Int J Fract
DOI 10.1007/s10704-010-9527-0
ORIGINAL PAPER
Long-term dynamic stability of discrete dislocations
in graphene at finite temperature
M. P. Ariza · M. Ortiz · R. Serrano
Received: 1 February 2010 / Accepted: 22 June 2010
© Springer Science+Business Media B.V. 2010
Abstract We present an assessment of the finitetemperature dynamical stability of discrete dislocations in graphene. In order to ascertain stability,
we insert discrete dislocation quadrupole configurations into molecular dynamics calculations as initial
conditions. In calculations we use Sandia National
Laboratories Large-scale Atomic/Molecular Massively
Parallel Simulator (LAMMPS) and the Adaptive Intermolecular Reactive Empirical Bond-Order (AIREBO)
potential. The analysis shows that the core structures
predicted by discrete dislocation theory are dynamically stable up to temperatures of 2,500 K, though they
tend to relax somewhat in the course of molecular
dynamics. In addition, we find that discrete dislocation
theory accurately predicts energies, though it exhibits
a slight overly-stiff bias.
Keywords Graphene · Discrete dislocations ·
Molecular dynamics · Dynamic stability ·
Finite temperature
M. P. Ariza (B) · R. Serrano
Escuela Técnica Superior de Ingenieros,
Universidad de Sevilla, 41092 Sevilla, Spain
e-mail: mpariza@us.es
R. Serrano
e-mail: rserrano@iat.es
M. Ortiz
Division of Engineering and Applied Science, California
Institute of Technology, Pasadena, CA 91125, USA
e-mail: ortiz@aero.caltech.edu
1 Introduction
Graphene is a one-atom-thick carbon material with
carbon atoms packed densely in a hexagonal honeycomb lattice arrangement, which has been observed
to be stable in two dimensions on a variety of substrates (Novoselov et al. 2004) or in free-standing form
(Meyer et al. 2007). Owing to its anomalous electronic
behavior and remarkable mechanical characteristics
(Novoselov et al. 2005; Silvestrov and Efetov 2007;
Castro Neto et al. 2009; Geim and Novoselov 2007;
Bunch et al. 2007), graphene has recently been identified as a promising novel semiconducting material
with numerous potential applications including chemical sensing instruments, flexible displays, biosensors,
nanomechanical devices, and others.
However, graphene sheets grown in the laboratory
are observed to contain a variety of defects (Rutter
et al. 2007; Hashimoto et al. 2004). Defects can also be
introduced extrinsically by a variety of means including electron-beam irradiation (Hashimoto et al. 2004;
Telling and Heggie 2007; Meyer et al. 2008), adatoms
(Ewels et al. 2002; Li et al. 2005), mono and multivacancies (Xu et al. 1993; Li et al. 2005; Jeong et al.
2008), and others. Conversely, thermal annealing or
chemical treatment can restore graphene to its pristine
defect-free state (Elias et al. 2009; Geim 2009). The
motion of defects has been suggested (Suenaga et al.
2007) to be responsible for the plastic deformation of
single-walled carbon nanotubes at high temperatures.
However, Meyer et al. (2008) have observed a number
123
M. P. Ariza et al.
of defect configurations and their real-time dynamics and have concluded that the dynamics of defects
in two-dimensional graphene sheets is different from
the dynamics of defects closed-shell structures such
as nanotubes or fullerenes. The presence of defects in
graphene modifies its physical and chemical properties and plays a key limiting role in applications. This
deleterious effect motivates the need for a fundamental mechanistic understanding of the equilibrium and
kinetic properties of defects in graphene.
Graphene defects have been analyzed by a variety of means. For instance, Jeong et al. (2008) have
studied the stability of dislocation dipoles with 5–7
core structure using density-functional theory. These
5–7 pairs have been observed to form complex defect
structures (Hashimoto et al. 2004). In addition to firstprinciples calculations, interatomic potentials have
also been widely used for modeling carbon structures in general and graphene in particular (Wirtz and
Rubio 2004; Falkovsky 2008; Mounet and Marzari
2005; Grüneis et al. 2002; Tersoff 1988; Brenner
1990; Aizawa et al. 1990; Stuart et al. 2000; Tewary
and Yang 2009). The simplest types of potential are
harmonic and are defined in terms of force constants (Tersoff 1988; Aizawa et al. 1990; Wirtz and
Rubio 2004; Tewary and Yang 2009). More general bond-order interatomic potentials include the
reactive empirical bond-order REBO potential developed by Brenner (1990). The addition of torsion,
dispersion, and non-bonded repulsion interactions to
the latter potential resulted in a new hydrocarbon potential (AIREBO) that is suitable for studying reactivity in
molecular condensed phases (Stuart et al. 2000).
