Molecular Dynamics Modeling of Mechanical Properties of Polymer Nanocomposites Reinforced by C₇N₆ Nanosheet

Abstract

Carbon-nitride nanosheets have attracted remarkable attention in recent years due to their outstanding physical properties. $C_7{N}_6$ is one of the hotspot nanosheets which possesses excellent mechanical, electrical, and optical properties. In this study, the coupled thermo-mechanical properties of the single nanosheet $C_7{N}_6$ are systematically investigated. Although temperature effects have a strong influence on the mechanical properties of $C_7{N}_6$ monolayer, thermal effects were not fully analyzed for carbon-nitride nanosheet and still an open topic. To this end, the presented contribution aims to highlight this important aspect and investigate the temperature influence on the mechanical stress-strain response. By using molecular dynamics (MD) simulation, we have found out that the $C_7{N}_6$ monolayer’s maximum strength decreases as the temperature increase from 300 K to 1100 K. In the current contribution, 5% to 15% volume fractions of $C_7{N}_6$/P3HT composite were employed to investigate the $C_7{N}_6$ reinforcing ability. Significantly, the uniaxial tensile of $C_7{N}_6/P3HT$ composite reveals that $10\%$ $C_7{N}_6$ can enhance the maximum strength of the composite to 121.80 MPa which is 23.51% higher than the pure $P3HT$ matrix. Moreover, to better understand the enhanced mechanism, we proposed a cohesive model to investigate the interface strength between the $C_7{N}_6$ nanosheet and P3HT matrix. This systematic study provides not only a sufficient method to understand the $C_7{N}_6$ thermo-mechanical properties, but also the reinforce mechanism of the $C_7{N}_6$ reinforced nanocomposite. Thus, this work provides a valuable method for the later investigation of the $C_7{N}_6$ nanosheet.

1. Introduction

Nowadays, two-dimensional (2D) materials [1,2,3] are among the most attractive research topics due to their excellent physical properties in comparison with conventional materials. Graphene (GN) is a typical 2D material that has high electrical [4], thermal conductivity [5,6], and mechanical [7] properties. GN has the simplest lattice structure while compared with other two dimensional materials like $C_3{N}_4$, $C_6{N}_6$, and $C_7{N}_6$. However, one of the shortages for GN is the thermo-mechanical stability. The stability of GN can reach only 480 ${}^{\circ }$C as reported in [8]. This limits the wide application of GN in some field. Apart from the Graphene monolayer, the carbon-nitride family monolayer has attracted wide attention from researchers in recent years due to the diverse lattice structures and their excellent electrical, and optical properties. Different from Graphene, graphitic carbon nitrides exhibit porous atomic lattices and low thermal conductivities [9]. Among the various classes of 2D materials, carbon-nitride nanomembranes have been among the most successful nanomaterials that have inherent semiconducting electronic characters [10]. The carbon-nitride compounds consisting of covalent-bond grids of carbon and nitrogen atoms are the most successful, which possess lower thermal conductivities than graphite and exhibit semiconducting properties [11]. Carbon-nitride nanomaterials exhibit a stiff and stable physical component owing to the formation of strong covalent bonds between the C-C and C-N bonds [10]. In this regard, the maximum strength of carbon-nitrides depends much on the strong covalent bonds (C-C, C-N). Previous works have found that $C_7{N}_6$ monolayer exhibits a strong elastic tensile modulus of 212 N/m, as outlined in [10].

