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

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3. Results and discussion

Condensation and growth of about one thousand different species have been simulated. Various regimes of temperature [FORMULA] = 2000, 3000, 5000 and 7000 K have been considered for three carbon to hydrogen abundance ratios [FORMULA] ([FORMULA]), 1 ([FORMULA]) and 10 ([FORMULA]). Results of these calculations are displayed in Figs. 4-10. Other cases with relative abundance ratios smaller than (or equal to) [FORMULA] have been also considered, but are not presented here. In fact, for weak values of this ratio, only very small clusters are created. For instance, at [FORMULA] K and for [FORMULA], hydrogen is essentially in its molecular form and carbon is locked in [FORMULA]. The longest carbon chain is [FORMULA] with a relative abundance of [FORMULA]. A comparison with relative abundance of [FORMULA] ([FORMULA]), clearly indicates that hydrogenation of [FORMULA] prohibits the formation of chains with [FORMULA]. Benzene is formed at an insignificant rate and consequently aromatic-like species are absent.

[FIGURE] Fig. 4. Logarithmic abundances of some typical clusters when [FORMULA] vs. distance r ([FORMULA]cm) for [FORMULA]K ...............; 3000 K - - - - - -; 5000 K - - - - -; 7000 K ______________________.

[FIGURE] Fig. 5. Logarithmic abundances of some typical clusters when [FORMULA] vs. distance r ([FORMULA]cm) for [FORMULA]K ...............; 3000 K - - - - - -; 5000 K - - - - -; 7000 K ______________________.

[FIGURE] Fig. 6. Logarithmic abundances of some typical clusters when [FORMULA] vs. distance r ([FORMULA]cm) for [FORMULA]K ...............; 3000 K - - - - - -; 5000 K - - - - -; 7000 K ______________________.

[FIGURE] Fig. 7. Sketch of the carbon cluster distribution simulated in the outer atmosphere of a carbon-rich star at temperatures of 2000 and 5000 K for a carbon to hydrogen ratio of 1. The relative proportions of the various compounds are given in the text.

[FIGURE] Fig. 8. Logarithmic abundances of [FORMULA], [FORMULA] and [FORMULA] linear species with respect to [FORMULA] plotted as functions of r for [FORMULA]K and [FORMULA].

[FIGURE] Fig. 9a and b. Logarithmic abundances of PAHs with respect to [FORMULA] plotted as functions of [FORMULA] (number of cycles) for [FORMULA]K and various carbon to hydrogen ratios: [FORMULA], 1 and 10. a fully hydrogenated, b dehydrogenated to 50%.

[FIGURE] Fig. 10. Logarithmic abundances of fully hydrogenated PAHs with respect to [FORMULA] plotted as functions of [FORMULA] (number of cycles) for [FORMULA]K and various carbon to hydrogen ratios: [FORMULA], 1 and 10.

