- Email: [email protected]
Effects of boron doping on mechanical properties and thermal conductivities of carbon nanotubes
Solid State Communications 152 (2012) 1973–1979
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effects of boron doping on mechanical properties and thermal conductivities of carbon nanotubes Mir Masoud Seyyed Fakhrabadi n, Akbar Allahverdizadeh, Vahid Norouzifard, Behnam Dadashzadeh Karaj Branch, Islamic Azad University, Karaj, Iran
a r t i c l e i n f o
abstract
Article history: Received 21 July 2012 Accepted 5 August 2012 by E.L. Ivchenko Available online 15 August 2012
The paper presents the elastic and failure behaviors as well as the thermal conductivities of pure and boron doped carbon nanotubes (CNTs). The molecular dynamic simulation technique is applied to investigate the effects of various doping concentrations, loading rates and temperatures on the mechanical characteristics of the mentioned nanostructures. The results show the decrement in the maximum strengths and thermal conductivities when the CNTs are doped with boron. For example, only 5% boron doping reduces the thermal conductivity more than 65%. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Carbon nanotubes C. Mechanical behaviors D. Thermal conductivities D. Boron doping
1. Introduction Carbon nanotubes (CNTs) have demonstrated excellent properties in many fields including electrical, mechanical, civil and material engineering as well as medical and biomedical applications. Hence, there has been tremendous growth in the researches conducted on these nanomaterials in the recent years. From mechanical and physical points of views, CNTs, in comparison with many nanomaterials, show higher strength to weight ratios and elastic modulus as well as higher electrical and thermal conductivities. Thus, they are considered exceptional choices in nanoengineering. Although, pure CNTs have remarkable characteristics, doping them with other materials leads to increase in their abilities for different applications. For example, electronic and optical properties of the boron and nitrogen doped CNTs dominate to those for the pure CNTs. Therefore, this is considered an effective technique to make the properties better in the recent years. Koretsune and Saito investigated the electronic properties of boron doped single-walled CNTs via application of the first principles technique based on the density functional theory [1]. In their research, the total energy, band structure, and density of states were evaluated. They proved that a narrower CNT required less energy cost to substitute a carbon atom with a boron one. Synthesis and Raman characterization of boron doped singlewalled CNTs were conducted by McGuire et al. [2]. In this paper,
n
Corresponding author. Tel./fax: þ 98 935 5928477. E-mail addresses: [email protected], [email protected] (M.M.S. Fakhrabadi). 0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.08.003
a research was conducted on the doped single-walled CNT bundles with varying amounts of boron via application of the pulsed laser vaporization method. The CNTs with 0.5–10% boron were fabricated by mixing elemental boron with carbon paste and the Co/Ni catalysts. Wang and Zhang investigated theoretically the different doping techniques of boron and nitrogen doped CNTs [3]. In their article, the boron and nitrogen doped (4,4)i and (8,0)i (i¼3–5) CNTs, BCyN (y¼1–4, 6 and 8), B3CyN3 (y¼2 and 4), B2CN2, as well as BN nanotubes were studied by B3LYP/6-31G * approach. The lowest energy structures of BxCyNx nanotubes of doped (8,0)i CNTs revealed that the boron and nitrogen atoms prefer to dope (8,0)i CNTs from bottom layers to top layers one by one. Ling-Na et al. applied the first-principles method to analyze the electronic band structure and the quantum transport properties of metallic CNTs with boron/nitrogen co-doping [4]. They discovered that both of the mentioned properties were very sensitive to the doping concentration of the boron and nitrogen pairs. They contended that the energy gaps increased with doping concentration increasing both along the axis and around of the CNT, because the mirror symmetry of the nanostructure was broken via application of boron and nitrogen doping. Panchakarla et al. investigated the boron and nitrogen doped CNTs and graphene sheets [5]. They claimed that doping with boron and nitrogen caused remarkable changes in the Raman spectra of the CNTs. In their article, they presented the synthesis, characterization and properties of the mentioned nanostructures. In another research, Ding et al. suggested a one-pot synthesis technique with the combination of boron doping and fabrication of mesoporous carbon via application of sol–gel approach using boric acid as the catalyst, dopant and pore-forming agent [6].
