Combined CI-MD approach in formulation of engineering moduli of single layer graphene sheet
Introduction
Research in graphene has attracted significant interest in materials science and nanotechnology due to its attractive properties [1], [2]. A single layer graphene sheet has the thickness of only one carbon atom which makes it the thinnest material [3] with a large specific surface area [4]. This feature of graphene combined with its exceptional mechanical properties makes it an ideal candidate for fluid separation [5], nano-filtration [6], [7] and for applications in nanoelectromechanical (NEMS) systems [8]. These future applications require a critical understanding of the exceptional mechanical properties of graphene, particularly its Young’s modulus for its applications deployment in NEMS and nano-level biological devices.
The elastic properties of free-standing graphene monolayer using an atomic force microscope (AFM) was measured by Lee et al. [9] who recorded the Young’s modulus of graphene sheet to be about 1 TPa. Similarly, Frank et al. [10] determined the mechanical strength of bulk graphene sheets suspended on photo lithographically defined trenches in silicon dioxide using AFM technique. They found that the bulk graphene sheet can withstand tensile force of the order of 10−7 N. Zhao et al. [11] investigated the mechanical properties of graphene under tensile loading using molecular dynamics (MD) simulation and orthogonal tight-binding techniques. The anisotropic mechanical properties of graphene sheets were predicted by Ni et al. [12] using MD simulation. Their results confirmed that the hexagonal arrangement of carbon atoms in graphene sheet attributes to the anisotropic mechanical properties. Tsai and Tu [13] investigated the mechanical properties of graphite flakes and single graphene layer using MD simulation. They found that the expansion and exfoliation of the graphite flakes can provide better reinforcement effect in nanocomposites. The mechanical properties of graphene under shear deformation were investigated by Min and Aluru [5] using MD simulation method. They predicted that the wrinkling behavior of graphene under shear deformation to be a significant factor for reduced shear strength of graphene at high temperatures. Zhang et al. [14] investigated the mechanical properties of bilayer graphene sheets coupled by sp3 bonding using MD simulation method. Their simulation suggested that these sp3 bonds exert a strengthening influence on the interlayer shear modulus and load transfer rate of graphene sheet.
Recent studies on MD simulation of graphene have focused on the effect of parameters such as defects and temperature on the mechanical properties of graphene. For instance, He et al. [15] studied the effect of Stone–Thrower–Wales (STW) defects on the mechanical properties of graphene. They found that when there is more than one STW defect, the mechanical properties of graphene depends on the tilting angle of STW defects. Shen [16] studied the torsional properties of graphene sheets using MD simulation by simulating a twist. It was found that the wider graphene sheets have better anti-torsion capability and thermal conductivity. Han et al. [17] investigated the ultimate strengths and failure types of asymmetric tilt grain boundaries with various tilt angles of graphene sheet with the aid of MD simulations and analytic theory. They found that the tensile strength of armchair-oriented graphene GBs shows a tendency to increase as the misorientation angle rises, while that of zigzag-oriented graphene GBs non-monotonically increases. Wang et al. [18] studied the fracture of single-layered graphene sheets with edge crack under simple tension using MD simulations. It is found that the existing edge crack weakens mechanical properties of SLGSs. Zhang et al. [19] introduced an atomistic simulation methodology, based on the energy release rate, as a tool to unveil the fracture mechanism of graphene at nanoscale. It was found that the increase of temperature leads to the reduction of fracture strength, fracture deformation, and the critical energy release rate of graphene.
Theoretical studies based on MD simulation has become more popular to study the compressive strength of nanomaterials when compared to that of laboratory based experiments [20], [21], [22]. This is due to the reason that MD simulation allows rapid reconstruction of defects, altering of chirality and system size [23], [24]. This is useful to understand the influence of system parameters on the mechanical properties of graphene sheet. Hence, MD simulation models can be used as a viable alternative compared to time consuming and expensive experiments for estimating mechanical properties at nanoscale [25]. In addition, MD simulation is capable of generating accurate solutions in predicting diverse properties of nanoscale system with minimal cost and high rapidity [26], [27]. However, the MD simulation does not provide information on relationship between the input parameters and the generated output. Artificial intelligence (AI) techniques can prove to be a useful tool for predicting the relationship between the input parameters and generated output. However, they cannot be used to predict system properties in nanoscale materials.
