A MOLECULAR DYNAMICS STUDY ON SLOW ION INTERACTIONS WITH THE POLYCYCLIC AROMATIC HYDROCARBON MOLECULE ANTHRACENE

For the intra-molecular potential we use the Brenner reactive bond-order potential, which is based on the pioneering work on (reactive) bond-order potentials by Abell (1985) and Tersoff (1986). Brenner (1990) devised a functional form for the potential, which properly describes radical structures and conjugation and which gives excellent results for a great number of hydrocarbon structures. The potential energy is defined as the sum of an attractive pair potential V_A(r_ij) and a repulsive pair potential V_R(r_ij) between the atoms i and j, where r_ij is the internuclear distance:

$$\begin{equation} E_{b}=\sum _{i}\sum _{j>i}V_{R}(r_{ij})-\overline{B}_{ij}V_{A}(r_{ij}). \end{equation} \tag{ 1 }$$

The many-body coupling between the atoms i and j, depending on the local environment of the bond, is included in the bond-order function $\overline{B}_{ij}$. $\overline{B}_{ij}$ explicitly depends on the angles between the ij bond and neighboring bonds ik and jl. Its functional form can be found in Brenner (1990). The pair potentials are defined as

$$\begin{eqnarray} V_R(r_{ij}) &=& f_{ij}\frac{D_{ij}^{e}}{S_{ij}-1} e^{-\sqrt{2S_{ij}}\beta _{ij}(r_{ij}-R_{ij}^{e})}, \end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray} V_A(r_{ij}) &=& f_{ij}\frac{D_{ij}^{e}S_{ij}}{S_{ij}-1} e^{-\sqrt{2/S_{ij}}\beta _{ij}(r_{ij}-R_{ij}^{e})}. \end{eqnarray} \tag{ 3 }$$

For S_ij = 2 the sum of these terms equals the Morse-potential (Morse 1929) with a potential well depth $D_{ij}^{e}$, an equilibrium internuclear distance $R_{ij}^{e}$ and a force constant β_ij. f_ij is a smooth cutoff function to limit the range to nearest neighbors only. The parameters employed here were determined by Brenner (1990) by fitting the free parameters to experimental data.

The interaction between fast atomic particles, i.e., between the impinging ion and a PAH constituent atom, can be approximated by a repulsive screened-Coulomb potential:

$$\begin{equation} V(r) = \frac{Z_1Z_2}{r}\Phi (r/a), \end{equation} \tag{ 4 }$$

where Z₁ and Z₂ are the atomic numbers of the collision partners, e is the elemental charge and r the internuclear separation of the particles. The screening function Φ accounts for the action of the electron distributions of the collision partners. The screening length a sets the length scale for this effect. We have chosen the functional form for Φ that was suggested by Ziegler et al. (1985) and that is most widely used for the description of energetic collisions of atomic particles:

$$\begin{equation} V(r)=\frac{Z_1Z_2}{r}\sum _{i=1}^{4}a_ie^{-b_ix}, \end{equation} \tag{ 5 }$$

with

$$\begin{equation} x=r/a_U,\quad a_U = \frac{0.8854a_0}{Z_1^{0.23}+Z_2^{0.23}}. \end{equation} \tag{ 6 }$$

a₀ = 0.0592 nm is the Bohr radius. The coefficients in the Ziegler–Biersack–Littmark (ZBL) potential are given in Table 1.

To numerically integrate the classical equations of motion, we have employed the particularly accurate Beeman algorithm (Beeman 1976):

$$\begin{eqnarray} r(t+\delta t)&=& r(t)+\delta v(t)+\frac{\delta t^2}{6} \left[4a(t)-a(t-\delta t)\right], \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} v(t+\delta t)&=& v(t)+\frac{\delta t}{6} \left[2a(t+\delta t)+5a(t)-a(t-\delta t)\right]. \end{eqnarray} \tag{ 8 }$$

Here v and a are the velocity and acceleration in three dimensions of the particle under study. The algorithm is not self-starting, so to initialize the integration, the velocity-Verlet algorithm (Verlet 1967) was used. The projectile is initialized at a cutoff distance where interaction is negligible and the interaction is tracked until the projectile reaches the cutoff distance again. For perpendicular impact on anthracene, the cutoff distance was chosen to be 35 AU from the molecular plane.

