In this paper, we present an approach for characterizing the interfacial region using the molecular dynamics (MD) simulations and the shear deformation model (SDM). The bulk-level mechanical properties of graphene-reinforced nanocomposites strongly depend on the interfacial region between the graphene and epoxy matrix, whose thickness is about 6.8–10.0 Å. Because it is a challenge to experimentally investigate mechanical properties of this thin region, computational MD simulations have been widely employed. By pulling out graphene from the graphene/epoxy system, pull-out force and atomic displacement of the interfacial region are calculated to characterize the interfacial shear modulus. The same processes are applied to 3% grafted hydroxyl and carboxyl functionalized graphene (OH-FG and COOH-FG)/epoxy (diglycidyl ether of bisphenol F (DGEBF)/triethylenetetramine (TETA)) systems, and influences of the functionalization on the mechanical properties of the interfacial region are studied. Our key finding is that, by functionalizing graphene, the pull-out force moderately increases and the interfacial shear modulus considerably decreases. We demonstrate our results by comparing them with literature values and findings from experimental papers.

## Introduction

Thermoset epoxy resins are lightweight and exhibit remarkable mechanical performance compared to thermoplastic polymers due to the strong and entangled covalent bonding between the epoxy resins and hardener. To date, the epoxy resins have been adopted as polymer matrix of nanoscale fillers such as nanoparticles, carbon nanotubes (CNTs), and graphene in various studies [16]. Among them, graphene nanofillers—a thin sheet of sp2 bonded carbon—are known to bring enhancements on various physical and chemical properties such as mechanical stiffness, strength, electrical and thermal conductivity, and glass transition temperature at relatively low graphene concentration in the composites [58]. Compared to its fierce rival, CNT, the graphene-based nanofillers are cost-effective due to its possibility of mass production and might outperform CNT fillers regarding enhancements in mechanical, thermal, and electrical properties [812].

One of the major factors that influence bulk-level mechanical properties of nanocomposites is the interfacial region, a thin area in the polymer adjacent to the nanofillers, which is structurally different from that of the bulk polymer as revealed from atomistic simulations and transmission electron microscopy (TEM) image [1,5,13]. Because it is a challenge to characterize mechanical properties of this thin region in experimental ways, whose thickness is about 6.8–10.0 Å, computational molecular dynamics (MD) simulations have been widely used [1,5,13]. However, the mechanical properties of the interfacial region differ from characterization methods; the inverse calculation method expects higher elastic properties [1] whereas the interfacial energy differentiation method expects lower elastic properties than that of the bulk matrix [14]. Furthermore, even though there have been numerous studies on the functionalized graphene, which is known to enhance filler dispersion and interfacial bonding, there has been little research on characterizing mechanical properties of the interfacial region in the functionalized graphene (FG)/epoxy systems.

In this paper, we obtained the pull-out force and shear modulus of the interfacial region through the pull-out simulations, and conducted the same simulated tests on 3% graphited hydroxyl and carboxyl functionalized graphene (OH-FG and COOH-FG)/epoxy systems to analyze influences of the functionalization on mechanical properties of the interfacial region. Our key finding is that, by functionalizing graphene, the pull-out force increases moderately and the interfacial shear modulus considerably decreases. Although it is widely known that graphene functionalization enhances the effective stiffness of nanocomposites, at nanoscale, the stiffness of the interfacial region decreased in the current study; this is elucidated through findings from experimental literatures. Section 2 describes MD simulation models and methods. In Sec. 3, we propose a shear deformation model (SDM) that is combined with MD simulation results to characterize the interfacial region. Section 4 presents results and discussions. Finally, conclusions are made in Sec. 5.

## Molecular Dynamics Simulations

In order to construct nanoscale molecular structures, calculate the elemental strain and atomistic potential energy, and conduct the pull-out simulations, commercial software for molecular dynamics simulations, materialsstudio 2017 package [15], was adopted. COMPASSII (condensed-phase optimized molecular potentials for atomistic simulation studies) force field was applied to describe interatomic interactions as it has been widely employed to simulate systems containing amorphous polymers and carbon nanofillers [16,17].

