{i}^{0.23})}{0.8853{a}_{0}},\end{eqnarray} \tag{ 4 }$$

a_ZBL is the so-called screening length, and a₀ is the Bohr radius [7].

Chen et al [82] used a similar model as Postma et al [46] for electronic stopping together with potentials for the nuclear stopping (ZBL and screened Bohr) to study ion-PAH collisions. In this way they calculated the relative contributions from both of these stopping processes over a wide energy range. As a result they determined a scaling law for estimating the cross section of single C knockout from PAH molecules of different sizes [82]. Figure 8 shows the contributions of energy transfer through electronic and nuclear stopping in collisions between He and coronene (C₂₄H₁₂) along straight trajectories perpendicularly through the molecular plane, at center-of-mass collision energies of 11 keV (top row) and 110 eV (bottom row) [82]. At 11 keV we see that energy transfer through electronic stopping (left column) is completely dominant, with the nuclear stopping (middle panel) only significantly contributing to the total energy for trajectories where the impact parameter $b\approx 0$ for any given atom in the target molecule [82]. As can be expected, the largest contribution from electronic stopping is from regions with the highest electron density, i.e. in the bonds. On the other hand, nuclear stopping is responsible for most of the energy transfer at 110 eV collision energies, with the contribution from electronic stopping there only reaching a few eV at most [82].

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In the stopping calculations discussed in the previous section, the nuclear and electronic stopping energies are both calculated for straight projectile trajectories through a target molecule with a static structure. For calculations of the electronic stopping, where the electronic density of the target is required, performing fully dynamic quantum chemical calculations of collisions involving larger molecular systems comes with a high computational cost. Nonetheless, there are models that combine classical molecular dynamics (MD) with nonadiabatic quantum chemical calculations to model collisions over a wide range of energies [83]. Nuclear stopping models on the other hand are well suited for implementation in purely classical MD simulations when using functions like the ZBL [7] or screened Bohr [81] potentials to describe these interactions. Classical MD simulations are for this reason particularly useful when modeling low center-of-mass energy collisions, in the energy range where energy transfer through nuclear stopping dominates over electronic stopping.

A number of classical force fields (also known as potentials) have been used to describe intramolecular bonds in simulations of impulse driven fragmentation and bond formation. A class of force fields that are frequently used for these types of simulations are the reactive many-body force fields. Examples of these are the reactive potential by Tersoff [84, 85], the two generations of the REactive Bond Order (REBO) potential by Brenner et al [86, 87], and the Adaptive Intermolecular REBO (AIREBO) potential by Stuart et al [88], which is an extension of the REBO force field that includes a description of dispersion forces. What separates these force fields from more simplistic two-body potentials, such as harmonic or Morse potentials, is in how they dynamically adjusts the bond strengths as functions of the (changing) positions of all atoms in the system during a simulation. All of these potentials are developed and parameterized for systems containing carbon and allow the C atoms to form either sp-, sp²-, or sp³-type bonds in a realistic manner depending on the number and type of the neighboring atoms. This means that bond cleavage and the formation of new bonds can be modeled dynamically using classical MD tools.

While the reactive force fields used in classical MD simulations allow for dynamic bond breakage and formation, they do not include any charge effects. The charge distribution of a molecule or molecular ion depends on how the electron density distribution changes as atoms move and bonds are rearranged. In classical MD, partial charges are generally assigned statically to atoms or massless dummy atoms when they are considered important. In the case of fullerene or PAH molecules, which are large systems with many delocalized electrons, the removal of a single electron has little effect on the stability and reactivity of the molecule [82, 89]. The classical simulations discussed in this review, which do not included charges, are thus used to study both neutral and singly charged systems.

Besides being used to model collisions and prompt knockout processes, MD simulations are also applied in studies of secondary processes such as statistical fragmentation [41, 76] and bond-forming reactions inside molecular clusters [55, 57, 58, 89].

Beyond the use of classical force fields to describe interactions and molecular structures in MD simulations, quantum chemical calculations play a crucial role in our understanding of impulse driven reactions. The computational cost of high level quantum chemical methods, e.g. DFT, and the large PAH and fullerene systems limit the use of such calculations mainly to molecular structure and quantum-state energy calculations. Nonetheless, the ability to accurately determine characteristics of molecules and fragments is important.

Parameterized and semi-classical quantum chemical methods can in some cases be a useful alternative to full DFT methods, offering many of the advantages of higher level theories, but at a much reduced computational cost. While this reduction does generally lead to lower accuracies, they still provide very useful approximate results for many systems. One such method that has been used to study PAHs and fullerenes is the self consistent charge density functional tight binding (SCC-DFTB) method [90, 91]. SCC-DFTB is a parameterized simplification of DFT and has a much lower computational cost [90, 91]. This means that SCC-DFTB can be used to perform MD simulations that are only slightly more expensive than the classical ones, while they also account for additional effects such as charge dynamics.

