1

Molecular Dynamics Simulation of Cross-linked Graphene–Epoxy

anocomposites

R.Rahman1, A. Haque

2

ABSTRACT: This paper focuses on molecular dynamics (MD) modeling of graphene reinforced

cross-linked epoxy (Gr-Ep) nanocomposite. The goal is to study the influence of geometry, and

concentration of reinforcing nanographene sheet (NGS) on interfacial properties and elastic

constants such as bulk Young’s modulus, and shear modulus of Gr-Ep nanocomposites. The

most typical cross-linked configuration was obtained in order to use in further simulations. The

mechanical properties of this cross-linked structure were determined using MD simulations and

the results were verified with those available in literatures. Graphene with different aspect ratios

and concentrations (1%, 3% and 5%) were considered in order to construct amorphous unit cells

of Gr-Ep nanocomposites. The Gr-Ep nanocomposites system undergoes NVT (constant number

of atoms, volume and temperature) and NPT (constant number of atoms, pressure and

temperature) ensemble with applied uniform strain field during MD simulation to obtain bulk

Young’s modulus and shear modulus. The stress-strain response was also evaluated for both

amorphous and crystalline unit cells of Gr-Ep system under uni-axial deformation. The cohesive

and pullout force vs. displacement response were determined for graphenes with different size.

Hence as primary goal of this work, a parametric study using MD simulation was conducted for

characterizing interfacial properties and elastic constants with different NGS aspect rations and

volume fractions. The MD simulation results show reasonable agreement with available

published data in the literature.

KEY WORDS: Molecular dynamics, graphene-epoxy nanocomposites, elastic properties.

ITRODUCTIO

Epoxy resins are a big class of compounds containing two or more epoxy groups, which can

react with many compounds (called curing agents) with chemical groups such as amines and

anhydrides. The resultants exhibit a series of excellent performance, i.e. high modulus and

fracture strength, low creep and high-temperature performance, and thus widely serve as

coatings, adhesives, composites, etc. in electronics and aerospace industries. It is considered to

be an important structural resin material particularly used in aerospace industry due to its low

molecular weight and good mechanical property. Because of its wide range potential in the field

of structural, automobile and aerospace engineering, researchers focused on determining bulk

mechanical property of cross-linked epoxy resin.

1,2Department of aerospace engineering, The University of Alabama, Tuscaloosa, AL 35487,USA

Email: rrahman@crimson.ua.edu

2

In order to obtain optimized formulations and technologic conditions, numerous

experiments need to be carried out. It is difficult to gather all the detail information from

experimental analysis. Theoretical analysis of epoxy based structure has been done in continuum

scale (such as micromechanics and finite element analysis) since last decade. However,

continuum level analysis omits detail structural information of epoxy resins. Typically the bulk

mechanical and thermal properties for continuum scale analysis are highly dependent on

experimental study. This limits the continuum scale study difficult at certain stage when detail

structural, thermodynamic or interfacial properties are required. In order to obtain these

properties theoretically, atomistic simulation has become very useful.

Wu and Xu [1] performed MD simulation of cross-linked Diglycidyl Ether Bisphenol A

(DGEBA) epoxy with isophorone diamine (IPD) curing agent and determined elastic constants,

unit cell dimension and density. Fan and Yuen [2] also carried out MD simulation for cross-

linked Diglycidyl Ether Bisphenol F (EPON 862) epoxy in presence of the curing agent

Triethylenetetramine (TETA). They outlined the MD simulation methodology in detail and

eventually, determined the Young’s modulus, glass transition temperature of the cross-linked

network. Recently, Bandyopadhay and co-researchers [3] studied the mechanical and thermal

properties of cross-linked epoxy polymer using atomistic modeling. Hence it is clear that

investigation of epoxy composites using MD simulation is quite popular among the researchers

for many years.

In recent years scientists are more interested in nanocomposites due to their enhanced

mechanical property [4]. Experimental as well as theoretical study of epoxy reinforced with

nanomaterials such as carbon nanotube (CNT), graphene nanoplate (GNP), nanoclay (NC) and

carbon nanofiber (CNF) etc are becoming important research areas. In order to investigate the

mechanical property of these epoxy based nanocomposites, MD simulation has become an

important tool for capturing more detailed information at atomic scale such as: all the atomic

configurations (level of cross-link, orientation of nanofillers, atomic arrangements of polymer

chains etc) of the polymer. Eventually, using different polymer structural configurations at the

molecular level, MD simulation is capable in evaluating average mechanical properties of

nanocomposites. MD simulation is also considered as an effective tool to evaluate interaction

energy between epoxy and nanofiller materials. This encourages many researchers in MD

simulation of epoxy based nanocomposites to characterize mechanical properties and interfacial

properties at the molecular level. In this context, Yu and others [4] performed a MD analysis of

epoxy (EPON 862) nanocomposite with alumina (Al2O3) as nanoreinforcement. Their study

revealed that the mechanical property is improved by embeding alumina nanofillers in the epoxy

matrix. Zhu [5] performed MD simulation on a unit cell consisting of a single wall carbon

nanotube (SWNT) and cross-linked EPON 862 epoxy resin to study their stress-strain behaviors.

Frankland and co-researchers [6] investigated the effect of polymer-nanotube cross-linking on

critical load transfer from carbon nanotubes to polymer matrix by using molecular dynamics

(MD) simulation. Zhu calculated [5] the stress-strain behavior of SWNT-epoxy nanocomposites

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based on molecular dynamics (MD) simulation scheme. This was done by deforming large unit

cell followed by calculating average stress in the entire unit cell. For large systems the average

stress calculated from MD simulations is comparable to the stress in continuum scale. Hence MD

simulation has become quite reliable tool for evaluating properties of large systems.

Besides alumina or CNT nanofilliers, graphene platelets have drawn attention as

filler/reinforcing material to enhance the mechanical, electrical and thermal property of

traditional polymer system [7]. It is to be noted that graphene sheets are comparatively cost

effective than CN. Graphene posseses very high stiffness which makes it very good candidate as

a nanofiller in polymer nanocomposites. Studying electrical, mechanical properties of graphene

and polymer-graphene nanocomposites is currently one of the most popular topics among

researchers because of its unexplored behavior in polymer matrix. Cho and co-workers [8]

predicted the elastic constants of graphite using molecular mechanics model and subsequently,

determined the elastic constants of graphite/epoxy nanocomposites using micromechanical

model based on Mori-Tanaka method. The analysis was not entirely independent of experimental

input. Rafiee and co-researchers studied the enhancement of mechanical properties of graphene-

epoxy nanocomposites in presence of low graphene content [9]. Yasmine and Daniel calculated

thermal properties and elastic modulus of graphite/epoxy nanocomposites based on experimental

study [7]. However, experimentally it is quite challenging to prepare graphene/polymer

nanocomposites instead of graphite/polymer nanocomposites due to its tendency to stay in

agglomerated form. As a result the experimentally calculated mechanical property always has

influence from several parameters such as: graphene dispersion, load transfer mechanism from

graphene to epoxy or vice versa, void percentage etc. Besides experimental analyses

micromechanics models are also capable of predict elastic properties of graphene

nanocomposites. Ji and coworkers [10] developed a micromechanics model for graphene-

polymer nanocomposite based on Mori-Tanaka model. However, effect of above mentioned

parameters are difficult to be taken in account for micromechanics models as well. Moreover,

micromechanics models consider the homogenized unit cell which is not necessarily the exact

scenario. The density of polymer chains may not be well distributed throughout the unit cell.