Fig. 1
Fig. 1
Close modal

### Interfacial Thickness Calculation.

A vertical density distribution profile was obtained from the dynamically crosslinked graphene-epoxy nanocomposite to define the thickness of the interfacial region. Detailed simulation procedures are explained in this section.

Fig. 2
Fig. 2
Close modal
Table 1

Compositional details and achieved crosslinking density of the matrices

Matrix11505073.0–80.3
Matrix21505074.7–76.1
Matrix11505073.0–80.3
Matrix21505074.7–76.1

The amorphous cell then went through a series of energy minimization processes. At first, the geometric structure of the amorphous cell was optimized through the conjugate gradient method. After stabilizing the geometric configurations, the repeated unit cell was subjected to a series of ensemble dynamics to minimize the total potential energy, while it avoids a local energy minimum at the same time. First, an constant number of particles, volume, and temperature (NVT) ensemble with Andersen thermostat was performed at 900 K for 9 ps with 0.3 fs time-step. At this stage, enough thermal energy was given to each atom to avoid being trapped to a local energy minimum. Next, another NVT ensemble was performed at 600 K for 15 ps with 0.5 fs time-step, and an NPT (constant pressure and temperature) ensemble with Berendsen barostat was performed at 298 K and atmospheric pressure for 50 ps, 1 fs time-step. The lattice size of this energy minimized amorphous cell was about $60×60×60Å$. The density distribution profile was then obtained along the normal direction of the graphene, which is z-direction.

### Pull-Out Simulations.

In order to comprehend interfacial dynamics and obtain mechanical properties of the interfacial region, pull-out simulations were performed through analogous methods suggested by several studies [2327]. The x-direction lattice parameter of the amorphous cell was elongated to 75.5 Å, which is a summation of the graphene length (60 Å) and van der Waals cut off distance (15.5 Å), in order to remove the nonbonded interaction between successive cells along the x-direction [23,24]. The top and bottom regions of the graphene/epoxy system in the z-direction, marked by rounded rectangles in Fig. 3, were fixed [23,26,27]. At every step, the graphene was pulled out to 1.5–3 Å, and the cell went through energy minimization by the conjugate gradient method [2325]. The ensemble equilibration was not performed in this step to avoid huge computational burden; in fact, there is no clear evidence that the ensemble equilibration guarantees results, which are compatible with experiments. Successive steps were repeated until the graphene was segregated from the epoxy matrix. Figure 4 shows the scheme of the pull-out simulation at different pull-out displacements. At each step, x-coordinate of the representative atom sets and the interfacial energy, which will be discussed in Sec. 3, were recorded in a study table.

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal

### Functionalized Graphene.

The same process was applied to the FG/epoxy systems: hydroxyl functionalized graphene (OH-FG)/epoxy system and carboxyl functionalized graphene (COOH-FG)/epoxy system. Both OH and COOH functional groups were grafted to pristine graphene with the same grafting percentage of 3%. For example, the pristine graphene used in this study consists of 1312 carbon atoms. Three percentage grafting means that about 41 out of 1312 carbon atoms are bonded to the functional groups. Figure 5 shows structural configurations of the pristine graphene, OH-FG, COOH-FG, and COOH-FG/epoxy system. The FG/epoxy systems were built in the same process discussed in Sec. 2.2, where the FGs replaced the pristine graphene.

Fig. 5
Fig. 5
Close modal

## Shear Deformation Model for Determination of Interfacial Properties

The basic concept of the SDM was applied to the pull-out simulation to obtain mechanical properties of the interfacial region. Figure 6 represents a configuration of the SDM, where $ug$ and $ui$ are the displacements of the matrix adjacent to the graphene and outer matrix, respectively. $T$ and $t$ are the thickness of the total interfacial region and graphene layer each, $L$ is the x-direction length of the matrix, $s$ is the coordinate axis starting in the middle of the embedded graphene, parallel to the x-direction.

