An embedded simulation approach for modeling the thermal conductivity of 2D nanoscale material

https://doi.org/10.1016/j.simpat.2014.02.003Get rights and content

Abstract

The thermal property of single layer graphene sheet is investigated in this work by using an embedded approach of molecular dynamics (MD) and soft computing method. The effect of temperature and Stone–Thrower–Wales (STW) defects on the thermal conductivity of graphene sheet is first analyzed using MD simulation. The data obtained using the MD simulation is then fed into the paradigm of soft computing approach, multi-gene genetic programming (MGGP), which was specifically designed to model the response of thermal conductivity of graphene sheet with changes in system temperature and STW defect concentration. We find that our proposed MGGP model is able to model the thermal conductivity of graphene sheet very well which can be used to complement the analytical solution developed by MD simulation. Additionally, we also conducted sensitivity and parametric analysis to find out specific influence and variation of each of the input system parameters on the thermal conductivity of graphene sheet. It was found that the STW defects has the most dominating influence on the thermal conductivity of graphene sheet.

Introduction

Graphene is a 2-dimensional nanomaterial which has attracted significant research interest in material science due to its attractive physical and mechanical properties [1], [2]. Ever since it was discovered [3], the exceptional qualities of graphene has been widely studied and investigated to explore its diverse possible applications in real world. These include applications in electric circuits such as graphene-based integrated circuits (ICs), field effect transistors (FETs) and solar cells [4], [5], [6]. In addition, it is an ideal candidate for potential applications in biomedical, chemical and industrial processes enhanced or enabled by the use of new graphene materials [7], [8], [9]. These applications of graphene requires a critical understanding of its thermal properties which are key to design future graphene based nano-electronic devices. In addition, the increasing demand to manufacture graphene based nano-components for applications in aerospace, defense and electronics is one of the major incentives to study the thermal properties of graphene.

Numerous studies have been undertaken to predict the thermal properties of graphene sheet. Cai et al. [10] deployed laser heating and monitoring of Raman G-peak to determine the room temperature thermal conductivity of graphene sheet synthesized using chemical vapor deposition (CVD) process. They found that graphene possess extremely high thermal conductivity of about 2500 W/mK at room temperature and decreases to about 1400 W/mK at 500 K. Similarly, Ghosh et al. [11] reported the thermal conductivity of graphene as grown across trenches in Si/SiO2 substrate layer. They found the thermal conductivity of graphene flakes to about 3080–5150 W/mK and possess phonon mean free path of 775 nm near room temperature. Theoretical studies have also been conducted to estimate the thermal conductivity of graphene sheets. Evans et al. [12] made use of equilibrium molecular dynamics (MD) to study the thermal conductivity of graphene sheet with smooth and rough edges. They found that the thermal conductivity is highest for graphene sheet with smooth edges, whereas it is equal for graphene sheet with armchair and zigzag edges. Zhong et al. [13] made use of non-equilibrium MD simulation to deduct the thermal conductivity of graphene sheets. They found that the room-temperature thermal conductivity decays monotonically with the number of the layers in graphene. Hao et al. [14] investigated the effect of mono-vacancies and Stone–Thrower–Wales (STW) defects on the thermal properties of graphene sheet using MD simulation. They found that though the defects cause a gradual decrease in the Young’s modulus of graphene sheet, their effect on thermal properties is more pronounced. Lan et al. [15] investigated quantum thermal transport properties of graphene sheet using tight-binding techniques. It was found from their analysis that the thermal transport in graphene sheet shows substantial dependence on the width due to edge reconstructions. It is also useful to note that theoretical analysis allows reconstruction of defects, altering of chirality and system size [16], [17] to understand the influence of system parameters on the thermal properties of graphene sheet. Hence, MD simulation models can be used as a viable alternative compared to time consuming and expensive experiments for monitoring thermal transport at nanoscale. MD simulation is capable of generating accurate solutions in predicting mechanical and thermal properties of nanoscale system with minimal cost and high rapidity. However, the MD simulation does not provide information on relationship between the input parameters and the generated output. On the other hand, though soft computing techniques (such as GP) can predict the relationship between the input parameters and generated output, they cannot be used to predict system properties in nanoscale materials.

