Interfacial modeling of flattened CNT composites with cyanate ester and PEEK polymers

Abstract

Flattened carbon nanotubes (flCNTs) are a promising form of composite material reinforcement because of their capacity for self-assembly and packing efficiency, which could ultimately lead to improved thermo-mechanical properties relative to current state-of-the-art composite materials. An important material design parameter for composite materials is the choice of polymer matrix, as characteristics of the reinforcement/polymer interface can have a significant effect on the bulk-level properties. Because flCNT-based composites are too expensive to develop via experimental trail-and-error approaches, the goal of this research is to use computational methods to drive their design via efficient polymer selection. Molecular dynamics modeling is used to predict the flCNT/polymer interface properties for a thermoplastic resin (polyether ether ketone - PEEK), and two thermosetting resins (fluorinated and non-fluorinated cyanate esters). For each polymer system, the interfacial interaction energy, flCNT shearing friction, and the transverse tension strength is predicted. While the PEEK and non-fluorinated cyanate esters demonstrate superior interaction energies (23.1% and 11.4% higher, respectively) compared to the fluorinated cyanate ester, the fluorinated cyanate ester has a significantly higher resistance to shearing with the flCNT surface (125% higher than PEEK and non-fluorinated cyanate ester). In pull-apart transverse tension simulations, the non-fluorinated cyanate ester system demonstrates the highest peak strength (8.53% higher than PEEK and fluorinated cyanate ester), while the fluorinated cyanate ester exhibits the highest toughness and stiffness (12.8% and 4.89% higher, respectively, than PEEK and non-fluorinated cyanate ester). Given equal weight, these predictions show that the fluorinated cyanate ester demonstrates the best overall compatibility with flCNTs.

Introduction

Carbon nanotube (CNT) reinforcement has been shown to have promise in next-generation composite materials for aerospace applications [1,2], especially for aerospace vehicles designed for deep space exploration. In particular, flattened CNTs (flCNTs) have the potential for self-assembly, alignment, and efficient packing for improved properties over current state-of-the-art composite materials [3,4]. Because flCNT-based composites are too difficult and expensive to develop via experimental trail-and-error approaches, computational methods are necessary to help guide their design. Direct computational simulation of flCNT/polymer interfaces can provide insight that greatly facilitates the development of these materials, particularly molecular simulation that directly incorporates the molecular structure into predictions of material behavior.

One of the important design parameters for flCNT/polymer composites with optimal mechanical properties is the choice of polymer resin. It is well-established that the characteristics of the nanoparticle/polymer interface can have a significant effect on the bulk-level properties, and multiscale computational modeling approaches can be used to accurately predict these effects [[5], [6], [7], [8], [9], [10]]. Therefore, multiscale computational modeling can help guide selection of polymers for flCNT/polymer systems for optimal molecular-level interfacial characteristics and thus bulk-level properties. Of particular interest are high-performance resins that are used for structural applications in aerospace vehicles.

Patil et al. [11] used molecular dynamics (MD) simulation to investigate the interaction and load transfer characteristics of two different flCNT/polyimide composite interfaces on the molecular level. They demonstrated that the presence of fluorinated groups in the polyimide molecule has a significant effect on the interfacial interaction energy, friction resistance, and separation resistance. Although this information is important for understanding the behavior of flCNT/polyimide interfaces, it is not known how it applies to other high-performance polymer systems that can be used for flCNT composites, including thermosets.

The objective of this research is to use MD modeling to predict the interfacial characteristics for flCNT composites systems with three different high-performance polymer systems: fluorinated cyanate ester, non-fluorinated cyanate ester, and polyether ether ketone (PEEK). The choice of these three polymers will provide insight into the effects of fluorine groups on flCNT/cyanate ester composites, and the relative difference in interfacial characteristics between thermosets and thermoplastics. These three systems were specifically chosen for this study because they are all high-performance systems that can be used for structural composite parts in future aerospace vehicles. In addition to their excellent mechanical properties, they demonstrate high-temperature durability. Thus, the choice of these three systems compliments the data previously established by Patil et al. [11].

In this paper, the MD modeling procedure of the composite systems will be discussed first, followed by a discussion of the results. It is important to note that there is no experimental validation of the simulations discussed herein. Because these simulations are intended to drive the design of new materials systems that cannot yet be fabricated, full experimental validation is not possible. If computational simulation is to be used to drive new material design, accurate model predictions will have to relied upon without full experimental validation. It is also important to note that the general MD modeling techniques employed herein have been previously experimentally validated [5,10,[12], [13], [14]].