1974
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
In this paper, the resultant materials were analyzed using TEM, small-angle and wide-angle XRD, Raman, nitrogen sorption, SEM, XPS and B MAS NMR measurement methods. Ishii et al. synthesized a large amount of boron doped multiwalled CNTs using hot-filament chemical vapor deposition (CVD) [7]. The synthesis was performed in a flask using a methanol solution of boric acid as a source material. The SEM, TEM and micro-Raman spectroscopy were applied to evaluate the structural properties of the obtained nanomaterials. Arutyunyan et al. investigated the boron and nitrogen doping of the CNTs
and carbon flakes using high resolution TEM and electron energy loss spectroscopy [8]. The results obtained from optical absorption spectroscopy measurements showed a monotonous increment in the bandgap value of the synthesized CNTs with the increment of the content of boron and nitrogen phases in the initial mixture of the synthesis compounds. Lee et al. studied the effects of substitutional boron on the oxidation of CNTs in air and oxygen plasma [9]. They doped multi-walled CNTs with boron at the initial loading of 1 and 2.5 wt% and investigated their oxidation properties in both air and oxygen plasma to evaluate the effects of substitutional boron in carbon oxidation and assess the theory that the preferential attack by atomic oxygen on basal plane was due to delocalized pelectrons. Kalijadis et al. doped glassy carbon with boron using chemical modification and irradiating the precursor polymer with boron ions [10]. A polymer with monotonous distribution of boron in the bulk was fabricated by chemical modification and the polymer was situated in a narrow region under the surface via irradiation with B3 þ. The obtained materials were carbonized at high temperatures and examined using X-ray diffraction, Raman spectroscopy, temperature programmed desorption (TPD) and hardness measurements. Chen et al. applied the first principles techniques to estimate the electronic band structure of metallic CNTs with boron and nitrogen co-doping [11]. They revealed the formation energies proposing that the boron and nitrogen co-doping configuration was the energetically stable structure. They also showed that the electronic structure properties depended on the doping concentration of the CNTs and doping positions. Fakhrabadi et al. investigated the mechanical properties and thermal conductivities of the nitrogen doped CNTs using molecular dynamics [12]. In their research, the elastic and failure properties as well as thermal conductivities of the mentioned nanostructures for different doping concentrations were studied. Mortazavi et al. employed the molecular dynamics simulations to study the tensile behavior of the graphene [13]. The accuracy of the obtained results
Fig. 2. Schematic view of the model to determine the thermal conductivities of the CNTs.
Fig. 1. Schematic views of the pure and boron doped CNTs.
Fig. 3. Stress–strain diagram of the pure and 4% boron doped CNTs.
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
was validated by comparing the simulation outcomes with the available experimental data. By performing uniaxial tension simulations, they studied the effects of different strain rates, chiralities, nanoribbon widths and number of atomic planes on the elastic and failure properties of the graphene. In another paper, Mortazavi and Azhi analyzed the effects of boron atom substitution on the thermal conductivity and mechanical properties of the single-layer graphene via application of the molecular dynamics technique [14]. They applied the uniaxial tension condition and revealed that this substitution slightly decreased the elastic modulus and tensile strength of the graphene. Mortazavi et al. studied the effects of nitrogen doping and curvature on the thermal conductivity of the graphene sheets using molecular dynamics simulations [15]. They applied the Tersoff potential in their research. It was revealed that just 1% concentration of nitrogen doping in the graphene decreased its thermal conductivity more than 50%. Some of the papers reviewed above proposed that doping may affect the characteristics of the CNTs such as their mechanical and thermal behaviors that should be investigated in detail. The mentioned influences may be considered advantages or disadvantages of this technique. In this paper, we are going to study them. The effects of various boron doping rates, temperatures and loading rates on the elastic and failure characteristics of the CNTs as well as the thermal conductivities are discussed in the following lines.
