Disclination ensembles in graphene
We consider graphene disclination networks (DNs) — periodic distributions of disclination defects. Disclinations manifest themselves as 4-, 5-, 7- or 8-member carbon rings in otherwise 6-member ring ideal 2D graphene crystal lattice. Limiting cases of graphene-like 2D carbon lattices without 6-memb...
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| Опубліковано в: : | Физика низких температур |
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| Дата: | 2018 |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України
2018
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Disclination ensembles in graphene / М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, А.Е. Romanov // Физика низких температур. — 2018. — Т. 44, № 9. — С. 1171-1179. — Бібліогр.: 54 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859649458244943872 |
|---|---|
| author | Rozhkov, М.А. Kolesnikova, А.L. Yasnikov, I.S. Romanov, А.Е. |
| author_facet | Rozhkov, М.А. Kolesnikova, А.L. Yasnikov, I.S. Romanov, А.Е. |
| citation_txt | Disclination ensembles in graphene / М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, А.Е. Romanov // Физика низких температур. — 2018. — Т. 44, № 9. — С. 1171-1179. — Бібліогр.: 54 назв. — англ. |
| collection | DSpace DC |
| container_title | Физика низких температур |
| description | We consider graphene disclination networks (DNs) — periodic distributions of disclination defects. Disclinations manifest themselves as 4-, 5-, 7- or 8-member carbon rings in otherwise 6-member ring ideal 2D graphene
crystal lattice. Limiting cases of graphene-like 2D carbon lattices without 6-member motives, i.e., pseudographenes, are also studied. The geometry and energy of disclinated 2D carbon configurations are analyzed with the
help of molecular dynamics (MD) simulation technique. A comparison of the obtained MD results with analytical
calculations within the framework of the theory of defects of elastic continuum is presented.
Розглянуто дисклінаційні сітки (DNs) — періодичні розподіли дисклінаційних дефектів у графені. Дисклінації проявляють себе як 4-, 5-, 7- або 8-членні вуглецеві кільця на відміну від 6-ланкових кілець, з яких складається двовимірна 2D ідеальна гратка графена. Також досліджено граничні ви-падки графеноподібних 2D вуглецевих граток без 6-ланкових кілець — так звані псевдографени. Геометрія та енергія дисклінованих 2D-вуглецевих конфігурацій аналізуються за допомогою метода молекулярної динаміки (MD). Наведено порівняння результатів MD моделювання та аналітичних розрахунків в рамках теорії дефектів пружного континууму.
Рассмотрены дисклинационные сетки (DNs) — периодические распределения дисклинационных дефектов в графене.
Дисклинации проявляют себя как 4- , 5-, 7- или 8-членные
углеродные кольца в отличие от 6-звенных колец, из которых
состоит двумерная (2D) идеальная решетка графена. Также
исследуются предельные случаи графеноподобных 2D углеродных решеток без 6-звенных колец — так называемые псевдографены. Геометрия и энергия дисклинированных 2D углеродных конфигураций анализируются с помощью метода
молекулярной динамики (MD). Представлено сравнение результатов MD моделирования и аналитических расчетов, проведенных в рамках теории дефектов упругого континуума.
|
| first_indexed | 2025-12-07T13:31:49Z |
| format | Article |
| fulltext |
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9, pp. 1171–1179
Disclination ensembles in graphene
М.А. Rozhkov1, А.L. Kolesnikova1,2, I.S. Yasnikov3, and А.Е. Romanov1
1ITMO University, 49 Kronverkskiy Pr., St. Petersburg 197101, Russia
E-mail: alexey.romanov@niuitmo.ru
2Institute of Problems of Mechanical Engineering RAS, 61 Bolshoj Pr., Vas. Ostrov, St. Petersburg 199178, Russia
3Togliatti State University, 14 Belorusskaya Str., Togliatti, 445020, Russia
Received 10 April, 2018, published online July 26, 2018
We consider graphene disclination networks (DNs) — periodic distributions of disclination defects. Discli-
nations manifest themselves as 4-, 5-, 7- or 8-member carbon rings in otherwise 6-member ring ideal 2D graphene
crystal lattice. Limiting cases of graphene-like 2D carbon lattices without 6-member motives, i.e., pseudo-
graphenes, are also studied. The geometry and energy of disclinated 2D carbon configurations are analyzed with the
help of molecular dynamics (MD) simulation technique. A comparison of the obtained MD results with analytical
calculations within the framework of the theory of defects of elastic continuum is presented.