Specific examples of the use of quantum chemical structure calculations are those determining the stability [40] and reactivity [57, 82] of knockout products, identifying reaction barriers and dissociation energies for molecules and molecular clusters [41, 52, 92, 93]. The results may be used for benchmarking or as input parameters for simpler theoretical models [89], such as positions of nuclei and electronic densities for use in stopping models (see section 3.2) [46, 53, 82].

Statistical fragmentation of fullerenes has been extensively studied in the laboratory following interactions with ions/atoms. A comprehensive review of that field is beyond the scope of the present work and for such aspects we refer the reader to [31, 32]. It is well established through these studies that internally hot fullerenes fragment by loss of C₂-units, sometimes in long evaporation sequences, depending on the energy deposited in the collisions. This is illustrated in figure 9, which shows a mass spectrum from 200 keV ${{\rm{C}}}_{60}^{+}$ + H₂ (${E}_{\mathrm{CM}}=550$ eV) collisions [94]. To the left of the intact ion peak ($M/q=60$) there is a distribution of peaks separated by two carbon masses. These are due to statistical fragmentation processes in which the molecules are significantly heated and where fragmentation predominantly proceeds through the lowest energy dissociation pathways (C₂-emission),

$$\begin{eqnarray*}&&{{\rm{C}}}_{60}^{+}\to {{\rm{C}}}_{60-2n}^{+}+n{{\rm{C}}}_{2}\quad (n=1,2,...).\end{eqnarray*}$$

A corresponding fragment mass distribution with the same pattern for doubly charged fullerenes is seen at lower mass-to-charge ratios in figure 9.

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PAHs have been much less studied than fullerenes by ion/atom impact, in particular in the keV collision energy regime [46–51, 53]. Postma et al reported the first keV ion-PAH collision study in which H⁺/He²⁺ ions collided with anthracene (C₁₄H₁₀). In figure 10 we show their mass spectrum for 30 keV He²⁺ + C₁₄H₁₀ collisions [46]. There are strong peaks corresponding to singly ($m/q=178$) and doubly charged ($m/q=89$) intact ions. These stem from distant electron transfer collisions in which small amounts of energy are deposited (see section 3.1). To the left of the C₁₄${{\rm{H}}}_{10}^{+}$ and C₁₄${{\rm{H}}}_{10}^{2+}$ peaks there are less intense peaks corresponding to H- and C₂H₂-losses (inset in figure 10):

$$\begin{eqnarray*}{{\rm{C}}}_{14}{{\rm{H}}}_{10}^{+/2+} & \to & {{\rm{C}}}_{14}{{\rm{H}}}_{10-n}^{+/2+}+n{\rm{H}}\quad (n=1,2,...)\\ {{\rm{C}}}_{14}{{\rm{H}}}_{10}^{+/2+} & \to & {{\rm{C}}}_{12}{{\rm{H}}}_{8-n}^{+/2+}+{{\rm{C}}}_{2}{{\rm{H}}}_{2}+n{\rm{H}}\quad (n=0,1,...).\end{eqnarray*}$$

These are the lowest energy dissociation pathways for neutral and moderately charged PAHs (see e.g. [15, 16]). The rich distributions of small hydrocarbon ions may be attributed to multi-fragmentation processes following frontal collisions in which large amounts of energy are transferred mainly in electronic stopping processes (see figure 2). Using a simple collision model described in section 3.2, Postma et al [46] found that the electronic stopping is almost equal for 30 keV He²⁺ and 10 keV H⁺ impact, still markedly different fragmentation patterns were observed in these two cases. The differences were attributed to double electron capture by the He²⁺ projectiles, a channel which is strongly suppressed for protons [46].

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Ławicki et al [47] investigated the effect of the projectile charge state in a wider range using collisions between He²⁺/O³⁺/Xe²⁰⁺ and pyrene or coronene (C₁₆H₁₀/C₂₄H₁₂). Singly up to quadruply charged intact molecular ions were clearly observed with Xe²⁰⁺ projectiles, demonstrating that it is possible to produce multiply charged PAHs close to their ultimate Coulomb stability limits [15] in distant electron transfer collisions. The experimental ionization yields compare favorably with those from the classical over-the-barrier model for infinitely thin circular metal disks [68].

