Introduction
Carbon is a chemical element with atomic
number of 6. It is solid at room temperature. Depending on the formation
conditions, carbon can be found in different allotropic forms (diamond,
graphite, fullerenes and nanotubes) and also as an amorphous material [1].
Different crystalline structures of carbon can be understood following the
electronic characteristics of this element. The carbon atoms present an
electronic structure given as: 1s22s22p2. The
nature of this configuration allows the so called hybridization phenomena
[2–4]. Therefore, we can find electronic structures such as: sp, sp2
and sp3. Carbon atoms in the sp2 hybridization state
gives rise to planar structures where each carbon atom is bonded to other 3
carbon atoms [5]. This carbon atoms arrangement produces the allotropic form
known as graphite. The icosahedra structure appears with the platonic solids
and has a great importance in the study of the physics of small clusters. These
structured configurations have been observed in different metallic elements.
These kinds of metals crystallize in FCC or HCP structures (cubic centered face
or hexagonal compact) [6]. One of the main characteristics of these small
particles is that they present faceted external surfaces and also in some
cases, quasispherical domes. The common growth is based in the initial
formation of a cuboctahedron or in the multiple twinned particles (MTPs)
limited for triangular {111} faces of tetrahedrons [6]. The minimization of the
surface energy can be obtained when the particle is approximate to a spherical
form.
Experimental
Assembling of triangles modules were constructed and
joined to form hexagon or pentagon forms (Figure 1). A growth is given in each
side of the pentagon with triangles to generate the walls of the fullerene
(Figure 2). In this wall, the triangles form an arrangement of hexagons and
they close with another pentagon. It is possible to build a particle that tends
to a spherical shape or a faceted form growing the pentagonal dome and the
walls of the model. A bigger growth is possible by increasing more triangles on
the dome of the particle.
A module was designed to simulate the union of three of
them to form a triangle (Figure 3) and with these modules to form a pentagon
and later a growth is given in each side of the pentagon with more modules
until icosahedra, star structure or decahedra are obtained.
A final energy minimization of some
clusters, using a geometry optimization method, was performed using the GULP
program [7]. The performance of the molecular dynamics results were tested by
comparing the energies of selected clusters with the constructed origami
models.
Results and Discussion
The carbon atom and their structural bonds have been
simulated using origami models with the form illustrated in Figure 1 and 2.
Arrangements of 6 carbon atoms give rise to structural hexagonal shapes (Figure
2c). Also, five carbon atoms arrangement generates the pentagonal structural
forms (Figure 2d). These structures, for example, can form a ring system of
hexagonal shapes which defines an atomic plane. Depending on the stacking of
these structural planes, the hexagonal
structure of graphite
can be obtained [1]. In sp2
Figure 1.
Primary folding which form the primary module.
Figure 2.
a) and b) Origami structural forms corresponding to the carbon atom and three
different bonds. c) and d) hexagonal and pentagonal structural forms.
Figure 3.
Origami construction of different clusters starting from triangular forms.
Figure 4.
a) 3D construction using origami structural models of the C60
Fullerene, b) its corresponding schematic atomic structure, and c) Construction
of the 3D model for the C240 Fullerene using the origami elementary
structural forms.
structures the bonds between the carbon atoms are
covalent in nature, however, the bonding between the stacked layers are Van der
Waals in nature. Carbon atoms in sp3 structures are bonded to 4
other carbon atoms in a 3D structure which gives rise to another allotropic
form, the diamond structure [8]. The Fullerenes are the third stable form of
carbon. These structures have hybridized states between the sp2 and
sp3 stable structures. The hybridized states allow the formation of
pentagonal and hexagonal forms which gives rise to 3-dimensional structures
[3]. The most common Fullerene is the so-called C60 (60 carbon
atoms) with a morphological shape similar to a soccer ball. Figure 4a
illustrates the 3D model of the C60 molecule constructed with the
elements defined in Figure 2. In this construction each pentagonal form is
surrounded with a ring of hexagonal shape structures. This is illustrated in
Figure 4b, where the schematic computer simulated C60 is compared
with the origami 3D model of this molecule. Figure 4c shows the 3D structure of
the C240 Fullerene obtained after adding two rings of hexagonal
forms to the initial pentagonal structural form [9]. This structure maintains
its spherical curvature shape similar to the C60 Fullerene.
Carbon nanotubes also present intermediate hybridizations
and can be considered as rolled graphite layers. The nanotubes can be open or
closed structures and the closure region can be considered as a semi-fullerene.
The structural geometry of the graphite nanotubes have been described in the
past [10]. They are constructed based on a rolled hexagonal carbon grid with a
cylindrical shape. The carbon atoms are located in each node grid with three
close carbon neighbors.
Depending on the rolled nature of the carbon grid, three
different types of nanotubes can be described: the armchair, the zigzag and the
chiral structural models [11]. 3D models can also been constructed based on the
carbon origami unit illustrated in Figure 2. All the nanotube types (armchair,
zigzag and chiral) can be obtained using this structural unit. Thus, for
example, Figure 5a illustrates the armchair nanotube, where the hexagonal
structural shapes are aligned along the main axis of the nanotube. It is
interesting to mention that the structure can be closed with a semi-fullerene.
