Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects
In this study, which is a continuation of our earlier work, the effect of double-atom vacancy defects (DAVD) on the elastic properties of single-layered graphene sheets (SLGS)-polymer nanocomposites is assessed by the multiscale modeling. According to the latter approach, the polymer matrix is model...
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nasplib_isofts_kiev_ua-123456789-1738762025-02-23T18:41:32Z Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects Расчет упругих свойств полимерных нанокомпозитов из однослойных листов графена с двойными атомарными вакансионными дефектами на основе мультимасштабного моделирования Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. Научно-технический раздел In this study, which is a continuation of our earlier work, the effect of double-atom vacancy defects (DAVD) on the elastic properties of single-layered graphene sheets (SLGS)-polymer nanocomposites is assessed by the multiscale modeling. According to the latter approach, the polymer matrix is modeled by finite element approach, while the SLGS and interphase layer are simulated at the atomistic scale by the molecular structural mechanics approach. In view of the Lennard–Jones potential concept, it is assumed that the SLGS and polymer matrix are related by van der Waals reciprocity. The fact that the elastic modulus of a polymer with 5% volume fraction of SLGS is increased by 17 times is demonstrated by numerous simulation results. It is also shown that the elastic modulus of SLGS-polymer nanocomposites with DAVDs is deteriorated with an increase in the number of DAVDs and improved with the increased volume fraction of SLGS. В рамках ранее предложенного авторами подхода на основе мультимасштабного моделирования исследуется влияние двойных атомарных вакансионных дефектов на упругие свойства полимерных нанокомпозитов из однослойных листов графена. Моделирование полимерной матрицы осуществляется методом конечных элементов, а интерфазный слой и однослойный лист графена моделируются на атомарном уровне в рамках молекулярной структурной механики. На основании концепции потенциала Леннарда Джонса предполагается наличие зависимости ван дер Ваальса между свойствами однослойных листов графена и полимерной матрицы. Результаты численных расчетов показывают, что модуль упругости полимера можно увеличить в 17 раз за счет добавления 5%-ной объемной доли однослойного листа графена. Установлено, что при увеличении количества двойных атомарных вакансионных дефектов модуль упругости полимерных композитов из однослойными листами графена снижается, в то время как с ростом объемной доли однослойных листов графена наблюдается его повышение. У рамках раніше запропонованого авторами підходу на основі мультимасштабного моделювання досліджується вплив подвійних атомарних вакансійних дефектів на пружні властивості полімерних нанокомпозитів з одношарових листів графена. Моделювання полімерної матриці виконано методом скінченних елементів, а інтерфазний шар і одношаровий лист графена моделюються на атомарному рівні в рамках молекулярної структурної механіки. На основі концепції потенціалу Леннарда Джонса припускається наявність залежності ван дер Ваальса між властивостями одношарових листів графена і полімерної матриці. Результати чисельних розрахунків показують, що модуль пружності полімеру можна збільшити в 17 разів за рахунок додання 5%-ної об ємної частки одношарового листа графена. Установлено, що з ростом кількості подвійних атомарних вакансійних дефектів модуль пружності полімерних композитів з одношаровими листами графена зменшується, в той час як зі зростанням об ємної частки одношарових листів графена відмічається його підвищення. 2018 Article Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects / Z.Q. Wang, Z.W. Yu, X.Y. Sun, H. Li, Y.J. Wang // Проблемы прочности. — 2018. — № 2. — С. 38-44. — Бібліогр.: 12 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173876 621.515/62-752 en Проблемы прочности application/pdf Інститут проблем міцності ім. Г.С. Писаренко НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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Научно-технический раздел Научно-технический раздел |