In the last decade, an investigation reveals that $C_{3}N$ carbon-nitride as a novel semiconductor has excellent mechanical properties and tensile strength can reach $35.2$ GPa.nm [12] which is very close to defect-free Graphene. In terms of energy storage and conversion, $C_{3}N$ displays a much smaller voltage hysteresis than graphene; this also indicates its promise as a new anode material for lithium-ion batteries [13]. With the consideration of the excellent mechanical properties, various graphitic carbon-nitrides are used in energy conversion/storage, catalysis [14], photocatalysis [15], and oxygen reduction systems [16,17,18]. From the perspective of stability, the $C_{3}N$ can be stable up to 550 ${}^{\circ }$C in air, which is more stable than GN with stability only up to 480 ${}^{\circ }$C [8]. Among several carbon-nitride monolayers, $C_7{N}_6$ was founded to be a new 2D structure composed of $s{p}^2$-hybridized carbon atoms forming hexagonal lattice [11]. An investigation from Wu et al. [11] have successfully found out that the $C_7{N}_6$ has high-temperature stability by Ab initio molecular dynamics simulations(AIMD) which indicates mechanical stability and maintains stability at high temperatures up to 1500 K for $C_7{N}_6$ monolayer. The AIMD simulations were conducted at 300, 500, and 1000 K for 20 ps long simulations which reveals that the $C_7{N}_6$ nanosheet can stay completely intact at all the studied temperatures [10]. Moreover, by solving the full Boltzmann transport equation, monolayer $C_7{N}_6$ was found to possesses good electronic transport properties and high lattice thermal conductivities (134.55 W/mK at 300 K) [11]. $C_7{N}_6$ has strong stability and stiff mechanical response owing to the covalent bond interaction from C-C and C-N bonds, as concluded in [9,10].

Current development in carbon-nitride fabrication technologies has also accelerated the exploration of nanomaterials with the combined theoretical investigation. A recent study by Hu et al. [19] shows that $C_7{N}_6$ monolayer demonstrates the high capacity and efficient reversibility for hydrogen storage in a realistic condition. However, to the best knowledge of the authors, the temperature influence for the $C_7{N}_6$ monolayer in the field of a mechanical response and the combined interface strength of $C_7{N}_6$ with $P3HT$ polymer matrix is an open topic. As well documented the P3HT is a semiconductor polymer which has good conductivity, electric property, and conversion efficiency [20,21,22]. However, the mechanical property of P3HT is not comparable to conventional polymer, since the fracture of material takes place at the strain $9\pm 1.2$% [23] of the mechanical loading. On the other hand, P3OT and P3DDT are from the same polymer family with a strain of $65.24\pm 2.5$%, $47\pm 3.1$% for a crack in tensile loading, respectively [24]. Hence, to increase the mechanical properties, many researchers intend to use conductive or mechanical reinforced particles dispersed in an elastic matrix as outlined in [25]. On behalf of the strong thermal stability and mechanical properties, $C_7{N}_6$ will be used as a reinforcement to enhance the P3HT polymer.

Hereby, $C_7{N}_6$ demonstrates good temperature stability, but how does this temperature influence the fundamental stress-strain relationship is still unknown. Furthermore, how strong the interface strength between $C_7{N}_6$ and $P3HT$ matrix is unclear yet. Those issues limit the further application and development of $C_7{N}_6$ in the field of the electrical and aerospace industry and fabrication. To address these issues, we systematically investigate firstly the mechanical properties of the $C_7{N}_6$ monolayer and the temperature influence for the stress-strain response. Secondly, a nanocomposite of $C_7{N}_6/P3HT$ with different volume fractions of $C_7{N}_6$ will be analyzed to investigate their mechanical properties as well as the enhancement of semiconductor P3HT polymer. Finally, a cohesive model is carried out to predict a non-bond interaction (normal interface strength).

2. Simulation Methodology

The structure of the monolayer $C_7{N}_6$ consists of 3640 atoms having a width of $164.75$ Å and height of $67.94$ Å as documented by VESTA [26] and sketched in Figure 1a. For better understanding, Figure 1b represents the single lattice structure of the $C_7{N}_6$ monolayer; every single lattice has 7 carbon atoms and 6 nitrogen atoms, where the brown atoms represent carbon atoms; the gray atoms represent nitrogen atoms. The basic load direction for the $C_7{N}_6$ monolayer in the X-direction along the edge during the uniaxial tensile test is demonstrated in Figure 1c. It is worthwhile to mention that the thickness of the $C_7{N}_6$ monolayer cannot be neglected. To this end, a thickness of $3.20$ Å is used to define the thickness along the Cartesian direction, see [12].