Keywords: graphene; pseudo-graphene; disclination; disclinated carbon ring; disclination quadrupole;
disclination network; molecular dynamics.
1. Introduction
With the discovery and mass fabrication of graphene
[1,2] and with a large number of experimental studies of
graphene structure, see for example [3–18], the theoretical
interest to 2D atomic crystals has grown considerably.
Along with the analysis of the properties of ideal graphene
lattice containing only 6-member carbon rings (hexagons) it
was found that various defects exist in graphene and
graphene-like carbon lattices, i.e. rings in the form of square,
pentagon, heptagon, or octagon [19–21]. Big efforts were
then spent to the understanding the behavior of graphene
with defect walls and chains, i.e., polycrystalline graphenes
with grain and intercrystallite boundaries [18–34]. In partic-
ular, the effect of the localized defects on the physical and
mechanical properties of graphene was analyzed. The stud-
ies of defects distributed throughout the graphene sheet have
so far been less developed [35–37]. In the limiting case of a
dense packing of pentagons with octagons or heptagons in
graphene, two 2D carbon modifications (pseudo-graphenes)
were described: pentagon–octagon (PO) graphene [35] and
phagraphene [37].
The main technique to model graphene and other 2D
crystals with defects and without them is molecular dy-
namics (MD) simulation, e.g. see Refs. 28, 33, 34. Within
MD approach, the information about equilibrium atomic
configurations and the energy of these configurations can
be delivered. The other known approach to investigate de-
fects in 2D crystals operates with the analytical methods of
the theory of defects in solids [38–43].
Important feature of defects, which are possible in
graphene lattice, is their intrinsic connection to disclinations
— defects of rotational type [44,45]. Using disclination no-
menclature, 4-, 5- or 7-, 8-member rings are viewed as
disclinated rings and are classified as the cores of positive or
negative wedge disclinations, respectively [45].
In the present work, we report on the results of model-
ing graphene and graphene-like carbon structures with dis-
tributed disclinated rings utilizing both methods of MD
simulation and theory of elasticity for 2D solid structures.
2. Background
Low-dimensional systems in the condensed matter phys-
ics have always provoked genuine interest among research-
ers. Wherein an analysis of their defective structure is a hot
topic in scientific periodicals. For example in Refs. 38, 41, 42
Natsik and Smirnov presented the theoretical study of the
properties of intrinsic dislocation- and crowdion-type struc-
tural defects in 2D crystals. The results obtained by using the
continual theory were improved by comparing with the re-
sults of numerical analysis by the methods of MD simulation
© М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, and А.Е. Romanov, 2018
М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, and А.Е. Romanov
of atomic structure of dislocations and crowdions in a hexa-
gonal lattice 2D crystals.
In addition to concept of dislocations, pioneering ideas
on the disclinations in 2D crystals have been outlined four
decades ago by Harris [46]. In this sense, the use of the
disclination concept in two-dimensional hexagonal graphene
lattice seems reasonable. Typical defects in 2D hexagonal
graphene lattice — square, pentagonal, heptagonal, octago-
nal carbon rings- and their ensembles such as internal
boundaries and two-dimensional distributions were success-
fully described by wedge disclinations [19,33,34]. In addi-
tion, disclinations can move 2D flat crystal into the third
dimension, thereby lowering the energy of elastic distor-
tions, as it occurs in fullerene macromolecules [47].