Multiply charged PAHs open up new statistical fragmentation pathways where more than one charged fragment is emitted in Coulomb-driven processes. Reitsma et al [51] measured coincidences between charged fragments emitted from doubly charged naphthalene (C₁₀${{\rm{H}}}_{8}^{2+}$) and found that the strongest fragmentation channel corresponds to C₁₀${{\rm{H}}}_{8}^{2+}\to$ C₇${{\rm{H}}}_{5}^{+}$ + C₃${{\rm{H}}}_{3}^{+}$, i.e. fragments with an odd number of carbons. The results were successfully interpreted in view of molecular structure calculations of dissociation energies and reaction barriers [51]. The kinetic energy releases in these processes are in the 2–3 eV range, which could be important for overcoming low barriers in molecular growth processes that may occur in, e.g., the interstellar medium [51].

The group in Lyon used the CIDEC technique (see section 2.2) to measure the internal energies associated with different fragmentation pathways for doubly charged anthracene [48, 49]. The results from one of these studies are shown in figure 11. The 2D-images in panel (a) display coincidence measurements of the ion mass-to-charge ratio (vertical axis) and the voltage applied to analyze the anions formed in F⁺ + C₁₄${{\rm{H}}}_{10}\,\to$ F⁻ + C₁₄${{\rm{H}}}_{10}^{2+}$ collisions (horizontal axis). The analyzer voltage is used to determine the kinetic energy loss of the F⁻ projectile (${\rm{\Delta }}E$), and then the excitation energy (${E}_{{\rm{exc}}}$) of the intact or fragmented C₁₄${{\rm{H}}}_{10}^{2+}$ ion through the relation

$$\begin{eqnarray}&&{E}_{{\rm{exc}}}={\rm{\Delta }}E+{E}_{{\rm{therm}}}-{E}_{{\rm{def}}},\end{eqnarray} \tag{ 5 }$$

where ${{\rm{E}}}_{{\rm{therm}}}$ is the internal energy of the neutral anthracene and

$$\begin{eqnarray}&&{E}_{{\rm{def}}}={I}_{1}({{\rm{C}}}_{14}{{\rm{H}}}_{10})+{I}_{2}({{\rm{C}}}_{14}{{\rm{H}}}_{10})-{I}_{1}({\rm{F}})-{\rm{EA}}({\rm{F}})\end{eqnarray} \tag{ 6 }$$

is the energy defect. In [48] Martin et al used ${{\rm{E}}}_{{\rm{def}}}=-1.6\,{\rm{eV}}$ from literature values of the first and second ionization energies for anthracene (I₁(C₁₄H₁₀) and I₂(C₁₄H₁₀)), and the first ionization energy (I₁(F)) and electron affinity (EA(F)) for fluorine. The panels to the left in figure 11 show the projections for four different 2D-spots on the excitation energy axis (see the bottom scale). From top to bottom (b–e) the distributions have maxima at 8.5 eV (intact ion), 13.8 eV (loss of two hydrogens), 10.4 eV (C₂H₂-loss), and 14.3 eV (loss of two C₂H₂). Thus about 10 eV is required to induce fragmentation on the experimental microsecond timescales. This is significantly higher than the lowest dissociation energy pathway (4.3 eV for C₂H₂-loss [49]), which reflects the large heat capacity of PAHs. They will thus eventually fragment unless there are sufficiently fast competing (radiative) cooling processes, as observed for singly charged anthracene in the small electrostatic ion storage ring, MINIring, in Lyon [96].

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One interesting aspect in this context is whether molecular hydrogen may be formed from internally heated native PAHs [16, 48, 93, 97]. This has been the subject of debate as this channel is impossible to distinguish from sequential emission of two hydrogen atoms (H + H) by means of mass spectrometry alone. A recent combined experimental and theoretical study suggests that keV ions may effectively induce H₂-formation from PAHs, but that the same process is not possible through absorption of single photons with energies below the Lyman limit [93]. However, this situation needs to be unambiguously clarified through additional experiments.

To the best of our knowledge, all experimental studies of keV ion impact on weakly bound PAH and fullerene clusters have so far been carried out at the ARIBE facility in Caen (see section 2.2). A recent review of this work is given in [98] and, there, studies of biomolecules, fullerenes, and PAHs are also included. We will now highlight some of the studies on PAHs and fullerenes that were performed at this facility [92, 95, 99, 100, 100].

In 2003, Manil et al [99] reported on the first measurements of keV ions (in this case Xe²⁵⁺) interacting with clusters of fullerenes. In contrast to previous results on the ionization of weakly bound atomic rare gas clusters [101], they found that the charge is rapidly redistributed among all individual fullerenes in the cluster. It was later shown that this charge communication is ultrafast and takes place on subfemtosecond timescales [59, 100, 102]. Further, Manil et al [99] showed that the excitation energy is rapidly shared among the cluster constituents such that the individual molecules become protected from damage in a surrounding environment.