Hence there is some discrepancy remains between experimental observation and micromechanics

model [9]. These issues could be taken into account in MD model. Eventually, mechanical

properties such as Young’s modulus, shear modulus and interfacial properties can be calculated

from the MD simulation.

Recently, it has been observed that the high mechanical performance of the polymer

composites not only depends on the inherent properties of the nanofiller, but also more

importantly depends on the optimizing the dispersion, interface energy and nanoscale

morphology within the polymer matrix [11]. It is difficult conduct all types of experiments to

study the interfacial characteristics of nano- composites. Therefore, atomistic simulation may be

useful methods to investigate the interfacial reinforcement mechanisms of nanocomposites. In

last few years several researchers investigated the interfacial bonding between nanofiller: CNT

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and matrix: polymer [5, 12]. They calculated the pullout strength, critical length of CNT in

matrix etc. However, for graphene both pullout (shear) and cohesive strength (normal) are

required. This cohesive interaction between a nanofiller surface and polymer is controlled by the

van der Waals energy. Aswathi [13] analyzed pullout as well as cohesive interaction in the

graphene/polyethylene system using molecular dynamics (MD) simulation. However, the

determination of graphene-epoxy interfacial property is yet to be analyzed.

So, in the current work the fundamentals goal was evaluating bulk elastic and interfacial

properties of graphene-epoxy nanocomposites using MD simulation by incorporating the effects

of aspect ratio, weight percentage of graphenes. The analysis was extended by addressing the

effect of graphene dispersion, atoms density in epoxy, difference between amorphous and

crystalline models etc.

THEORY

Molecular dynamics uses Newton’s equations of motion to calculate the trajectory of

atoms in any system. It is a deterministic method unlike Monte Carlo method. The advantage of

MD simulation is its capability of updating a system in future time step based on the current state

of the system. At each step, the forces on atoms are computed and combined with current

position as velocity to update positions and velocities of atoms in shorter timesteps by using

flowing system of equations:

( )1 2 3 N

(1)

, , ... (2)

i i i

i

i

m r f

E r r r rf

r

=

∂= −

∂

ii

Here,

( )1 2 3 N

:Mass of th atom

: Position of th atom

: Force on th atom

, , ... : Pottential energy in N-atoms system

i

i

i

m i

r i

f i

E r r r r

−

−

−

Molecular dynamics simulation consists of several important parts. Each part has its own

importance as a building block of MD simulation scheme. These are discussed as follows:

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Molecular interactions:

Typically molecular interactions are divided into two major parts: Intramolecular and

intermolecular interactions. These are specified in terms of potential which is called force field.

Components of force field are most defined semi-empirically by using ab-initio calculation

scheme. In this work the Compass class-2 or pcff force field is used to specify the molecular

interactions. Polymer consistent force field (pcff) force field is known to be appropriate for MD

simulation of organic or inorganic polymeric systems [2]. The class-2 potential energy can be

written as [14]:

( )1 2 3 N, , ... (3)

bond angle cross relation improper Coulomb VdWE r r r r E E E E E E−= + + + + +

Figure 1. Different components of forcefield

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( )1 2 3 N, , ... : Total potential energy

: Bond energy

: Angle energy

: Improper or out-of-plane energy

: Coulombic energy

: Energy due to Van der Waals interaction

bond

angle

improper

Coulomb

VdW

E r r r r

E

E

E

E

E

In the above Equation (3) the first term corresponds to bond energy, second term

corresponds to bond-angle energy, third term resembles cross correlation or dihedral-angle

energy, fourth term represents improper-angle energy term and the last term is related to non-

bonded energy. The last term refers to energy due to van der Walls interaction. These first three

terms are all together called as “Intramolecular interaction energy” and the last term is

“Intermolecular interaction energy”.

Fundamental algorithms in MD simulation:

MD simulation primarily concerns about solving Newton’s equations of motions for N -

atoms system. At each timestep the total energy or Hamiltonian of the system is written as:

( ) ( )

( )

1 2 3 N 1 2 3 N

2N

1 2 3 N

1

, , .. , , ... (4)

, , ..2

: Momentum of -th atom

i

i i

i

H K p p p p E r r r r

pK p p p p

m

p i

=

= +

=∑

Based on the total energy conservation criteria the equations of motion of the N-atoms

system can be defined in terms of a system of ordinary differential equations:

( )1 2 3 N

0 (5.1)

, , ...0 (5.2)

ii

i

i

i

pr

m

E r r r rp

r

− =

∂+ =

∂

i

i

Let’s assume the above system of equations is obtained at time t . Finite difference

method is used to update the current trajectory after solving the equations of motion. At the

current state, acceleration of each atom ( )i ia t r=ii

is calculated by combining the Equations (5.1)

and (5.2) as:

( )1 2 3 N, , ...0 (6)i i

i

E r r r rm r

r

∂+ =

∂

ii

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This acceleration is combined with positions and velocities at time t to calculate the

positions ( )ir t and velocities ( )iv t of the atoms at time ( )t t+ ∆ and so on. This is commonly

known as “Time integration (TI) scheme” in MD simulation. There are several types of TI

schemes available. Verlet algorithm is the most widely used among all the available methods.

This algorithm uses the positions and accelerations at time ( )t t−∆ to predict the position and

velocities at time ( )t t+ ∆ . The advantage of Verlet algorithm is: it’s simple, fast and memory

efficient. So, Verlet algorithm is highly used in large systems. The positions are calculated at

( )t t+ ∆ by using velocities and accelerations at t by using following equation [15]:

( ) ( ) ( ) ( )21 (7)

2i i i ir t t r t tv t t a t+ ∆ = + ∆ + ∆

The velocities at time 1

2t t + ∆

is calculated using the velocities and accelerations at

time t :

( ) ( )1 1 (8)

2 2i iv t t v t ta t + ∆ = + ∆

Hence, using these velocities and positions the forces on each atom is computed at

current positions. This gives ( )ia t t+ ∆ . Finally, the velocities at ( )t t+ ∆ are calculated:

( ) ( ) ( ) ( ) ( )1 5 1 (9)

3 6 6i i i i iv t t v t ta t ta t ta t t+ ∆ = + ∆ + ∆ − ∆ −∆

Ensemble average in MD simulation:

As discussed earlier the main objective of MD simulation is to replicate a material’s

structure at atomic scale and predict macroscopic properties without directly depending on

experiments. A large system of atoms (approximately: 2310 ) might represent a system equivalent

to macroscopic level. At each timestep t the force as well as acceleration on each atom will be

calculated in order to update for further timesteps. Finally, any property AverageΩ of this large

system can be calculated by averaging ( ) ( ),i ip t r t Ω for very long time tτ = →∞ [15]:

( ) ( )

( ) ( )

Average

0

1lim , (10)

, : Functions of

i i

t

i i i

p t r t dt

p t r t f

τ

τττ→∞= =

Ω = Ω

Ω

∫

8

There are different types of ensemble averages used in MD simulation. Among these first

three types of ensembles are widely used in MD simulations. Based on different conditions these

are:

- Constant energy, constant volume (NVE)

- Constant temperature, constant volume (NVT)

- Constant temperature, constant pressure (NPT)

- Constant temperature, constant stress (NST)

- Constant pressure, constant enthalpy (NPH)

In the current work the systems were equilibrated by using both NPT and NVT conditions.