Fig. 6
Fig. 6
Close modal
Let us assume the interfacial region is isotropic so that only the shear modulus $Gi$ and the Poisson's ratio $νi$ of this region can completely define its mechanical properties. Also, deformation of the matrix is assumed to follow linear elastic behavior; this will be demonstrated in Sec. 4.3.2. The following equation is the definition of the shear strain:
$γ=dudz=τisGi$
(1)
where $τi$ is the interfacial shear stress applied to the graphene and the interfacial region and $Gi$ is the interfacial shear modulus. We integrate Eq. (1) between upper and lower boundary of the lower interfacial region
$∫uguidu=τisGi∫−t2−T2dz$
(2)
$ug−ui=τis2Gi(T−t)$
(3)
and rearrange Eq. (3) with respect to $τi$ to get
$τis=2GiT−tugs−uis$
(4)
By integrating Eq. (4) from $s=−L−x/2$ to $s=L−x/2$ and multiplying by matrix width $b$
$b∫−L−x2L−x2τisds=2bGiT−t∫−L−x2L−x2ugs−uisds$
(5)
The left side of Eq. (5) is the same as half of the pull-out force; hence, it can be expressed as
$Fi=4bGiT−t∫−L−x2L−x2ui−ugds$
(6)

where $Fi$ is the pull-out force. According to the shear lag model, the interfacial shear stress follows a distribution shown in Fig. 7 [28]. The pull-out force can be obtained from the interfacial shear stress integrated with the contact area. However, it also can be easily obtained from the interfacial energy under the assumption of linear elasticity.

Fig. 7
Fig. 7
Close modal
The interfacial energy $Eint$ is presented by the following equation:
$Eint=Etotal−(Ematrix+Egraphene)$
(7)
where $Etotal$ is the total energy of amorphous cell; $Ematrix$ is the energy of matrix; $Egraphene$ is the energy of the graphene. The pull-out force is the energy differentiated with respect to the displacement
$Fi=limΔx→0ΔEintΔx$
(8)

where $x$ is the pull-out displacement. The left side of Eq. (6) then can be numerically calculated from the interfacial energy profile.

Twenty-four representative atoms were selected (m1….m24 at the upper boundary and m25…m48 at the lower boundary of the interfacial region) where they were arranged at equal intervals, as shown in Fig. 6. By charting x-coordinates of the representative atoms at every pull-out step, $ug−ui$ can be calculated numerically so that we can obtain the integral at the right side of Eq. (6). Now, the only unknown in Eq. (6) is $Gi$. Hence, the interfacial shear modulus $Gi$ is expressed as
$Gi=T−t4bFi∫−L−x2L−x2ug−uids$
(9)
As expressed in Eq. (9), the relative difference between the numerator and denominator determines the shear modulus of the interfacial region.

## Results and Discussions

The elastic moduli of the pure crosslinked epoxy model were compared with those listed in the literature, as shown in Table 2. Both Young's modulus and shear modulus of the present study are about 10–20% higher than the literature values. For example, according to Kim et al. [18], Young's modulus and shear modulus are obtained as 3.76 and 1.37$GPa$, respectively on 62.5% crosslinking density model, which are 10% and 11% lower than the present results of 80% crosslinking density. However, results in the present study are reasonable considering that the values obtained from MD simulations in the literature show enhanced mechanical properties at a high crosslinking density [1,18,29].

Table 2

Comparison of mechanical properties of DGEBF (EPON862)-TETA epoxy resin with the values in literature

Simulation
Present studyYu et al. [1]aKim et al. [18]bFan and Yuen [29]cExperiment [29]
Young's modulus (GPa)4.23.362.30–3.763.753.43
Shear modulus (GPa)1.541.220.82–1.37
Simulation
Present studyYu et al. [1]aKim et al. [18]bFan and Yuen [29]cExperiment [29]
Young's modulus (GPa)4.23.362.30–3.763.753.43
Shear modulus (GPa)1.541.220.82–1.37
a

Crosslinking density of 50%.

b

Crosslinking density range from 16.66% to 62.50%.

c

### Interfacial Thickness Definition.

Figure 8 is the density distribution profile along the z-axis. The peak in the middle represents the relatively high density of the graphene compared to that of the bulk matrix. There are slightly higher density regions on both sides of the peak in the middle, which are noted by dashed boxes in Fig. 8. The thickness of the interfacial region was assumed to be 8.5 Å each, which is bigger than the red boxes for two reasons: (1) to match the thickness with the literature values (6.8–10.0 Å) [1,5] and (2) to clarify the displacement difference between upper and lower end of the interfacial region.