In the theory of defects in 3D solids, two types of lin-
ear defects, namely, dislocations as carriers of transla-
tional deformation modes and disclinations that are re-
sponsible for rotational deformation modes, are
distinguished [44]. Despite the fact that the concept of
disclinations was introduced by Vito Volterra into me-
chanics in solids in 1907, the approach based on the anal-
ysis of rotational deformation modes in real crystals actu-
ally revealed itself only at the end of the last century [45].
It should be noted that the disclination approach is effec-
tive for describing the properties of 3D crystals in the
form of small particles and microcrystals with pentagonal
symmetry [48,49].
Fig. 1. (Сolor online) Volterra’s procedure for the formation of wedge disclinations in 2D hexagonal crystal lattice: (a) negative
disclination and associated 7-member ring; (b) negative disclination and associated 8-member ring; (c) positive disclination and asso-
ciated 5-member ring; (d) positive disclination and associated 4-member ring. Minimal magnitude of disclination strength in hexagonal
lattice is ω = π/3. Negative and positive disclinations are denoted by empty and black triangles, respectively (adopted from [34]).
1172 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9
Disclination ensembles in graphene
Volterra’s procedure for the formation of wedge
disclinations in 2D hexagonal crystal lattice is presented
in Fig. 1. Wedge disclinations are formed by inserting or
removing a wedge of 60 or 120 degrees from a hexagonal
lattice [34,44–46], leaving in the vertex of this wedge, 7-,
5-, 8- or 4-member carbon rings, in another words,
disclinated rings. The strength (or charge) of the wedge
disclination ω is determinate by the magnitude of the
wedge angle: ω = – π/3, + π/3, – 2π/3, + 2π/3.
It is known, that single disclination introduces global dis-
tortion in the crystal lattice, see for example Refs. 44, 45,
and, according to the continual theory of disclinations, its
energy for plane elasticity depends quadratically on the
characteristic size of the crystal [44]. In particular, energy E
of the wedge disclination in the center of elastically isotropic
disk obeys formula [44]:
2 2
0
1
8
E D R= ω , (1)
where ω is the strength of the disclination, 0R is the radius
of the disk, (1 )/2D G= + ν π for a 2D disk [38,40,44], G
is the shear modulus in units Force/Length, and ν is Pois-
son ratio.
Disclinations in solids are realized in the form of self-
screening ensembles, i.e., dipoles and quadrupoles [44,45]. In
graphene, self-screening ensembles of disclinations can be
present in the form of grain boundaries and intercrystallite
boundaries, i.e. in the form of linear defects, see in details in
Refs. 18, 27, 33, 34, 40.
3. Numerical and analytical methods used
In this paper, we utilize the method of molecular dy-
namics (MD) simulation as a numerical approach, the re-
sults of which are also used as an input for analytical mod-
elling in the framework of the disclination theory. We
would like to answer the question whether we can estimate
the energy of disclination configurations in graphene using
formulas for the energies of screened disclinational config-
urations without any additional MD simulation. In this
sense, the sharing and comparison of the results of two
independent methods such as theoretical and numerical
approaches give an algorithm for choosing a method for
solving a particular class of problems when describing
graphene-like configurations.
MD simulations of ideal graphene and graphene with
disclinated carbon rings were performed with LAMMPS
software package [50]. The interatomic interactions were
described by the adaptive AIREBO potential [51]. The
post-processing and images of equilibrium atomic struc-
tures were produced with software package OVITO [52].
The MD simulation was performed at zero temperature,
and Polak–Ribiere version of the conjugate gradient algo-
rithm for energy minimization was used [53].