The code was validated by reproduction of the atomization energies calculated by Brenner (1990). The results are shown in Appendix A. A deviation is only found for ethynylbenzene which is probably due to a typo in Brenner's work. Our value of 68.4 eV is in agreement with more recent studies (Che et al. 1999).

The projectile–target interaction was validated by simulating head-on binary collisions for which an analytical solution for the energy transfer between projectile and target is known (Ziegler et al. 1985):

$$\begin{equation} T_m = \frac{4M_pM_t}{(M_p+M_t)^2}E_0 \equiv \gamma E_0, \end{equation} \tag{ 9 }$$

where M_p and M_t are the projectile and target masses respectively, E₀ is the initial projectile kinetic energy and T_m is the maximum transferred energy of the projectile to the target particle. Excellent agreement was found. The simulation geometry used in the following consists of an anthracene molecule and a projectile atomic particle as displayed in Figure 1.

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Figures 7 and 8 (Appendix B) show a series of frames of a He projectile colliding head-on with a C atom in the anthracene molecule. The frames are separated in time by 1000 AU = 2.42 · 10⁻¹⁴ s. The He kinetic energies for these trajectories are E_kin = 40 eV (Figure 7) and 50 eV (Figure 8), respectively. Clearly at 40 eV, the molecule is left vibrationally excited, but intact whereas at 50 eV the direct hit to a C atom leads to its ejection.

Figure 2 displays the atom numbering scheme for the anthracene molecule used in the following. For a better insight into the interaction process, first a number of characteristic collision geometries are investigated. Table 2 shows results for the situation sketched in Figure 1. The molecule is oriented in the yz-plane and a 100 eV He projectile initially moves toward the molecule in x-direction. The time step was 10 AU = 0.24 fs. Relevant quantities are the projectile kinetic energy loss ΔE_kin, the kinetic energy of the knocked-out fragment(s) E_frag and the vibrational energy of the remaining molecule E_vib. The latter is defined as E_vib = ΔE_kin − E_frag − E_trans, with E_trans being the translational energy of the remaining molecule after the collision (usually on the order of 1–2 eV). For symmetry reasons only results for impact on C3, C4, C4a, C10, H3, H4, and H10 and at the midpoint of the respective bonds are presented. For each impact site, Table 2 lists ΔE_kin (Column 2), E_vib (Column 3), E_frag (Column 4) and the outcome of the collision (Column 5). Clearly, a substantial amount of the projectile energy is carried away by the knocked-out fragment.

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The results for 1 keV He projectiles are shown in Table 3 (time step 1 AU = 0.024 fs). In collisions with such energetic projectiles, much less of the projectile kinetic energy is transferred elastically to the remaining molecule. Instead, in head-on collisions an energetic recoil atom is produced. In collisions with a C atom in the molecule, for example, it is obvious that the C atom is knocked out with a substantial amount of energy, while the hydrogen that was attached to it is more or less stationary and stays in the vicinity of the molecule.

Tables 4 and 5 display the results for 100 eV and 1 keV H impact, respectively (time step 1 AU = 0.024 fs). For through-bond trajectories, no bond scission is observed in either case. Head-on collisions always lead to knock out of the respective atom.

In this context it is of interest to investigate the collision dynamics as a function of projectile atom energy, in particular in the energy range where direct atom knock-out sets in. The threshold kinetic energy transfer $T_0=\Delta E_{{\rm kin}}^{{\rm threshold}}$ is reached when the hitting atom obtains sufficient momentum to get liberated from the anthracene molecule. The quantity T₀ is often referred to as the vacancy formation energy. T₀ can be determined for a given projectile and impact site by simulation of the respective collision as a function of the projectile kinetic energy.

In Figure 3 the projectile kinetic energy dependence of ΔE_kin, E_vib, E_frag and E_trans for collisions of H (blue) and He (red) with the anthracene carbon atom number 4a is depicted. The dashed vertical lines indicate the threshold kinetic energy for direct knock-out $E_{{\rm kin}}^{{\rm threshold}}$. Note that prompt bond scission already sets in at projectile energies a few eV below $E_{{\rm kin}}^{{\rm threshold}}$. ΔE_kin (Figure 3, top) exhibits an almost linear increase with E_kin as expected from classical mechanics. For low projectile kinetic energies, ΔE_kin is almost entirely transferred into E_vib (≈95% for H, ≈90% for He), which accordingly exhibits an almost linear dependence on E_kin as well. Because of the much larger mass of the anthracene molecule as compared to the projectile atom, molecular translational energies E_trans stay below 2.5 eV and 2 eV for He and H respectively and increase almost linear with E_kin.