Fig. 8
Fig. 8
Close modal

### Pull-Out Simulation.

The purpose of the pull-out simulation is to calculate the right side of Eq. (9) numerically so that we can obtain the interfacial shear modulus $Gi$. The pull-out force $Fi$ on the numerator was calculated from the energy increment divided by the pull-out displacement while the integration on the denominator was obtained from the displacement of the representative atom sets. Before that, we have to demonstrate the deformation of the matrix is within the linear elastic zone. Therefore, results of the pull-out simulation will be presented in following five sections: pull-out force calculation, demonstration of the linear elasticity, integration of the interfacial region displacement, characterization of the interfacial shear modulus and comparison of graphene/epoxy and FG/epoxy systems.

#### Pull-Out Force Calculation.

Figure 9 shows the interfacial energy along the pull-out distance. The interfacial energy increases linearly from pull-out displacement 10 Å to 50 Å. This tendency coincides with CNT pull-out simulations reported by Li et al. [23], where they divided the graph into three stages: (1) initial ascent stage, (2) subsequent platform stage, and (3) final descent stage [23]. The length of both first and third stages is similar to van der Waals cutoff distance, which is also coincident with the literature [23]. According to Eq. (8), the slope of stage 2 is equal to a pull-out force $2.24nN$.

Fig. 9
Fig. 9
Close modal

#### Demonstration of the Linear Elasticity.

We assumed that the deformation of the matrix is within the linear elastic zone until the pull-out displacement reaches to 35 Å. In Sec. 4.3.5, the pull-out force was calculated on stage 2, within a domain of 10–50 Å pull-out displacement. However, to demonstrate the linear elasticity of the interfacial region, we conservatively set the boundary of linear elasticity as 35 Å. The graphene in the graphene/epoxy system was removed not to apply any force to the matrix, as shown in Fig. 10(b). Then NPT ensemble was conducted to the system for 500 ps, and the average of displacements of the 48 representative atom sets was investigated, as shown in Fig. 10(c). The average of displacements decreases as the NPT ensemble proceeds, converging to minus six around 300 ps. Note that the negative convergence value is caused by the cell shrinkage during the NPT simulation. Because the graphene was removed, the cell shrinkage is unavoidable during the NPT simulation. As shown in Fig. 10(d), a remarkable drop in x-length from 70 Å to 50 Å is observed while little increase is observed in y- and z-length. Since the x-length of the cell decreased to 50 Å, the average of displacements converged to a negative value.

Fig. 10
Fig. 10
Close modal

As the graphene is the only medium that applies the shear force to the matrix, removing the graphene means no more shear force is applied to the matrix. The NPT ensemble dynamics to the system at 1 atm is analogous to a situation where the shear force is just removed from the system. Because the linear elasticity means that the deformed shape caused by an external force fully recovers once the force is removed, the convergence of the average of displacements implies that the system is within the linear elastic zone.

#### Integration of Interfacial Region Displacement.

Translations in the x-direction of the 48 representative atom sets are shown in Fig. 11. Fluctuation in the early pull-out stage becomes stable as the pull-out simulation proceeds. From the data in Fig. 11, the integration in Eq. (9) can be numerically calculated.

Fig. 11
Fig. 11
Close modal

#### Characterization of Interfacial Shear Modulus.

Now, everything needed for characterizing $Gi$ in Eq. (9) has been acquired. By substituting 20.4 Å to T, 3.4 Å to t, and 60 Å to b, $Gi$ was obtained at each pull-out step. Figure 12 represents the interfacial shear modulus from the pull-out displacement 15 Å to 35 Å included in stage 2. The dotted line in Fig. 12 indicates the average value, 2.34$GPa$. There are fluctuating $Gi$ values over the average up to the pull-out displacement of 20 Å. These points may be attributed to a nonuniformity of the displacements (i.e., $ugandui$) of the epoxy matrix and small values of the displacement integration in Eq. (9). The numerator in Eq. (9), pull-out force, remained constant during the stage 2 pull-out steps. However, the displacement integration in the denominator of Eq. (9) fluctuated around 2–15 Å2 with an average of 9 Å2, which led to relatively large fluctuation on the interfacial shear modulus. These fluctuations diminished as the displacement of the interfacial region (or the displacement integration) increased; this will be discussed in Sec. 4.3.5.