To find the energies of disclination ensembles in
graphene in the framework of the analytical approach one
can use the results of Ref. 54 for energy NE of N
disclinations in an elastic disk. In Fig. 2 the geometrical
scheme for calculation of energy NE is shown. In such a
geometry NE is expressed by the following formula:
____________________________________________________
2 2(1 ) 1
8 2
N
i i
N
i
R rGE
R
ω+ ν = − + π
∑
2 2 2 2 2
2 2 2 2 2
2 2 2 4 2
1 1
( 2 cos )
( 2 cos ) ln
2 cos
N N
i j i j ij i j
i j i j j j ij i j
i j i i j i j ij
R r r r r r r
r r r r R r r
r r R r r R R= = +
+ − θ
+ ω ω + − θ + − − +
− θ +
∑ ∑ , (2)
_______________________________________________
where iω is a strength of the i-disclination; ir is a dis-
tance between center of disk and i-disclination; ijθ is an
angle between radiuses of i- and j- disclinations; R is a
radius of the disc. In Eq. (2) we take into account that the
disk is infinitely thin, i.e., is a 2D solid.
4. Results and discussion
The essence of our MD simulation is as follows. In
Fig. 3, MD modelled graphene-like sheets with the most
dense networks of disclinations are presented. These 2D
crystals cannot be called “graphene”, because they have
Fig. 2. Schematics for calculating energy of the disclination en-
semble in an elastic disk.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9 1173
М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, and А.Е. Romanov
either very few 6-member atomic rings characteristic for
hexagonal graphene lattice (Fig. 3(a)), or do not have them
at all (Figs. 3(b)–(d)). They can be better classified as
graphene-like carbon structures or pseudo-graphenes.
One of the crystals composed with 5- and 7-member car-
bon rings (Figs. 3(a),(b)) is phagraphene [37] (Fig. 3(a)), in
which 6-member rings, usual for graphene, are required for
joining disclinated rings. The disclination strengths ω in
phagraphene and crystal “5–7 B” (Fig. 2(a),(b)) are + π/3 and
– π/3. Pseudo-graphene, composed with 5- and 8-member
carbon rings (Fig. 2(b)), is pentagon–octagon (PO) graphene
[35], where ω = + π/3 and – 2π/3. Pseudo-graphene, com-
posed with 4- and 8-member carbon rings is presented in
Fig. 3(c). In this case, ω = +2π/3 and – 2π/3. In our classifica-
tion, previously adopted for structural units in graphene and
linear defects composed of them [33,34], these crystals have
the designations “5–7 A”, “5–7 B”, “5–8–5 D” and “4–8”
(Fig. 3).
The pseudo-graphenes, considered here, can be con-
structed using the linear defects of 2D hexagonal lattice. For
example, phagraphene can be constructed from the favorite
symmetric grain boundaries “docked” to each other [27,40]
and PO graphene can be composed from linear defects first
described in Ref. 18 and then also modelled in Ref. 34, and
pseudo-graphene “4–8” can be composed from linear de-
fects “4–8”, introduced and described in Ref. 34.
In Fig. 4, the differences between the average energies
per atom for the pseudo-graphenes ae and the ideal
graphene 0
ae are presented: 0
a a ae e e∆ = − . In diagram,
zero energy is the energy per atom for the ideal graphene
0
ae . On the one hand, when aρ and 0
aρ are the atomic den-
sities of pseudo-graphene and the ideal graphene, corre-
spondingly, the energy 0 0
a a a ae e e∆ = ρ − ρ is the difference
in the energies of the pseudo-graphene and graphene per
unit area of the crystal. Therefore e∆ can be treated as the
average energy per unit area of the disclination network
(DN) DNe embedded into the graphene crystal (Fig. 3).
The energy DNe can be also found with the analytical
formulas of disclination theory, i.e., Eq. (2), for each of
studied pseudo-graphenes. To do this, the self-screened
DN should be chosen. If ensemble of N disclinations sat-
isfies the following conditions: zero disclination charge
and zero disclination dipole moment, then it is self-
screened configuration, and its energy does not depend
on the external screening parameter R. The simplest self-
screening disclination ensembles are quadrupoles in the
forms of a rectangle or line, and their energies are known,
Fig. 3. (Сolor online) Pseudo-graphene crystals with disclination
networks (DNs). Red circles denote carbon atoms. Empty and black
triangles denote negative and positive disclinations, respectively.
Fig. 4. Energy of the modeled pseudo-graphenes.