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Molecular excitation exhibits a clear peak at $E_{{\rm kin}} \approx E_{{\rm kin}}^{{\rm threshold}}$, the threshold projectile kinetic energy for carbon knock-out. These values (He: 43 eV, H: 99 eV) are strongly dependent on the projectile mass. Independent from the projectile mass is the quantity $\Delta E_{{\rm kin}}^{{\rm threshold}} = T_0$ which amounts to approximately 27 eV (see horizontal line in Figure 3, top). For solid targets T₀ is often referred to as "threshold displacement energy"). It depends not only on the site but also on the orientation and therefore is a rather ill-defined quantity, still experimentally undetermined for free PAHs. Experimental and theoretical determinations of T₀ on solid carbon have yielded numbers ranging from 5 eV for amorphous graphite (Cosslett 1978), over 7.6–15.7 eV for carbon nanotubes (Füller & Banhart 1996), to 15–20 eV for graphitic nanostructures (Banhart 1997). In their BCA study, Micelotta et al. chose a conservative value of T₀ = 7.5 eV for most of their calculations, which is in line with the nanotube data. Very recently, however, from first principles density functional theory MD calculations on electron collisions with graphene, a displacement energy T₀ = 22.03 eV was obtained (Kotakoski et al. 2010). Calculations using the tight binding approximation found T₀ = 23 eV (Zobelli et al. 2007), whereas classical MD simulations of ion bombardment on graphene with interactions based on the Brenner potential yielded T₀ = 22.2 eV (Lehtinen et al. 2010). Using the first principles value T₀ = 22.03 eV, experimentally observed cross sections for electron induced knock-out from graphene sheets could be very well reproduced (Meyer et al. 2012).

Tables 6 and 7 display T₀ for perpendicular head-on collisions of H and He projectiles on the different C sites in anthracene obtained from our MD simulation. For comparison, the values for coronene (C₂₄H₁₂) and graphene are displayed as well (graphene was simulated as a sheet of 200 C atoms with the impact site in the center). Clearly, very similar values are observed for the two PAHs and for graphene, i.e., the recently obtained threshold displacement energies for graphene can be used as a reference for our data. Our results for graphene are markedly higher than the experimentally determined T₀ = 22.03 eV. To some extent, this discrepancy is due to the long range nature of the interaction potentials, which implies an interaction between the incoming ion/atom and typically more than one C atom in the target. Energy is thus also transferred to neighboring C atoms and not only to the knock-out atom. Furthermore, the minimum value for T₀ depends also on the orientation of the system. Direct initialization of the knock out atom with appropriate momentum in the MD code used here gives T_d ≈ 21.3 eV, which is close to the reference data. It can thus be concluded, that the T₀ values for PAHs displayed in Tables 6 and 7 are realistic.

For projectile kinetic energies exceeding the knock-out threshold the excitation of the molecule decreases asymptotically to the vacancy energy E_vac. The vacancy energy is a quantity that weakly depends on the C-atom location in the molecule and on the size of the PAH. For C4a in anthracene, E_vac = 16.0 eV whereas for a coronene inner ring atom, E_vac = 15.3 eV. In the E_vib plot in Figure 3, the vacancy energy (E_vac) is indicated by a horizontal line.

It is obvious that—particularly close to threshold—the removal of a carbon atom from a PAH molecule implies E_vib > E_vac. The implication of these findings is that the E_vib of a PAH after a direct C knock out depends on E_kin. It starts at the threshold at T₀ and decreases asymptotically with E_kin to reach E_vac.

Differential cross sections for elastic energy transfer from a projectile atom to an anthracene molecule are obtained by averaging over many collision events where the target molecule is randomly oriented and where the impact parameter is randomly chosen. To this end Monte Carlo simulations of H and He projectiles interacting with the anthracene molecule were performed. These Monte Carlo simulations were performed by implementing a parallelized version of the code on the "Millipede" computer cluster of the Center for High Performance Computing and Visualization of the University of Groningen.