For higher carbon to hydrogen ratio, the situation is quite different and much more interesting. However in the following, in order to convey the discussion more directly, we essentially discuss the model with [FORMULA]. Other modeling results are given but by way of comparison. For stellar temperature [FORMULA]K and [FORMULA], the [FORMULA] ratio is very high, [FORMULA], due to the fact that i/ [FORMULA] is fully hydrogenated in [FORMULA] ([FORMULA] whereas [FORMULA]), ii/ [FORMULA] is more easily photofragmented from the carbon chains than [FORMULA]. Cyclopropenylidene ([FORMULA]) and diacetylene ([FORMULA]) are among the most important species: [FORMULA], [FORMULA] - [FORMULA]. The chains [FORMULA] and [FORMULA] are indeed abundant up to [FORMULA] (Fig. 8), [FORMULA] (resp. [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]), [FORMULA] (resp. [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]). However very large chains with [FORMULA] are also present, though in smaller proportions (Fig. 8), [FORMULA] (respectively [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]), [FORMULA] (respectively [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]). For pure carbon monocyclic structures ([FORMULA]), we obtain [FORMULA] (respectively [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]) and [FORMULA] (respectively [FORMULA] for [FORMULA] and [FORMULA] for [FORMULA]). As described by Hunter et al. (1994), carbon chains with [FORMULA] are very floppy and can eventually lead to rigid polyhedral structures, such as fullerenes by structural rearrangement (note in particular the Fig. 7 displayed in their paper). With reference to fullerenes, we suggest another route leading to curved or spheroidal structures by direct association and reprocessing of [FORMULA] rings (Goeres & Sedlmayr 1991), the latter species being indeed found abundantly in the present model ([FORMULA]). Total abundance of linear clusters with a number of carbon atoms, N, larger than 60 relatively to total abundance of linear and ring-shaped clusters with [FORMULA], that is [FORMULA], is of the order of [FORMULA], which indicates that production of fullerenes can be appreciable, even though linear chains and rings are clearly dominant. Rate of benzene is found to be high, [FORMULA], and relative abundance of naphtalene with respect to benzene is equally high, [FORMULA]. These aromatic species are slightly dehydrogenated, [FORMULA], [FORMULA]. Thus small PAHs are only partially stripped of their hydrogens. On the other hand aromatic-like compounds with functional groups attached to them appear fairly abundant, [FORMULA], [FORMULA]. Besides, for the coronene, we have [FORMULA] (Fig. 9a). In fact, the latter species appears in a highly dehydrogenated state, [FORMULA] (Fig. 9b) [except when the carbon to hydrogen ratio is smaller than unity, for [FORMULA] we find [FORMULA]]. As a general rule, larger PAHs are fully dehydrogenated, i.e. appear in the form of a small piece of graphite. On the other hand, regular PAHs are more abundant than irregular congeners; for instance the coronene appears more abundant than its neighbours, [FORMULA]. Likewise the abundance of medium-sized PAHs with a functional group attached to them is relatively low, [FORMULA]. The abundance of true PAHs is given in Fig. 9a, together with that of the dehydrogenated congeners (Fig. 9b). Summing up the abundances of true PAHs (regular and non-regular), of the corresponding species but dehydrogenated and of the intermediary forms gives the sp2 rate (Table 5a). Estimation of the sp2/sp ratio ([FORMULA] for [FORMULA]) shows that the aromatic-like compounds are not dominant compared to other forms of carbon, for instance the linear chains, [FORMULA] and [FORMULA].

The third type of carbon, sp3-hybridized, is stocked either in the form of linear (non-branched) alkanes when [FORMULA] or in the form of compact (diamond-like) compounds when [FORMULA]. The sp3/sp ratio is of the order of 20 for [FORMULA] (Table 5b), indicating that sp3 could possibly compete with sp in the inner region of circumstellar envelopes. In fact, the sp-hybridized (acetylenic or cumulenic) structures are energetically the most stable ones, but tetrahedral (sp3) structures are in a sense globally less fragile [To dissociate a sp3-hybridized carbon at the periphery of any medium-sized carbon cluster, it is necessary to break two or three [FORMULA] bonds against one bond in sp-hybridized open structures. This statement strongly contrasts with the population of clusters which very likely exists in the outer regions of circumstellar envelopes and in the interstellar medium, estimated to be essentially composed of sp2-hybridized structures (PAHs) (Allain et al. 1996a,b). But a simple scenario can still be envisaged in order to increase the sp2/sp ratio. As a matter of fact, clusters produced in the inner regions of circumstellar envelopes are very likely reprocessed by interstellar UV radiation field when driven in the outermost regions. As already noted above, the carbon chains are very floppy and can easily lead to small piece of graphite by structural rearrangement [Besides, the inverse process, leading from graphite to chain, is extremely difficult to realize].


Table 5a. [FORMULA] ratio estimated for [FORMULA] at [FORMULA]K


Table 5b. [FORMULA] ratio estimated for [FORMULA] at [FORMULA]K

However, more likely, small graphitic entities bound together by mono and di-acetylenic segments can also appear via this type of process, ultimately leading by gradual sticking to a very intricate and strongly dehydrogenated network of mixed sp2 and sp hybridized carbons.

With regard to the diamond-like structures, we can also conjecture that these ones can be more and more dehydrogenated and graphitized when arriving in the outermost regions ([FORMULA]) of the circumstellar envelope by interstellar UV radiation.