Fig. 5. Self-screened disclination quadrupoles. Parallelogram (a),
special cases of parallelogram: a rectangle (b), a rhombus (c), a quad-
rate (d), and line quadrupoles (e), (f) as degenerate parallelograms.
1174 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9
Disclination ensembles in graphene
e.g., see Refs. 44, 45. Additional analysis of Eq. (2)
shows that the most general self-screening ensembles,
i.e., those with the energies that do not depend on the
external parameter R, are quadrupoles in the form of par-
allelograms and their particular cases (Fig. 5). These
quadrupoles can be recognized in graphene structures as
repetitive self-screening ensembles, and hence their ener-
gies should be used to calculate the energy of disclination
networks DNE as a whole.
The energies of quadrupoles, shown in Fig. 5, have the
following algebraic representations:
____________________________________________________
(a) for the parallelogram (Fig. 5(a))
42
2 1
par 1 2 2 2 2
1 2 1 2 1 1 2 1 2 1
16(1 ) ln
4 ( 2 cos )( 2 cos )
rGE r
r r r r r r r r
+ ν ω
= +π + − θ + + θ
4 2 2
2 2 1 2 1 2 1
2 1 2 12 2 2 2 2 2
1 2 1 2 1 1 2 1 2 1 1 2 1 2 1
16 2 cos
ln 2 cos ln
( 2 cos )( 2 cos ) 2 cos
r r r r r
r r r
r r r r r r r r r r r r
+ − θ
+ + θ
+ − θ + + θ + + θ
; (3a)
(b) for the rectangle (Fig. 5(b))
( )
2 2
1
rec 1 1 1 1
(1 )
ln 4 (1 cos ) ln(1 cos ) (1 cos ) ln(1 cos )
2
G r
E
+ ν ω
= − − θ − θ − + θ + θ =
π
2 2 2 22
2 21 2 1 2
1 22 2
1 2
(1 ) ln ln
a a a aG a a
a a
+ ++ ν ω
= + π
; (3b)
(c) for the rhombus (Fig. 5(c))
2 22
2 21 2
rhomb 1 22 2 2 2
1 2 1 2
4 4(1 ) ln ln
2 ( ) ( )
r rGE r r
r r r r
+ ν ω
= + π + +
; (3c)
(d) for the square (Fig. 5(d))
2 22 2
1
qudr
2 (1 )(1 )
ln 2 ln 2
G aG r
E
+ ν ω+ ν ω
= =
π π
; (3d)
(e) for the line quadrupole (Fig. 5(e))
2 22
2 21 2 1 2
1 2 1 22 2 2 2
1 2 1 2 1 2
4 4 ( )(1 ) ln ln 2 ln
2 ( ) ( ) ( )
lq
r r r rGE r r r r
r r r r r r
++ ν ω
= + − = π − − −
2 2 2 22
2 22 1 2 1 2 1
1 2 1 2 1 2 2 12 2
1 2 2 1
(1 ) ln ln 2 ln , ,
a a a a a aG a a a a r r a a
a a a a
− − ++ ν ω
= + + > > π −
; (3e)
(f) for the line quadrupole (Fig. 5(f))
2 22 2
1 4 (1 )(1 )
ln 2 ln 2lq
G aG r
E
+ ν ω+ ν ω
= =
π π
. (3f)
_______________________________________________
Formulas (3(b), (d)–(f) were originally given in Ref. 44.
For each crystal with a periodic DN, a suitable
disclination quadrupole can be determined for calculating
DN energy per unit area DNe . For example, for
phagraphene (Fig. 3(a)) this is the disclination quadrupole
in the form of the parallelogram (Fig. 5(a)), for structure
“5–7 B” (Fig. 3(b)) this is the rhombus (Fig. 5(c)), for
structure “5–8–5 D” (Fig. 3(c)) this is the line quadrupole
(Fig. 5(f)), and for structure “4–8” (Fig. 3(d)) this is the
square (Fig. 5(d)).