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Научно-технический раздел Научно-технический раздел Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects Проблемы прочности |
| description |
In this study, which is a continuation of our earlier work, the effect of double-atom vacancy defects (DAVD) on the elastic properties of single-layered graphene sheets (SLGS)-polymer nanocomposites is assessed by the multiscale modeling. According to the latter approach, the polymer matrix is modeled by finite element approach, while the SLGS and interphase layer are simulated at the atomistic scale by the molecular structural mechanics approach. In view of the Lennard–Jones potential concept, it is assumed that the SLGS and polymer matrix are related by van der Waals reciprocity. The fact that the elastic modulus of a polymer with 5% volume fraction of SLGS is increased by 17 times is demonstrated by numerous simulation results. It is also shown that the elastic modulus of SLGS-polymer nanocomposites with DAVDs is deteriorated with an increase in the number of DAVDs and improved with the increased volume fraction of SLGS. |
| format |
Article |
| author |
Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. |
| author_facet |
Wang, Z.Q. Yu, Z.W. Sun, X.Y. Li, H. Wang, Y.J. |
| author_sort |
Wang, Z.Q. |
| title |
Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects |
| title_short |
Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects |
| title_full |
Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects |
| title_fullStr |
Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects |
| title_full_unstemmed |
Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects |
| title_sort |
multiscale modeling-based assessment of elastic properties of slgs-polymer nanocomposites with double-atom vacancy defects |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2018 |
| topic_facet |
Научно-технический раздел |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/173876 |
| citation_txt |
Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer Nanocomposites with Double-Atom Vacancy Defects / Z.Q. Wang, Z.W. Yu, X.Y. Sun, H. Li, Y.J. Wang // Проблемы прочности. — 2018. — № 2. — С. 38-44. — Бібліогр.: 12 назв. — англ. |
| series |
Проблемы прочности |
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UDC 539.4
Multiscale Modeling-Based Assessment of Elastic Properties of SLGS-Polymer
Nanocomposites with Double-Atom Vacancy Defects
Z. Q. Wang, Z. W. Yu, X. Y. Sun,
1
H. Li, and Y. J. Wang
2
College of Aerospace and Civil Engineering, Harbin Engineering University, Harbin, China
1 sunxiaoyu520634@163.com
2 906871278@qq.com
ÓÄÊ 539.4
Ðàñ÷åò óïðóãèõ ñâîéñòâ ïîëèìåðíûõ íàíîêîìïîçèòîâ èç îäíîñëîéíûõ
ëèñòîâ ãðàôåíà ñ äâîéíûìè àòîìàðíûìè âàêàíñèîííûìè äåôåêòàìè íà
îñíîâå ìóëüòèìàñøòàáíîãî ìîäåëèðîâàíèÿ
Ç. Ê. Âàíã, Ç. Â. Þ, Ê. ß. Ñóí, Õ. Ëè, È. Æ. Âàíã
Êîëëåäæ àýðîêîñìè÷åñêîãî è ãðàæäàíñêîãî ñòðîèòåëüñòâà, Õàðáèíñêèé èíæåíåðíûé óíèâåð-
ñèòåò, Õàðáèí, Êèòàé
 ðàìêàõ ðàíåå ïðåäëîæåííîãî àâòîðàìè ïîäõîäà íà îñíîâå ìóëüòèìàñøòàáíîãî ìîäåëèðî-
âàíèÿ èññëåäóåòñÿ âëèÿíèå äâîéíûõ àòîìàðíûõ âàêàíñèîííûõ äåôåêòîâ íà óïðóãèå ñâîéñòâà
ïîëèìåðíûõ íàíîêîìïîçèòîâ èç îäíîñëîéíûõ ëèñòîâ ãðàôåíà. Ìîäåëèðîâàíèå ïîëèìåðíîé
ìàòðèöû îñóùåñòâëÿåòñÿ ìåòîäîì êîíå÷íûõ ýëåìåíòîâ, à èíòåðôàçíûé ñëîé è îäíîñëîéíûé
ëèñò ãðàôåíà ìîäåëèðóþòñÿ íà àòîìàðíîì óðîâíå â ðàìêàõ ìîëåêóëÿðíîé ñòðóêòóðíîé
ìåõàíèêè. Íà îñíîâàíèè êîíöåïöèè ïîòåíöèàëà Ëåííàðäà–Äæîíñà ïðåäïîëàãàåòñÿ íàëè÷èå
çàâèñèìîñòè âàí äåð Âààëüñà ìåæäó ñâîéñòâàìè îäíîñëîéíûõ ëèñòîâ ãðàôåíà è ïîëèìåðíîé
ìàòðèöû. Ðåçóëüòàòû ÷èñëåííûõ ðàñ÷åòîâ ïîêàçûâàþò, ÷òî ìîäóëü óïðóãîñòè ïîëèìåðà
ìîæíî óâåëè÷èòü â 17 ðàç çà ñ÷åò äîáàâëåíèÿ 5%-íîé îáúåìíîé äîëè îäíîñëîéíîãî ëèñòà
ãðàôåíà. Óñòàíîâëåíî, ÷òî ïðè óâåëè÷åíèè êîëè÷åñòâà äâîéíûõ àòîìàðíûõ âàêàíñèîííûõ
äåôåêòîâ ìîäóëü óïðóãîñòè ïîëèìåðíûõ êîìïîçèòîâ èç îäíîñëîéíûìè ëèñòàìè ãðàôåíà
ñíèæàåòñÿ, â òî âðåìÿ êàê ñ ðîñòîì îáúåìíîé äîëè îäíîñëîéíûõ ëèñòîâ ãðàôåíà íàáëþ-
äàåòñÿ åãî ïîâûøåíèå.