Prediction of mechanical properties:

MD simulation is useful in calculating mechanical and thermal properties of a system. It

is quite possible to observe stress-strain response of a system at atomic scale. The stress

components ijσ (both tensile and compressive) can be calculated from virial expression:

( )N

1

1 T T

ij i i i ij ij

i i j

m v v r fVol

σ= <

= − +

∑ ∑

In this above expression:

: Volume of the simulation boxVol

im : Mass of thi atom

iv : Velocity of thi atom

N : Total number of atoms

ijr : Distance between thi and thj atoms

T

ijf : Force exerted on thj atom by thi atom

Hence, for series of six uniaxial strains [ ]11 22 33 12 23 31

Tε ε ε ε ε ε=ε the unit cell

(simulation box) undergoes tensile and compressive deformations in all six directions. This leads

us to calculate the corresponding stress tensors by using virial expression.

Every time the unit cell is deformed in just one direction while other strain components

are fixed to zero. So, the stiffness elements can be written as:

(11)

9

21ijkl

ij kl

UC

V ε ε∂

=∂ ∂

Here, U is total energy and V is the volume of the unit cell. This can be written in terms of stress

components as:

ij

ijkl

kl

Cσ

ε

∂=∂

This term can be written in finite difference form. Hence, each components of stiffness matrix

consists of difference between a tensile and a compressive stress component:

ij ij

ijkl

kl

Cσ σ

ε

+ −−=

∆

The most simplified form of this equation in order to calculate stiffness components is:

2

i iij

j

Cσ σ

ε

+ −−=

∆

The advantage of using this first derivative form instead of second derivative form is to

reduce the amount of error. In finite difference forms, the higher order terms cause more error

than the first order term. Hence, the first derivative is used to calculate stiffness matrix (equation

[16]. The deformation process is considered to be strain controlled; the change in strain

components jε∆ is 2 jε due to tensile and compressive deformation. So, the Lame’s constants

,λ µ were calculated from the stiffness matrix. This lead to evaluating the bulk elastic properties:

( ) ( )

( )

11 22 33 44 55 66

44 55 66

3 2

1 2

3 3

1

3

E

G

C C C C C C

C C C

λ µµ

λ µµ

λ

µ

+=

+

=

= + + − + +

= + +

Besides computing the bulk elastic properties it is also possible to obtain stress-strain

relationship using MD simulation. The purpose is to perform a numerical experiment with the

unit cell by applying deformation on the unit cell in any direction and calculate the average

stress. Typically any uniaxial deformation causes atoms in the system to move along the change

in dimension of the simulation box. The stress due to any uni-axial deformation can be

represented as an average of the principle stresses or “hydrostatic stress” calculated from the

Young’s modulus

Shear modulus

(12)

(13)

(14)

(15)

(16)

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formula in equation (11). So, for any small strain iiε (this is simply ε for amorphous systems),

the average stress in the system is written as:

( )11 22 33

11 22 33

(17)3

, , : Virial stresses in principle directions

: Hydrostatic stress

σ σ σσ

σ σ σ

σ

+ +=

A generic scheme of evaluating stress-strain response is iteratively applying strain to the

system followed by MD runs. After every strain it is important to ensure that the system is stable

as the atoms are remapped into new positions. Lack of proper relaxation causes incorrect stress-

strain response.

Calculating interaction of amorphous structure on a crystal surface:

MD simulation calculates total potential energy ( )1 2 3 N, , ...E r r r r of a N -atoms system.

This leads to calculating force on each atom in the system. Similar to the previous discussion, it

is possible to evaluate interaction between a polymer and nanostructure at atomic scale.

There are two different types of interaction between graphene and epoxy polymer matrix.

One is normal separation force (cohesive force) CohesiveF and another is shear separating force

(pullout force) PulloutF . These forces are calculated from the change in interaction energy between

the nanostructure and polymer system is simply calculated as:

( )Interaction Total Polymer-without nanostructure Nanostructure (18)E E E E= − +

Here, TotalE is the total potential energy obtained from MD simulation of combined

polymer-nanostructure system. Once the nanofiller is moved away from the polymer matrix it

experiences reaction forces CohesiveF or PulloutF . The system is non-periodic along the direction of

movement in order to minimize the effect from surrounding neighborhood unit cells.

Considering graphene as nanofiller, the equations for calculating ultimate strength are:

Figure 2. Uniaxial deformation of amorphous system

Deformation

in X-direction

11

Max

CohesiveCohesive

Max

PulloutPullout

(19.1)

u

u

c

F

L W

F

l W

σ

σ

=×

= (19.2)

: Length of graphene

: Width of graphene

: Critical length at failurec

L

W

l

SIMULATIO DETAILS

The objective of the current work is to calculate the mechanical and interfacial properties

of graphene reinforces epoxy polymer nanocomposite. Elastic properties are calculated in both

amorphous and crystalline unit cells of graphene-epoxy nanocomposite (G-Ep-Nc). Interfacial

properties are calculated semi-periodic unit cells.

Amorphous model

MD simulation was conducted in unit cells consisting of cross-linked EPON 862 resin. A

typical amorphous unit cell contains epoxy-curing agent in the ratio of 12:4 [2]. pcff (Compass

class 2) force field was used in defining the bonded and non-bonded interactions [14]. A brief

description of MD simulation is provided below:

Initial amorphous unit cell with cross-linked epoxy polymer:

The EPON 862 monomer has two reactive sites. During cross-linking process either one

or both of these reactive sites of EPON 862 (epoxide groups) create new bonds with any of the

four reactive sites in TETA (nitrogen groups: 2NH,NH ) hardener if they come close enough to

each other within a certain distance. Typically this distance is 4A0 – 10A

0 [4]. This leads to

generate cross-linked networks of epoxy polymer (Figure 3). A typical amorphous cross-linked

epoxy network consists of 12 EPON 862 neat resin molecules and 4 TETA curing agent

molecules. Hence, each crosslinked epoxy repeat unit consists of 157 atoms. The initial cell

volume is approximately 87300.00 30A with average initial density approximately 0.9-1.0

gm/cc. Later, graphene sheets were embedded into this amorphous epoxy unit cells to predict

mechanical properties of epoxy-graphene nanocomposites.