For [FORMULA]K, we found [FORMULA] and [FORMULA], both resulting from the fact that [FORMULA] is fully hydrogenated in [FORMULA] (but [FORMULA]). On the other hand, [FORMULA], [FORMULA] indicating that [FORMULA], weakly hydrogenated, is a free radical. But again as [FORMULA], [FORMULA] is fully hydrogenated, [FORMULA]. Carbon chains [FORMULA] and [FORMULA] are very short, for instance [FORMULA] and [FORMULA]. Temperatures [FORMULA]K are thus sufficient to impede the formation of linear [FORMULA] and [FORMULA] species possessing a number of carbon atoms of the order of - or larger than - 10. Likewise monocyclic rings are not observed ([FORMULA]). Besides, benzene and naphtalene are abundant ([FORMULA], [FORMULA], but coronene is not present.

For moderately high temperatures, [FORMULA] 5000-7000 K, typically corresponding to R Coronae Borealis stars (Schönberner, 1989), the scenario drastically changes. In the vicinity of the star, hydrogen and carbon largely remain in a atomic state. For instance, at [FORMULA]K and [FORMULA], [FORMULA], [FORMULA], [FORMULA]. A very noticeable fact, however, is that as soon as self-shielding due to these clusters becomes efficient, formation and growth of medium-sized species appear very rapidly (Figs. 4-6) [As is well known, R CrB stars show deep minima in the light curves usually explained by formation of dust which blocks the photospheric flux very efficiently (Waters et al. 1990)]. Nevertheless, the compounds which are formed differ in a qualitative manner from those obtained at lower temperatures (2000 K). The linear chains are now very short ([FORMULA], but [FORMULA]) even though monocyclic rings are still present, [FORMULA]. Larger values of n are not represented for the monocycles. Besides, benzene produced by isomerization of linear [FORMULA] and subsequent hydrogenation is very abundant, [FORMULA] (even though gradually transformed into naphtalene by addition of [FORMULA]). Then benzene and naphtalene act as nucleation points and promote formation of big PAHs by pumping of atomic carbon ([FORMULA]). In clear contrast with the situation encountered in the low temperature regime (Fig. 10):

i/ PAHs are produced here at a very high rate ([FORMULA] against [FORMULA] at 2000 K).

ii/ these PAHs are generated in both regular and irregular - but compact - forms ([FORMULA] against 0.02 at 2000 K).

iii/ dehydrogenated PAHs are totally absent whereas at 2000 K dehydrogenated PAHs are dominant over fully hydrogenated species.

Furthermore, the very high sp2/sp ratio, of the order of 0.6 (Table 5a), shows that sp2 hybridized clusters (aromatic-like species) now compete with sp-hybridized species (linear chains and monocycles). Eventually nanodiamond-like structures are still present as noticed by the relatively high sp3/sp ratio (0.3).

The next stage is then to proceed from these medium-sized species, which are formed in the vicinity of the star over a fairly short distance [FORMULA], to microscopic grains ([FORMULA] 0.01 - 0.1 µm in size). Very crude calculations show that medium-sized entities such as fully hydrogenated PAHs with 20-50 atoms can cluster together to generate microscopic grains over a distance of the order of [FORMULA] [assuming an attractive potential of the order of several tenths of an eV; for coronene the calculated potential is 0.2-0.3 eV]. Following a very simple scenario, we obtain a hierarchy of clusters which gradually coagulate in order to produce very fluffy aggregates in which hydrogenated graphitic islands are loosely connected together by Van der Waals forces. These grains are surrounded by a relatively inert atmosphere composed of the most abundant species, that is [FORMULA]. It is also likely that a few diacetylenic molecules and individual PAH units are loosely coupled to the carbonaceous particles in an exohedral position. On the other hand, both laboratory experiments and theoretical arguments suggest that grains composed of more or less individualized structural units, the latter ones possessing relatively large optical gaps ([FORMULA]eV for coronene), are transparent in the visible. Besides, larger PAHs have comparatively smaller optical gaps making use of the relationship [FORMULA] (eV), where M is the number of rings in the structure (see Robertson 1991). At larger distances from the star, say when [FORMULA], the grains (formed in the innermost regions of the circumstellar envelope according to the present scenario) are UV-processed by the interstellar radiation. Consequently they become opaque in the visible by reduction of the optical gap due to dehydrogenation and subsequent graphitization (i.e. enlargement of the PAH structural units).

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Online publication: July 7, 2000