Rapid charge and energy distribution have also been observed for clusters of e.g. biomolecules [103–106] and PAHs [92, 95]. In figure 12, we show an example for 11.25 keV He⁺ and 360 keV Xe²⁰⁺ ions colliding with anthracene monomers and clusters [95]. In the rightmost panels of figure 12, we show the mass region for intact monomers and clusters, which display decreasing intensity distributions as functions of cluster size for both projectiles. These distributions are rather similar but stem from markedly different processes. For He⁺, the clusters are mainly heated through electronic stopping of ions interacting closely with a few of the molecular building blocks. The excess energy is rapidly shared among the cluster constituents and the cluster cool by the evaporation of neutral anthracene monomers. This results in singly charged intact PAH monomers which are significantly colder than for collisions with a single anthracene molecule isolated in vacuum. As a consequence, the monomer fragmentation yield reduces by a factor ten for collisions with clusters compared to monomers, as shown in the middle and left upper panels in figure 12, respectively. Thus, the individual molecules are protected from damage by the surrounding (cluster) environment [95].

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For Xe²⁰⁺ (lower panel in figure 12), the collisions are dominated by distant electron transfer processes and small amounts of energy transfers directly in the collisions (see section 4.1). In this case, the monomers are not protected from damage as the fragmentation yields in the lower middle panel of figure 12 are similar to the ones for a target of monomers (lower left panel). One possible explanation is heating by Coulomb explosions of multiply charged clusters following rapid charge redistribution. In such cases a substantial part of the potential energy is converted into internal energy of emitted singly charged monomers [95]. This effect has been shown to be important for multiply charged fullerene dimers [59, 100].

The first theoretical studies of prompt single atom knockout from fullerene molecules predates the first experimental findings. In 1994, Cui et al presented a study on particle radiation damage to C₆₀ molecules using classical MD simulations [108]. They modeled the C₆₀ using the Tersoff potential to describe the bonds in the molecule together with the ZBL potential for close range interactions where nuclear scattering was important. Cui et al did not include projectiles in their simulations. Instead they gave the PKA a velocity at the beginning of the simulation, as if that atom had been struck by a projectile. From these simulations they predicted the knockout driven formation of C₅₈, C₅₉, and endohedral C@C₅₉ in collisions between C₆₀ and energetic particles [108]. They also deduced an average displacement energy of 29 eV for dislocating or completely removing a C atom from C₆₀ [108].

This study was followed by work from Ehlich et al [109], Larsen et al [14], and Tomita et al [110] who used classical MD simulations to explicitly model collisions between noble gas atoms and C₆₀. These groups all identified knockout driven fragmentation of C₆₀ as an important mechanism in low center-of-mass energy collisions. The addition of a projectile atom to the simulations enabled them to calculate internal energy distribution for the C₆₀ molecules and any large C_x fragments following the collision [14, 109, 110].

Experimental evidence of nonstatistical fragmentation of fullerene molecules was first reported by Larsen et al [14] and later by Tomita et al [110]. In these experiments they let C₆₀ ions, both cations and anions, collide with nobel gas targets. However, they only directly observed knockout, i.e. ${{\rm{C}}}_{59}^{+}$ fragments, with the ${{\rm{C}}}_{60}^{-}$ projectiles [14, 110]. Both the C₆₀ anions and cations were produced using the same electrospray ionization source, but the negative ions were shown to have lower internal energies than the positive ions [14, 110]. The higher internal energy of the ${{\rm{C}}}_{60}^{+}$ ions prior to the collisions meant that the fragments also had higher internal energies than those from ${{\rm{C}}}_{60}^{-}$ projectiles. The result of this difference was that the ${{\rm{C}}}_{59}^{+}$ fragments that were formed from the collisions with ${{\rm{C}}}_{60}^{-}$ projectiles were more likely than those from ${{\rm{C}}}_{60}^{+}$ projectiles to survive long enough to be detected [14, 110]. Using internal energy distributions from their MD simulations, Larsen et al further calculated theoretical mass spectra that included both effects of prompt non-statistical fragmentation and the subsequent statistical fragmentation [14]. These compared well with their experimental results and showed that knockout was an important first fragmentation step in collisions with fullerenes at sufficiently low energies.

In addition to positively charged fragments, Tomita et al also detected ${{\rm{C}}}_{59}^{-}$ and ${{\rm{C}}}_{58}^{-}$ fragments produced by prompt single and double C atom knockout [110]. Figure 14 shows zoom-ins of negative ion mass spectra obtained by Tomita et al for collisions between ${{\rm{C}}}_{60}^{-}$ and He and Ne targets at center-of-mass energies of 276 eV and 1350 eV, respectively [110]. The detection of negatively charged fragments in the collisions shows that knockout may, under certain conditions, result in fragments with very low internal energies [110]. This is because most of the excitation energy is deposited to and carried away by the atom knocked out in the collisions. Collisions leading to more significant heating of the molecules also leads to the loss of one or more electrons, and as a result, the positively charged fragments that are detected [14, 110].