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Graphene sheet: Graphene sheet is a 2D structure consists of carbon atoms bonded by sp2

hybridized electrons. The carbon atoms are arranged in a hexagonal pattern with the shortest

distance of 1.42 A0 between atoms with bond angle 120

0 . Single layer graphene possesses very

high mechanical properties. The average Young’s modulus of single layer zigzag graphene sheet

with length 20.18 nm and width 4.18 nm is 1.033 TPa [17].

Typically graphene sheets are obtained from graphite oxide (GO). Hence, in the atomistic

model the carbon atoms at the edge of the graphene were kept terminated by adding hydrogen

atoms [18] (Figure 5).

Figure 3. Cross-linked epoxy polymer

Figure 4. Amorphous unit cell with cross-linked epoxy

Reactive sites

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Graphene embedded in epoxy amorphous cell: The graphene sheets with hydrogen terminated

edges were embedded in the 3D periodic amorphous unit cells for evaluating bulk elastic

properties. In this study, 1%, 3% and 5% of graphene by weight was considered in the

amorphous unit cell. The density of amorphous graphene-epoxy unit cells was kept in the range

of 0.7-1.0 gm/cc. However, this changes during the MD equilibration process. The number of

epoxy monomers was determined in such a way which ensures expected weight percentage of

graphene in the unit cell. Table 1 shows four material configurations with relevant graphene

aspect ratio and weight percentage:

Figure 5. Graphene sheets with two different aspect ratios: Type (a): Aspect ratio (AR): 5.0

(approximately) and Type (b): Aspect ratio (AR): 13.0 (approximately)

Type-(a)

Type-(b)

Figure 6. Amorphous cells with embedded graphene sheets

Graphene sheet

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Crystalline model

Crystalline model of graphene-epoxy nanocomposite consists of a sandwiched structure

of graphene lattice and amorphous epoxy matrix. Graphene is considered to be a crystal with

symmetry P3m [19]. The amorphous graphene-epoxy model is isotropic whereas crystalline

model is anisotropic. This model is important in analyzing effect of graphene stacking on the

mechanical property. Combined stack of graphene layers are considered as graphite which

possesses different mechanical property than graphene. Three different crystalline models are

constructed based on agglomerated or dispersed graphene. The objective is to study the effect of

single graphene, dispersed graphene and agglomerated graphene on mechanical property. Each

crystalline graphene-epoxy unit cell contains 3% graphene. Average graphene-epoxy spacing is

less than 2A0. Summary of crystalline model is given in the following table:

Configuration Unit cell dimension (A0) umber of graphene

plates

CRYS-GnEp-I a=9.84, b=19.02, c=1056.42 1

CRYS-GnEp-II a=9.84, b=19.02, c=2116.24 3 (separated)

CRYS-GnEp-III a=9.84, b=19.02, c=3104.21 3 (stacked)

Material

Configuration

Aspect ratio (AR) Weight

percentage

of graphene

umber of

epoxy

molecules

Unit cell

dimension (A0)

G-Ep-Nc-I AR=5 (Type-a) 1% 67 a=b=c=65

G-Ep-Nc-II AR>=10 (Type-b) 1% 262 a=b=c=90

G-Ep-Nc-III AR=5 (Type-a) 3% 22 a=b=c=38.8

G-Ep-Nc-IV AR>=10 (Type-b) 3% 90 a=b=c=62

G-Ep-Nc-V AR=5 (Type-a) 5% 13 a=b=c=33

G-Ep-Nc-VI AR>=10 (Type-b) 5% 53 a=b=c=52.3

Table 1. Number of epoxy molecules and graphene in a unit cell

Table 2. Unit cell configuration for crystalline model

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Graphene-epoxy interface study model

In this section we are interested in studying the load transfer mechanism between

graphene sheet and epoxy polymer matrix at atomistic level. Both normal and shear separation

forces were calculated on a slowly displaced graphene sheet in normal and shear directions

respectively.

The normal and shear movements lead to mode-I and mode-II debonding phenomena.

Initially the space between epoxy and graphene sheet is kept less than 2A0. It is quite important

to maintain certain boundary condition while the graphene is displaced. For mode-I case the

graphene-epoxy unit cell is non-periodic in Z-axis because the cohesive interaction along Z-

direction needs to be purely influenced by the epoxy-graphene system in the current unit cell.

Similarly, for mode-II case the system was nonperiodic in xy-plane due to transverse movements

of graphene. Eventually the reaction forces on graphene due to mode-I or mode-II separation

were calculated. This leads us to obtain forces vs. displacement curve for both mode-I and mode-

II cases. In this study of interfacial properties the weight percentage of graphene is considered to

be higher (> 3%) in order to emphasize the effect of graphene on van der Waals interaction

between epoxy (with density 1.0 gm/cc) and graphene. Average displacement rate of graphene

was 0.0001A0 per femto-second. Summary of unit cell configurations are mentioned in following

Table 3:

Figure 7. Crystalline model of graphene-epoxy nanocomposite

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Configuration Graphene sheet

dimension

(length x widths)

(A0)

2

Aspect ratio

(AR)

Weight

percentage of

graphene

Interaction

force type

Mode-I-small 39.36 x 19.02 2.07 6.8% Normal

Mode-I-big 118.08 x 19.02 6.20 17.7% Normal

Mode-II-small 39.36 x 19.02 2.07 3.3% Shear

Mode-II-big 118.08 x 19.02 6.20 9.7% Shear

Figure 8. Mode-I separation of graphene from epoxy polymer

Figure 9. Mode-II separation of graphene from epoxy

polymer

Table 3. Unit cell configuration for interface model

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Computational details

The overall methodology for evaluating mechanical property of epoxy nanocomposite

consists of several steps such as: structure optimization, equilibration, and calculation of stiffness

property. Before proceeding through all these steps the force field has to be defined in order to

establish the interaction protocol among atoms.

Force field

As mentioned earlier, pcff (Compass class-II) force field was used to define the bonded

and non-bonded interactions among atoms [20] (Equation (3)). The cutoff distance of 8-10 A0

was used in defining the long range interaction by 6/9-Lennard-Jones potential. Columbic

potential was calculated using Ewald summation method.

Cross-linking and structure optimization

The initial amorphous structure of epoxy polymer consists of epoxy monomers (EPON

862) and curing agents TETA. Initially, the atoms in the amorphous cell were subjected to

molecular dynamic simulation with the pcff force field under NVT condition at 600K

temperature. The purpose is to rearrange the atom positions by adding high velocity under high

temperature. During this time the reactive cross-linking sites (Figure 3) were taken into

consideration. Typically, the cutoff distance was less than 10 A0. Hence, if the reactive sites of

epoxy monomer come close (within the cutoff distance) to any of the reactive sites from TETA

monomer, a single bond was created between the reactive sites. Based on several trials, the most

frequent cross-linked unit is represented in Figure 4. So, this structure was used as repeat unit for

cross-linked epoxy molecule in later sections of the [4]. This configuration indicates 50% cross-

linking.