During the uniaxial tensile test of the monolayer, the structure atoms number should be big enough to overcome the fluctuation which will influence the final results. For more details about the specific structure data of the $C_7{N}_6$ lattice, the atomic structure with VASP POSCAR format is provided in Appendix A.

2.1. Potential and Uniaxial Tensile for $C_7{N}_6$ Monolayer

In this contribution, the optimized tersoff potential was proposed by Lindsay and Broido for the formulation of the interactions of the carbon-carbon atoms [27]. The potential parameters for the carbon-nitrogen interactions were adopted from the research by Klnacl [28]. The accuracy of the predictions derived from the MD simulations strongly correlates to the appropriate selection of the potential to define the atomic interactions [12], since the potential plays a key role in the accuracy calculation during the simulation. The optimized tersoff potential is the most accurate potential for molecular dynamics simulation of the mechanical and thermal transport along with $s{p}^2$ carbon structure [29]. Therefore, the BNC.tersoff potential file was chosen for $C_7{N}_6$ monolayer tensile. Hereby, Tersoff parameters for B, C, and BN-C hybrid-based graphene-like nanostructures can be found from BNC.tersoff [30]. To this end, the N-N, N-C, and C-C bond parameters are used. For more detail about the potential, please see Appendix B.

As a result of the porous structure of $C_7{N}_6$, the uniaxial tensile test is carried out along X (Cartesian a) and Y (Cartesian b) directions to check the anisotropic mechanical response. In this study, a molecular dynamics method was carried in the simulation by LAMMPS (Large-scale Atomic/Molecular Massively Parallel Simulator) open-source package [31]. A periodic boundary condition was defined along X, and Y-direction; however, a non-periodic and shrink-wrapped boundary was defined in Z-direction. To avoid any sudden bond stretching or void formation due to the loading conditions, the atomic positions at every step of the loading were rescaled according to the applied changes in the simulation box size [12]. Moreover, the time increment and pair style were defined as $0.5$ femtoseconds (fs) and tersoff respectively under the metal unit system. Next, a Gaussian distribution initial velocity was used to initialize the structure before the simulation. Then a cooling procedure was carried out for the structure from 500 K to 300 K by Nose Hoover barostat and thermostat ($NPT$, see [32]) method for 30 picoseconds (ps). Besides, an equilibrium process should be taken for the structure within the target temperature (300 K) for 30 ps in an $NPT$ ensemble. Next, several constant strain rates ($2.0\times {10}^8$ s${}^{-1}$, $6.0\times {10}^8$ s${}^{-1}$, $2.0\times {10}^9$ s${}^{-1}$, and $6.0\times {10}^9$ s${}^{-1}$) are carried out during the $C_7{N}_6$ uniaxial test in $NPT$ ensemble. Every 1000 fs, the virial stress [33] was averaged 2 times to print the engineering stress-strain response. To fully understand the temperature influence for the stability of $C_7{N}_6$, we also test a monolayer at different temperatures (300 to 1100 K) within the control of the $NPT$ ensemble. Those temperature comparisons are also carried out in a Y-direction within a constant strain rate of $2.0\times {10}^8$ s${}^{-1}$. Apart from this, other steps are the same as mentioned above.