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Around this time, Kunert and Schmidt presented a theoretical study of atomic ions colliding with C₆₀ molecules [111]. Using a method called nonadiabatic quantum molecular dynamics (NA-QMD) [83] they combined classical MD simulations with time-dependent DFT calculations to directly and self-consistently model both electronic and vibronic excitations. In this way they studied collisions with several different projectile ions at a wide range of energies and observed knockout driven fragmentation of the fullerene molecules. This study also showed that at collision velocities above about 0.25 a.u. (equivalent to center-of-mass energies of about 6 keV, 18 keV, and 59 keV for He⁺, C⁺, and Ar⁺ projectile ions, respectively) the stopping models from SRIM [7, 8] overestimated the energies transferred in collisions [111].

The first report of C₅₉ ions being directly detected from collisions with C₆₀ cations came over a decade later from experiments at lower energies [43]. Here ${{\rm{C}}}_{60}^{+}$ ions were made to collide with He or Ne at center-of-mass energies of 50 eV and 240 eV, respectively. From collisions with He, there was no direct evidence of prompt single C knockout detected [43]. On the other hand with the Ne target there was a clear, but weak, signal of ${{\rm{C}}}_{59}^{+}$ in the mass spectrum [43]. Even at these low collision energies, where electronic stopping is weak [82], the fragment mass spectra from fullerenes are dominated by products from statistical fragmentation [43]. While C₆₀ are large molecules with high dissociation energies, their three dimensional structures cause any atom that penetrates the molecules to interact with two layers of atoms. Any C atom that is knocked out by the projectile may also in itself become a secondary projectile that can inflict further damage to the rest of the molecule [43]. Furthermore, the colliding atom or the PKA (primary knock-on atom) may be captured by C₆₀ forming an endohedral fullerene or a C@C₅₉ molecule [108]. In the former case, the available center-of-mass energy, which may be several tens of eV, will be converted into internal energy of the the combined system [43, 111]. The latter cannot be unambiguously identified by means of mass spectrometry alone as it has the same mass-to-charge ratio as the intact C₆₀.

Experimental mass spectra from 110 eV collisions between He and PAH cations of different sizes are shown in figure 15 [40]. Here the collision energy is low enough such that significant amounts of energy is transferred from the projectile to the target almost entirely through nuclear stopping [82] (see also figures 2 and 8). When this is the case the interactions will initially be localized to single atoms in the molecules, with relatively little overall heating. The mass spectra recorded at 110 eV center-of-mass energy are very different from those where PAH molecules are strongly heated by electronic stopping in much faster keV collisions (see e.g. the top-left panel in figure 12). There are strong mass peaks corresponding to losses of about 12 amu, which are highlighted in gray in figure 15. This is from PAH fragments where a single C atom, and possibly a number of H atoms, has been knocked out of the molecule in the collision and where the fragment has survived long enough to be detected [40].

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The peaks for mass loss of about 24 amu in figure 15 are mainly due to statistical fragmentation in the form of C₂H_x-loss. The corresponding events stem from close collisions where the energy transfer to the individual atoms were too small for knockout, but sufficient for statistical fragmentation on the experimental microsecond timescales. For anthracene, about 10 eV is required for statistical fragmentation (C₂H₂-loss etc) on the microsecond timescale [48]. Peaks due to fragments that have lost three or more C atoms, are either due to pure statistical processes, or to secondary statistical fragmentation processes following knockout of a C atom.

The intensities of the CH_x-loss and C₂H_x-loss peaks in figure 15 reflect the relative contributions of nonstatistical versus statistical fragmentation processes [40, 82]. For small PAH molecules like anthracene, the strongest feature in the mass spectrum is from C₂H_x-loss, but when the PAH size increases the CH_x-loss peak begins to dominate. This is due to the increasing heat capacities, and that the dissociation energies and the total excitation energies are rather independent of the PAH size [40, 82]. The reason for the latter is that the interactions with the electron cloud and nuclei is mainly localized along the projectile's trajectory through the PAH molecule (see figure 8). In all but the most extreme edge-on trajectories, the energy deposited in a collision with a given center-of-mass energy will thus be nearly the same regardless of PAH size. Because of this, the absolute cross sections for the knockout of a single C atom from a PAH scales with the number C atoms in the molecule, while at the same time the rate of statistical fragmentation depends on the internal temperature of the molecule or fragment after the collision (which is lower for large PAH systems for a given energy input). So not only is a large PAH less likely to fragment statistically than a small molecule, but the same is also true for fragments due to prompt C knockout [40, 82] (although there are some smaller isomeric differences [107]). Therefore in the case of very large PAH molecules—such as those believed to be abundant in the interstellar medium and with, on average, about 50–200 C atoms [24]—nonstatistical fragmentation may be an important starting point for molecular evolution [40, 82].