The molecule was positioned at the origin of the simulation geometry with a fixed orientation or with random orientations. The ion started a fixed distance of 35 AU away from the origin to ensure negligible interaction. A great many ion trajectories with random impact parameters were generated in a window of 40 AU by 40 AU. For every ion trajectory the kinetic energy transfer to the molecule was recorded. This allowed for the determination of differential cross sections for energy transfer for a number of projectile kinetic energies of astrophysical interest.

Figure 4 displays differential cross sections for energy transfer (projectile kinetic energy loss) of He atoms of various kinetic energies interacting with anthracene for ion trajectories perpendicular (top panel) to the plane of the molecule and for random orientations (bottom panel), respectively. The differential cross section for energy transfer is defined as

$$\begin{equation} \frac{d\sigma }{dE}=\frac{N(E)}{\mathcal {L}\Delta E}, \end{equation} \tag{ 10 }$$

where N(E) is the number of events with energy transfer E, $\mathcal {L}$ is the (time-)integrated luminosity (number of particles per surface area), and ΔE the bin width. The data were binned using bins of 0.1 eV width.

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Total cross sections for classes of collisions with energy transfers exceeding a defined threshold can now be computed easily. The values are computed by integrating the differential cross sections using an appropriately chosen lower cut-off value T_co. For a better comparison to the BCA results (Micelotta et al. 2010b), it is instructive to choose their knockout energy value of T_co = 7.5 eV. As already shown, this value falls short of the more realistic knock-out energy, obtained from the MD simulation. However, it is in the range of activation energies for anthracene dissociation. Figure 5 displays absolute cross sections for He impact perpendicular to the PAH plane (top panel) and for random orientations (bottom panel) as a function of projectile energy. In addition, cross sections for T_co = 4.6 eV are displayed, as this value is the activation energy for the important C₂H₂ loss process. Also shown are the respective data for anthracene targets from which the H atoms were removed, as well as for 14 single non-interacting C atoms, mimicking the anthracene carbon skeleton. The latter data are compared to the BCA results (solid line), representing 14 times the cross section for the above threshold (7.5 eV) energy transfer to an isolated C atom target.

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For perpendicular orientation, BCA and MD give almost identical results for the case of 14 C atoms and T_co = 7.5 eV. When the MD code is run for a target molecule with the H atoms removed, i.e., with a molecular carbon skeleton, the total cross sections already exceed the BCA data by about 50% at 55 eV collision energy and T_co = 7.5 eV. This indicates that the results of the BCA framework cannot immediately and straightforwardly be extrapolated to molecular structures. MD results for the entire anthracene molecule peak at He collision energies of about 20 eV while the BCA data peaks around 70 eV. At T_co = 7.5 eV, these MD calculations exceed the BCA data by more than a factor of two. For T_co = 4.6 eV, the discrepancy amounts to a factor of 3.

Clearly, cross sections drop, when going from a perpendicular collision geometry to random orientations, mainly because of the smaller geometric cross section. In the extreme case of in-plane collisions, projectile interactions only directly affect PAH constituents facing the projectile, while the others are shadowed. This effect is obviously absent in the BCA and MD cases for atomic C. Micelotta et al. therefore corrected BCA cross sections by a factor 0.5 to account for orientational averaging. Including this correction, the discrepancy between MD for the entire molecule and BCA for single atoms would be almost identical in both panels.

In the next section, the cross sections will be used to estimate PAH dissociation cross sections due to excitation by the projectile.

In recent collision studies on anthracene (and other PAH) cations with neutral He at center of mass energies of 110 eV, Stockett et al. observed single C atom loss for all PAHs under study (M. H. Stockett et al. 2014, private communication), which clearly is the experimental signature of the direct knock-out process investigated here. From this experimental study as well as from the MD results presented here, it is however obvious, that direct C knock out is not the main pathway leading to PAH destruction. In photoionization (Jochims et al. 1994), low energy electron impact (Deng et al. 2006), or collision induced dissociation (Arakawa et al. 2000) studies, typically H, H₂, and C₂H₂ loss from singly charged PAH cations are identified as the most important channels and typically no signatures of single C atom loss are observed. These channels and their respective energetics are appropriate for the studies here, as under astrophysically relevant conditions ions rather than neutrals impact on the PAHs, typically leaving the latter positively charged. It is straightforward to assume a statistical process which leads to a scenario in which an excited molecule is subject to competition between two relevant de-excitation mechanisms: infrared (IR) photon emission and dissociation of the molecule.