A different type of analysis of dislocations in graphene has been presented by Ariza and Ortiz (2010).
This analysis relies on the theory of discrete dislocations (Ariza and Ortiz 2005; Ramasubramaniam
et al. 2007) in order to derive closed form analytical solutions of displacement fields and energies of
graphene sheets containing arbitrary distributions of
dislocations. The approach combines lattice harmonics, the theory of eigendeformations and the discrete
Fourier transform in order to formulate expressions of
the stored energy of defective graphene that are analytically tractable. Despite this analytical tractability, the
approach accounts seamlessly for both the long wavelength elastic properties of graphene sheets as well as
for its fine lattice structure. In particular, the theory
predicts dislocation core structures and core energies
123
that are in accord with more refined analyses, including
first principles and molecular dynamics calculations,
and with experiment. The theory also lends itself to the
exact evaluation of limits of interest that are not accessible to direct simulation, such as the limit of dilute
dislocation densities.
Despite these appealing features, the theory of discrete dislocations makes a number of uncontrolled
approximations that must be carefully assessed and
verified. In particular, owing to its reliance on force
constants, it is not possible to determine from within
the theory if the predicted defect structures are dynamically stable over long times at finite temperature. In
this work, we ascertain this question by recourse to
fully-nonlinear molecular dynamics calculations. Specifically, we take the discrete dislocation solutions for
a variety of dislocation configurations as initial conditions for a molecular dynamics calculation based on the
AIREBO potential (Stuart et al. 2000). The calculations
are carried out using the Sandia National Laboratories
LAMMPS code (Plimpton 1995). We find that the discrete dislocation core structures are indeed dynamically
stable over times larger than the relaxation time for the
thermalization of the lattice.
2 The lattice complex of graphene
A general theory of discrete dislocation in crystal lattice, and its specialization to graphene, has been developed elsewhere Ariza and Ortiz (2005), Ariza and Ortiz
(2010), but will stand a brief review in the interest of
completeness. Following Ariza and Ortiz (2005), we
regard the graphene lattice as a cell-complex C, i.e., as
a collection of cells of different dimensions equipped
with discrete differential operators and a discrete integral. In particular, the graphene complex is two-dimensional and consists of: atoms, or 0-cells; atomics bonds,
or 1-cells; and hexagonal cells, or 2-cells, Fig. 1. For
ease of indexing, we denote by Ep (C) the collection of
all cells of dimension p = 0, 1, 2 in the graphene cell
complex C. These cells supply the support for defining
functions, or forms, of different dimensions. Thus, of
dimension p assign vectors to each cell of dimension p
of the lattice. In particular, we refer a function defined
over the atoms as a 0-form, a function defined over the
atomic bonds as a 1-form and a function defined over
the hexagonal cells as a 2-form. As we shall see, forms
provide the vehicle for describing the behavior of the
Long-term dynamic stability of discrete dislocations
Fig. 1 The oriented 0, 1 and 2-cells of graphene grouped by type
2-forms, defined over the hexagonal cells, to vectors.
The discrete differential operators thus defined may be
regarded as the discrete counterparts of the familiar
grad, curl and div of vector calculus. In particular: the
differential of 0-forms is the discrete counterpart of the
grad operator; the differential of 1-forms is the discrete
counterpart of the curl operator; and the differential of
2-forms is the discrete counterpart of the div operator from vector calculus. It is readily verified from the
definition of the discrete differential operators that
d 2 = 0,
Fig. 2 Diagram for the definition of the discrete differential
operators of graphene
graphene lattice, including its displacements, eigendeformations and dislocation densities.
In order to define the discrete differential operators
of the lattice, we begin by oriented all cells, Fig. 1.
Suppose that ω is a 0-form defined over the atoms and
let eab be an atomic bond defined by atoms a and b, cf.
Fig. 2. Suppose, in addition, that eab is oriented from
a to b. Then, the differential dω(eab ) of ω at eab is
dω(eab ) = ω(eb ) − ω(ea ).
(1)
Suppose now that ω is a 1-form defined over the atomic
bonds and let eabcdef be an hexagonal cell bounded by
the atomic bonds eab , ebc , ecd , ede , eef and ef a , cf.