2.2. $C_7{N}_6/P3HT$ Nanocomposite Mechanical Test

In the second part of this work, the mechanical properties of the $C_7{N}_6$ reinforced nanocomposite are further analyzed. To this end, a specifically designed volume fraction composite structure is built by Materials Studio (MS). Herein, three types of volume fractions ($5\%$, $10\%$, $15\%$) of $C_7{N}_6$ are randomly inserted within a size of $69.5\times 69.5\times 69.5$ Å ($X\times Y\times Z$) cubic. Those models are depicted in Section 3.3. Then the contained coordinates and force field parameters files are transferred into LAMMPS in Linux system to generate the data file before the simulation. During the uniaxial tensile test of the nanocomposite, a periodic boundary condition was defined for the structure along with three Cartesian directions within a unit of real unit system [34], for more detail of the unit sees LAMMPS manual. Furthermore, the simulation accuracy was predefined as $0.0001$ within Ewald [35]. A hybrid pair style was used to describe the atoms pairs since the polymer matrix and the $C_7{N}_6$ monolayer were described by Lennard-Jones 6-9 potential with a cutoff of $12.0$ Å and tersoff potential, respectively. Another bond, angle, dihedral, and improper style was defined as class2 (second generation force field [36]). To better understand the mechanical response of nanocomposite, a small-time increment of $0.2$ femtoseconds (fs) is used. Then an initial equilibrium stage will be carried out by $NPT$ with 500 K for 50 picoseconds. After the nanocomposite is fully equilibrated, the system is cooling down from 500 K to a target temperature of 300 K for 50 picoseconds by $NPT$ ensemble. Thereafter, we equilibrate the structure at 300 K for 80 ps to guarantee its full relaxation at this target temperature of 300 K. Next, we start applying the load from the initial state during the uniaxial tensile test. Then, the structure will be stretched along X-direction under an $NVT$ [37] ensemble to control the temperature at 300 K for 2000 ps. Every 2 ps, the mechanical response will be printed 1 time to output the fundamental stress and strain data.

2.3. Cohesive Model for $C_7{N}_6$—$P3HT$ Interface Investigation

In this section, a cohesive model is built to investigate the non-bond interaction between $C_7{N}_6$ monolayer and polymatrix. In this regard, we have proposed an analytical model and combined molecular dynamics (MD) simulation with the analytical model of [38] to investigate the normal strength between the $C_7{N}_6$ monolayer and polymer matrix. For molecular dynamics (MD) result, the non-bond interaction can be formulated by Lennard-Jones 6-9 potential P

$$P\left(r\right)=2\epsilon {\left(\frac{\sigma }{r}\right)}^9-3\epsilon {\left(\frac{\sigma }{r}\right)}^6$$

(1)

as proposed in [39], where r describes the current distance, $\epsilon$ represents equilibrium energy depth and $\sigma$ depicts the equilibrium distance between two atoms formulated by i and j without bond connection. Following [40], $\sigma$ and $\epsilon$ can be defined as

$${\sigma }^{ij}=\sqrt[6]{\frac{{\left({\sigma }^i\right)}^6+{\left({\sigma }^j\right)}^6}{2}}\mathrm{and}{\epsilon }^{ij}=\frac{2\sqrt{{\epsilon }^i{\epsilon }^j}{\left({\sigma }^i\right)}^3{\left({\sigma }^j\right)}^3}{{\left({\sigma }^i\right)}^6+{\left({\sigma }^j\right)}^6},$$

(2)

${\sigma }^i$, ${\epsilon }^j$ are i and jth type atoms-distance and energy. In the presented contribution, several specific model parameters were listed in

Table 1. Parameters for atoms in the cohesive model. The mole ratio represents the type of atoms for each unit or each single lattice structure.

	Types	Mass (g/mol)	$\mathit{\sigma }$ (Å)	$\mathit{\epsilon }$ (kcal/mol)	Mole Ratio	Comment
$C_7{N}_6$	c	12.011	0.054	4.010	7/13	generic $S{p}^3$ carbon
	n	14.007	0.096	3.830	1/13	generic $S{p}^2$ carbon
	n2	14.007	0.050	4.010	3/13	generic nitrogen
	n3	14.007	0.015	3.720	2/13	generic nitrogen
P3HT	c3a	12.011	0.068	3.915	6/25	generic $S{p}^3$ carbon
	s2a	32.064	0.125	4.047	1/25	generic sulphur
	h1	1.008	0.032	2.878	14/25	generic hydrogen
	c4	12.011	0.062	4.010	4/25	generic $S{p}^2$ carbon

which was generated by the PCFF [39] force field in the toolbox of LAMMPS.