Three polymer systems were simulated in this study: polyether ether ketone (PEEK), non-fluorinated cyanate ester (principal component in Primaset PT-30), and fluorinated cyanate ester (principal component in AroCy F-10). The molecular structures are shown in Fig. 1, Fig. 2, Fig. 3, respectively. All molecular dynamics simulations were conducted with the LAMMPS software [15]. All atomistic visualizations were created with the OVITO software, unless otherwise specified [16]. PCFF-IFF was used for all simulations because it is particularly well-suited for modeling interfaces accurately and efficiently [17,18]. Moreover, PCFF-IFF has a capability of assigning accurate partial charges on molecules as well as capturing dihedral energies between atoms [18,19].

Example structures of CNTs and flCNTs are shown in Fig. 4. The flattened geometry is generally energetically favorable for CNTs with diameters greater than 4.2 nm [3]. Below this diameter, CNTs may partially flatten in response to applied external pressure, but they will not completely collapse. At small diameters, CNTs may stay round even in the presence of flattened CNTs [4].

Similar to the flCNT/polyimide topology simulated by Patil et al. [11], which was developed to both predict characteristic flCNT/polyimide interface features and to efficiently provide input data for meso-scale modeling [[20], [21], [22]], each flCNT was modeled as two graphene sheets to represent the flattened region of a flCNT. Fig. 5 shows the complete molecular structure. The rounded ends of the flCNT (dashed lines in Fig. 5) were not modeled for computational efficiency as they can be more efficiently accommodated in meso-scale modeling [[20], [21], [22]].

To create the complete molecular structure, a flCNT structure was first created using an in-house Python script with PCFF-IFF parameters and then run for 50 ps at 300 K using the NVT ensemble (Fig. 6). The flCNT had a surface area of 5156.99 Å² (x = 100.86 Å, y = 51.13 Å). Each carbon atom of the flCNT had two dummy atoms attached which functioned as virtual $\pi$ orbitals [18]. The flCNT was replicated in the z-direction at the start of the run for a total of two flCNTs (Fig. 6). The total number of atoms was 23,616 with 11,808 atoms per flCNT (3936 are carbon atoms, the remainder were dummy atoms).

The overall process for inserting the polymer molecules into the composite structure was the same for each of the three polymer systems. A range of monomer mass fractions from 0.225 to 0.796 was simulated. The mass fraction is defined as the ratio of molecular mass of the monomer/polymer to molecular mass of the total system. For PEEK and the fluorinated cyanate ester, the numbers of monomers per layer (two layers per model) were 48, 64, 80, 100, 125, 150, 180, 216, 252, and 294. For the non-fluorinated cyanate ester, the numbers of monomers per layer were 36, 48, 64, 80, 100, 125, 150, 180, 216, and 252. The total numbers of atoms in the models varied between 24,000 and 46,000, depending on the monomer type and monomer mass fraction. A Lennard-Jones cutoff of 10.0 Å was used until the nanocomposites were fully densified. The weighting coefficients for non-bonded Lennard-Jones and Coulombic interactions were set to 0 for 1–2 interactions, 0 for 1–3 interactions, and 1 for 1–4 interactions. Thus, for atoms directly bonded and one bond away the non-bonded interactions were turned off, but the atoms two bonds away from each other experienced non-bonded interactions at full-force. The model construction process is shown in Fig. 7 for a representative system.

In step 1, A low-density simulation box was constructed with the monomers, as shown in Fig. 7, for each monomer type and mass fraction. Each dimension of the simulation boxes was gradually decreased at a rate of 10 Å/ns until the respective bulk densities were achieved for each system (PEEK: 1.30 g/cc, fluorinated cyanate ester: 1.497 g/cc, nonfluorinated cyanate ester: 1.25 g/cc). These density values were chosen simply as initial guesses and the final densities were achieved in subsequent steps. The NVT ensemble at 300 K and a time step of 1 fs was used. The simulation boxes were fixed in all three directions and the “fix wall/reflect” command was used to prevent atoms from crossing the fixed boundaries. The x- and y-directions of the simulation boxes were fixed at 100 Å and 50 Å, respectively. The dimensions of the densified polymer models in the x- and y-directions were kept slightly smaller than the flCNT dimensions to prevent the loss of atoms and bonds when placed into the larger box with periodic boundary conditions.

In step 2, the densified polymer models were inserted into the two gaps between the flCNT layers using the “read_data” command in LAMMPS. The entire system was flattened in the z-direction at a rate of 50 Å/ns until the mass density of the composite satisfied the inverse rule of mixtures with the mass densities of the flCNTs and polymer. Thus, the corresponding simulation box length in the z-direction was determined using

$$l_z=\left({10}^{24}\frac{{\text{Å}}^3}{{\text{cm}}^3}\right)\frac{M}{l_x\cdot l_y}\left(\frac{m_{flCNT}}{{\rho }_{flCNT}}+\frac{1-m_{flCNT}}{{\rho }_{poly}}\right)$$

where l_x and l_y are the simulation box dimensions in the x- and y-directions in Å, respectively; M is the total mass of the system in g; m_flCNT is the mass fraction of the flCNT; ρ _flCNT is the mass density of the flCNTs (2.23 g/cm³); and ρ _poly is the mass density of the polymer phase. The NVT ensemble at 300 K was used for this step of the simulation process.