Setting up the molecular dynamic simulation

After forming the cross-linked repeat unit further step is to perform MD equilibration

prior to calculating mechanical properties. In the current work there are three different kinds of

models being used. Thus MD equilibration might not be the same in all these cases. Hence the

MD simulation scheme is explained subjective to specific type of model:

MD summary for amorphous models: The initial amorphous models were built with average

density 0.7-1.0 gm/cc. These graphene-epoxy amorphous unit cells were firstly optimized by

conjugate gradient method in order to achieve the stable initial structural configuration. The

systems were subjected to molecular dynamic equilibration under NPT ensemble (constant

number of atoms, pressure and temperature) for 5000 thousand steps of MD simulation with 1.0

fs time step was carried out at temperature of 298 K and 1.0 atm pressure. The objective was to

equilibrate the internal pressure of the system as and obtain a stable volume. Nose-Hoover

18

barostat was used to add damping parameter into the fluctuating pressure [15]. After that, MD

equilibration was carried out under NVT condition at 0.1 K with Nose-Hoover thermostat for

0.5-3.0 ps. This equilibration stabilizes the thermal fluctuation by scaling the velocity at low

temperature. Once the system is equilibrated it was subjected to uniaxial deformation. Low

temperature was preferred in order to avoid thermal noise in the analysis. The atom positions and

velocities were updated by Verlet time integration scheme (Equation (7-9)). Each time step size

was averagely 0.5-1.0 fs found to be compatible to all the amorphous graphene-epoxy systems.

At this stage the amorphous systems are subjected to either uniaxial deformation in order

to obtain stress-strain response or series of static deformations in all directions in order to

calculate bulk Young’s modulus and shear modulus (Equations (12-16)). For uniaxial

deformation the unit cells were subjected to deform with strain rate of 0.1% in every 0.3 ps for

1.5 ps except the case for G-Ep-Nc-II. This system was deformed with strain rate of 0.01% in

evergy 0.1 ps for 0.5 ps. After each deformation the atoms were remapped into the deformed unit

cell and their velocities were rescaled under NVT condition at 0.1 K. At each step the time-

average hydrostatic stress in the system was calculated using Equation (17).

Static deformation method requires a series of deformations in xx-yy-zz-xy-yz-xz

directions. Prior to every deformation the energy in the unit cells was minimized by conjugate

gradient based energy minimization algorithm with energy and force tolerance in the order of

10-8 and 10

-10 respectively. Based on different systems the global energy minima were searched

for 30-1000 steps. In every direction the unit cells were subjected to ± 10-6 % strain in all the

directions. After each deformation average stress in the system was calculated by Equation (11).

MD summary for crystalline models: Crystalline models were also required to be equilibrated

under different conditions. During MD runs the atoms on graphene sheet are supposed to sustain

initial 2D arrangement based on p6/mmm symmetry. These atoms were not constrained in order

to let the system behave in a more realistic manner. Initially the energy was minimized followed

by NVT based MD runs for 5000 steps. The system was then cooled down to 0.1 K for averagely

500 - 30000 steps depending on the system. Typical time step size was 0.8-1.0 fs. The positions

and velocities of the atoms were updated using Verlet time integration scheme.

Once the system is equilibrated it was subjected to deform in xx-yy-zz directions. The

epoxy weight percentage was 3% in all the crystalline unit cells. Hence the orthorhombic

simulation box was quite large along Z-axis compared to X and Y-axes (Table 2). So, the shear

deformation not justified in the current cases. The unit cells were deformed in every 0.1 ps

followed by NVT based equilibration with 1.0 fs time step size. Stain rates varied within the

range 10-2 %- 10

-7 %. Nose-Hoover thermostat was used to add a damping in the systems for 0.05

ps. In addition to MD, the atoms in the systems were also subjected to Langevin dynamics [15]

with a damping term of 0.01 ps. This was done in order to reduce any unexpected vibration in the

graphene.

19

MD summary graphene-epoxy interface models: The MD scheme in the graphene-epoxy

interface models is a bit similar to the ones in crystalline models. However, the boundary

condition is unlike to a typical periodic system (Figure 9 and 10). Prior to transverse or

longitudinal movements of graphene plate energy is minimized in all the systems by conjugate

gradient method 1000 steps. Energy and force tolerance were in the order of 10-10. Average

movement rate of graphene was 0.01 A0/fs for 0.5 – 3.0 ps in transverse direction (pullout) and

0.001-0.0001 A0/fs for 1.5 ps in longitudinal direction (cohesive). The positions and velocities of

the atoms were updated using verlet time integration scheme under NVT condition. Temperature

of the system was kept within the range of 0.1-1.0 K in order to avoid unexpected vibration and

swirling of graphene sheet.

RESULTS AD DISCUSSIO

Elastic properties

The molecular dynamic simulations were carried out using open source molecular

dynamics code LAMMPS. The elastic properties such as bulk Young’s modulus and shear

modulus were calculated using static deformation method (Equations (12-16)). It is important to

analyze the initial structural properties prior to any further calculation. Two types of RDF were

considered. One is between graphene atoms and epoxy molecules and another is between any

pair of atoms in the system.

According to Figures 10 and 11, the RDF is representing the most likely pair-wise

distance between atoms from graphene and atoms from epoxy molecules. Within first 2.5-2.8 A0

from graphene surface there is no possibility to find any epoxy molecule. This resembles the

repulsion between graphene and epoxy due to the van der Waals interaction within a cutoff

radius less than 3 A0. For both of the cases, the most epoxy dense region is at about 4.3-4.5 A

0

from the graphene sheets. This might be due to the attraction between graphene and epoxy

molecules. More generally we can conclude that the average binding occurs between graphene

and epoxy at 40% of the cutoff distance in the attraction region.

The RDF was also calculated for the overall system. This may represent the distribution

of average atom-atom distance among all the epoxy molecules and graphene sheet itself. As all

these unit cells consist of amorphous epoxy and graphene, it is expected to have RDF of similar

pattern in all the graphene-epoxy systems as well as only epoxy system (EPON 862). It is clearly

observed in Figures 12 and 13. For both kinds of graphene the RDF resembles most possible

bond length of 3.9 A0 (approximately) with maximum 10 times and minimum 2 times probability

of finding this bond length in graphene-epoxy polymer systems than in ideal gases.