Fig. 2. Then, the differential dω(eab ) of ω at eabcdef is
dω(eabcdef ) = −ω(eab ) + ω(ebc )
which is the discrete counterpart of the identities curl
◦ grad = 0 and div ◦ curl = 0.
The Discrete Fourier Transform (DFT) provides a
natural tool for the analysis of discrete forms, cf., e.g.,
Babuska et al. (1960), Ariza and Ortiz (2005). In order
to define the DFT, we begin by grouping cells of the
same dimension by type, Fig. 1. Thus, cells of the same
type are translations of each other and have the same
complement of neighbors, or environment. According
to this definition, graphene has two types of atoms,
three types of atomic bonds and one type of hexagonal cell. The fundamental property of cells of the same
type is that they are arranged as simple Bravais lattices, Fig. 3. Thus, the atoms of graphene define two
simple Bravais lattices, the atomic bounds define three
simple Bravais lattices, and the hexagonal cells define
one simple Bravais lattice. We recall that the DFT of a
function f defined on a simple Bravais lattice Zn is
f (l)e−iθ·l ,
(5)
fˆ(θ ) =
l∈Zn
where the angle variables θ range over [−π, π ]n . The
DFT admits an inverse given by
1
f (l) =
(6)
fˆ(θ )eiθ·l dθ,
(2π )n
[−π,π ]n
−ω(ecd ) + ω(ede )
−ω(eef ) + ω(ef a ).
(4)
(2)
Finally, ω is a 2-form defined over the hexagonal cells.
Then, its differential is the vector
ω(e2 ).
(3)
dω =
e2 ∈E2 (C )
Thus, the differential operator maps: 0-forms, defined
over the atoms, to 1-forms, defined over the atomic
bonds; 1-forms, defined over the atomic bonds, to
2-forms, defined over the hexagonal cells; and
and has properties similar to those of the Fourier transform, including a discrete Parseval identity and and a
discrete convolution theorem.
3 Eigendeformation theory of discrete lattice
dislocations
It is possible to fashion a theory of discrete dislocations
in crystals from the classical theory of eigendeformations, cf., e.g., Mura (1987). In the present setting, the
123
M. P. Ariza et al.
Fig. 3 The simple Bravais
lattices defined by the
atoms, atomic bonds and
hexagonal cells of graphene
(a)
theory rests on the fundamental property of crystals
that certain uniform deformations leave the crystal lattice unchanged and, hence, should cost no energy. The
entire class of lattice-invariant deformations is characterized by a classical theorem of Ericksen (1979) as the
set of unimodular affine mappings with integer lattice
coordinates. An energy that satisfies this property by
construction is
1 B(e1 , e1 )(du(e1 )
E(u, β) =
2
e1 ∈E1 (C ) e1 ∈E1 (C )
− β(e1 )), (du(e1 ) − β(e1 ))
1
≡ B(du − β), (du − β)
2
(7)
where the sums take place over the atomic bonds of
the crystal lattice and: u(e0 ) is the atomic displacement of atom e0 ; du(e1 ) is the deformation of atomic
bond e1 ; β(e1 ) is the eigendeformation at bond e1 ;
and B(e1 , e1 ) are bond-wise force constants. In (7),
the local values β(e1 ) of the eigendeformation field are
constrained to defining lattice-invariant deformations.
By this restriction and the form of the energy (7), uniform lattice-invariant deformations du cost no energy,
as desired. We note that, owing to the discrete nature of
the set of lattice-invariant deformations, the energy (7)
is strongly nonlinear. In particular, the reduce energy
E(u) = inf E(u, β)
β
(8)
is piecewise quadratic with zero-energy wells at all uniform lattice-invariant deformations.
The lattice-invariant deformations considered in this
work are shown in Fig. 4a. The deformation shears
the graphene
lattice thought a displacement of length
√
|b| = 3a. The corresponding slip systems are shown
in Fig. 4b. Every bond can shear in the direction normal
to itself and in two additional directions at 60◦ to the
normal, namely,
123
(b)
(c)
(a)
(b)
Fig. 4 a Fundamental lattice-preserving shear deformations of
graphene considered in this work; b Resulting slip planes and
Burgers vectors, defining the operative slip systems of graphene
√
3 R(−π/3)dx(e1 ),
√
b2 (e1 ) = 3 R(−π/2)dx(e1 ),
√
b3 (e1 ) = 3 R(−2π/3)dx(e1 ),
b1 (e1 ) =
(9a)
(9b)
(9c)
where R(θ ) denotes a two-dimensional rotation
through an angle θ .