Once the models were assembled, they were annealed to give the molecules more time and energy to achieve more desirable configurations. At this point the pair style was changed over from “lj/class2/coul/cut 10” to “lj/class2/coul/long 10” in LAMMPS. When predicting interfacial properties such as friction or adhesion, it is important to include long-range dispersion forces [23]. The “kspace style” was set to “pppm” with a relative accuracy of $1\times {10}^{-6}$; the “pppm style” uses Hockney's particle-particle particle-mesh solver. The glass transition temperature (T_g) of PEEK is 143 °C or 416.15 K, the T_g of the fluorinated cyanate ester is 270 °C or 543.15 K, and the T_g of non-fluorinated cyanate ester is 400 °C or 673.15 K [24]. To best allow the molecules to reach a more favorable configuration, the temperature was ramped to at least 50 K above the respective T_g values. The temperature was ramped up to 500 K for the PEEK models, 600 K for the fluorinated cyanate ester models, and 700 K for the non-fluorinated cyanate ester models over 100 ps using the NVT ensemble. The temperature was then decreased by 50 K per ns using the NVT ensemble until 300 K was reached. The models were allowed to equilibrate for 2 ns at 300 K and 1.0 atm using the NPT ensemble. This step was taken to relax the systems and minimize residual stresses, as equilibration allows the molecules to stabilize and reach a minimum potential energy state.

After equilibration, two interfacial characteristic metrics (interaction energy, and friction force) between monomers and flCNTs were assessed. The interaction energy was assessed by assigning the monomer atoms to one group and the flCNT atoms to another group and using the “compute group/group” command (“kspace yes”). For this calculation, attraction is defined as a negative potential energy and repulsion is defined as a positive potential energy. The reference point for zero potential energy is defined by the pair styles in use. Each nanocomposite was simulated for 500 ps at 300 K and 1.0 atm using the NPT ensemble; the interaction energy between the monomers and flCNTs was computed as a running average over the 500 ps.

The friction force calculations were implemented as described by Patil et al. [11]. To prescribe the motion of the flCNTs, the centers of mass of the top and bottom flCNTs were each tethered by separate springs to separate points within the simulation box. The top flCNT (referred to hereafter as the fixed flCNT) was tethered via a spring to a fixed point. The bottom flCNT (referred to hereafter as the sliding flCNT) was tethered via a spring to a point moving at a constant velocity in the x-direction. The spring constant of both springs was set to 1 kcal/(mol Ų). The constant velocity was defined independently for each simulation. The range of velocities simulated were 0.1 Å/ps to 0.9 Å/ps by 0.1 Å/ps increments (10–90 m/s), and 1 Å/ps to 5 Å/ps by 1 Å/ps increments (100–500 m/s). The positions of the sliding flCNT atoms and polymer atoms were updated using NVE integration (constant volume and energy). The fixed flCNT atoms were thermostatted at 300 K and 1.0 atm using NPT integration. The friction force in the x-direction between the sliding flCNT and polymer was computed using the “compute group/group” command (“kspace yes”). The friction force in the x-direction between the fixed flCNT and polymer was computed similarly. The former are the forces reported herein. These forces were computed as a running average over the 1 ns runtime.

After the step 2 equilibration (Fig. 7), the models were polymerized in step 3. PEEK undergoes linear polymerization (Fig. 8) whereas cyanate ester systems undergoes cyclotrimerization (see Fig. 9). Please note that these polymerization reactions do not reflect the true polymerization reaction mechanisms; they are adaptations to simplify the polymerization process. The LAMMPS command “fix bond/react” was used to polymerize the models [25]. With this approach a set of pre- and post-reaction templates is used to describe the molecular structure before and after a set of chemical reactions.