Therefore, there is a need to develop an integrated MD based AI simulation technique for modeling the material properties of graphene sheet. The new integrated approach combines advantages of accuracy and low cost of MD simulation with the explicit model formulation of AI techniques. These methods require input training data which can be obtained from the MD simulations which is based on a specific geometry and temperature. Considering input data, the AI technique can then be able to generate an explicit mathematical model that describes the output parameter (for e.g. engineering moduli) based on specific input parameters (such as number of atoms, system temperature, and defects). Additionally, among the various available AI techniques, we focused on GP because it has the ability to learn the process itself and evolves an equation that explains the underlying process behavior without assuming any form of a model structure [28]. Multi-gene genetic programming (MGGP), is primarily used in the current study. It is to the best of author’s knowledge that limited work exists on the application of MGGP based model on evaluating mechanical properties of the nanoscale system.
The proposed GP approach is employed to investigate the effect of aspect ratio (AR), temperature, number of atomic planes and vacancy defects on the engineering moduli of armchair and zigzag graphene sheets. An explicit relationship is obtained with respect to aspect ratio (AR), temperature, number of atomic planes and vacancy defects. The performance of the proposed model is evaluated against the actual data obtained from literature. Further the parametric and sensitivity analyses conducted is used to validate the robustness of the proposed model by unveiling dominant input parameters and hidden non-linear relationships. It is also worthwhile to note that all MD simulations reported in this work is based on application of Brenner’s second generation potential [29] for modeling inter-atomic interactions in graphene sheet. Since the value of elastic modulus depends on the deployed inter-atomic potential function, one can expect variation in the reported engineering moduli should a different potential function be employed.
Section snippets
Molecular dynamics simulation
The classical molecular dynamics (MD) simulation approach is deployed at first to model the tensile and shear loading of the graphene sheet using LAMMPS software package [30]. The Brenner’s second generation reactive empirical bond order potential (REBO) [29] is used to describe the interaction between the carbon atoms in the graphene sheet. The Brenner’s potential is ideal for simulating a system consisting of large number of hydrocarbon atoms while maintaining the accuracies of semi-empirical
Tensile and shear loading of graphene sheet
The system is first thermally equilibrated in an NVT ensemble to release any residual stresses. The temperature stability of the system is attained by using the Nose–Hoover thermostat [36], [37]. Following equilibration, tensile and shear loading is effected by applying a constant outward displacement (strain rate = 0.001 ps−1) on the atoms at the lateral ends of the graphene sheet as shown in Fig. 1a and b respectively. The system is allowed to relax after every 1000 time steps such that the
Multi-gene genetic programming
In this proposed approach, the data obtained from the MD simulation is further fed into CI method, multi-gene genetic programming (MGGP) cluster. The key difference between genetic programming (GP) and the MGGP is that, in the latter, the model participating in the evolution is a combination of several sets of genes/trees [48], [49]. GP based on Darwin’s theory of ‘survival of the fittest’ finds the optimal solution by mimicking the process of evolution in nature. Genetic programming (GP) is
Evaluation of the performance of models
The results obtained from the MGGP models for the two graphene sheets are illustrated in Fig. 9, Fig. 10 on the training and testing data for evaluation of Young modulus and Shear modulus respectively. Performance of the proposed models is evaluated using five metrics: square of the correlation coefficient (R2), mean absolute percentage error (MAPE (%)), RMSE, relative error (%) and multi-objective error function (MO). The five metrics used the following relationships:
Sensitivity and parametric analysis
Sensitivity and parametric analyses about the mean was conducted for the validation of our MGGP model. The sensitivity analysis (SA) percentage of the tensile modulus to each input parameter was determined using the following formulas:where fmax(xi) and fmin(xi) are, respectively, the maximum and minimum of the predicted output over the ith input domain, where the other variables are equal to their mean values.
Table 14, Table 15 shows the sensitivity results
Conclusions
The present work introduces an evolutionary approach of MGGP in simulating the engineering moduli of single layer graphene sheet based on aspect ratio, temperature, number of atomic planes and vacancy defect concentration ratio. MD simulation based on REBO potential was used to describe inter-atomic potential energy in graphene sheet. The results show that the predictions obtained from the proposed models are in good agreement with the actual data of Min and Aluru [5]. Based on the sensitivity
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The first two authors made equal contribution in this work and are both equally considered as first author.