In Fig. 6, the square DNs originated from Fig. 3(d) are
presented for various motive periods. In Fig. 7, the average
energies per atom for graphene with periodic alternating
DNs, see Figs. 6(a)–(e), are shown. Crystal marked “4–8 g0”
is a pseudo-graphene “4–8”. It follows from the diagram that,
with the exception of the tightly packed structure “4–8 g0”,
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9 1175
М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, and А.Е. Romanov
the average energy per atom in the crystal depends weakly on
the period of the DN.
In Fig. 8, the average energy per unit area for structures
in Fig. 6, as a function of the square of the DN period is
shown. The energies are normalized to the energy of a
tightly packed structure “4–8 g0”. The dependences are
found from MD simulations (Fig. 8(a), blue dots), calculat-
ed with Eq. (3d) (Fig. 8(a), red line), and calculated with
Eq. (2) adopted to 4 quadrupoles (Fig. 8(b), grey line). In
disclination scheme (Fig. 8(b)), the area related to the
quadrupole when calculating the DN energy is highlighted.
Along with the investigation of the quadratic DNs shown
in Fig. 6, we studied networks containing quadrupoles of
disclinations with charges ω = +2π/3 and – 2π/3, which size
is the smallest possible in a graphene crystal. In Fig. 7, the
single disclination quadrupole in the graphene crystal and its
possible formation scheme are given. It can be seen that
such a single quadrupole has the shape of a rhombus, be-
cause the sizes of the 4-member and 8-member rings, which
are the nuclei of disclinations with ω = +2π/3 and – 2π/3,
respectively, are significantly different. The elastic distor-
tions induced by the quadrupole in the graphene lattice de-
cay rapidly over a distance of the order of the average size of
the quadrupole (Fig. 9(b)).
Fig. 6. (Сolor online) Networks of 4-member and 8-member car-
bon rings in graphene as a periodic structures of disclinations of
stregth ω = +2π/3 and –2π/3.
Fig. 7. Energy per carbon atom for graphenes with periodic struc-
tures of alternating disclinations, given in Fig. 6. Crystal marked
“4–8 g0” is a pseudo-graphene “4–8”.
Fig. 8. (Сolor online) Energy per unit area for structures “4–8”, as a
function of the square of the disclination network (DN) period. The
blue dots correspond to the energies calculated with the help of MD
simulation; red (1) and grey (2) lines correspond to the dependenc-
es, calculated analytically taking into account 1 and 4 disclination
quadrupoles, correspondingly. The energies are normalized to the
energy of a tightly packed structure “4–8 g0” shown in Fig. 5(a). In
disclination scheme (b), the area related to the quadrupole used in
analytical calculation of DN energy, is highlighted.
1176 Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9
Disclination ensembles in graphene
In Fig. 10 the periodic structures of small disclination
quadrupoles containing 4- and 8-member rings, named
“4–8” quadrupoles, are shown. As can be seen from
Fig. 10(a), tightly packed quadrupoles take the form of
the quadrats. Quadrat quadrupoles “4–8” of large size are
also observed in the DNs shown in the Figs. 6(b)–(e).
Here they are also tightly packed, i.e. the size of the
quadrupole is DN half-period.
With increasing period of the quadrupole network (QN)
their transformation is observed: they evolve from square
(Fig. 10(a)) to parallelogram (Figs. 10(b)–(e). This is ex-
plained by considering the QN as composed of linear
chains of quadrupoles “4–8” and 6-member carbon rings. It
has been shown in the Ref. 34 that the intercrystallite
boundary composed of only quadrupoles 4–8 does not in-
troduce a misorientation of the neighboring crystals. Ad-
ding 6-member rings between the quadrupoles in the
intercrystallite boundary “4–8” leads to the appearance of
the misorientation angle in interval 0°–60°. This is similar
to the phenomenon found in the study of the “5–7” grain
boundaries in graphene [40].
In Fig. 11, the diagram of the energy per atom for
graphene with periodic QNs, given in Fig. 10, is shown. As
expected, with an increase of the period of QN, the average
energy per atom of the disclinated crystal decreases.