The zigzag and chiral nanotubes are illustrated in Figure 5b and 5c,
respectively. In the zigzag structure, the hexagonal structural shapes are
aligned perpendicular to the main axis of the nanotube. However, in the chiral
structure the structural hexagonal shapes make an angle of 30° with the main
axis of the structure. The closure of this structure requires the use of two
pentagonal structural shapes.
Some examples of structures constructed with the origami
technique are shown in Figure 6. These structures were constructed joining the
different triangles from the edge or from the vertex side (see Figure 3). The
Figure 6a shows the 3D structure of a regular icosahedron. An icosahedron is
any polyhedron having 20 faces, but usually a regular icosahedron is
characterized by its equilateral triangles faces. This structure has 30 edges
and 12 vertices. The regular icosahedron has six five-fold symmetry axes
defined by the straight line that join the opposed vertexes, fifteen two-fold
symmetry axes defined by the straight line that join the centers of opposed
edges and a symmetry center. The previous symmetry elements define one of the
groups of symmetry of the icosahedrons, the denominated Ih according
to the Schöenflies notation. Figure 6b shows a modified icosahedron, where one
of the faces of the corresponding regular icosahedron is rotated, resulting in
Figure 5.
Origami models of some nanotubes; a) Armchair nanotube, b) Zigzag and c) Chiral
nanotubes.
Figure 6.
Model for some clusters using the origami elementary structural forms; a)
Icosahedron, b) modified Icosahedron, c) decahedron and d) star.
Figure 7.
Stability energy of the different clusters using molecular simulations.
parallel faces and reducing the number of
five-fold symmetry axes. Therefore, rotating one of the faces allows us to
obtain a new structure.
Some of the origami models were repeated with rigid
spheres structures and their stability has been assessed. This methodology has
been employed to test the possible stability of the origami designed
structures. Some simple simulations based on the Molecular Dynamics were
carried out using the program GULP. This program allows us to calculate the
total energy of a specific system, also to explore some physical properties.
Figure 7 shows the models of rigid spheres and the corresponding formation
energy. The first model is the fullerene C60 that is similar with
the Figure 4a model, where the vertexes represent the atoms of C. As it can be
observed, the fullerene C60 only shows two main facets, the first
one, where the face of the pentagon is exposed and it is surrounded by
hexagons, and the second one, where the face of the hexagon is exposed and it
is surrounded alternate by pentagons and hexagons. From energetic point of
view, the fullerene is a most stable structure, its formation energy is higher
than to the other models (-398.5 eV). With the origami technique some models
were made where the main unit was the triangle, for the case of rigid spheres
models the tetrahedron unit was used. This was with the purpose of transform
the origami models to solid structures. In this case we used rigid spheres
models with atoms of Au. These models are solid and a great similarity exists
with the topology of the origami models.
The first model was the icosahedron Au55, which
presents 20 tetragonal faces, from any edge that is rotated to the center; the
five-fold symmetry axes will be seen. The formation energy of this structure
was -138.6 eV, smaller than the case of the fullerene C60. A small
modification of this structure was carried out. Two faces of the icosahedron
were rotated; resulting in a modified icosahedron with two parallel faces (the
next model). As it can be observed, the difference resides in the body or
central part. This model is similar to the Figure 6b. The origami technique can
be used to construct infinity of models by only joining the corresponding
triangles. The modified icosahedrons have formation energy similar to the
regular icosahedron. This indicates that both structures can be stables in
experimental samples. It is necessary to mention that the second icosahedron
(modified icosahedron) could be similar to the Ino decahedron (the next in
descending order); however their difference resides mainly in the central body
region. The Ino decahedron form an atomic arrangement of (100) planes. From the
energetic point of view, the Ino decahedron is more stable (-188.6 eV), for
this reason it is more frequency founded in Au particles [12].
Another of the models constructed by origami
technique and rigid spheres model was the star (similar to the Figure 6d). This
model is very similar to the Marks decahedron (the next in the order); however,
the difference resides in the central body part. The Marks decahedron has a
body formed by three columns of atoms, while the star model only has two. In
spite of its similarity, removing a column of atoms of Marks decahedron to form
the star model gives a non stable structure, on the contrary, the star model is
very unstable and its presence in experimental samples is uncertain. However,
the Marks decahedron has formation energy similar to the icosahedrons, and this
structure can be obtained under similar conditions to the regular icosahedrons.
Conclusions
The origami method introduced in this
communication allows the construction of 3D models for carbon nanotructures
such as: fullerenes and nanotubes and also allows the construction of 3D models
for defects or junctions of these types of structures. This technique allows
the visualization of the topological characteristics of fullerenes, nanotubes,
and defects in nanostructured solids.
Acknowledgments.
The authors gratefully thank A. Aguilar for the technical assistance and
DGAPA-UNAM IN-101709 for financial support.
References and Notes
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The
Chemical Educator, Vol. 14, No. 5,
Published on Web 12/31/2009, 10.1333/s00897092248a, © 2009 The Chemical
Educator