Fig. 12
Fig. 12
Close modal

#### Comparison of Pristine Graphene/Epoxy and Functionalized Graphene/Epoxy Systems.

The same process was applied to 3% grafted OH-FG and COOH-FG/epoxy systems. As shown in Fig. 13, FG/epoxy systems showed higher pull-out forces than the pristine graphene/epoxy system as the functional groups enhance van der Waals interaction between FGs and matrix. Among the FGs, the pull-out force of the COOH-FG/epoxy system was higher than that of the OH-FG/epoxy system. Figure 14 shows how the functional groups develop an interlaced structure with the epoxy matrix. The COOH-group develops a more entangled structure with the matrix than the OH-group, which enables higher van der Waals interaction in the COOH-FG system. In the same manner, because there are no functional groups in the graphene/epoxy system, the smallest van der Waals interaction is expected for the graphene/epoxy system among the others.

Fig. 13
Fig. 13
Close modal
Fig. 14
Fig. 14
Close modal

A sharp decrease of the interfacial shear modulus was observed on FG/epoxy systems in Fig. 15(c). This is because of an increment on the displacement of the interfacial region in FG/epoxy systems, as shown in Fig. 16, where more deformation of the interfacial region was observed in FG/epoxy systems than the graphene/epoxy system. The displacement of the interfacial region can be expressed numerically as the displacement integration, the denominator of Eq. (9). Average of the displacement integration is about 9 Å2 for graphene/epoxy, 27 Å2 for OH-FG/epoxy and 194 Å2 for COOH-FG/epoxy system.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal

Note that both the pull-out force and the displacement integration increased when graphene was functionalized. The pull-out force is proportional to the interfacial shear modulus because it is in the numerator of Eq. (9), whereas the displacement integration is inversely proportional to the interfacial shear modulus because it is the denominator of Eq. (9). Therefore, the functionalization has both increasing effect and decreasing effect on the interfacial shear modulus. However, the pull-out force increased less than two times whereas the displacement integration increased even more than 20 times at COOH-FG/epoxy system. This led to the overall decrease of the interfacial shear modulus of the FG/epoxy systems. More decrease of the interfacial shear modulus in the COOH-FG/epoxy system compared to the OH-FG/epoxy system can be explained in the same way.

Comparing the pristine graphene/epoxy (Fig. 12), OH-FG/epoxy, and COOH-FG/epoxy systems (Figs. 15(a) and 15(b)), the fluctuation of the $Gi$ values decreased in the order of graphene/epoxy, OH-FG/epoxy, and COOH-FG/epoxy systems. It is due to the relatively larger displacement integration. Although the displacement integration still fluctuated around 170–220 Å2 in the COOH-FG/epoxy system, the fluctuation of the interfacial shear modulus was minimal because the displacement integration was the highest among the three systems.

Current simulation results are compared with literature values, as shown in Table 3. Yu et al. defined the thickness of the interfacial region between nanoparticle and epoxy matrix as 6.8 Å and inversely derived a stiffness matrix of the interfacial region using Mori-Tanaka micromechanics [1]. Because they used a nonfunctionalized nanoparticle in their atomistic simulations, their result can be compared to graphene/epoxy system in this study; they show moderate consistency. Arash et al. [14] obtained Young's modulus of the interfacial region between CNT and poly (methyl methacrylate) matrix from the double differentiation of interfacial energy. In their study, information of the interfacial shear modulus is not given, but we can estimate it by applying two assumptions: (1) the interfacial region is isotropic and (2) Poisson's ratio is 0.3. With these assumptions, we can obtain the interfacial shear modulus as 0.28–0.9 $GPa$, which is lower than the results in this study. A possible reason for the inconsistency is that they used a thermoplastic polymer that has lower mechanical properties than the epoxy. Also, the definition of the interfacial region is ambiguous because they obtained the interfacial Young's modulus only from the interfacial energy.

