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Astron. Astrophys. 359, 799-810 (2000)

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2. Methods

Between nanoscopic carbon clusters such as [FORMULA], [FORMULA], [FORMULA], etc. and microscopic carbonaceous grains different kinds of medium-sized hydrocarbons may exist. In the present study, we only consider

i/ aromatic molecules (2D structures) in which the [FORMULA] atoms are sp2-hybridized. As a very typical example, we can cite the PAHs.

ii/standard alkanes, alkenes and alkynes.

iii/ the preceding structures i/ and ii/ but in various states of dehydrogenation. The different species are then labelled "species-like" (Fig. 1). For total dehydrogenation of chains with a number of carbon atoms larger than 10, pure carbon monocyclic rings are assumed to be the ground state geometry (Weltner & Van Zee 1989) and fullerenes when [FORMULA] (Hunter et al. 1994) [note that for pure carbon clusters with heteroatoms belonging to the first and second rows of the periodic table, it has been found that the closure of the chain in a monocyclic ring is delayed at values of n larger than 10 (Pascoli & Lavendy 1998a,b), but we do not consider this case here]. In fact, for each species, a large number of isomers can be present in a metastable state [For instance the linear chains can coexist with monocycles when [FORMULA] and with fullerenes when [FORMULA], Fye & Jarrold 1997)]. Finally intermediary forms for the chains can also exist, given their floppiness. The energetic separations between all these intermediary forms can be smaller than 1-2 eV and one form can pass to another one in the ambient radiation field.

[FIGURE] Fig. 1. A sample of hydrogenated carbon clusters considered in this study.

The way all the different types of structures i/, ii/, iii/ are built when competition exists between them, has not yet been examined even though this point appears to be very important to understand the nucleation process of growth of carbon clusters in diluted media (for instance, in carbon-rich stellar atmospheres). In the following, [FORMULA] designates a cluster with N carbons and P hydrogens. The corresponding abundances are denoted [FORMULA]. For instance, when a cluster [FORMULA] captures a carbon atom, it becomes the cluster [FORMULA]. The relative velocity, v, between the carbon atom and the cluster [FORMULA] is assumed to be determined by the Maxwell-Boltzmann statistics. The effective cross-section, [FORMULA], for addition of a carbon atom to a [FORMULA] cluster is given by




with the most probable relative velocity


In the latter expression, T designates the gas temperature and [FORMULA], the reduced carbon mass.

Similar relations are used for addition of a hydrogen atom to all species or, more specifically, a [FORMULA] or a [FORMULA] radical to alkyne-like species. The aggregation process by addition of a [FORMULA] or [FORMULA] atom, a [FORMULA] or [FORMULA] radical is limited by opposite mechanisms of chemical attack or photodestruction due to radiation field. Fragments produced by photodissociation are listed in Table 1. We assume that the star radiates as a black body at temperature [FORMULA].


Table 1. Fragments for each type of compounds

The mean field equation governing the change in density of any cluster [FORMULA], with time is


When [FORMULA], however, the whole set of reactions tabulated by Baulch et al. (1992), involving small hydrogenated carbon clusters or radicals without heteroatoms, has also been incorporated.

The summations are over reactions which either create (+) or destroy (-) the species [FORMULA]. These equations are not independent and are linked by the following conservation equations


where [FORMULA] and [FORMULA] are the total number of carbon and hydrogen atoms, respectively (the summations are performed on all the species considered).

For a given [FORMULA], the second member of Eq. (4) includes the sticking of a carbon or a hydrogen atom, represented respectively by the first and second terms, the sticking of a [FORMULA] or [FORMULA] radical to alkyne-like clusters (respectively third and fourth terms). Chemical attack by hydrogen atoms of hydrogenated clusters, which can be important at higher temperatures, is also taken into account for all species (fifth term) (except for alkyne-like species for which abstraction of H by this process is much more difficult to realize). A mean activation barrier of 0.7 eV has been adopted for stripping (see Kiefer et al. 1985; Frenklach & Feigelson 1989). The photodissociation of a [FORMULA], [FORMULA], [FORMULA] ([FORMULA]) or [FORMULA] group is represented by the eleventh term. The kernels [FORMULA], [FORMULA], [FORMULA] and [FORMULA] represent respectively the probabilities of aggregation of [FORMULA], [FORMULA], [FORMULA], [FORMULA]; likewise [FORMULA] denotes the probability of stripping and [FORMULA] the probability of photodissociation relative to the reaction [FORMULA].