5. Summary and Conclusions
Resulting from our research, we formulate the following.
(i) The average energy of graphene with alternating
disclination networks (DNs) remains practically unchanged
with increasing DN period. The exceptions are the crystals
with the densest DNs. These crystals contain a minimal
number of 6-member carbon rings typical for ideal graphene,
or do not have them at all. It is correct to call such 2D carbon
crystals pseudo-graphenes. Pseudo-graphenes are low-energy
Fig. 9. (Сolor online) Quadrupole of wedge disclinations of
strength ω = +2π/3 and –2π/3 in graphene. (a) The sequential pro-
cess of quadrupole formation by inserting two carbon atoms into
a graphene lattice. (b) Disclination scheme of the quadrupole.
Fig. 10. (Сolor online) Networks of disclination qudrupoles in
pseudo-graphene and graphene-like structures.
Fig. 11. Energy per atom for graphene-like 2D structures with
periodic ensembles of disclination quadrupoles, given in Fig. 10.
Crystal marked “4–8 g0” is a pseudo-graphene “4–8”.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9 1177
М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, and А.Е. Romanov
containing disclination defects configurations. The energies
of pseudo-graphenes “5–7 A” and “5–7 B” exceed the energy
of an ideal graphene by only 0.28–0.38 eV/atom.
(ii) For an approximate estimate of the energy of
graphene with embedded alternating DNs, one can use the
analytical formulas for the energy of a single disclination
quadrupole in the form suitable for a given DN.
Acknowledgement
The work was supported by the Russian Foundation for
Basic Research (grant No 18-01-00884).
_______
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___________________________
Ансамблі дисклінацій у графені
М.А. Рожков, А.Л. Колеснікова,
І.С. Ясніков, А.Е. Романов
Розглянуто дисклінаційні сітки (DNs) — періодичні роз-
поділи дисклінаційних дефектів у графені. Дисклінації про-
являють себе як 4-, 5-, 7- або 8-членні вуглецеві кільця на
відміну від 6-ланкових кілець, з яких складається двовимірна
2D ідеальна гратка графена. Також досліджено граничні ви-
падки графеноподібних 2D вуглецевих граток без 6-ланкових
кілець — так звані псевдографени. Геометрія та енергія дис-
клінованих 2D-вуглецевих конфігурацій аналізуються за
допомогою метода молекулярної динаміки (MD). Наведено
порівняння результатів MD моделювання та аналітичних
розрахунків в рамках теорії дефектів пружного континууму.
Ключові слова: графен, псевдографен, дисклінація, дисклі-
новане вуглецеве кільце, дисклінаційний квадруполь, сітка
дисклінацій, молекулярна динаміка.
Ансамбли дисклинаций в графене
М.А. Рожков, А.Л. Колесникова,
И.С. Ясников, А.Е. Романов
Рассмотрены дисклинационные сетки (DNs) — периодиче-
ские распределения дисклинационных дефектов в графене.
Дисклинации проявляют себя как 4- , 5-, 7- или 8-членные
углеродные кольца в отличие от 6-звенных колец, из которых
состоит двумерная (2D) идеальная решетка графена. Также
исследуются предельные случаи графеноподобных 2D угле-
родных решеток без 6-звенных колец — так называемые псев-
дографены. Геометрия и энергия дисклинированных 2D угле-
родных конфигураций анализируются с помощью метода
молекулярной динамики (MD). Представлено сравнение ре-
зультатов MD моделирования и аналитических расчетов, про-
веденных в рамках теории дефектов упругого континуума.
Ключевые слова: графен, псевдографен, дисклинация, дискли-
нированное углеродное кольцо, дисклинационный квадруполь,
сетка дисклинаций, молекулярная динамика.