Êëþ÷åâûå ñëîâà: ãðàôåí, äâîéíûå àòîìàðíûå âàêàíñèîííûå äåôåêòû, óïðóãèå ñâîé-
ñòâà, ìóëüòèìàñøòàáèðîâàíèå, íàíîêîìïîçèò.
Introduction. Graphene is a new type of low-dimensional carbon material, which
consists of carbon atoms in a honeycomb lattice structure. The latter provides high strength,
high thermal conductivity and stiffness of graphene, as reported by numerous researchers
[1–3]. The nanoindentation technology applied to graphene by Lee et al. [3] made it
possible to evaluate its elastic properties using an inherent suspended graphene sheet (GS)
and of a series of open holes on a Si substrate: the stress of 130 GPa and elastic modulus of
about 1 TPa were attained. This finding strongly suggests that graphene is a perfect
enhancing material for the next generation of composites.
The outstanding mechanical properties are demonstrated by nanocomposites based on
polymer with exfoliated graphene sheets as enhancing factors have been demonstrated, e.g.,
by Rafiee et al. [4], who reported that inclusion of graphene sheets amounting to 1% of the
composite weight resulted in significant improvement of its stiffness and the strength. As a
© Z. Q. WANG, Z. W. YU, X. Y. SUN, H. LI, Y. J. WANG, 2018
38 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 2
reinforcement material, the magnitude’s order obtains significance of the underlying
benefits of graphene. For the GS weight fraction in the epoxy matrix equal to 0.1%, the
authors [5] have revealed the 52%-increase in the nanocomposite buckling load value. In
addition, the graphene fillers in the composite improved its resistance to fatigue crack
propagation by almost two orders of magnitude, as compared with the base epoxy material
[6].
In order to study the effect of GS presence on the polymer mechanical properties, the
numerical simulation and theoretical studies have been conducted. By combining the
Mori–Tanaka method [7] with the molecular mechanics approach, Cho et al. [8]
investigated the mechanical properties of epoxy matrix reinforced with graphene sheets.
The elastic modulus of the latter was calculated through the molecular force field approach,
which formed the basis of the proposed micromechanical model. Awasthi et al. [9]
investigated the nanoscale load transfer between a graphene sheet and polyethylene by
applying the MD simulation. The further calculations strongly indicated that the
nanocomposite elastic modulus varied with the GS aspect ratios, i.e., length-to-width
relative values, rather than with their absolute values (sizes).
The effect of double-atom vacancy defects (DAVDs) on the elastic properties of
single-layered graphene sheets (SLGS) has been estimated via a multiscale model in the
recent study of the authors of this paper [10]. The same approach is applied in the present
study to determine how the number of DAVDs affects the elastic properties of SLGS-
polymer nanocomposites. For this purpose, the finite element (FE) approach is used to
model the polymer matrix, while the molecular structural mechanics (MSM) is applied to
the simulation of SLGS and interphase layer at the atomistic scale.
1. The Proposed Model. To establish a representative volume element (RVE)
composed of a cubic body of the matrix with an embedded SLGS [16], a potential-based
method is used, because the conventional theoretical frame is to assess the macroscopic
mechanical properties of composites. This article forecasts the stable performance of
SLGS-polymer composites with DAVDS in a similar way by a new MSM/FE multiscale
modeling method with three stages. The components of the three-phase RVE are: the outer
polymer matrix, the interphase, and the transition layer between the polymer matrix and
graphene.
