## 2. MethodsBetween nanoscopic carbon clusters such as , , , 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 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 (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 and with fullerenes when , 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.
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,
designates a cluster with where with the most probable relative velocity In the latter expression, Similar relations are used for addition of a hydrogen atom to all species or, more specifically, a or a radical to alkyne-like species. The aggregation process by addition of a or atom, a or 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 .
The mean field equation governing the change in density of any cluster , with time is When , 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 . These equations are not independent and are linked by the following conservation equations where and are the total number of carbon and hydrogen atoms, respectively (the summations are performed on all the species considered). For a given , 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 or 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 , , () or group is represented by the eleventh term. The kernels , , and represent respectively the probabilities of aggregation of , , , ; likewise denotes the probability of stripping and the probability of photodissociation relative to the reaction . The kernels , , and can be written in the form where The determination of the quantity
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
is independent of where is the mean geometrical
area of the clusters with a skeleton composed of
The photodestruction rate corresponding to the reaction is given by where is the absorption cross
section (cm 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 . In the mean field approximation, we can write The coefficients represent the relative probabilities of photodetachment of a functional group from the compound . These quantities are listed in Table 3 for alkane-like, alkene-like and alkyne-like species. The geometrical cross sections, , are defined above. At , we have .
The energetics of the various photodestruction reactions is determined by the dissociation energies (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 , a number of structures have been optimized in order to determine i/ the ground state geometry for given ii/ the dissociation energy of any functional group belonging to the series listed in Table 1.
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
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 , 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
© European Southern Observatory (ESO) 2000 Online publication: July 7, 2000 |