Low Temperature Physics/Fizika Nizkikh Temperatur, 2018, v. 44, No. 9 1179
http://lammps.sandia.gov/
https://doi.org/10.1063/1.481208
http://www.ovito.org/
https://doi.org/10.1134/S1063783416060342
https://doi.org/10.1134/S1063783416060342
1. Introduction
2. Background
3. Numerical and analytical methods used
4. Results and discussion
5. Summary and Conclusions
Acknowledgement
|
| id | nasplib_isofts_kiev_ua-123456789-176255 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0132-6414 |
| language | English |
| last_indexed | 2025-12-07T13:31:49Z |
| publishDate | 2018 |
| publisher | Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| record_format | dspace |
| spelling | Rozhkov, М.А. Kolesnikova, А.L. Yasnikov, I.S. Romanov, А.Е. 2021-02-04T07:53:05Z 2021-02-04T07:53:05Z 2018 Disclination ensembles in graphene / М.А. Rozhkov, А.L. Kolesnikova, I.S. Yasnikov, А.Е. Romanov // Физика низких температур. — 2018. — Т. 44, № 9. — С. 1171-1179. — Бібліогр.: 54 назв. — англ. 0132-6414 https://nasplib.isofts.kiev.ua/handle/123456789/176255 We consider graphene disclination networks (DNs) — periodic distributions of disclination defects. Disclinations manifest themselves as 4-, 5-, 7- or 8-member carbon rings in otherwise 6-member ring ideal 2D graphene crystal lattice. Limiting cases of graphene-like 2D carbon lattices without 6-member motives, i.e., pseudographenes, are also studied. The geometry and energy of disclinated 2D carbon configurations are analyzed with the help of molecular dynamics (MD) simulation technique. A comparison of the obtained MD results with analytical calculations within the framework of the theory of defects of elastic continuum is presented. Розглянуто дисклінаційні сітки (DNs) — періодичні розподіли дисклінаційних дефектів у графені. Дисклінації проявляють себе як 4-, 5-, 7- або 8-членні вуглецеві кільця на відміну від 6-ланкових кілець, з яких складається двовимірна 2D ідеальна гратка графена. Також досліджено граничні ви-падки графеноподібних 2D вуглецевих граток без 6-ланкових кілець — так звані псевдографени. Геометрія та енергія дисклінованих 2D-вуглецевих конфігурацій аналізуються за допомогою метода молекулярної динаміки (MD). Наведено порівняння результатів MD моделювання та аналітичних розрахунків в рамках теорії дефектів пружного континууму. Рассмотрены дисклинационные сетки (DNs) — периодические распределения дисклинационных дефектов в графене. Дисклинации проявляют себя как 4- , 5-, 7- или 8-членные углеродные кольца в отличие от 6-звенных колец, из которых состоит двумерная (2D) идеальная решетка графена. Также исследуются предельные случаи графеноподобных 2D углеродных решеток без 6-звенных колец — так называемые псевдографены. Геометрия и энергия дисклинированных 2D углеродных конфигураций анализируются с помощью метода молекулярной динамики (MD). Представлено сравнение результатов MD моделирования и аналитических расчетов, проведенных в рамках теории дефектов упругого континуума. The work was supported by the Russian Foundation for Basic Research (grant No 18-01-00884). en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Физика низких температур Низькотемпературна фізика пластичності та міцності Disclination ensembles in graphene Ансамблі дисклінацій у графені Ансамбли дисклинаций в графене Article published earlier |
| spellingShingle | Disclination ensembles in graphene Rozhkov, М.А. Kolesnikova, А.L. Yasnikov, I.S. Romanov, А.Е. Низькотемпературна фізика пластичності та міцності |
| title | Disclination ensembles in graphene |
| title_alt | Ансамблі дисклінацій у графені Ансамбли дисклинаций в графене |
| title_full | Disclination ensembles in graphene |
| title_fullStr | Disclination ensembles in graphene |
| title_full_unstemmed | Disclination ensembles in graphene |
| title_short | Disclination ensembles in graphene |
| title_sort | disclination ensembles in graphene |
| topic | Низькотемпературна фізика пластичності та міцності |
| topic_facet | Низькотемпературна фізика пластичності та міцності |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/176255 |
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