By defining the distance between $C_7{N}_6$ monolayer and polymer matrix with the parameter h, we introduce the cohesive energy per unit area as

$$W=\sum _{i=1}^{n}2\pi {\rho }_p^i{\rho }_c{\int }_h^{\infty }{P}^i\left(r\right)r(r-h)dr,$$

(3)

in terms of the n types of atoms in the polymer unit, here we have 4 types (see Table 1). ${\rho }_c$ is the $C_7{N}_6$ area density $(3.252\times {10}^{19}$ m${}^{-2})$ which represents the number of atoms per unit area of $C_7{N}_6$. This can be evaluated based on our structure with given height and width as plotted in Figure 1a. ${\rho }_p^i$ is the volume density (number of polymer molecules per unit volume) of the i-th type of polymer atoms. Mass density of $P3HT$ was tested $1.025\times {10}^3$ kg/m${}^3$ (${\rho }_{exp}=1.1$ g/cm${}^3$ [41,42]). Mass of every $P3HT$ unit ($-C_{10}S{H}_{14}-$) was evaluated at $2.76\times {10}^{-25}$ kg. Then ${\rho }_p$ can be evaluated by dividing of the mass density with respect to the mass of the unit polymer. Therefore, ${\rho }_p=3.99\times {10}^{27}$ m${}^{-3}$, see [43]. For different types of atoms in Table 1, the ${\rho }_p^i$ is equal to ${\rho }_p$ multiply by mole ratio. Hence, cohesive stress can be defined as

$${\sigma }_{coh}=\frac{\partial W}{\partial h}=-\sum _{i=1}^{n}2\pi {\rho }_p^i{\rho }_c{\int }_h^{\infty }{P}^i\left(r\right)rdr.$$

(4)

By taking the necessary derivation and reformulations we end up with the following compact form

$${\sigma }_{coh}=\sum _{i=1}^{n}2\pi {\rho }_p^i{\rho }_c{\epsilon }^{iA}{\left({\sigma }^{iA}\right)}^2\left[\frac{3{\left({\sigma }^{iA}\right)}^4}{4h^4}-\frac{2{\left({\sigma }^{iA}\right)}^7}{7h^7}\right],$$

(5)

herein A represents $C_7{N}_6$ atoms. Next, we replace the distance $h\to h_0+d$, where $h_0=\sqrt[3]{\frac{8}{21}}\sigma$ is the equilibrium distance at ${\sigma }_{coh}=0$ and d represents the separated distance, thus the cohesive stress yields

$${\sigma }_{coh}=\sum _{i=1}^{n}2\pi {\rho }_p^i{\rho }_c{\epsilon }^{iA}{\left({\sigma }^{iA}\right)}^2\left[\frac{3{\left({\sigma }^{iA}\right)}^4}{4{(h_0+d)}^4}-\frac{2{\left({\sigma }^{iA}\right)}^7}{7{(h_0+d)}^7}\right].$$

(6)

The key goal here is to employ the cohesive model (presented above) at the interface between $C_7{N}_6$ and $P3HT$ and compare it with the proposed molecular dynamics MD simulations. The basic structure can be presented in Section 3.4. The model was constructed with a size of $60.2\times 60.2\times 65.9$ Å ($X\times Y\times Z$) and 23,790 atoms. In this section, a non-equilibrium molecular dynamics(NEMD) was carried out to predict the interfacial strength. Firstly, a non-periodic boundary condition with a shrink-wrapped boundary was defined in Z-direction since a normal velocity of $1.0$ Å/ps was loaded in Z-direction. Other Cartesian axes ($X,Y$) were also defined as periodic boundary conditions. Furthermore, the top monolayer will be separated within 10 nm and the time increment is defined as 0.2 fs. The bottom part, which is below 20 Å along Z-direction, is fixed in all directions to get the interfacial strength. The pair, angle, dihedral, and improper style are the same as the previous definition in Section 2.2. During the separation process, the system is fixed at a target temperature of 300 K within the control of the $NVT$ ensemble.