20

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

3.00

0.00 2.00 4.00 6.00 8.00 10.00

g(r

)

Distance from graphene to epoxy molecules ( A0)

G-Ep-Nc-IG-Ep-Nc-IIIG-Ep-Nc-V

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

14.00

16.00

18.00

0.00 2.00 4.00 6.00 8.00 10.00

g(r

)

Distance from graphene to epoxy molecules ( A0)

G-Ep-Nc-II

G-Ep-Nc-IV

G-Ep-Nc-VI

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

0.00 2.00 4.00 6.00 8.00 10.00

g(r

)

Average atom-atom distance in the system (A0)

G-Ep-Nc-I

G-Ep-Nc-III

G-Ep-Nc-V

EPON 862

Figure 11. RDF between graphene (type-(b)) and epoxy

Figure 10. RDF between graphene (type-(a)) and epoxy

Average gap between graphene

and epoxy = = 2.5 A0

Average gap between graphene and

epoxy = 2.8 A0

Figure 12. RDF of entire system with graphene (type-(a))

21

The unit cell configurations for type-(a) and type-(b) graphene are already mentioned in

Table 1. Hence the calculated mechanical properties using static deformation method are

mentioned in Table 4:

From the Table 4 the maximum Young’s modulus is observed in the case of G-Ep-Nc-I.

The predicted Young’s modulus and shear modulus of graphene-epoxy nanocomposite clearly

show increased values compared to neat resin in most of the cases. It appears that modulus of

nanocomposites with lower graphene aspect ratio is comparatively higher than the same with

higher aspect ratio. The bulk elastic properties are helpful in comparing the Young’s modulus

calculated from the stress-strain response.

Stress-strain response

Amorphous systems: Besides calculating the bulk Young’s modulus by static deformation

method it is also possible to calculate it from stress-strain response. The cells were subjected to

series of uniaxial tensile strains (0.1%). Hence, the system responded to these strains by

-2.00

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.00 2.00 4.00 6.00 8.00 10.00

g(r

)

Average atom-atom distance in the system (A0)

G-Ep-Nc-II

G-Ep-Nc-IV

G-Ep-Nc-VI

EPON 862

Material

Configuration

Aspect ratio (AR) Weight percentage

of graphene

Bulk

Young’s

modulus: E

(Gpa)

Bulk shear

modulus: G

(Gpa)

G-Ep-Nc-I AR>=5 1% 4.56 1.73

G-Ep-Nc-II AR<=10 1% 3.45 1.16

G-Ep-Nc-III AR>=5 3% 3.98 1.37

G-Ep-Nc-IV AR<=10 3% 1.16 0.391

G-Ep-Nc-V AR>=5 5% 2.98 1.07

G-Ep-Nc-VI AR<=10 5% 1.34 0.45

Epoxy (EPON 862)

[21, 22]

N/A N/A 2.5 0.95

Table 4. Elastic properties calculated from static deformation method

Figure 13. RDF of entire system with graphene (type-(b))

22

increasing overall stress (Equation (17)). Calculated bulk Young’s modulus is mentioned in

Table 4:

Material

Configuration

Aspect ratio

(AR)

Weight

percentage of

graphene

Bulk Young’s

modulus from

stress-strain

response

Bulk Young’s

modulus from

micromechanics

model [10] G-Ep-Nc-I AR>=5 1% 5.00 2.96

G-Ep-Nc-II AR<=10 1% 4.27 2.63 G-Ep-Nc-III AR>=5 3% 3.98 3.6

G-Ep-Nc-IV AR<=10 3% 2.04 2.71

G-Ep-Nc-V AR>=5 5% 3.56 4.33

G-Ep-Nc-VI AR<=10 5% 1.77 4.69

EPON-862 (from

MD simulation)

N/A N/A 3.42 N/A

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Str

ess

(G

Pa

)

Strain (A0/A0)

G-Ep-Nc-I

G-Ep-Nc-III

G-Ep-Nc-V

EPON 862

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Str

ess

(G

Pa

)

Strain (A0/A0)

G-Ep-Nc-II

G-Ep-Nc-IV

G-Ep-Nc-VI

EPON 862

Figure 14. Stress-strain response of amorphous system with type-(a) graphene

Figure 15. Stress-strain response of amorphous system with type-(b) graphene

Table V. Young’s modulus calculated from stress-strain response

The bulk Young’s modulus is predicted

reasonable agreement with the prediction from

proposed by Ji [10]. The micromechanics model is based on homogenized po

Hence the prediction shows a monoto

increasing volume fraction of nanofiller. But in real

in this manner. There are some factors such as percentage of void, dispersion

etc cause decreasing in Yong’s modulus.

decrease in the Young’s modulus of graphene

increases. They considered graphene

dimension in nanometer scale. The unit cells in the current work may be represented according to

their case by scaling down to angstroms. Thus weight fraction of graphene is scaled up ten times

relative to the one considered in the paper by

times. The graphene is closely comparable

on this hypothetical comparison

graphene has good agreement with th

Atom density in the unit cell also plays an important role in this

(Figures 10 and 11) comparatively larger gap between graphene

weight fraction for both types of graphenes: G

in all the amorphous cells are represented in Figures 16 and 17

of G-Ep-Nc-I and G-Ep-Nc-II compared to

system.

0.02

0.04

0.06

Ato

m d

en

sity

(ato

ms/

vo

lum

e)

Figure 16. Atom density in amorphous system with type

23

bulk Young’s modulus is predicted from stress-strain response is observed to have a

reasonable agreement with the prediction from graphene-polymer micromechanics model

he micromechanics model is based on homogenized po

prediction shows a monotonically increasing nature in bulk Young’s modulus with the

of nanofiller. But in real, Young’s modulus may not

There are some factors such as percentage of void, dispersion quality of graphene

cause decreasing in Yong’s modulus. The work by Rafiee [9] shows first increase and then

decrease in the Young’s modulus of graphene-epoxy nanocomposites as the volume fraction

They considered graphene sheet with aspect ratio of 19 (approximately)

The unit cells in the current work may be represented according to

their case by scaling down to angstroms. Thus weight fraction of graphene is scaled up ten times

o the one considered in the paper by Rafiee. This also leads to scale up the stress by ten

The graphene is closely comparable to the one with AR=13 (type-(b)) in this paper. Based

on this hypothetical comparison, the Young’s modulus at 1%, 3% and 5% cases for type

graphene has good agreement with the experimentally calculated [9].

Atom density in the unit cell also plays an important role in this regard.