Table 3

Comparison of interfacial shear modulus with literature values

Present study
GrapheneOH-FGCOOH-FGYu et al. [1]Arash et al. [14]Pure epoxy
Shear modulus (GPa)2.340.720.141.22–3.611.54
Young's modulus (GPa)3.36–9.460.73–2.344.2
Present study
GrapheneOH-FGCOOH-FGYu et al. [1]Arash et al. [14]Pure epoxy
Shear modulus (GPa)2.340.720.141.22–3.611.54
Young's modulus (GPa)3.36–9.460.73–2.344.2

The decrease of the interfacial shear modulus in FG/epoxy systems does not imply that the total shear modulus would decrease or the nanocomposites would become less stiff. In fact, the interfacial region appears to be small compared to the whole nanocomposite considering that usually less than a few weight percent of the graphene is added to a polymer matrix, which means that a decreasing effect on the interfacial shear modulus is marginal. Instead, the functionalization still has positive effects on the stiffness such as better dispersion, which may lead to an overall increase of stiffness parameters of the nanocomposites. Furthermore, the less stiffened interfacial region of the FG/epoxy system plays a role as a bumper between the graphene and the bulk matrix, which may lead to a remarkable increase in fracture toughness. This tendency is found from an experiment done by Cha et al. [4]. They conducted mechanical testing on melamine functionalized CNT (M-CNT)/epoxy composite. They reported a remarkable enhancement on fracture toughness (95–100%) of the M-CNT/epoxy composite compared to that of neat epoxy, while Young's modulus and tensile strength increased only 64% and 22% [4]. This result is consistent with the interpretation of this study. As a result, graphene functionalization enhances toughness and ductility of the nanocomposites while it degrades stiffness of the interfacial region.

## Conclusions

A combined MD-SDM approach for characterizing the shear modulus of the interfacial region was developed. The pull-out force was calculated for the pristine graphene/epoxy, OH-FG/epoxy, and COOH-FG/epoxy systems. The pull-out force of OH-FG/epoxy system and COOH-FG/epoxy system were 5% and 54% higher than that of the graphene/epoxy system. The interfacial shear modulus of OH-FG/epoxy system and COOH-FG/epoxy system decreased 56% and 85%, respectively, compared to that of graphene/epoxy system. This implies that the graphene functionalization makes the interfacial region less stiff. The proposed combined MD-SDM method can be applied for various interfacial regions in nanocomposites between nanofillers and bulk matrix.

## Acknowledgment

This work was supported by National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) and SNU Undergraduate Research Program. Authors are grateful for their support.

## Funding Data

• National Research Foundation of Korea (2017R1A2B4004996).

• Seoul National University (Undergraduate Research).