3. Results and Discussion

3.1. $C_7{N}_6$ Mechanical Properties and Thermal Stability Test

The material fracture depends on many working conditions. The material strength under different temperatures indicates the stretchability for thermal resistance and stability. To better understand the temperature influence for the target carbon-nitride monolayer, we have tested the mechanical response of $C_7{N}_6$ at different temperatures (300 to 1100 K). Such a temperature domain is reliable as confirmed in [11] resulting in good mechanical stability (stable to 1500 K) and maintains the stability of $C_7{N}_6$ by using Ab initio molecular dynamics simulations (AIMD). Here, we take the temperature into consideration, since we want to investigate the temperature’s influence on the mechanical response of a single $C_7{N}_6$ monolayer. The fundamental stress-strain curve of $C_7{N}_6$ for different temperatures is shown in Figure 2. It can be observed that the uniaxial tensile strength of $C_7{N}_6$ gets decreasing with the increase of temperature from 300 K to 1100 K while loading in both X and Y-direction. The mechanical response of Y-direction illustrates more ductile failure behavior as shown in Figure 2b, while the stress-strain response in X-direction exhibits an earlier failure before yielding, see Figure 2a. Furthermore, it is worthwhile to mention that $C_7{N}_6$ monolayer possesses high-temperature stability. From the stability perspective of carbon-nitride, the $C_{3}N$ can be stable up to 550 ${}^{\circ }$C (823.15 K) in air, which is more stable than $GN$ with stability only up to 480 ${}^{\circ }$C (753.15 K) [8]. Interestingly, comparing with $C_{3}N$, our investigation finds out that the $C_7{N}_6$ can be stable up to 1100 K in both directions (X, and Y). This conclusion confirmed the previous report from Wu et al. [11] that the mechanical stability of $C_7{N}_6$ monolayer can reach high-temperature (maximum 1500 K). The maximum strength of $C_7{N}_6$ for X and Y-direction are $9.388$, $8.942$ GPa.nm, respectively, under 1100 K (see

Table 2. Temperature influence comparing for $C_7{N}_6$ monolayer in different loading directions. The maximum tensile stress unit is GPa.nm.

Loading Direction		X		Y
Temperatures (K)	Stress $\left({\mathbf{\sigma }}_{\mathit{xx}}\right)$	Strain $\left({\mathbf{\epsilon }}_{\mathit{xx}}\right)$	Stress $\left({\mathbf{\sigma }}_{\mathit{xx}}\right)$	Strain $\left({\mathbf{\epsilon }}_{\mathit{xx}}\right)$
300	18.101	0.146	19.908	0.161
500	15.815	0.123	17.306	0.152
700	12.997	0.108	14.465	0.135
900	11.317	0.101	12.391	0.126
1100	9.388	0.090	8.942	0.087

), and the corresponding strains are $0.090$ and $0.087$, respectively. For the evaluation of the true stress or strength of $C_7{N}_6$ monolayer, the depicted results are supposed to be divided by the monolayer thickness ($3.20$ Å) as documented in [12].

In order to evaluate anisotropicity in the $C_7{N}_6$ mechanical properties, uniaxial tensile simulations were conducted along with the two perpendicular planar directions, see Figure 3b. With the point of the vesicular texture of two dimensional $C_7{N}_6$ from Figure 1b, the result from Figure 3b indicates that $C_7{N}_6$ produces isotropic mechanical response along with X, and Y-direction. The initial linear stress-strain relations from Figure 3b coincide closely with the loading along with X, and Y-directions, which reveals convincingly isotropic elasticity for the modulus. The isotropic or anisotropic elasticity modulus of the nanosheet is base on the modulus comparing armchair and zigzag orientation. When the modulus of the armchair and zigzag is the same, then the nanosheet can be regarded as isotropic elasticity. This formulation can be traced back to an explanation from the previous work at here [44]. However, comparing the maximum tensile strength along X, and Y-direction of $C_7{N}_6$ monolayer, the result shows an anisotropic response, the difference strength can reach 1.81 GPa.nm.