In figure 16 we show experimental (black) and simulated (red) absolute fragmentation cross sections for 5 keV anthracene cations colliding with He, Ne, Ar, and Xe targets from [42]. In the same figure, we also show simulated knockout fragmentation cross sections (blue). Prompt knockout is only responsible for about half of the total destruction cross section with He target atoms. The other half comes from anthracene molecules that have been heated to at least 10 eV internal energy in nuclear scattering processes that did not lead to knockout [48]. For heavier targets the contribution from non-statistical fragmentation increases—more than 80% of the total cross sections with Ne, Ar, and Xe is from such processes. With very large targets, e.g. Xe, the destruction cross section is close to the geometrical size of the molecules. This means that all penetrating collisions lead to fragmentation and the molecule is no longer partially transparent to the target atoms as is the case for e.g. He [42].

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Postma et al performed classical MD simulations to study non-statistical fragmentation in collisions of H or He atoms with anthracene, coronene, or sheets of graphene [12]. They used the ZBL potential to model nuclear scattering and the REBO potential to describe the bonds in the target systems. Using these simulations, Postma et al determined absolute cross sections for single C knockout as well as the displacement energy for the removal of a C atom from the molecules in direct face on collisions. They found that the displacement energy for removing a C atom from PAH molecules was similar to that for graphene, with a slight dependence on the location of the C atom in the PAH molecule. Further, their displacement energies had a weak dependence on the projectile species and was slightly higher for He (26.7 eV) than for H (25.5 eV) on coronene [12]. This dependence was attributed to interactions between the projectile and atoms other than the one that is knocked out from the target—with the larger He atom interacting more strongly with other atoms in the PAH than an H atom does. Overall the displacement energies determined by Postma et al [12] using classical MD simulation were only slightly larger than those determined for the removal of single C atoms from graphene by electron impact yielding results of 22.03 eV (ab initio calculations [112]) and 23.6 eV (experiments [113, 114]).

Stockett et al reported on the first measurements of the energy threshold for single carbon knockout from PAH molecules in collisions with He [13]. Cross sections for single C knockout as a function of center-of-mass collision energy from this study are shown in the top panel of figure 17. Classical MD simulations were used for comparison and showed the same energy dependence for the knockout cross section as in the experiment, but with a shift of $\delta =8.5\,{\rm{eV}}$ in the measured threshold energy. The mean MD displacement energy for anthracene, pyrene, and coronene was found to be 27.0 ± 0.3 eV, in good agreement with the values by Postma et al for anthracene [12]. However, using the experimental results to rescale the theoretical displacement energies (bottom panel of figure 17), Stockett et al deduced a semi-empirical mean displacement energy of 23.3 ± 0.3 eV for (randomly oriented) He-PAH collisions [13], which agrees well with the values for electron impact on graphene [112–114].

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We have carried out a comparison of the three reactive potentials used to describe intramolecular bonds covered in this review—the Tersoff, REBO, and AIREBO potentials. Here we compare the effect that these three different models have on the displacement energy for removing individual C and H atoms from coronene. To determine the displacement energies independent of projectile type, we perform the simulations without using a projectile. Instead we displace a single atom (the PKA) along a predetermined initial trajectory. The lowest kinetic energy required to promptly remove that atom from the molecule is then taken to be the displacement energy for that specific initial path. The displacement energy of a given atom in coronene is determined along 500 evenly distributed trajectories.

Due to its symmetry, there are only four unique atom positions of the 36 atoms in coronene—three for C and one for H (see figure 18). The displacement energies of each position were determined over all angles and used to calculate distributions of the displacement energy for the entire coronene molecule. The simulations were performed using the LAMMPS MD package [115, 116] and the Tersoff, REBO, and AIREBO potentials as defined in the software. Parameters for describing the C–C, C–H, and H–H interactions using the Tersoff potential are from [42].

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Figure 18 shows the angular distributions of the displacement energy for the four unique atom positions in coronene calculated with the Tersoff potential. The color and size of each point in the figure represents the displacement energy of the enclosed atom along that trajectory. The highest displacement energies are found for trajectories that follow the bonds to the nearest neighboring atoms and can surpass 100 eV in a few extreme cases. For the two positions where a C atom is bound to three other C atoms, the lowest displacement energies are obtained along trajectories perpendicular to the molecular plane. With the Tersoff potential the value for these is on average about 22 eV. The lowest displacement energies for removing a C atom is found for the peripheral atoms that are bound to an H atom, which can be as low as 17 eV along trajectories close to parallel to, but away from, the molecular disk. The displacement energy for knocking out an H atom is more uniform in its angular distribution at about 9 eV. The values of the displacement energy differ between the different force fields, but their angular dependent behavior remains largely the same as shown in figure 18 and agree well with results from simulations of graphene [117].

