Chemistry Reference
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cut out of hexagons. The fullerenes were obtained by overlapping and gluing given
hexagons (Beaton
1995
). In the projection method the pentagons of the fullerenes
are constructed from the honeycomb lattice by removing a triangular region (Dres-
selhaus et al.
1996
). The nanotubes can be imagined as rolled up parallelograms or
nanoribbons (Dresselhaus et al.
1996
). In all of the previous graphene patterns the
initial structures have more carbon atoms than the final fullerene or nanotube. Thus
they can not be used as self organizing patterns.
2.3.3
Geometrical Patterning for Self Organizing Processes
We are looking for patterns which have the same number of carbon atoms as the
final structure and they can be transformed into the desired arrangement in a self
organizing way.
In biology the information is encoded in the amino acid sequence. This sequence
determines the structure and properties of the protein. In our previous publications
(László and Zsoldos
2012a
,
b
,
2014
) the final structure of the fullerenes and nanotubes
was coded in the geometrical structure of the graphene patterns. The conditions of
the coding were the followings: (1) A pattern contains only hexagons. (2) There are
some fourth neighbouring atoms on the perimeter which can approach each other
during their heat motion by constructing new pentagonal or hexagonal faces. (3) After
the formation of new faces other carbon atoms will be in appropriate positions to
produce new bonds. Repeating steps (2) and (3) we obtain the final structure selected
by the initial pattern.
In Figs.
2.1
-
2.17
we present 17 examples of isolated pentagon C
n
fullerenes for
each possible even values of n in the ranges (
n
100). These
patterns fulfil the above mentioned conditions. Applying the isolated-pentagon rule
(IPR) there is only one isomer for C
60
,C
70
,C
72
and C
74
. For (62
=
60) and (70
≤
n
≤
68) there
is not any fullerene having the IPR. As n increases, the number of isomers increases
significantly. For (
n
≤
n
≤
100) it
reaches the value of 450 (Fowler and Manolopoulos
1995
). In Figs.
2.1
-
2.17
also
the Schlegel diagrams of the corresponding fullerenes are presented with the cut out
pattern in the Schlegel diagram and the same cut out pattern in the graphene sheet.
For the notation of the fullerenes and the spiral codes we used the definitions given
in reference (Fowler and Manolopoulos
1995
).
In tight binding molecular dynamics simulations the patterns of Figs.
2.1
and
2.2
developed into the desired C
60
and C
70
fullerenes (László and Zsoldos
2012a
,
b
,
2014
). In these publications the possibility of transforming into the desired structures
was presented several fullerene and nanotube patterns. These processes were con-
trolled by applying appropriate environmental temperature conditions in the frame
work of the Nosé-Hoover thermostat.
=
76) the number of isomers is two only, but for (
n
=