## References

1.
Yu
,
S.
,
Yang
,
S.
, and
Cho
,
M.
,
2009
, “
Multi-Scale Modeling of Cross-Linked Epoxy Nanocomposites
,”
Polymer
,
50
(
3
), pp.
945
952
.
2.
Yang
,
S.
,
Choi
,
J.
, and
Cho
,
M.
,
2012
, “
Elastic Stiffness and Filler Size Effect of Covalently Grafted Nanosilica Polyimide Composites: Molecular Dynamics Study
,”
ACS Appl. Mater. Interfaces
,
4
(
9
), pp.
4792
4799
.
3.
Alian
,
A. R.
,
Kundalwal
,
S. I.
, and
Meguid
,
S. A.
,
2015
, “
Multiscale Modeling of Carbon Nanotube Epoxy Composites
,”
Polymer
,
70
, pp.
149
160
.
4.
Cha
,
J.
,
Jun
,
G. H.
,
Park
,
J. K.
,
Kim
,
J. C.
,
Ryu
,
H. J.
, and
Hong
,
S. H.
,
2017
, “
Improvement of Modulus, Strength and Fracture Toughness of CNT/Epoxy Nanocomposites Through the Functionalization of Carbon Nanotubes
,”
Compos. Part B
,
129
, pp.
169
179
.
5.
,
C. M.
,
Jensen
,
B. D.
,
,
A.
,
Odegard
,
G. M.
,
Koo
,
A.
, and
Liang
,
R.
,
2013
, “
Molecular Modeling of EPON-862/Graphite Composites: Interfacial Characteristics for Multiple Crosslink Densities
,”
Compos. Sci. Technol.
,
76
, pp.
92
99
.
6.
Rahman
,
R.
, and
Haque
,
A.
,
2013
, “
Molecular Modeling of Crosslinked Graphene–Epoxy Nanocomposites for Characterization of Elastic Constants and Interfacial Properties
,”
Compos. Part B
,
54
, pp.
353
364
.
7.
Shiu
,
S.-C.
, and
Tsai
,
J.-L.
,
2014
, “
Characterizing Thermal and Mechanical Properties of Graphene/Epoxy Nanocomposites
,”
Compos. Part B
,
56
, pp.
691
697
.
8.
Kim
,
H.
,
Miura
,
Y.
, and
Macosko
,
C. W.
,
2010
, “
Graphene/Polyurethane Nanocomposites for Improved Gas Barrier and Electrical Conductivity
,”
Chem. Mater.
,
22
(
11
), pp.
3441
3450
.
9.
Ramanathan
,
T.
,
Abdala
,
A. A.
,
Stankovich
,
S.
,
Dikin
,
D. A.
,
Herrera-Alonso
,
M.
,
Piner
,
R. D.
,
,
D. H.
,
Schniepp
,
H. C.
,
Chen
,
X.
,
Ruoff
,
R. S.
,
Nguyen
,
S. T.
,
Aksay
,
I. A.
,
Prud'homme
,
R. K.
, and
Brinson
,
L. C.
,
2008
, “
Functionalized Graphene Sheets for Polymer Nanocomposites
,”
Nat. Nanotechnol.
,
3
(
6
), pp.
327
331
.
10.
Rafiee
,
M. A.
,
Rafiee
,
J.
,
Wang
,
Z.
,
Song
,
H. H.
,
Yu
,
Z. Z.
, and
Koratkar
,
N.
,
2009
, “
Enhanced Mechanical Properties of Nanocomposites at Low Graphene Content
,”
ACS Nano
,
3
(
12
), pp.
3884
3890
.
11.
Rafiee
,
M. A.
,
Rafiee
,
J.
,
Srivastava
,
I.
,
Wang
,
Z.
,
Song
,
H. H.
,
Yu
,
Z. Z.
, and
Koratkar
,
N.
,
2010
, “
Fracture and Fatigue in Graphene Nanocomposites
,”
Small
,
6
(
2
), pp.
179
183
.
12.
Rafiee
,
M. A.
,
Rafiee
,
J.
,
Yu
,
Z. Z.
, and
Koratkar
,
N.
,
2009
, “
Buckling Resistant Graphene Nanocomposites
,”
Appl. Phys. Lett.
,
95
(
22
), p.
223103
.
13.
Mansfield
,
K. F.
, and
Theodorou
,
D. N.
,
1991
, “
Atomistic Simulation of a Glassy Polymer/Graphite Interface
,”
Macromolecules
,
24
(
15
), pp.
4295
4309
.
14.
Arash
,
B.
,
Wang
,
Q.
, and
,
V. K.
,
2014
, “
Mechanical Properties of Carbon Nanotube/Polymer Composites
,”
Sci. Rep.
,
4
, p.
6479
.
15.
Accelrys and BIOVIA,
2017
, “
Materials Studio
,” Accelrys/BIOVIA, San Diego, CA.
16.
Li
,
Y.
,
Wang
,
S.