Polymerization of PEEK requires only one set of pre- and post-reaction templates while the cyclotrimerization of the cyanate esters requires three sets of pre- and post-reaction templates. Fig. 9 shows the three reactions (labelled I, II and III). Each reaction requires one set of pre-, and post-mapping templates. PEEK can only bond once between monomers because of the linear nature of the polymerization, whereas, with cyclotrimerization up to three bonds between monomers are formed until the reaction is complete. The first set of templates for the cyanate esters created the first bond between two monomers to create a dimer (reaction I, Fig. 9), the second set created the bond between the dimer and the third monomer/polymer to create a trimer (reaction II, Fig. 9), and the third set created the third bond which closed the ring (reaction III, Fig. 9). Having three sets of templates with progressive bonding allows for flexibility of potential bond sites and allows for network connectivity even if the full ring will not be formed. As will be discussed in the results section, this scheme allows for a model to be fully networked, if not fully converted (all possible reactive bonds formed). In a fully networked system, the entire polymer is connected via chemical bonds. In a fully converted system, every possible bond which could be created has been created. As discussed previously, fully converted systems are unrealistic with epoxies [6,13,[26], [27], [28], [29], [30]] because of the amorphous nature of the material presence of steric hindrance, and it expected that this limitation applies to cyanate esters as well.

Polymerization took place over 200 ps for all three polymer systems. All atoms not participating in a reaction were thermostatted using NVT at 300 K at 1 fs timesteps. Bonds could be created every time step between atoms separated by a maximum cut-off distance of 7 Å. Atoms participating in a reaction were stabilized for 500 time steps using an NVE/limit time integrator which limits the movement of atoms in a single time step (0.03 Å for PEEK and fluorinated cyanate ester atoms, and 0.04 Å for the non-fluorinated cyanate ester atoms). The reacting atoms had their temperatures rescaled to 300 K before they were returned to the all-atom thermostat. The probabilities governing bond formation of PEEK and the first step of the cyclotrimerization were set such that bonds were formed gradually over the course of the 200 ps. For the cyanate esters, the second and third reaction probabilities were set higher than the initial step to prioritize completing the cyclotrimer rings. The probabilities were 10⁻⁴ for PEEK; 10⁻⁴, 10⁻² and 10⁻¹ for the first, second, and third reactions for the non-fluorinated cyanate ester, respectively; 10⁻⁵, 10⁻³ and 10⁻¹ for the first, second, and third reactions for fluorinated cyanate ester, respectively. It is important to note that the polymerization was performed at 300 K for all three systems. Due to the small size of the monomers, polymerization at elevated temperatures was not required.

The conversion density was calculated using

$$\text{conversion}\left(\text{\%}\right)=\frac{N_{createdbonds}}{2\cdot N_{reactivesites}\cdot N_{monomers}}\cdot 100\%$$

where N_createdbonds is the total number of converted bonds in the simulation box that have been created during the polymerization process (obtained via fix bond/react command in LAMMPS), N_{reactivesites} is the number of total reactive sites per monomer (1 for PEEK, 2 for fluorinated cyanate ester, 3 for non-fluorinated cyanate ester), N_monomer is the total number of monomers per layer in the simulation box, and the factor of 2 in the denominator corresponds to the two polymer layers per simulation box.

The extent of networking was determined by

$$\text{network}\left(\text{\%}\right)=\frac{2\cdot N_{monomers}-N_{clusters}+2}{2\cdot N_{monomers}}\cdot 100\%$$

where N_clusters is the number of polymer clusters (independent groups of bonded atoms) in the model. For the nanocomposites presented in this study, a fully networked system had six clusters (four flCNT layers and two polymer layers) because the six layers of the nanocomposite were not connected via covalent bonds. Thus, the addition of two in Equation (3) ensures that a system with two polymer clusters has an extent of conversion of 100%. The OVITO software package (using the “Cluster analysis” modifier) can compute the extent of networking using a LAMMPS data file of the composite system. The only requirement of the OVITO method is that the force field used to generate the data file must be a fixed-bond force field. OVITO can compute the number of clusters based off the bond topology present in the data file.

After polymerization, the models were annealed to give the molecules more time and energy to achieve more desirable configurations. The temperature was ramped up to 500 K for the PEEK models, 600 K for the fluorinated cyanate ester models, and 700 K for the non-fluorinated cyanate ester models over 100 ps using NVT. The temperature was then decreased by 50 K per ns using the NVT ensemble until 300 K was reached. The models were then allowed to equilibrate for 2 ns at 300 K and 1.0 atm using the NPT ensemble. The equilibrated, polymerized models (see Fig. 10, Fig. 11, Fig. 12 and step 4 in Fig. 7) were then assessed for interaction energy and friction force using the same methodology as described above for the monomer models. Also, the polymerized models were assessed for transverse strength. Specifically, the simulation boxes were subjected to applied strain along the z-axis at a strain rate of 2 × 10⁸ s⁻¹ until complete separation occurred at one of the polymer/flCNT interfaces. The overall simulation box stresses and strains were recorded during the simulations. The corresponding stiffness, peak strength, and toughness were calculated. The stiffness was computed as the slope of the linear-elastic region of the stress-strain curves. The peak strength was calculated as the maximum stress experienced during the simulation. The toughness corresponded to the area under the stress-strain curves. An R script was used to compute these quantities from the stress-strain data.