We note from (7) that energy vanishes if the eigendeformations are compatible, i.e., if β = dv for some
atomic displacement field v. Indeed, in that case the
energy is minimized for u = v and the minimum energy
is zero. Hence, the energy at equilibrium, or stored
energy, i.e., the energy that remains stored in the crystal when the displacement field is equilibrated, can only
depend on the degree of incompatibility of the eigendeformations. A measure of that incompatibility is provided by the discrete dislocation density
α = dβ,
(10)
which may be regarded as the discrete curl of the
eigendeformations. Thus, α provides a discrete counterpart of Nye’s dislocation density tensor field (Nye
1953). Since the eigendeformations β are defined on
the atomic bonds, it follows that the discrete dislocations of graphene are defined on the hexagonal cells
Long-term dynamic stability of discrete dislocations
Fig. 5 Basis for the
discrete dislocations of
graphene corresponding to
the lattice-invariant shears
defined in Fig. 4
1 Γ (l − l )α(l ), α(l)
2
2 2
of the lattice, i.e., the discrete dislocation density α
assigns a Burgers vector to every hexagonal cell of the
graphene lattice. From the fundamental property (4) of
the discrete differential operator if follows that
E(α) =
dα = 0,
where the matrix Γ (l) gives the interaction energy
between Burgers vectors (9a) at the origin and at position l on the simple Bravais lattice defined by the hexagonal cells. Alternatively, in terms of the DFT we have
the representation
1
E(α) =
Γˆ (θ )α(θ ), α ∗ (θ ) dθ.
(14)
(2π )2
(11)
i.e., the net Burgers vector of the discrete dislocation
density must vanish. This condition is the discrete counterpart of the classical divergence-free property of the
Nye’s dislocation density tensor field, which may in
turn be regarded as a conservation of Burgers vector
property. A basis for the free-abelian group of the discrete dislocation densities generated by the eigendeformations defined in (9a) and Fig. 4 is shown in Fig. 5.
Every pair of Burgers vectors in Fig. 5 defines an elementary dipole, and an arbitrary discrete dislocation
density may be obtained through an integer linear combination of elementary dipoles.
A theorem of Ariza and Ortiz (2005) shows that perfect lattices, including the graphene lattice (Ariza and
Ortiz 2010), possess a Helmholtz–Hodge decomposition. By this discrete Helmholtz–Hodge decomposition, it follows, in particular, that α = 0 if and only if
β = dv for some displacement field v, i.e., if and only
if the eigendeformations are compatible. Thus, the discrete dislocation density does indeed provide a measure
of the incompatibility of the eigendeformations. It also
follows from the discrete Helmholtz–Hodge decomposition that α is determined by β up to an arbitrary displacement field. From these properties it may be shown
(Ariza and Ortiz 2005)
inf E(u, β) = E(α),
u
(12)
i.e., that the stored energy of a crystal may be written
as a function of the discrete dislocation density. By the
quadratic dependence of the energy (7) on the displacement field it follows that the stored energy must be of
the form
l∈Z l ∈Z
1
≡ Γ ∗ α, α,
2
(13)
[−π,π ]2
Explicit expressions for the influence function Γ in
terms of the force constants of the lattice are given in
Ariza and Ortiz (2010). Again we note that, despite
is harmonic appearance, the stored energy (13) is rendered strongly nonlinear by the constraint that the local
Burgers vectors α(l) must be integer linear combinations of the basic Burgers vectors (9a). This strong nonlinearity renders the determination of low-energy dislocation structures mathematically challenging.
4 Long-term dynamic stability at finite
temperature
The eigendeformation energy (7) relies on a harmonic
approximation, i.e., assumes that the energy is quadratic in the elastic bond deformations du − β. This is
an uncontrolled approximation, i.e., no error bounds are
known at present relative to a fully nonlinear atomistic potential, and, hence, the accuracy of the approximation must be carefully verified. Ariza and Ortiz
(2010) have obtained closed-form analytical solutions
for the displacement field and the energies of general
periodic distributions of dislocations, and compared
the predictions of the theory with experiment and full
123
M. P. Ariza et al.
Fig. 6 Discrete lattice Z in the angle-variable Brillouin zone
[−π, π ]2 corresponding to periodic functions
atomistic calculations for dipolar and quadrupolar dislocation arrangements. In particular, the calculation of
the displacement fields and energies of periodic distributions of dislocations can be reduced to finite sums.