Based on the mechanical response of $C_7{N}_6$ at Figure 3a,b, we can conclude that $C_7{N}_6$ can be used as a sensor material since $C_7{N}_6$ has not just only strong temperature stability but also very fine linear mechanical response during the uniaxial tensile. Comparing the previous tested stress-strain curve of typical two-dimensional materials, e.g., Graphene [44], $C_{3}N$ [12], and $C_3{N}_4$ [44] monolayer, we can conclude that the $C_7{N}_6$ monolayer with many porosities shows obviously elastic modulus since the $C_7{N}_6$ monolayer has a very fine linear mechanical response before the fracture of structure.

Next, different strain rates ($2.0\times {10}^8$ s${}^{-1}$ to $6.0\times {10}^9$ s${}^{-1}$) in Figure 3a are investigated to further study its influence on the mechanical properties of $C_7{N}_6$. The strength of the $C_7{N}_6$ monolayer gets increases as the strain rates increase. Hereby, the stress drops dramatically after reaching maximum strength. The maximum strength of $C_7{N}_6$ was tested at $17.70$ GPa.nm at the strain rate of $2.0\times {10}^8$ s${}^{-1}$ and 300 K during the uniaxial tensile test, this strength is a little bit different from previous density functional theory (DFT) calculations of $14.10$ GPa.nm [10] by Vienna Ab initio Simulation Package (VASP) [45,46,47]. On the one hand, a larger time increment choosing might be a reason for the difference. Furthermore, classical molecular dynamics (CMD) simulations offer the possibility to study larger systems at elevated temperatures, but CMD requires accurate interatomic potentials to formulate the C-N, and C-C bond for carbon-nitride. In our study, the C-N bond and C-C bond are likely to be in line with the loading direction during the tensile of the $C_7{N}_6$ monolayer (Figure 1a). In other words, those two types of bonds bear the load parallel together. However, the maximum strength of $C_7{N}_6$ has no much sensitivity for different strain rates, since most maximum strength is around 18–20 GPa.nm at the strain around 0.15. This also represents that $C_7{N}_6$ has not just only a stable property at high temperature but also a stable tensile strength with different strain rates.

3.2. $C_7{N}_6$ Fracture Analysis

During the $C_7{N}_6$ monolayer fracture deformation analysis, it can be found that the first debond of $C_7{N}_6$ is the C-C bond at the strain of 0.268 during the $C_7{N}_6$ uniaxial tensile test along with the X-direction at 300 K. However, an obvious crack shape of $C_7{N}_6$ took place at the strain of 0.274; this can be depicted in Figure 4. After the crack point, the crack extends rapidly to the edge within a short strain of 0.004.

From Figure 4 and Figure 5, we can find that the fracture resistance ability along the Y-direction is weaker compared with loading in the X-direction. Since the first fracture takes place at the shorter strain of 0.165; however, a fracture appearing along the X orientation is at the strain of 0.274. This result reveals that the $C_7{N}_6$ monolayer has a stronger loading resistance in the X-direction. Figure 5 reveals that the crack takes place first in the bottom edge of the $C_7{N}_6$ monolayer. Then, a new crack takes place in the right bottom edge of the structure. It can be seen from Figure 5c,d that the crack extends from the left and right sides to the middle side along the top edge. For the same initial crack of the sheet materials, the stress intensity defector of the edge is much bigger than the internal crack for the fundamental plane stress case. Therefore, the fracture will happen much possible on the edge. That is the reason why the tensile crack extends from the edge to the middle as shown in Figure 4 and Figure 5 fracture propagation. Moreover, the crack of $C_7{N}_6$ extends in a crook path instead of the straight line like $C_{3}N$ as outlined in [48]. This represents that the mechanical property of $C_7{N}_6$ is not fully brittle as the stress-strain result shown in Figure 3. During the analysis of the $C_7{N}_6$ ruptured sample along with those two orientations, we confirmed that the first debonding of the investigated monolayer in this study taking place at the C-C bond. This means that the maximum tensile strength of $C_7{N}_6$ relies much on the maximum bond strength of the C-C bond no matter how in X, and Y-directions.