) comparatively larger gap between graphene and epoxy in the case of 5%

weight fraction for both types of graphenes: G-Ep-Nc-V and G-Ep-Nc-V. Average a

represented in Figures 16 and 17 explains higher Young’s modulus

II compared to other cases due to distinguishably denser atoms in the

G-Ep-Nc-I G-Ep-Nc-III G-Ep-Nc-V

0

0.02

0.04

0.06

Figure 16. Atom density in amorphous system with type-(a) graphene

strain response is observed to have a

micromechanics model

he micromechanics model is based on homogenized polymer matrix.

bulk Young’s modulus with the

t always increase

quality of graphene

first increase and then

epoxy nanocomposites as the volume fraction

sheet with aspect ratio of 19 (approximately) with

The unit cells in the current work may be represented according to

their case by scaling down to angstroms. Thus weight fraction of graphene is scaled up ten times

e up the stress by ten

in this paper. Based

cases for type–(b)

. The RDFs show

and epoxy in the case of 5%

Average atom density

explains higher Young’s modulus

other cases due to distinguishably denser atoms in the

During deformation process it is important to observe the

of energy. The applied strain causes change in atom positions, velocities and overall molecular

structure. An equilibrated system responds to deformation by increasin

However it is expected to increase in the potential energy of the system as the deformation

process is quasi-static. Potential

mol Bond Angle Dihedral ImproperE E E E E= + + +

comparatively larger contribution from the molecular energy than van der Waals energy.

Molecular energy is expected to be increasing

molecular energy clearly explains the elongation in t

0.02

0.04

0.06

Ato

m d

en

sity

(ato

ms/

vo

lum

e)

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

3.00E+04

E_

mo

l (

kca

l/m

ole

)

Figure 17. Atom density in amorphous system with type

Figure 18. Molecular energy in amorphous system with type

24

During deformation process it is important to observe the response of the system

. The applied strain causes change in atom positions, velocities and overall molecular

structure. An equilibrated system responds to deformation by increasing its

it is expected to increase in the potential energy of the system as the deformation

Potential energy consists of molecular energy

mol Bond Angle Dihedral ImproperE E E E E ) and van der Waals energy [20]. Potential energy has

comparatively larger contribution from the molecular energy than van der Waals energy.

Molecular energy is expected to be increasing (Figure 18 and 19). The increase in slope of this

molecular energy clearly explains the elongation in the molecular topology with

G-Ep-Nc-II G-Ep-Nc-IV G-Ep-Nc-IV

0

0.02

0.04

0.06

0 0.05 0.1 0.15Strain (A 0/A 0)

G-Ep-Nc-I

G-Ep-Nc-III

G-Ep-Nc-V

Figure 17. Atom density in amorphous system with type-(b) graphene

Figure 18. Molecular energy in amorphous system with type-(a) graphene

the system in terms

. The applied strain causes change in atom positions, velocities and overall molecular

g its overall energy.

it is expected to increase in the potential energy of the system as the deformation

consists of molecular energy (

otential energy has

comparatively larger contribution from the molecular energy than van der Waals energy.

he increase in slope of this

he molecular topology with applied strain.

0.2

(b) graphene

(a) graphene

25

Kinetic energy in all the cases has least effect on the total energy during deformation

period. As the MD during was under NVT condition at low temperature (0.1 K) it is obvious to

have rescaling of atom velocities in order to be consistent with the temperature. Thus kinetic

energy was scaled down and eventually became very small in order to sustain the stability of the

system. This is interestingly important for quasi-static deformation process because the change in

potential energy of the atoms has a significant correlation with the deforming molecular

topology.

Besides the energy we can analyze the “Mean square displacement” (MSD) of the

graphene-epoxy system during deformation process. MSD in uniaxial direction during quasi-

static deformation process should represent the diffusion of atoms into the newly elongated unit

cell. So clearly the slope of this MSD is expected to be non-zero throughout this time (Figures 20

and 21).

0.00E+00

2.00E+04

4.00E+04

6.00E+04

8.00E+04

1.00E+05

1.20E+05

0 0.05 0.1 0.15 0.2

E_

mo

l (k

cal/

mo

le)

Strain (A0/A0)

G-Ep-Nc-II

G-Ep-Nc-IV

G-Ep-Nc-VI

0.00E+00

5.00E-01

1.00E+00

1.50E+00

2.00E+00

2.50E+00

3.00E+00

3.50E+00

4.00E+00

4.50E+00

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Me

an

sq

ua

re d

isp

lace

me

nt

(A 0

)

Strain (A 0/A 0)

G-Ep-Nc-I

G-Ep-Nc-III

G-Ep-Nc-V

Figure 19. Molecular energy in amorphous system with type-(b) graphene

Figure 20. MSD of amorphous system with type-(a) graphene

26

Crystalline systems: Amorphous systems are quite useful in calculating the bulk elastic

properties of the graphene-epoxy polymer nanocomposites. Graphene also has a tendency to

restack and form graphite due to van der Waals interaction between layers. Hence the effect of

graphite formation from graphene on elastic properties needs to be addressed in the MD

simulation. For all three different cases mentioned in Table 2 the stress-strain response in xx, yy,

zz directions are shown separately.

Configuration umber of

graphene plates

E11 (GPa) E22 (GPa) E33 (GPa)

CRYS-GnEp-I 1 1.89 1.94 0.9

CRYS-GnEp-II 3 (separated) 4.81 8.76 0.32

CRYS-GnEp-III 3 (stacked) 3.99 5.67 3.80

0.00E+00

1.00E+00

2.00E+00

3.00E+00

4.00E+00

5.00E+00

6.00E+00

7.00E+00

8.00E+00

9.00E+00

1.00E+01

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

Me

an

sq

ua

re d

isp

lace

me

nt

(A0)

Strain (A0/A0)

G-Ep-Nc-II

G-Ep-Nc-IV

G-Ep-Nc-VI

0.00

0.01

0.01

0.02

0.02

0.00 0.05 0.10 0.15 0.20 0.25 0.30

Str

ess

(G

Pa

)

Strain (A0/A0)

sig_zStress along Z- direction

Figure 21. MSD of amorphous system with type-(b) graphene

Table 6. Young’s modulus calculated from stress-strain response for crystalline model

27

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.00 0.02 0.04 0.06 0.08 0.10 0.12

Str

ess

(G

Pa

)

Strain (A0/A0)

sig_x

sig_y

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.00 0.20 0.40 0.60 0.80 1.00

Str

ess

(G

Pa

)

Strain (A0/A0)

sigma_z

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.20 0.40 0.60 0.80 1.00

Str

ess

(G

Pa

)

Strain (A0/A0)

sigma_x

sigma_y

Stress along X-direction

Stress along Y-direction

Stress along Z-direction

Stress along X-direction

Stress along Y-direction

Figure 22. Stress-strain response in xx, yy and zz directions for CRYS-GnEp-I

Figure 23. Stress-strain response in xx, yy and zz directions for CRYS-GnEp-II

28

The scales were different for different cases due to varying strain rates in order to sustain

stability during deformation process. Hence the stress-strain plots are shown separately in

Figures 22, 23 and 24. The Young’s modulus is calculated from the linear region of the stress-

strain curve and tabulated in Table 6. In the crystalline model graphene sustains its stiffness

along x and y direction. This is a primary difference between the amorphous model and the

crystalline model. Weight percentage of epoxy remains 3% in all these three crystalline

structures and the volume fraction of graphene increases as graphene forms graphite. The

objective is to emphasize on effect of graphene restacking on the elastic properties.