Total displacement energy distributions calculated with each of the three force fields for all atoms are shown in figure 19. Here the total distributions over all displacement angles are shown in red and the contribution from angles, θ, less than 60 degrees from perpendicular to the molecular plane are shown in blue. All three total distributions show a strong peak at energies around 25–30 eV which is mainly the result of knockout trajectories out of the molecular plane (in blue) and from peripheral C atoms that have low displacement energies away from the molecular disk (see figure 18). The Tersoff and REBO potentials give very similar results, peaking around 27 eV with the AIREBO potential giving on average somewhat higher displacement energies as well as a more pronounced tail towards high energies.

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The mean displacement energies for the removal of a carbon atom along any initial trajectory (red distributions in figure 19) are 35.4 eV, 45.1 eV, and 48.3 eV for the Tersoff, REBO, AIREBO potentials, respectively. The corresponding values for trajectories out of the molecular plane ($\theta \lt 60^\circ$, blue distributions in figure 19) are 26.1 eV, 30.2 eV, and 33.2 eV. These out of plane values are in good agreement with the displacement energies predicted by Postma et al for trajectories perpendicular to the molecular plane ($\theta =0^\circ$) when using the REBO potential [12].

Intact fullerenes and PAH molecules are stable and not particularly reactive molecular systems. However, when an atom is promptly removed from one of these molecules the result is often a highly reactive fragment. Figure 20 shows an example of a circumcoronene molecule where a single C atom has been removed from the innermost hexagonal ring. The left panel in this figure shows the structure of this relaxed C₅₃H₁₈ fragment seen face-on. Around the defect caused by the removal of a C atom we see that there are three C atoms left with dangling bonds, two identical atoms labelled as 1 and a single atom labelled as 2. All three of these sites are highly reactive and the right panel in figure 20 shows the structure of the system when a phenyl group (C₆H₅) forms bonds with one of the atoms in the fragment at position 1 [82]. The binding energy of the phenyl radical to this position is about 4 eV for a neutral system and 4.4 eV if the combined C₅₉H₂₃ system is positively charged [82]. These binding energies are significantly higher than when the phenyl group is bound to an intact circumcoronene molecule at the same position, at 0.2 eV and 0.9 eV for the neutral and cation systems, respectively [82]. Chen et al found similar trends for several reaction partners—the reactivity of PAH molecules is significantly increased when an atom is removed by knockout [82]. This is consistent with results from a theoretical study on the role of defects in the reactivity of graphene [11].

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When non-statistical fragmentation takes place inside of molecular clusters, newly formed radicals may react with neighboring molecules and efficient molecular growth can take place. Delaunay et al detected widespread molecular growth in experiments at the ARIBE facility where ions collided with PAH clusters [58]. Starting with loosely bound clusters of pyrene (C₁₆H₁₀) they detected covalently bound molecules containing up to more than 30 C atoms formed in collisions with ions at center-of-mass energies above 10 keV. For light projectiles (e.g. He) this energy range is well into the regime where electronic stopping dominates, but by varying the mass and velocity of the projectiles they concluded that nuclear scattering processes were responsible for the observed growth [58]. Here, the fast redistribution of the excess energy among all cluster constituents is crucial as it allows the molecular growth products to be formed sufficiently cold to survive on the experimental timescale (see section 4.2).

The left panels of figure 21 shows experimental mass spectra measured by Delaunay et al for collisions between pyrene clusters and 11 keV He⁺ ions (bottom row) and 12 keV Ar²⁺ ions (top row) [58]. In both mass spectra we see the range of masses that include the intact molecular monomer (C₁₆${{\rm{H}}}_{10}^{+}$) and loosely bound dimers ([C₁₆H₁₀]${}_{2}^{+}$) that remain after the collisions. However, with 12 keV Ar²⁺ projectiles we also see a wide range of peaks coming from products with more than 16 C atoms. These features were not unique to the collisions with Ar²⁺, but decreased in intensity with lighter and faster projectiles. With the lightest projectiles, like He⁺, these features are mostly absent [58].