Once the energy transferred to the molecule (E_vib) is known, it is possible to determine a dissociation probability for the molecule if the rates for dissociation and IR-emission are known (Tielens 2005). The dissociation rate can be determined using an Arrhenius relation,

$$\begin{eqnarray} k_{{\rm diss}} &=& k_0\; \mbox{exp}\left[-\frac{E_0}{k_BT_{{\rm eff}}}\right],\\ T_{{\rm eff}} &\approx & 2000\left(\frac{E_{{\rm vib}}}{N_C}\right)^{0.4}\left(1-0.2\frac{E_0}{E_{{\rm vib}}}\right),\nonumber \end{eqnarray} \tag{ 11 }$$

where T_eff is the effective temperature of the system, N_C the number of carbon atoms in the molecule, and E₀ the binding energy in eV of the fragment produced in the dissociation process. k_B is Boltzmann's constant. E_vib is the excitation energy in eV.

Dissociation competes with successive emission of IR photons of energy (typically 0.16 eV for a C-C vibrational mode) at a rate k_IR, which in turn reduces the molecular excitation energy. This implies that the dissociation probability changes with each IR photon emission. The probability for the excited molecule to dissociate between the nth and (n + 1)th IR photon emission equals

$$\begin{equation} \phi _n = p_{n+1}\prod _{i=0}^{n}(1-p_i), \end{equation} \tag{ 12 }$$

where the probability for dissociation at each step is p_i. The unnormalized probability for dissociation at every step p_i equals

$$\begin{equation} p_i(E_i) = \frac{k_{{\rm diss}}(E_i)}{k_{{\rm IR}}(E_i)},\quad \mbox{where}\quad E_i = E_{{\rm vib}}-i\epsilon, \end{equation} \tag{ 13 }$$

i.e., the ratio of the rate constants for both processes. The total (unnormalized) probability for dissociation of the molecule at the end of the emission chain is the sum of the probabilities for dissociation at each photon emission step. Micelotta et al. assume a constant, average p_i = p_av and write the unnormalized total dissociation probability after a maximum of n_max photon emissions as (Micelotta et al. 2010a)

$$\begin{equation} P(n_{{\rm max}}) = \sum _{n=0}^{n_{{\rm max}}}\phi _n \equiv (n_{{\rm max}}+1)p_{{\rm av}} = \frac{k_0\exp [-E_0/kT_{{\rm av}}]}{\left[k_{{\rm IR}}/(n_{{\rm max}}+1)\right]}. \quad \end{equation} \tag{ 14 }$$

The average temperature T_av associated with an average probability p_av is taken as the geometric mean $\sqrt{T_{{\rm in}}\times T_{n_{{\rm max}}}}$, where T_in is the effective temperature immediately after excitation of the molecule by the ion and $T_{n_{{\rm max}}}$ is the effective temperature after n_max IR photon emissions corresponding to an internal energy of (T_E − n_max). In line with Micelotta et al. (2010a), for anthracene we furthermore assume n_max ≈ 3, k_IR = 100 s⁻¹ (Jochims et al. 1994) and k₀ = 1.4 × 10¹⁶ s⁻¹ (Ling & Lifshitz 1998). Of crucial importance is the determination of the Arrhenius energy E₀. By fitting the Arrhenius rate to experimental data from Jochims et al. (1994), Micelotta et al. (2010a) find E₀ = 3.65 eV, which is lower than the ≈4.2 eV C₂H₂ binding energy determined for small PAHs (Ling & Lifshitz 1998). For better agreement with astrophysical data, they determine an alternative value of E₀ = 4.6 eV (Micelotta et al. 2010a). Note that the determination of E₀ is fundamentally difficult because C₂H₂ loss competes with H and H₂ loss, which have very similar activation energies.