1.1. SLGS Model. Li and Chou [11] brought in the atomistic FEA approach, coined as
the molecular structural mechanics (MSM) and implemented it via ANSYS commercial
code to calculate the GSs elastic properties with account of vacancy-type defects. Since the
detailed description of this approach can be found elsewhere [10, 11], only the minimal
important details are given in this paper. Finite element nodes correspond to carbon atoms,
and in the graphene sheet, the interatomic potentials between the covalently joint carbon
atoms are modeled by the structural beams, in this method. Using continuous and discrete
energy dependences linking the key effects of molecular mechanics related to torsion,
bending, bond stretching and the consistent beam strain energies, the beams’ elastic and
geometric parameters are attained. Here k� (torsion), kr (stretching), and k� (bending)
interatomic force field constants can be interrelated with the beam geometric and elastic
properties as follows:
k
E A
L
r
beam� , (1)
k
E I
L
beam
� � , (2)
k
G J
L
beam
� � , (3)
Multiscale Modeling-Based Assessment of Elastic Properties ...
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 2 39
where Gbeam and Ebeam are the beam shear and elastic moduli, respectively, J is the
beam polar moment of inertia, I is the beam moment of inertia, L is the beam length
(bond distance between carbon atoms), and A is the cross-sectional area (which is
presumed to have a circular shape. Equations (1)–(3) are constructed to derive the elastic
properties of the beam factors and the effective diameter d as follows:
d
k
kr
� 4
�
, (4)
E
k L
k
beam
r�
2
4� �
, (5)
G
k k L
k
beam
r
�
2
28
�
��
. (6)
A C–C bond length of L� 0.1421 nm was applied in Eqs. (4)–(6) with k� �
� � �878 10 10. N�nm/rad 2, kr � � �652 10 7. N/nm, and k� � � �278 10 19. N�m/rad 2 for the
purpose of computing the beam cross-sectional parameters, which yields the beam diameter
d � 0.147 nm, elastic modulus Ebeam � 5.49 TPa, and shear modulus Gbeam � 0.871 TPa.
In addition, Poisson’s ratio � � 0.3 is assumed based on Fig. 1 demonstrates the numbers
of the DAVDs from zero to seven and the SLGS with DAVDs investigated in this work.
1.2. Polymer Matrix Model. Continuum-based FE formulation plays an important part
in analyzing the outer polymer matrix and the interphase layer. An isoparametric cubic
factor, which is applied for modeling of the matrix, is defined by eight nodes with three
degrees of freedom, each node including three translations in the z, y, and x directions.
Polymethyl methacrylate (PMMA) is the polymer matrix chosen in the research. It is
presumed that the elastic modulus of this isotropic amorphous polymer is 2.5 GPa, in
compliance with its experimental range between 3.8 and 2.24 GPa attained in [12].
Poisson’s ratio of 0.35 is used for the PMMA polymer matrix.
Z. Q. Wang, Z. W. Yu, X. Y. Sun, et al.
40 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 2
Fig. 1. SLGS with the number of DAVDs ranging from zero to seven.
1.3. Interphase Layer Model. The Lennard–Jones potential represents the SLGS and
polymer matrix interactions, which correspond to vdW without a link:
V
r r
LJ �
�
�
� �
�
�
�
�
�
�
�
�
�4
12 6
�
� �
, (7)
where r is the interatomic distance, � and � represent the potential and the distance,
respectively, between two balanced atoms. The values of � and � are 38655 10 13. � � N/nm
and 0.34 nm, respectively, as for carbon atoms. When the distance was shorter compared
with the LJ potential’s shortcut radius, an elastic beam element was used to simulate the
vdW interaction equivalent to the extensive strain energy Ur , provided by Eq. (2) between
C atoms of SLGS and H atoms of the polymer matrix. Thus, the respective equation can be
derived as follows:
V U k r rLJ r� � �
1
2
0
2( ) . (8)
The following equation can be obtained by expanding the Lennard–Jones potential
into a Taylor series with the first two terms:
k
r r
� �
�
�
�
�
�4 156 42
12
14
6
8
�
� �
. (9)
Meanwhile, the force constant k can be attained through Eq. (4) as follows:
k
EA
r
� . (10)
For the earlier described beam section, the beam element radius amounts to
0.146618 nm. By re-arranging Eqs. (9) and (10) and using Poisson’s ratio of 0.3, we can
attain the beam effective elastic modulus.
Poisson’s ratio and the elastic modulus of the equivalent beam of SLGS, interphase
and polymer matrix applied in the current work have been defined in our earlier work [11].