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In connection to the experiments, Delaunay et al also performed classical MD simulations of the collisions and bond-forming reactions [58]. In that work the entire collision processes between keV ions and the PAH clusters—and the following bond-forming reactions—could be followed in simulations extending over picosecond timescales. Using the AIREBO potential to describe the molecular bonds and dispersion forces in the cluster, and the ZBL potential for modeling the collision interactions, these simulations were able to reproduce the detailed features in the experimental mass spectra as seen in the right panels of figure 21. SCC-DFTB simulations were then used to test the accuracy of the classical MD simulations with regards to bond formation and the effect of charge on the molecules formed. The results showed that the classical MD simulations gave structures more rigid than those found with quantum chemical methods (regarding preservation of bond angles). However, the reactivity of the fragments and the ability to form larger molecules was confirmed by the SCC-DFTB simulations [58].

Similar bond-forming reactions have also been detected in clusters of fullerene molecules. In a study of 22.5 keV He²⁺ ions colliding with C₆₀ clusters, Zettergren et al detected the formation of covalently bound dumbbell shaped ${{\rm{C}}}_{118}^{+}$ and ${{\rm{C}}}_{119}^{+}$ molecules [52, 57]. These were produced when the ion knocked out one or two atoms from molecules within the cluster. The remaining ${{\rm{C}}}_{59}^{+}$ or ${{\rm{C}}}_{58}^{+}$ fragments could then react with neighboring molecules. Coincidence measurements showed that the ${{\rm{C}}}_{118}^{+}$ and ${{\rm{C}}}_{119}^{+}$ products were strongly correlated with the detection of intact ${{\rm{C}}}_{60}^{+}$ ions, the result of penetrating collisions that lead to multiple ionization of the cluster. Zettergren et al also used classical MD simulations with the Tersoff potential to study the reactions between ${{\rm{C}}}_{59}^{+}$ fragments and intact C₆₀ that lead to the formation of ${{\rm{C}}}_{119}^{+}$ [57]. A sequence from simulations of dumbbell ${{\rm{C}}}_{119}^{+}$ forming in a cluster at different times is shown in figure 22. These simulations revealed that the barrier for reactions was 1 eV or less [57], almost two orders of magnitude lower than that for forming bonds between two intact C₆₀ molecules [118].

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In light of the high reaction barrier for forming covalently bound ${{\rm{C}}}_{120}^{+}$, it is surprising that Zettergren et al detected a strong signal for ions with this mass in the same type of penetrating collisions as dumbbell ${{\rm{C}}}_{118}^{+}$ and ${{\rm{C}}}_{119}^{+}$ [57]. A possible explanation is that [C@${{\rm{C}}}_{59}{]}^{+}$ is formed when a C atom is displaced within a molecule in the collision. This defective fullerene ion, while having the same mass-to-charge ratio as ${{\rm{C}}}_{60}^{+}$, could be as reactive as a ${{\rm{C}}}_{59}^{+}$ fragment and thus easily form covalent bonds with a neighboring molecule. Hence a covalently bound ${{\rm{C}}}_{120}^{+}$, or [C@${{\rm{C}}}_{119}{]}^{+}$, molecule might also form with a low reaction barrier in the same way as ${{\rm{C}}}_{118}^{+}$ and ${{\rm{C}}}_{119}^{+}$.

In a followup study, Yang et al used SCC-DFTB MD simulations and DFT calculations to study the formation of dumbbell ${{\rm{C}}}_{118}^{+}$ and ${{\rm{C}}}_{119}^{+}$ [89]. Yang et al used these quantum chemical methods to better describe the formation of dumbbell shaped fullerenes and showed that SCC-DFTB produced results in good agreement with more advanced DFT structure calculations at the B3LYP/6-31G(d) level of theory (for these types of systems) as seen in figure 23 [89].

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Dumbbell ${{\rm{C}}}_{118}^{+}$ and ${{\rm{C}}}_{119}^{+}$ molecules were also detected by Gatchell et al in collisions between 22.5 keV alpha particles and mixed cluster of C₆₀ and coronene [55]. However, they did not detect any bond-forming reactions between C₆₀ and coronene, or between two coronene molecules/fragments. They used the Tersoff potential in MD simulations to study the reactivity of C₅₉ and C₂₃H₁₂ fragments (of coronene) in interactions with intact C₆₀ and intact coronene molecules. These simulations showed that the mixed coronene-fullerene and coronene–coronene reactions had significantly higher reaction barriers than fullerene–fullerene when only a single atom is removed from the system [55].

The reactions that lead to dumbbell shaped fullerenes are of a very different type than the growth processes previously observed when 200 keV Xe²⁰⁺ ions impacted very large fullerene clusters [119]. There the collisions led to very strong local heating along the path of the projectile forming a plasma of C atoms and small fragments. The plasma then began to cool and to self-assemble or coalesce with neighboring molecules resulting in a broad distribution of larger fullerenes [119]. The competition between these growth processes and the statistical decay of hot reaction products led to both bottom-up and top-down fullerene formation processes being observed in these experiments [119].