The kernels [FORMULA], [FORMULA], [FORMULA] and [FORMULA] can be written in the form


where V is defined by expressions similar to (3) for addition of [FORMULA], [FORMULA] and [FORMULA].

The determination of the quantity [FORMULA] is a rather difficult task for any type of molecule-molecule reaction. However, reactions between radicalar compounds are exothermic and therefore not very temperature dependent. We can thus assume that [FORMULA] is independent of T for each reagent cluster and can be written in the form


where [FORMULA] is the mean geometrical area of the clusters with a skeleton composed of N carbons (Table 2a). The coefficients [FORMULA] indicate if addition of a hydrogen, a carbon atom or any other radicals is possible by testing the number of dangling bonds present in the structure (Table 2b). Similar relations to (6) and (7) for stripping reactions are used but i/ the coefficients [FORMULA] are taken from Table 3 (in order this time to ensure that no stripping can occur for fully dehydrogenated species, ii/ the expression (7) is multiplied by an activation factor written, as usual, in the form [FORMULA] with [FORMULA]0.7 eV (see above). In addition to radicalar reactions listed above, some important reactions, but which proceed with an activation barrier, have been also included for aromatic species. Formation of benzene by cyclotrimerization reactions of acetylene has been considered, but as is well known, we found this reaction is not efficient - or rather, benzene formation by this pathway requires a time prohibitively longer than the dynamic time scale for stellar wind [In fact the catalytic action of Fe could possibly lower this barrier, but this reaction is not easy to quantify, see Schröder et al. 1991]. A much more prominent - and low-energy - route to obtain a benzene ring is to go from the linear [FORMULA] structure (which forms abundantly from linear clusters [FORMULA]) to the corresponding cyclic isomer (fully dehydrogenated benzenic ring). DFT calculations give a smaller value of 1.11 eV for the activation barrier for conversion of the [FORMULA] linear into the cyclic form. On the other hand subsequent contamination by hydrogen stabilizes the cycle with creation of sp2 sites. This important route is also included in the present calculations [This pathway could also be envisaged for linear [FORMULA] which easily isomerizes into a fully dehydrogenated naphtalene-like structures, the latter ones being next stabilized by sticking of hydrogen atoms]. Starting from a benzenic ring, dual process with consecutive addition of [FORMULA] and [FORMULA] to PAHs (activation barrier 1.85 eV) is also relevant (Stein 1978). The latter reaction is efficient for production of naphtalene from benzene due to the high abundance of diacetylene found in all cases. Moreover, its relative inefficiency for larger PAHs is due to the fact that free [FORMULA] (or [FORMULA]) is rapidly locked in the carbon chains and photofragmentation of this radical from these chains is not easy. Another possibility (not considered here) would be to add uniquely [FORMULA], but this route leads to a sequence of non compact structures, such as polyhelicene or polyacene components which are seemingly absent from the interstellar infrared emission spectra (Léger et al. 1989) [Conversely, an excess of [FORMULA] radicals could lead from a seed of naphtalene to acenaphthylene with a five-membered ring and subsequently to a sequence of curved structures].


Table 2a. Mean geometrical areas for alkane, alkene, alkyne-like and aromatic-like compounds. N represents the number of carbon atoms in the structure and [FORMULA] the number of closed rings.
a) reduced by a factor 10 for [FORMULA] and [FORMULA]
b) deduced from reactions tabulated by Baulch et al. (1992)


Table 2b. Relative probability, [FORMULA], of sticking of a hydrogen or carbon atom or any other small radicals to the coumpound [FORMULA].
a) when [FORMULA], aromatic-like species are considered.
b) when [FORMULA], branched structures (diamond-like) are assumed.