For the purpose of attaining the transition layer’s thickness in a SLGS-polymer composite,
Hu et al. carried out the molecular mechanics calculations, which yielded the equilibrium
distance between C atoms of the nanotube and H atoms of the polymer from 0.2851 to
0.5445 nm. The mean value of this range, i.e., 0.40 nm, is used in our study as the transition
layers thickness. The SLGS-polymer nanocomposites’ multiscale model is demonstrated in
Fig. 2.
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 2 41
Multiscale Modeling-Based Assessment of Elastic Properties ...
Fig. 2. The multiscale model of SLGS-polymer nanocomposite.
The SLGS volume fraction ( fSLGS ) is an important variable in determining the
composite mechanical properties and is defined as
f
W h h h
A
SLGS
SLGS vdW SLGS vdW
cell
�
� �( )( )
,
2 2
(11)
where hvdW is the thickness of the transition layer whose value is 0.40 nm, hSLGS is the
thickness of SLGS whose value is 0.34 nm, fSLGS is SLGS volume fraction, WSLGS is
the width of SLGS, and Acell is the unit cell’s cross-sectional area. The RVE applied here
is constant as mentioned before. Therefore, the polymer matrix is the same as the SLGS’s
length in this study, whose value is 6.399 nm, and the width of SLGS WSLGS is 6.252 nm.
The input data containing the constituent properties are listed in Table 1. The elastic
modulus of the nanocomposite was calculated as follows [10]:
E
F A
L L
cell� �
�
� �
, (12)
where F is the total force(in units of nN) applied to RVE edge, �L is the elongation of
RVE, and L is the original length (in nm).
2. Results and Discussion. The elastic modulus of SLGS-polymer nanocomposites is
42.5 GPa, which is 17 times higher than that of purely polymer matrix when the SLGS
volume fraction is 5%. These results demonstrate that SLGS is a superior filler to improve
the elastic property in nanocomposite. Although the experimental results corroborated the
feasibility of the proposed multiscale model concerning SLGS’s enhancement abilities,
there are systematical deviations between the experimental data and numerical simulation
results: the former are much lower than latter. These discrepancies can be attributed to:
(i) the neglect of involvement of different SLGS layers in the load transfer; (ii) the
wrinkled (wavy) structure of SLGS in the polymeric matrix, which is observed at different
temperatures differs from a flat rectangular shape presumed in the model; (iii) DAVDs in
SLGS reduce their mechanical properties; (iv) an arbitrary orientation of fibers in tests does
not comply with the unidirectional one assumed in the numerical simulations.
42 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 2
Z. Q. Wang, Z. W. Yu, X. Y. Sun, et al.
T a b l e 1
Input Data for the FEA of SLGS-Polymer Nanocomposite
Parameter Value
Elastic modulus of equivalent beam of SLGS [N/nm2] 5 488 10 6. � �
Elastic modulus of polymer matrix [GPa] 2.5
Elastic modulus of interphase layer Solve Eqs. (10) and (11)
Poisson’s ratio of the equivalent beam of SLGS 0.3
Poisson’s ratio of polymer matrix 0.35
Poisson’s ratio of interphase layer 0.3
Length of SLGS [nm] 6.399
Width of SLGS [nm] 6.252
Thickness of SLGS [nm] 0.34
Length of polymer matrix [nm] 6.399
Thickness of interphase layer [nm] 0.4
Cross-sectional area of the unit cell [nm2] 8 039. fSLGS
The effect of DAVDs number variation (from zero to seven) on the elastic properties
of SLGS-polymer nanocomposites is depicted in Fig. 3. It is shown that the elastic modulus
of defect-free SLGS-polymer nanocomposite drops down with an increase in the number of
DAVDs by a linear correlation, namely by approx. 5.8% per defect. As seen from Fig. 3,
the elastic modulus of SLGS-polymer nanocomposites exhibits a linear deterioration trend
with the SLGS volume fraction: 1%-reduction of SLGS volume fraction results in a
10.6%-drop of the elastic properties of SLGS-polymer nanocomposite.
Conclusions. The effect of double-atom vacancy defects (DAVDs) on the elastic
properties of single-layered graphene sheets (SLGS)-polymer nanocomposites, is assessed
by the proposed multiscale model.
