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The superlative properties and potential applications of synthetic
carbon materials – particularly fullerenes [1], nanotubes
[2] and graphene [3] – illustrate their unique scientific
and technological importance and have motivated substantial
research efforts in recent years. Recent investigations of
‘‘exotic’’ carbon allotropes – including the successful synthesis
of carbyne [4] and graphdiyne [5], and the prediction of
T-carbon [6,7] – illustrate the continuing interest in all-carbon
chemistries beyond already known (and well studied) allotropes
such as fullerenes, nanotubes, and graphene [8].
Among the remaining predicted forms of carbon allotropes
[9], graphyne has been the subject of little yet continuing
interest among structural, theoretical, and synthetic scientists
due to its unknown electronic, optical, and mechanical
properties [10–15], as well as proposed practical strategies of
synthesis [14,16–18]. Although currently, large homogenous
sheets of graphyne with long-range ordered (periodic) internal
structures have not yet been reported, there are increasing
efforts in the synthesis and assembly of a class of
molecules known as dehydrobenzoannulenes (DBAs), a precursor
and subunit of graphyne [14,18]. Indeed, new synthetic
methods in annulene chemistry [19,20] now enable the
assembly of a diverse array of DBA topologies, leading to an
increasing potential and inevitable synthesis of graphyne.
For example, recently the first successful synthesis of thin
films of graphdiyne was achieved on copper substrates via a cross-coupling reaction using hexaethynylbenzene [5]. There
has been particular interest in terms of electronic properties,
motivating previous theoretical, experimental and quantumscale
studies [13,21–23]. However, the elastic and mechanical
properties, critical to successful implementation, have yet to
be explicitly determined. In addition, the search for newmodifications
of carbon has produced several new classes of macrocycles
that feature conjugated all-carbon backbones
without annellated benzene rings and display highly interesting
properties [24]. These systems – including dehydroannulenes,
expanded radialenes, radiaannulenes, expanded
‘‘Platonic’’ objects, and alleno-acetylenic macrocycles – may
well serve as precursors to graphyne in the near future [24].
In the interim, the mechanical characterization approach
outlined herein can equally be applied to such molecular substructures
to exploit the combinatorial features of mixed carbon
networks (such as unique molecular architectures based
on expanded dehydroannulenes and expanded radialenes
scaffolds).
The atomistic-level characterization techniques described
herein are equally applicable to small, graphyne-like DBA
substructures and can be immediately applied to various possible
carbon geometries. To provide immediate quantitative
comparison, there is an extensive catalogue of work currently
available regarding the mechanics of carbon nanotubes and
graphene facilitating a direct assessment of the mechanical
performance of graphyne.
Naturally occurring carbon exists in only two allotropes,
diamond and graphite, which consist of extended networks
of sp3- and sp2-hybridized carbon atoms, respectively. Other
ways to construct carbon allotropes are theoretically possible
by altering the periodic binding motif in networks consisting
of sp3-, sp2- and sp-hybridized carbon atoms [24–26].
Specifically, graphyne is a two-dimensional structure of sp–
sp2-hybridized carbon atoms (Fig. 1a), and thus graphyne
can be thought of as simply replacing one-third of the carbon–
carbon bonds in graphene by triple-bonded carbon linkages.
The presence of these acetylenic groups in these
structures introduces a rich variety of optical and electronic
properties that are quite different from those of graphene or
carbon nanotubes. Although significantly large molecular
segments of graphyne have been experimentally synthesized
[14], large regular sheets of graphyne have yet to be
achieved.
2. Methodology
A series of full atomistic calculations of mechanical test cases
is implemented here by classical molecular dynamics (MD) to
derive a simplified set of parameters to mechanically characterize
monolayer graphyne. Similar approaches have been
used for the characterization of carbon nanotubes [27] and
graphene systems [28]. The test suite implemented consists
of the following three loading cases: (i) a stacked assembly
of two sheets to determine the adhesion energy per unit area,
c, as well as effective sheet thickness, dvdW; (ii) uniaxial tensile
loading to determine in-plane stiffness, or Young’s modulus,
E, and; (iii) out-of-plane bending to determine the bending
stiffness per unit width, D. The test suite is applied using a
graphyne sheet of approximately 100 · 100A°
in dimension
as depicted in Fig. 1b. The open carbon edges of graphyne
are not chemically stable in ambient environment, and we
terminate them covalently with hydrogen atoms. A relatively
small material model of finite size (non-periodic boundary
conditions) was chosen partly due to the potential synthesis
of graphyne, which arise from annulene chemistry (that is,
DBA precursors). As such, graphyne, unlike graphene, may
be fabricated piece-wise from molecular building blocks,
and achieve specimens similar in scale to our current model
system and thus allowing for a direct comparison in finitesize
systems.
The full atomistic investigations utilize the ReaxFF potential
for carbon–carbon interactions [29,30]. The first-principles-
based ReaxFF force field has been shown to provide
an accurate account of the chemical and mechanical behavior
of hydrocarbons, graphite, diamond, carbon nanotubes,
and other carbon nanostructures [31–33] while it is capable
of treating thousands of atoms with near quantum-chemical
accuracy. Other reactive force fields have also been used
in recent studies of the mechanics of carbon materials and
may be similarly suitable for graphyne. These include the
AIREBO potential [34,35] and the long-range carbon bond-order
potential, LCBOPII [36], both successfully implemented
in previous studies of graphene [37,38], for example. Graphyne
offers a more challenging system for such potentials, as
the force field must accurately capture the possibilities for
bond alternation and conjugation between the acetylene
and the aromatic units absent in pristine graphene systems.
The version of the ReaxFF force field used here is that reported
by Chenoweth et al. [29]. The time step is chosen
to be on the order of a fraction of femtoseconds (0.2 ·
1015 s). It is noted that such a small time step was implemented
to ensure the stability of the simulations and reflect
the relatively high vibrational frequency of the triplebonded
acetylenic groups. All full atomistic simulations
are subject to a microcanonical (NVT) ensemble, carried
out at a temperature of 10 K to limit temperature fluctuations,
thereby approximating molecular mechanics. Temperature
control was achieved using a Berendsen thermostat
[39], with a damping parameter of 100 fs (500 time steps)
limiting temperature fluctuations to approximately ±1 K
during dynamic simulation. All MD simulations are performed
using the massively paralyzed modeling code LAMMPS
(Large-scale Atomic/Molecular Massively Parallel
Simulator1) [40] capable of running on large computing
clusters. Energy minimization runs of the system are performed
using a conjugate-gradient algorithm with an energy-
convergence criterion implemented in the LAMMPS
code. A tolerance of relative energies between minimization
iterations is set at 0.0 with a force tolerance of 108 to ensure
a sufficiently minimized system. As a result, energy
minimization is terminated via a line-search criterion triggered
by nominal atomic movement between iterations.
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