For instance, if the periodic cell of the discrete dislocation distribution is defined by lattice vectors (A1 , A2 ),
the energy then follows explicitly as
1 1
Γˆ ()α̂(), α̂ ∗ (),
(15)
|Z|
2
∈Z
where Z is the intersection of the reciprocal lattice
of (A1 , A2 ) and the angle-variable Brillouin zone
[−π, π ]2 . The corresponding displacement fields follow likewise from similar finite sums (Fig. 6).
Based on these explicit closed-form solutions, Ariza
and Ortiz (2010) have provided a detailed assessment
of the core structures and energies of discrete dislocations in dipolar and quadrupolar arrangements and
for the force-constant model of Aizawa et al. (1990).
The eigendeformation field of a quadrupole is shown in
Fig. 7 and consists of slip over intervals on two parallel
planes. Sample deformed configurations of the periodic
cell predicted by discrete dislocation theory are shown
in Fig. 8 (reproduced from Ariza and Ortiz (2010) for
completeness). The discrete-dislocation cores exhibit
pentagon-heptagon ring (5–7) core structures similar to
those found in dipoles. This structure is consistent with
Fig. 8 Quadrupole configurations. Discrete dislocations solutions using the Aizawa et al. (1990) potential (reproduced from
Ariza and Ortiz 2010). a 448 atom period cell. b 1,144 atom
periodic cell
the observations of Hashimoto et al. (2004) of pairs of
pentagon-heptagons attached to a missing row of atoms
in a zig–zag chain in electron-beam irradiated singlewalled carbon nanotube of large diameter. Jeong et al.
(2008) performed density functional theory calculations
of graphene sheets containing zig–zag chains of vacancies of different lengths. After atomic relaxation, they
observed the formation of two 5–7 pair defects at both
ends of the missing chain. This structure arises when
Fig. 7 Periodic
quadrupolar arrangement of
discrete dislocations, unit
periodic cell. Distribution of
eigendeformations βi (e1 )
defining one quadrupole,
consisting of two constant
and opposite Burgers
vectors over a zig–zag chain
of 1-cells
(a)
123
(b)
Long-term dynamic stability of discrete dislocations
Fig. 9 Quadrupole
configuration. LAMMPS
(Plimpton 1995) molecular
dynamics calculations using
the AIREBO (Stuart et al.
2000) potential.
Time-averaged
configurations. a 1,000 K;
b 1,500 K; c 2,000 K;
d 2,500 K
the number of vacancies is eight or more. In addition,
dislocation core energy predicted by discrete dislocation theory may be compared to the formation energy of
Stone-Wales defects (Stone and Wales 1986). This comparison shows that the core energies and bond-rotation
angles predicted by discrete dislocation theory are in the
ball park of full atomistic models (Kaxiras and Pandey
1988; Xu et al. 1993; Li et al. 2005; Los et al. 2005) and
experimental observation (Meyer et al. 2008).
This agreement notwithstanding, the harmonic
approximation underlying the energy (7) precludes
ascertaining from within the theory whether the predicted defect structures are dynamically stable at finite
temperature. In order to ascertain this question, we verify that the discrete dislocation configurations remain
stable when used as initial conditions in a finite-temperature molecular dynamics calculation using a full atomistic potential. To this end, we resort to Sandia National
Laboratories Large-scale Atomic/Molecular Massively
Parallel Simulator (LAMMPS) (Plimpton 1995) using
the (AIREBO) potential (Stuart et al. 2000). The characteristic thermalization time required for the system
to reach equilibrium may be estimated from the twodimensional heat equation as
3 kB
N
(16)
2 KA
where N is the number of atoms, kB is Boltzmann’s constant, KA is the two-dimensional thermal
conductivity of graphene. This two-dimensional thermal conductivity may in turn be estimated from
the experimentally reported three-dimensional thertc =
mal conductivity KV as KA = KV h, where h is a
nominal thickness of the graphene sheet. For a periodic cell of N = 3,200 atoms, with kB = 1.381 ×
J , K = 5 × 103 W (Fuhrer et al. 2010) and
10−23 K
V
mK
h = 0.334 nm (Baskin and Meyer 1955), equation (16)
gives tc ≈ 0.04 ps. The initial discrete dislocation configuration is allowed to relax over a time of 1 ps, which,
according to the preceding estimate, amply suffices for
the system to reach thermal equilibrium. Figure 9 compiles time-averaged atomic configurations of a quadrupole at temperatures of 1,000, 1,500, 2,000 and 2,500 K,
obtained using a time step of 10−4 ps. As may be seen
from the figure, the predicted discrete core structure is
dynamically stable up to 2,500 K.