The photodestruction rate [FORMULA] corresponding to the reaction [FORMULA] is given by


where [FORMULA] is the absorption cross section (cm2) and [FORMULA] is the emittance of photons at frequency [FORMULA] (cm-2s-1Hz-1). Quantum efficiency is taken equal to the unity. The limit of integration, [FORMULA], represents the threshold frequency, directly expressed as a function of the dissociation energy [FORMULA]. The latter quantity is the minimal energy required to dissociate the cluster [FORMULA] into the cluster [FORMULA] and a fragment [FORMULA]. At [FORMULA], the photon distribution law is given by the Planck function (Allen 1963): [FORMULA]. The amount of radiation which is absorbed and re-emitted at IR wavelengths as due to the photodissociation reaction [FORMULA] can be expressed as follows


Absorption of UV radiation by elements having a low ionization potential and a relatively high fractional abundance, namely Na (I.P. = 5.14 eV), Mg (I.P. = 7.64 eV), Al (I.P. = 5.98 eV), Ca (I.P. = 6.11 eV) and Fe (7.90 eV) has also been taken into account. Photoabsorption cross-sections for these elements in the energy range 5-7 eV are adapted from calculations performed by Reilman & Manson (1979).

Eventually, a extra factor is included for geometrical dilution of radiation at distances larger than [FORMULA]. In the mean field approximation, we can write


The coefficients [FORMULA] represent the relative probabilities of photodetachment of a functional group [FORMULA] from the compound [FORMULA]. These quantities are listed in Table 3 for alkane-like, alkene-like and alkyne-like species. The geometrical cross sections, [FORMULA], are defined above. At [FORMULA], we have [FORMULA].


Table 3. Relative probability, [FORMULA], of photodetachment of a functional group [FORMULA] from the compound [FORMULA].
a) When [FORMULA] and [FORMULA], aromatic-like species are favored over alkene-like structures. For aromatic-like entities the coefficients µ are determined separately in each individual case and no analytic expression is given.
b) When [FORMULA] and [FORMULA], alkanes are assumed to exist in compact form. As for aromatic-like species, the coefficients µ are separately determined in each individual case.

The energetics of the various photodestruction reactions is determined by the dissociation energies [FORMULA] (Table 4). These parameters were computed employing the package GAUSSIAN94 (Frisch at al. 1995). Density functional theory with B3LYP functional has been chosen. For each species, i.e. each couple [FORMULA], a number of structures have been optimized in order to determine

i/ the ground state geometry for given N and P.

ii/ the dissociation energy of any functional group belonging to the series listed in Table 1.


Table 4. Dissociation energies, [FORMULA] (eV), for a few typical reactions calculated with DFT/B3LYP method (Frisch et al. 1995)

These operations were carried out for a number of species and fits have been derived for the other neighbouring congeners (Figs. 2, 3). In fact, many isomers exist for each species [FORMULA] and very often the energetic separations between them are smaller than 1-2 eV. In particular, the carbon chains appear very floppy and can easily pass from a perfect linear form to a strongly bent or a (mono- or poly-) cyclic structure (Weltner & Van Zee 1989). The coexistence of many (metastable) isomers is possible in the gas. For instance, the cyclic geometry is favored for [FORMULA], but this form can coexist with other isomeric configurations such as the linear geometry, the fully dehydrogenated naphtalene, the fully dehydrogenated azulene, etc., under various proportions following the energetics of these species; even though the presence of hydrogen preferentially stabilizes the polycyclic forms. Nevertheless, here, the dissociation energies are computed for species in the ground state configuration. For aromatic-like compounds, a similar procedure was carried out even though the situation is more complex. Computations show that, for given N and P, compact forms are preferred. In the case of complete aromatic clusters composed of closed rings (PAHs), we have selected two possiblities: i/ photodetachment of a hydrogen atom ii/ photoframentation of a [FORMULA] group (acetylene). For incomplete clusters, three possibilities are envisaged: i/ photodetachment of a hydrogen atom, a [FORMULA] or [FORMULA] radical, iii/ photodetachment of a carbon atom when the latter one is in a free position, i.e. when a carbon atom is attached to the main structure by only one bond. Finally, the set of Eqs. (4) and (9) has been solved for a stellar wind with standard mass loss rate, [FORMULA] solar mass by year. Constant expansion velocity, [FORMULA] km/s, and spherical symmetry with adiabatic cooling for the gas ejection, are likewise assumed for simplification (Habing 1996); even though in the vicinity of the stellar surface turbulence and, possibly magnetic field, can produce very intricate gas motions (Pascoli 1997).

[FIGURE] Fig. 2. Dissociation energies plotted as functions of N (number of carbon atoms) for the reactions [FORMULA] in alkyne-like structures.

[FIGURE] Fig. 3. Dissociation energies plotted as functions of [FORMULA] (number of aromatic cycles) for the reactions [FORMULA] in aromatic-like structures.

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Online publication: July 7, 2000