The Monte Carlo calculations yield the distributions of energy E_vib transferred to the anthracene molecule, and thus the initial effective temperature T_in, for every single collision. Together with E₀ = 4.6 eV and setting n_max = 3 this in turn allows one to determine the anthracene dissociation probability on an event-by-event basis. The dissociation probability is used to calculate total dissociation cross sections by integrating the differential cross sections (for energy transfer of the ion to the molecule) multiplied by the dissociation probability for each energy transfer:

$$\begin{equation} P_{{\rm diss}}(T_E) = \frac{k_0\exp [-E_0/kT_{{\rm av}}]}{[k_{{\rm IR}}/(n_{{\rm max}}+1) + k_0\exp [-E_0/kT_{{\rm av}}]]}. \end{equation} \tag{ 15 }$$

The dissociation probability is relatively constant over the range of energy transfers at approximately 0.6.

The total dissociation cross section can now be determined for every collision energy. To do so, trajectories leading to direct knock out are not counted separately for two reasons. First of all, direct knock out is only a quantitatively small dissociation channel and second, this channel is associated with relatively high excitation energies, leading to subsequent dissociation processes. Figure 6 shows the total dissociation cross sections for perpendicular impact (top panel) and for random molecular orientations (bottom panel). Fortunately, these total dissociation cross sections are not very sensitive to the precise value of the integration lower cut-off of 4.6 eV or 7.5 eV, making the choice of the cut-off value less critical. The reason is the fact that the dissociation probability is close to zero near this threshold.

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Micelotta et al. (2010b) have determined PAH dissociation cross sections using the BCA and assuming ion induced C atom knock-out as the underlying mechanism. Knock out was assumed to occur if the transferred energy exceeded a displacement threshold of 7.5 eV. We have already shown that MD calculations reveal a significantly larger displacement energy, but how do the resulting fragmentation cross sections compare?

The knock-out results of BCA calculations for the 7.5 eV threshold from Figure 5 are displayed as a solid line in Figure 6 for comparison. Once again, these values have to be corrected by 0.5 for orientational averaging in the case of randomly oriented targets. It can be seen that for perpendicular impact the MD cross sections exceed the BCA data by almost a factor of 2 for 40 eV projectile energies whereas the curves almost converge at high projectile velocities. For random orientation, cross sections are again markedly lower and the maximum is now observed for projectile energies around 100 eV. Here, the dissociation cross section is more than twice as high as the BCA data (corrected for orientational averaging). Differences are smaller for lower and higher projectile energies.

In Table 8 we have made a comparison between the average energy transfer of He projectiles to the anthracene molecule as obtained from our simulations and the BCA of Micelotta et al. (2010b). The average energies obtained in this work are calculated using the lower limit of integration or cut-off of T_co = 4.6 eV instead of the somewhat artificially chosen lower limit of T_co = 7.5 eV for direct single carbon knock out, but as the range of energy transfers grows larger this difference is felt less and less. At projectile energies of 55 eV for He the average energy transfers are already very similar.

The astrophysical implications of ion induced PAH destruction in interstellar shocks have been investigated in great detail by Micelotta et al. (2010b). Based on BCA calculations assuming a displacement energy of 7.5 eV, they show that for a PAH containing 50 C atoms, destruction only sets in at shock velocities of about 100 km s⁻¹. For a displacement energy of 15 eV, this value shifts to about 125 km s⁻¹. In the light of the MD calculations, it is clear that the BCA systematically underestimates cross sections for above threshold energy transfer. However, even the displacement energy of 15 eV falls short of the ≈22 eV obtained by MD calculations. We conclude that the results presented here will not significantly change the shock velocity onset for PAH destruction. Electron collisions on the other hand are already very important for shock velocities of 75 km s⁻¹ (Micelotta et al. 2010b). The main astrophysical implication of the present study thus is the "outcome" of the collision. Due to the high displacement energies, direct C atom knockout is not a very strong destruction pathway. Destruction proceeds mainly through C₂H₂ loss from the periphery of the PAH. If the PAH is not fully destroyed, C₂H₂ loss can initiate isomerization and cage formation. The small relevance of knock-out collisions on the other hand leaves little room for PAHs with internal vacancies. N atom incorporation at such reaction sites is thus not an obvious pathway toward nitrogenated PAHs.