A comparison of the time averaged molecular
dynamics and discrete dislocation configurations at
zero temperature are shown in Fig. 10. As may be
seen from these comparisons, a general relaxation of
the initial core structures is observed in the molecular
dynamics calculations, but the relaxation is modest and
strongly localized. In particular, the initial core structure remains unchanged and remains stable over long
periods of time. In addition, the potential energy of
78.01 eV of a quadrupolar arrangement of discrete dislocations in a 1,144 atom periodic cell may be compared with the values 92.43 and 68.76 eV obtained
from molecular dynamics before and after relaxation,
respectively. The energy discrepancies may be regarded
as modest in consideration of the approximations made
in discrete dislocation theory and the accuracy limitations of empirical potentials.
123
M. P. Ariza et al.
Fig. 10 Quadrupole
configuration. Comparison
of: LAMMPS (Plimpton
1995) molecular dynamics
calculations using the
AIREBO (Stuart et al. 2000)
potential at zero temperature
(dark color); and discrete
dislocations solutions using
the Aizawa et al. (1990)
potential (light color).
a 448 atom period cell.
b 1,144 atom periodic cell.
c 3,200 atom periodic cell.
d Zoom of 3,200 atom
periodic cell
(a)
(c)
5 Summary and conclusions
We have presented an assessment of the finite-temperature dynamical stability of discrete dislocations
in graphene. The assessment is based on inserting
the discrete dislocation configurations into molecular dynamics calculations as initial conditions. In
particular, we use Sandia National Laboratories Largescale Atomic/Molecular Massively Parallel Simulator
(LAMMPS) (Plimpton 1995) and the Adaptive Intermolecular Reactive Empirical Bond-Order (AIREBO) potential (Stuart et al. 2000). The analysis shows
that the core structures predicted by discrete dislocation theory are indeed dynamically stable up to
temperatures of 2,500 K, though they tend to relax
somewhat. In addition, discrete dislocation theory is
somewhat stiff and over predicts core energies, though
the size of the discrepancy is modest and no worse
than discrepancies arising from different empirical
potentials.
Despite the robustness and perhaps better-thanexpected accuracy of discrete dislocation theory, a
question of interest concerns the formulation of convergent schemes that relax discrete dislocations in
accordance with a full atomistic potential. A particular
scheme that preserves the advantages of discrete dislocation theory, and in particular the ability to use Green’s
functions, was proposed by Gallego and Ortiz (1993).
In this scheme, the force constants that define the
123
(b)
(d)
energy in the discrete dislocation theory are obtained by
linearization of an empirical potential. The fully nonlinear solution is then obtained by the method of forces,
i.e., by appending unknown forces to the discrete dislocation energy so as to equilibrate the lattice with respect
to the fully nonlinear atomistic potential. Because of
the good starting accuracy of the discrete dislocation
theory, the corrective forces decay very rapidly away
from the core of defects and, hence, represent highly
localized corrections.
Finally, the extension of discrete dislocation theory
to dynamics and finite temperature may be effected,
e.g., by recourse to Langevin dynamics and Metropolis
equilibrium thermodynamics. In both these cases, the
dislocation density can be evolved by means of energydecreasing eigendeformation flips. Each flip consists
of the addition or subtraction of an elementary dipole to
the graphene lattice, Fig. 5. Conditions that ensure that
the flip is energy-decreasing are provided in Ramasubramaniam et al. (2007). In this manner, a discrete dislocation dynamics can be effectively formulated. These
and other enhancements of the theory suggest fruitful
avenues for further research.
Acknowledgments We gratefully acknowledge the support
of the Ministerio de Educación y Ciencia of Spain (DPI200605045), the support of the Consejería de Innovación of Junta de
Andalucía (P06-TEP1514) and the support of the Department of
Energy National Nuclear Security Administration under Award
Number DE-FC52-08NA28613 through Caltech’s ASC/PSAAP
Long-term dynamic stability of discrete dislocations
Center for the Predictive Modeling and Simulation of High
Energy Density Dynamic Response of Materials.
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