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.00 0.00 0.00 0.01 0.01 0.01 0.01

Str

ess

(G

Pa

)

Strain (A0/A0)

sigma_z

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0.00 0.20 0.40 0.60 0.80 1.00 1.20

Str

ess

(G

Pa

)

Strain (A0/A0)

sig_x

sig_y

Stress along Z-direction

Stress along X-direction

Stress along Y-direction

Figure 24. Stress-strain response in xx, yy and zz directions for CRYS-GnEp-III

29

The in plane Young’s modulus ,xx yyE E are highly dependent on the effect of graphene

volume fraction because graphene’s in plane property (Young’s modulus is approximately 1 TPa

0.00E+00

5.00E+03

1.00E+04

1.50E+04

2.00E+04

2.50E+04

0 0.02 0.04 0.06 0.08 0.1

E_

mo

l (k

cal/

mo

le)

Strain (A0/A0)

Molecular energy due to Z-deformation

Molecular energy due to Y-deformation

Molecular energy due to X-deformation

0.00E+00

5.00E+04

1.00E+05

1.50E+05

2.00E+05

2.50E+05

3.00E+05

3.50E+05

4.00E+05

0.00 0.20 0.40 0.60 0.80 1.00

E_

mo

l (k

cal/

mo

le)

Strain (A0/A0)

Molecular energy due to Z-deformation

Molecular energy due to Y-deformation

Molecular energy due to X-deformation

0.00E+00

1.00E+05

2.00E+05

3.00E+05

4.00E+05

5.00E+05

0.00 0.20 0.40 0.60 0.80 1.00

E_

mo

l (k

cal/

mo

le)

Strain (A0/A0)

Molecular energy due to Z-deformation

Molecular energy due to Y-deformation

Molecular energy due to X-deformation

Figure 25. Molecular energy for CRYS-GnEp-I

Figure 26. Molecular energy for CRYS-GnEp-II

Figure 27. Molecular energy for CRYS-GnEp-III

30

[17]). However, the off-plane Young’s modulus of the graphene-epoxy nanocomposites zzE is

mostly controlled by the van der Waals interaction between graphene-epoxy or graphene-

graphene. The increase in xxE and yyE from CRYS-GnEp-I to CRYS-GnEp-II in Table 6 indicates the

effect of increase in graphene reinforcement. As graphenes are restacked and form graphite (CRYS-

GnEp-II to CRYS-GnEp-III), the in plane Young’s modulus ( ,xx yyE E ) drops whereas zzE increases

because graphene-graphene van der Waals interaction starts playing significant role. The molecular

energy is increased as the system was deformed under strains in x and y directions (Figures 25, 26 and

27). However, lower in-plane modulus for CRYS-GnEp-I can be explained by very small change in the

slope of molecular energy curve (Figure 25) with respect to applied strain. The change in molecular

energy due to deformation in z-direction seems to be significantly less responsive to the applied strain. In

all these cases molecular energy has trivial role on zzE . This is because the major contribution of van der

Waals interaction in the total potential energy. The average van der waals energy for these three cases are

-2474.83 kcal/mole, -5405.64 kcal/mole and -5789.92 kcal/mole. Van der Waals energy decreases

significantly from CRYS-GnEp-I to CRYS-GnEp-II due to epoxy molecules between two graphenes.

Effect of restacking is observed as non-bonding energy is increased by -384.28 kcal/mole from CRYS-

GnEp-II to CRYS-GnEp-III.

Interfacial property calculation: Interface between graphene and epoxy plays a significant role in

load transfer mechanism from graphene to epoxy or vice versa. It is possible to determine the

normal and shear force-displacement response this interface. As model was controlled by

displacement, certain reaction force was observed in the graphene sheet. Normal displacement of

graphene leads us to obtain cohesive law between graphene and epoxy whereas shear

displacement helps us to study the pullout mechanism (Figures 8 and 9).

Configuration Graphene sheet

dimension

(length x

widths) (A0)

2

umber of

carbon

atoms in

graphene

Interaction

force type

Ultimate

strength

(MPa)

Displacement

at separation

or critical

length (nm)

Mode-I-small 39.36 x 19.02 248 Normal 0.03 0.00075

Mode-I-big 118.08 x 19.02 766 Normal 9.93x10-3

0.023

Mode-II-small 39.36 x 19.02 248 Shear 0.12 0.025

Mode-II-big 118.08 x 19.02 766 Shear 0.91 0.165

Table 7. Unit cell configuration for interface model

31

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

0 0.05 0.1 0.15 0.2 0.25

Fo

rce

(p

ico

Ne

wto

ns)

Displacement (nm)

Mode-I-big

0.00E+00

5.00E-02

1.00E-01

1.50E-01

2.00E-01

2.50E-01

0 0.0005 0.001 0.0015 0.002 0.0025 0.003

Fo

rce

(p

ico

Ne

wto

ns)

Displacement (nm)

Mode-I-small

0.00E+00

5.00E-03

1.00E-02

1.50E-02

2.00E-02

2.50E-02

3.00E-02

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40

Fo

rce

(p

ico

Ne

wto

ns)

Displacement (nm)

Mode-II-big

Mode-II-small

Critical lengths

Figure 28. Force vs displacement curve for Mode-I-big graphene-epoxy system

Figure 29. Force vs displacement curve for Mode-I-small graphene-epoxy system

Figure 30. Force vs displacement curve for Mode-II-big and Mode-II-small graphene-epoxy systems

Displacement at failure=0.023 nm

(approximately)

Displacement at failure =0.00075 nm

(approximately)

32

The cohesive or normal force-displacement curve is obtained in Figures 28 and 29 for

graphenes with large and small surface areas respectively. Ultimate cohesive strength is

comparatively higher in the Mode-I-small because of smaller surface area of graphene. In both of

the cases, maximum force at separation is approximately 2.3x10-1 pN. But the displacement at

separation is expectedly higher for Mode-I-big than Mode-I-small. Hence the fracture energy or

work of separation is higher for bigger graphene sheets. This causes higher interfacial stiffness

for larger graphenes.

Unlike mode-I failure, the mode-II or pullout strength is higher for longer graphene

(Mode-II-big). From Figure 30 the interfacial stiffness in pullout direction seems to be similar

for both small and large graphenes. For Mode-II-small the early drop in the pullout force

indicates failure. However, the pullout force was not observed to be monotonically decreasing

like Mode-II-big. It clearly indicates the effect of atoms. For Mode-II-big the number of atoms is

three times larger than that of Mode-I-small. Hence the effect of thermal fluctuation is higher for

smaller graphene. As a result the pullout force vs. displacement curve is smoother for larger

graphene.

COCLUSIO

In this paper mechanical properties of graphene reinforced epoxy composite are predicted

using MD simulation. The predicted results showed reasonable agreement with available

experimental data and theoretical prediction in the literature. Besides mechanical properties, the

MD simulation also provided some meaningful information on epoxy-graphene interaction

energy. This work successfully applies MD based frame work to graphene based polymer

nanocomposites which can be further extended.

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