3.3. $C_7{N}_6/P3HT$ Composite Deformation

Next, we show the benefit of employing $C_7{N}_6$ within a nanocomposite of $C_7{N}_6/P3HT$ in this benchmark test. The fundamental stress-strain curves of the $C_7{N}_6$ with different volume fractions are presented in Figure 6a. Those curves are the outcome of a Gaussian smooth algorithm (SMOOTHDATA) from MATLAB to reduce the noise from the data. Among several volume fraction strength comparisons, we find out that the $C_7{N}_6$ as reinforcement can not only enhance the maximum strength for the composite but also make the pure matrix more ductile.

Hereby, the strength of only pure $P3HT$ matrix reaches zero when stretched at a strain of 0.3. However, the strain increase to around 0.6 for enhanced nanocomposite, as shown in Figure 6a. Worthy to mention that $C_7{N}_6$ can enhance the strength of the $P3HT$ matrix to around 120 MPa. The maximum tensile strength (121.80 MPa) of the composite can be gotten with $10\%$ $C_7{N}_6$ $90\%$$P3HT$; this composite can keep a maximum strength from a strain of 0.1 to 0.2 during the tensile. Comparing with the maximum tensile strength of pure $P3HT$ with 98.61 MPa after smoothing, $10\%$ $C_7{N}_6$ can enhance the maximum tensile strength of the composite to $23.51\%$.

3.4. Interfacial Mechanical Properties

For further investigation of the proposed combination of $C_7{N}_6/P3HT$ composite, we carried out a 10 nm separated distance from the polymer. Here a comparison between the theoretical cohesive model and the proposed model with classic molecular dynamics is carried out in Figure 7a. The maximum traction stress for the MD simulation result was evaluated at 248.86 MPa. The difference between MD and analytical traction results (236.95 MPa) is 11.91 MPa which is closed to the analytical result. However, the normal strength of $C_7{N}_6$ separated from the matrix is larger than the maximum strength of pure $P3HT$. In other words, a stronger non-bond strength was constructed at the interfacial part. This can enhance strength for the $C_7{N}_6/P3HT$ nanocomposite. A crack will take place at the weakest part of the structure. Based on our finds, we can assume that the crack will not happen initially in the monolayer and matrix interface region since a stronger strength is constructed by the Van der Waals interaction.

4. Conclusions

In this study, we systematically investigated the thermo-mechanical properties of the $C_7{N}_6$ monolayer. We found out that $C_7{N}_6$ monolayer has strong stability at a high temperature(1100 K). By loading the structure in X and Y-directions, the isotropic behavior was confirmed at a constant strain rate of $2.0\times {10}^9$ s${}^{-1}$. The uniaxial tensile test of $C_7{N}_6/P3HT$ composite reveals that $10\%$$C_7{N}_6$ can enhance the maximum strength of the composite to 121.80 MPa which is $23.51\%$ higher than the pure $P3HT$ matrix strength. Finally, a cohesive model was carried out to predict the non-bond interaction between $C_7{N}_6$ monolayer and P3HT polymer matrix. As an outcome of this construction, a stronger non-bond strength was achieved at the interface. This failure model will be extended towards the phase-field approach to fracture in line with [49,50,51] in the future analysis.

Furthermore, it should be noticed that the reinforcing mechanism of the $C_7{N}_6$ reinforced nanocomposite is complicated. The interface strength is only one of the important parts for the enhancement of the $C_7{N}_6$/P3HT composite. Our result has found that the $C_7{N}_6$ monolayer can enhance the mechanical property of $C_7{N}_6$/P3HT by strong interface strength. Other reinforcing factors need to be further investigated in future work.