It is shown that the elastic modulus of defect-free SLGS-polymer nanocomposite
drops down with an increase in the number of DAVDs by a linear correlation, namely by
approx. 5.8% per defect. As seen from Fig. 3, the elastic modulus of SLGS-polymer
nanocomposites exhibits a linear deterioration trend with the SLGS volume fraction:
1%-reduction of SLGS volume fraction results in a 10.6%-drop of the elastic properties of
SLGS-polymer nanocomposite.
The limitations of the proposed approach, which resulted in systematic overestimation
of the elastic modulus, can be reduced to: (i) the neglect of involvement of different SLGS
layers in the load transfer; (ii) the wrinkled (wavy) structure of SLGS in the polymeric
matrix, which is observed at different temperatures differs from a flat rectangular shape
presumed in the model; (iii) DAVDs in SLGS reduce their mechanical properties; (iv) an
arbitrary orientation of fibers in tests does not comply with the unidirectional one assumed
in the numerical simulations.
The above problematic issues are envisaged to be tackled in the further theoretical and
numerical simulation studies.
Acknowledgments. This work was supported by the National Natural Science foundation
of China (Nos. 11602066, 11472086, and 11532013) and the National Science Foundation
of Heilongjiang Province of China (QC2015058 and 42400621-1-15047), the Foundation
Research Funds for the Central Universities.
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 2 43
Multiscale Modeling-Based Assessment of Elastic Properties ...
Fig. 3. The elastic modulus of SLGS-polymer nanocomposites with different SLGS volume fraction
versus the number of DAVDs (varying from zero to seven).
Ð å ç þ ì å
Ó ðàìêàõ ðàí³øå çàïðîïîíîâàíîãî àâòîðàìè ï³äõîäó íà îñíîâ³ ìóëüòèìàñøòàáíîãî
ìîäåëþâàííÿ äîñë³äæóºòüñÿ âïëèâ ïîäâ³éíèõ àòîìàðíèõ âàêàíñ³éíèõ äåôåêò³â íà
ïðóæí³ âëàñòèâîñò³ ïîë³ìåðíèõ íàíîêîìïîçèò³â ç îäíîøàðîâèõ ëèñò³â ãðàôåíà. Ìîäå-
ëþâàííÿ ïîë³ìåðíî¿ ìàòðèö³ âèêîíàíî ìåòîäîì ñê³í÷åííèõ åëåìåíò³â, à ³íòåðôàçíèé
øàð ³ îäíîøàðîâèé ëèñò ãðàôåíà ìîäåëþþòüñÿ íà àòîìàðíîìó ð³âí³ â ðàìêàõ ìîëåêó-
ëÿðíî¿ ñòðóêòóðíî¿ ìåõàí³êè. Íà îñíîâ³ êîíöåïö³¿ ïîòåíö³àëó Ëåííàðäà–Äæîíñà ïðè-
ïóñêàºòüñÿ íàÿâí³ñòü çàëåæíîñò³ âàí äåð Âààëüñà ì³æ âëàñòèâîñòÿìè îäíîøàðîâèõ
ëèñò³â ãðàôåíà ³ ïîë³ìåðíî¿ ìàòðèö³. Ðåçóëüòàòè ÷èñåëüíèõ ðîçðàõóíê³â ïîêàçóþòü,
ùî ìîäóëü ïðóæíîñò³ ïîë³ìåðó ìîæíà çá³ëüøèòè â 17 ðàç³â çà ðàõóíîê äîäàííÿ
5%-íî¿ îá’ºìíî¿ ÷àñòêè îäíîøàðîâîãî ëèñòà ãðàôåíà. Óñòàíîâëåíî, ùî ç ðîñòîì ê³ëü-
êîñò³ ïîäâ³éíèõ àòîìàðíèõ âàêàíñ³éíèõ äåôåêò³â ìîäóëü ïðóæíîñò³ ïîë³ìåðíèõ êîì-
ïîçèò³â ç îäíîøàðîâèìè ëèñòàìè ãðàôåíà çìåíøóºòüñÿ, â òîé ÷àñ ÿê ç³ çðîñòàííÿì
îá’ºìíî¿ ÷àñòêè îäíîøàðîâèõ ëèñò³â ãðàôåíà â³äì³÷àºòüñÿ éîãî ï³äâèùåííÿ.
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Received 08. 05. 2017
44 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 2
Z. Q. Wang, Z. W. Yu, X. Y. Sun, et al.
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/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
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(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
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/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
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/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
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/MultimediaHandling /UseObjectSettings
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/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
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/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|