Issue 
A&A
Volume 540, April 2012



Article Number  A1  
Number of page(s)  19  
Section  Interstellar and circumstellar matter  
DOI  https://doi.org/10.1051/00046361/201117623  
Published online  13 March 2012 
Online material
Appendix A: Aromatic cluster characterisation
In the study of polycyclic aromatic hydrocarbon (PAH) molecules and polycyclic aromatic clusters in hydrogenated amorphous carbon solids (aC:H or HAC) we need to characterise the constituent aromatic structures in terms of their number of carbon atoms, n_{C}, number of aromatic, sixfold rings, N_{R}, or coordination number, m. It has been shown that N_{R} is a function of the band gap, E_{g}, of a given hydrogenated amorphous carbon, i.e., N_{R} = [5.8/E_{g}(eV)] ^{2}, and that E_{g} can be expressed as a function of X_{H}, i.e. E_{g} ~ 4.3X_{H}, the aromatic coherence length, L_{a}(nm) = [0.77/E_{g}(eV)] , a measure of the aromatic domain size, also depends on the band gap (e.g., Robertson 1986, 1991).
In Sect. 2.2.1 we introduced an “aromatic cluster parameter”, Z, which is a function of N_{R} and is simply the number of constraints, N_{con}, per carbon atom for the given cluster, i.e., (A.1)where n_{C} is the number of carbon atoms per aromatic cluster. For rather small and compact aromatic clusters such as pyrene, perylene, etc. Jones (1990) showed that and for linear aromatic systems such as naphthalene and anthracene that Z = (5N_{R} + 7)/(4N_{R} + 2). For the linear clusters the expressions are valid for all N_{R}. However, for compact clusters the expressions give a reasonable fit for N_{R} ≤ 10 which is applicable to most of the aromatic clusters that occur in amorphous hydrocarbons (Robertson 1986) and therefore, by inference, in our eRCNs. Additionally, as pointed out by Robertson (1986), the compact clusters will be more stable that the row clusters.
The relevant expressions for linear and compact aromatic clusters are shown in Table A.1. The compact cluster expression given in the upper part of the table is in fact that for bilinear clusters, i.e., two parallel and connected linear structures. The middle part of the table gives the expressions for clusters consisting of a given number of rows of linear structures, N_{row}, where in this case the values of N_{R} are restricted to N_{R} ≥ pN_{row} where p is an integer ≥ 1. The expressions for compact clusters that give reasonable fits for all N_{R} are given in the lower part of the table.
Expressions for (polycyclic) aromatic cluster structures.
The expressions for the number of carbon atoms per aromatic cluster, n_{C}, and the coordination number m_{coord} of the cluster are plotted in Fig. A.1. The dashed lines in this figure show the simple fits to the data for general aromatic clusters, which are valid for the most compact clusters containing up to about 10rings but deviate for larger systems (the expressions for compact clusters are taken from the upper portion of Table A.1). The areas indicated by the vertical rows of dots show the values for all possible aromatic clusters with a given number of rings, i.e., the values for all possible isomers. The upper limits to the dots are for the most extended linear acenes (naphthalene, anthracene, tetracene, pentacene, etc.). Note that the simple expressions, represented by the dashed lines, give a very good representation of moderately compact clusters over the entire range of N_{R} considered here.
Fig. A.1
Number of carbon atoms n_{C} (upper curves, diamonds) and coordination number m_{coord} (lower curves, triangles) for the most compact aromatic clusters possible (solid lines). The dashed lines show the simple fits to the data for general aromatic clusters. 

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The coordination number per carbon atom in the aromatic cluster is plotted in Fig. A.2. In this figure the large diamonds show the values for the most compact aromatic clusters possible, i.e., for the large clusters these are the equivalent to the coronene, ovalene, circumcorenene, etc. structures. The small diamonds and their connecting dashed lines show the values for the multilinear acenes with N_{row} = 1, 2, 3, 4 and 5 (from top to bottom). The small diamonds mark the actual cluster values and the dashed lines are calculated using the multilinear acene expressions (given in the middle of Table A.1). The horizontal dotted line indicates the limiting values (0.5) for simple linear acenes (N_{row} = 1). Note that the compact cluster expression in the upper portion of Table A.1 is equivalent to the multilinear acene expression with N_{row} = 2. The thick solid line gives a fit for the most compact aromatic clusters calculated using the expressions in the lower portion of Table A.1.
Fig. A.2
Ratio of the coordination number m to the number of carbon atoms n_{C} in aromatic clusters, i.e., the maximum possible number of peripheral aromatic cluster C–H bonds per carbon atom. See the text for a full description of the plotted data. 

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Appendix B: eRCN model extension possibilities
The mathematical basis of the RCN and eRCN models (e.g., Phillips 1979; Döhler et al. 1980; Thorpe 1983; Angus & Jansen 1988; Jones 1990) is a pair of equations with three interdependent variables, viz., (B.1)and (B.2)Solutions to these equations are found by combining or “collapsing” two of the variables into a single one, i.e., X_{sp3}/X_{sp2} ≡ R, and solving for R as a function of X_{H}. It is therefore evident that the addition of any new structural component into the system introduces a new variable, which renders the resulting equations unsolvable because we then have two equations and four variables. This limitation could be circumvented in several ways:

1.
By a collapsing of three of the variables into one, i.e.,X_{sp}/(X_{sp3} + X_{sp2}) = P. However, this leads to ambiguities because of the degeneracy between the variables in the denominator and therefore to no useful or physicallyrealistic solutions.

2.
By fixing the concentration of the new component. This is a useful strategy but requires that the atomic fraction of the new component, W, is wellconstrained by measurements and also that the number of involved atoms is invariant and independent of the network structure, i.e., no atomic loss, or transformation, of component W is allowed. New carbon atom hybridisation or polyatomic structures, such as sp^{1} C atoms or fivefold C_{5} aromatic rings, do not fulfil this condition and therefore cannot be treated in this way. The addition of heteroatoms, such as O and N, should be possible using this approach, provided that their atomic abundances remain fixed. However, fixing the abundance of O and/or N atoms, and then treating their contribution as a constant, does not correctly take into account their accommodation into the structure and does not lead to realistic or useful solutions.

3.
By the elimination of one of the variables. For example, in order to introduce sp^{1} C atoms it is possible to eliminate either the X_{sp3} or the X_{sp2} component and then solve as a function of X_{H} by combining the remaining two variables. However, in neither of these particular cases does this result in physicallymeaningful solutions, which is not surprising since this treatment imposes unrealistic network constraints, i.e., networks consisting of only sp^{1} and sp^{2} atoms, with 2 and 3fold coordination, or of only sp^{1} and sp^{3} atoms, with 2 and 4fold coordination. The readers are left to verify this limitation for themselves.
In conclusion, it does not yet appear possible to add another carbon atom or heteroatom component to the eRCN model in a physicallyrealistic way.
© ESO, 2012
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