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S. Taioli
most striking properties of graphene is its Young’s modulus to density ratio, possibly
the highest achieved so far. We discuss here a systematic method for finding novel
energetically stable structures characterized by sp 2 -hybridized carbon atoms with
decreasing density but almost unchanged specific mechanical characteristics with
respect to graphene. In this way, lower structural weight could be achieved which is
an important request, e.g. in aerospace applications.
5.3.1 Structure Search Method
The geometry of graphene can be generated by using the two-dimensional packing
of congruent discs touching each other in three points under the following constraints (for further details, see [74]):
– No two discs overlap;
– Each disc is in contact with at least another disc;
– For any choice of two discs in the packing, there is always a path connecting
them through mutual contacts;
– Angles between the segments connecting two disc centres must be smaller than
π rad (local stability);
With these constraints in place, one can search for novel structures with specific
characteristics comparable to graphene and also for the least dense arrangement of
discs in the plane which is an important issue on its own.
5.3.1.1 Graphene and Graphene Daughter
The packing of graphene, reproduced in the left-hand side of Fig. 5.12a, has a
density equal to π/(3
√
3) ∼ 0.6046 and can be used to create the graphene net,
reported in the left panel of Fig. 5.12b, by positioning a carbon atom at the centre
of each disc. One can generate a novel architecture by replacing each disc in the
graphene packing with three discs having a radius
1
1+2/
√
3
smaller than that of
graphene, which leads to a less dense packing π(7
√
3) − 12 ∼ 0.390675, as shown
in the right panel of Fig. 5.12a. This process is known as “augmentation” [75], and
the “graphene daughter” obtained by applying this procedure is reported in the right
panel of Fig. 5.12b (called gr11 in [76]).
5.3.1.2 Tilene Parent and Tilene
By considering tilings with polygons having a number of sides larger than triangles,
one can obtain the packing associated with, e.g. the square-octagon tiling. In the
left panel of Fig. 5.13a, we show the tiling and in the left panel of Fig. 5.13b the
S. Taioli
most striking properties of graphene is its Young’s modulus to density ratio, possibly
the highest achieved so far. We discuss here a systematic method for finding novel
energetically stable structures characterized by sp 2 -hybridized carbon atoms with
decreasing density but almost unchanged specific mechanical characteristics with
respect to graphene. In this way, lower structural weight could be achieved which is
an important request, e.g. in aerospace applications.
5.3.1 Structure Search Method
The geometry of graphene can be generated by using the two-dimensional packing
of congruent discs touching each other in three points under the following constraints (for further details, see [74]):
– No two discs overlap;
– Each disc is in contact with at least another disc;
– For any choice of two discs in the packing, there is always a path connecting
them through mutual contacts;
– Angles between the segments connecting two disc centres must be smaller than
π rad (local stability);
With these constraints in place, one can search for novel structures with specific
characteristics comparable to graphene and also for the least dense arrangement of
discs in the plane which is an important issue on its own.
5.3.1.1 Graphene and Graphene Daughter
The packing of graphene, reproduced in the left-hand side of Fig. 5.12a, has a
density equal to π/(3
√
3) ∼ 0.6046 and can be used to create the graphene net,
reported in the left panel of Fig. 5.12b, by positioning a carbon atom at the centre
of each disc. One can generate a novel architecture by replacing each disc in the
graphene packing with three discs having a radius
1
1+2/
√
3
smaller than that of
graphene, which leads to a less dense packing π(7
√
3) − 12 ∼ 0.390675, as shown
in the right panel of Fig. 5.12a. This process is known as “augmentation” [75], and
the “graphene daughter” obtained by applying this procedure is reported in the right
panel of Fig. 5.12b (called gr11 in [76]).
5.3.1.2 Tilene Parent and Tilene
By considering tilings with polygons having a number of sides larger than triangles,
one can obtain the packing associated with, e.g. the square-octagon tiling. In the
left panel of Fig. 5.13a, we show the tiling and in the left panel of Fig. 5.13b the