,
Wang
,
Q.
, and
Xing
,
M.
,
2018
, “
Enhancement of Fracture Properties of Polymer Composites Reinforced by Carbon Nanotubes: A Molecular Dynamics Study
,”
Carbon
,
129
, pp.
504
509
.
17.
Jung
,
H.
,
Choi
,
H. K.
,
Kim
,
S.
,
Lee
,
H.-S.
,
Kim
,
Y.
, and
Yu
,
J.
,
2017
, “
The Influence of N-Doping Types for Carbon Nanotube Reinforced Epoxy Composites: A Combined Experimental Study and Molecular Dynamics Simulation
,”
Compos. Part A
,
103
, pp.
17
24
.
18.
Kim
,
B.
,
Choi
,
J.
,
Yang
,
S.
,
Yu
,
S.
, and
Cho
,
M.
,
2015
, “
Influence of Crosslink Density on the Interfacial Characteristics of Epoxy Nanocomposites
,”
Polymer
,
60
, pp.
186
197
.
19.
Clancy, T. C., Frankland, S., Hinkley, J., and Gates, T., 2009, “
Molecular modeling for calculation of mechanical properties of epoxies with moisture ingress
,”
Polymer
,
50
(12), pp. 2736–2742.
20.
Kim
,
B.
,
Choi
,
J.
,
Yang
,
S.
,
Yu
,
S.
, and
Cho
,
M.
,
2017
, “
Multiscale Modeling of Interphase in Crosslinked Epoxy Nanocomposites
,”
Compos. Part B
,
120
, pp.
128
142
.
21.
Khare
,
K. S.
, and
Khare
,
R.
,
2012
, “
Directed Diffusion Approach for Preparing Atomistic Models of Crosslinked Epoxy for Use in Molecular Simulations
,”
Macromol. Theory Simul.
,
21
(
5
), pp.
322
327
.
22.
Han, Y., and Elliott, J., 2007, “
Molecular dynamics simulations of the elastic properties of polymer/carbon nanotube composites
,”
Computational Materials Science
,
39
(2), pp. 315–323.
23.
Li
,
Y.
,
Liu
,
Y.
,
Peng
,
X.
,
Yan
,
C.
,
Liu
,
S.
, and
Hu
,
N.
,
2011
, “
Pull-out Simulations on Interfacial Properties of Carbon Nanotube-Reinforced Polymer Nanocomposites
,”
Comput. Mater. Sci.
,
50
(
6
), pp.
1854
1860
.
24.
Coto
,
B.
,
Antia
,
I.
,
Barriga
,
J.
,
Blanco
,
M.
, and
Sarasua
,
J.-R.
,
2013
, “
Influence of the Geometrical Properties of the Carbon Nanotubes on the Interfacial Behavior of Epoxy/CNT Composites: A Molecular Modelling Approach
,”
Comput. Mater. Sci.
,
79
, pp.
99
104
.
25.
Yu
,
B.
,
Fu
,
S.
,
Wu
,
Z.
,
Bai
,
H.
,
Ning
,
N.
, and
Fu
,
Q.
,
2015
, “
Molecular Dynamics Simulations of Orientation Induced Interfacial Enhancement Between Single Walled Carbon Nanotube and Aromatic Polymers Chains
,”
Compos. Part A
,
73
, pp.
155
165
.
26.
Jin
,
Y.
,
Duan
,
F.
, and
Mu
,
X.
,
2016
, “
Functionalization Enhancement on Interfacial Shear Strength Between Graphene and Polyethylene
,”
Appl. Surf. Sci.
,
387
, pp.
1100
1109
.
27.
Liu
,
F.
,
Hu
,
N.
,
Zhang
,
J.
,
Atobe
,
S.
,
Weng
,
S.
,
Ning
,
H.
,
Liu
,
Y.
,
Wu
,
L.
,
Zhao
,
Y.
, and
Mo
,
F.
,
2016
, “
The Interfacial Mechanical Properties of Functionalized Graphene–Polymer Nanocomposites
,”
,
6
(
71
), pp.
66658
66664
.
28.
Nairn
,
J. A.
,
1992
, “
A Variational Mechanics Analysis of the Stresses Around Breaks in Embedded Fibers
,”
Mech. Mater.
,
13
(
2
), pp.
131
154
.
29.
Fan
,
H. B.
, and
Yuen
,
M. M.
,
2007
, “
Material Properties of the Cross-Linked Epoxy Resin Compound Predicted by Molecular Dynamics Simulation
,”
Polymer
,
48
(
7
), pp.
2174
2178
.