Therefore, there is a need to develop a new embedded simulation approach consisting of MD simulation and soft computing techniques for modeling the material properties of nanoscale materials such as graphene. The new embedded approach combines powerful advantages of accuracy and low cost of MD simulation with the explicit model formulation of the soft computing 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 soft computing methods can then be able to generate meaningful solutions for the complicated problems [18], [19]. Additionally, among the various soft computing methods described above, evolutionary approach, namely, MGGP offers the advantage of a fast and cost-effective explicit formulation of a mathematical model based on multiple variables with no existing analytical models [20], [21]. It is to the best of author’s knowledge that limited or no work exists on the application of soft computing models on evaluating thermal properties of the nanoscale system. Additionally, the potential future applications of graphene in electronics industry requires a thorough understanding and investigation of various input parameters on the thermal conductivity of graphene sheet.

Therefore, the main purpose of the present study is to investigate the thermal conductivity of graphene sheet. Standard MD simulation approach is employed to investigate the effect of temperature and STW defects on the thermal conductivity of graphene sheet. Data generated from the MD simulations is fed into the paradigm of GP for the formulation of function expressions. The performance of these models is evaluated against the data generated from the MD simulations.

Section snippets

MD simulation methodology

This sections explains the MD simulation methodology adopted to determine the thermal conductivity of a single layer graphene sheet (hereafter referred to as graphene). The data generated from the MD simulation is used to provide the input data to the soft computing cluster (Fig. 1) for training and generation of results. Brenner’s second generation bond order function (REBO) [22] is used to describe the covalent bonding of the carbon atoms in graphene. The REBO potential is ideal for

Proposed multi-gene genetic programming (MGGP) approach

To understand the concept of the MGGP method, we will first discuss about genetic programming (GP) in brief. GP based on Darwin’s theory of ‘survival of the fittest’ finds the optimal solution by mimicking the process of evolution in nature Koza [36]. Due to which, GP has been extensively applied for solving symbolic regression problems of various systems. GP have same framework to that of GA, but, there is still a major difference between GP and GA. The GA is parameter optimization method,

Evaluation of the performance of the models

The results obtained from the two MGGP models are illustrated in Fig. 9, Fig. 10 on the training and testing data respectively. Performance of the proposed models is evaluated using the five metrics: the square of the correlation coefficient (R2), the mean absolute percentage error (MAPE), the RMSE, relative error (%) and multi-objective error function (MO) given byR2=i=1n(Ai-Ai)(Mi-Mi)i=1n(Ai-Ai)2i=1n(Mi-Mi)22MAPE(%)=1niAi-MiAi×100RMSE=i=1N|Mi-Ai|2NRelativeerror(%)=|Mi-Ai|Ai×100

Sensitivity and parametric analysis of the models

Sensitivity and parametric analysis about the mean is conducted for the validation of our proposed MGGP model. The sensitivity analysis (SA) percentage of the outputs to each input parameter is determined using the following formulas:Li=fmax(xi)-fmin(xi)SAi=Lij=1nLj×100where 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.

Fig. 11 shows the plots of the sensitivity

Conclusions and future work

The present work discusses the experimental and MD based studies conducted for the evaluation of thermal conductivity of various graphene structures. Alternatively, we illustrate the MGGP approach and explored its ability in simulating the thermal conductivity characteristic based on temperature and number of defects of the two graphene structures: zigzag and armchair. The results show that the predictions obtained from the proposed embedded models are in good agreement with the actual data.

Acknowledgement

This work was partially supported by the Singapore Ministry of Education Academic Research Fund through research grant RG30/10, which the authors gratefully acknowledge.

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    The first two authors made equal contribution in this work and are both equally considered as first author.

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