2. Atomistic simulation techniques In this section, the concepts of the applied atomistic simulation method are presented. The simulations are performed using LAMMPS
1975
(Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics software. The well-known Tersoff potential [16] in Eq. (1) is applied to model the covalent bonds between atoms either C–C or C–B. The constants corresponding with carbon and boron atoms to apply in the Tersoff potential are adopted from [16] E¼
1XX V 2 i j a i ij
ð1aÞ
V ij ¼ f C ðr ij Þ½f R ðr ij Þ þ bij f A ðr ij Þ
ð1bÞ
8 > < 1, f C ðrÞ ¼ 12 12 sin p2 rR D , > : 0,
ð1cÞ
r o RD RD o r oR þ D r 4 R þD
f R ðrÞ ¼ A expðl1 rÞ
ð1dÞ
f A ðrÞ ¼ B expðl2 rÞ
ð1eÞ
n n
bij ¼ ð1 þ b zij Þð1=2nÞ
zij ¼
X
ð1fÞ m
f c ðr ik Þgðyijk Þexp½l3 ðr ij r ik Þm
ð1gÞ
k a i,j
gðyÞ ¼ gijk 1 þ
c2 d
2
c2 2
!
½d þ ðcos ycosy0 Þ2
ð1hÞ
where fR denotes a two-body term and fA represents the three-body interactions. The summations in the formula are over all neighbors j and k of atom i within a cutoff distance¼RþD. The paper includes uniaxial tensile loading of the pure and boron doped CNTs and their thermal conductivities. The length and chirality of the CNTs considered in this paper are, respectively,
Fig. 4. Elongation and failure of the 4% doped CNT under a tensile load.
1976
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
10 nm and (10,10). In addition, periodic boundary conditions are assumed for the molecular dynamics simulations via application of LAMMPS software. Fig. 1 illustrates schematically both pure and boron doped CNTs. In this figure, the brown spheres represent boron atoms doped in the CNT structure. In uniaxial tensile loading, the lower sides of the CNTs are fixed while the upper sides are applied different strain rates in various simulations representing the applied load. It is worth noting that the upper side is only allowed to move in the vertical direction. The effects of various loading rates and temperatures are investigated on the elastic and failure properties of the pure and boron doped CNTs. The boron atoms are considered with different concentrations to comprehend various results. More results help us to predict the behaviors better and more accurate. To investigate the thermal conductivities of the CNTs and effects of various boron concentrations on them, Fourier conduction relation (Eq. (2)) is used. In this equation, qz is the heat flux, k is the thermal conductivity, A is the cross sectional area and dT/dz is the temperature gradient along the longitudinal axis
(in this paper, z) qz ¼ kA
dT dz
ð2Þ
The unknown values in Eq. (2) (qz and dT/dz) are computed from MD simulation and then the thermal conductivities can be evaluated. For the models studied in this paper, two sides of the CNTs are mechanically fixed and the two hot and cold regions are considered in two ends (Fig. 2). The temperatures of the two regions do not vary during the simulation. The required parameters in Eq. (2) are obtained from the middle atoms. The effects of various boron doping concentrations on the thermal conductivities are studied in detail.
3. Results and discussion This section includes the results of molecular dynamics simulation described in the previous section. It is divided into
Fig. 5. (a) Rupture stress vs. boron doping concentration and (b) rupture stress ratio vs. concentration.
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
two sub-sections; one for the mechanical properties and another for the thermal conductivities.
3.1. Mechanical properties As shown in Fig. 1, the lower layers of the CNTs are fixed and the upper layers are allowed to move only in the vertical direction. Fig. 3 illustrates the stress–strain diagrams of the pure and 4% boron doped CNTs under uniaxial loading conditions. The figure proposes that boron doping decreases the ultimate strength of the CNT. Fig. 4 displays the elongation and failur of the 4% boron doped CNT under the uniaxial tensile loading condition. The figure shows that the nanostructure tolerates the load up to maximum value and then fails due to the crack propogation. Fig. 5(a) and (b) depicts, respectively, the variation of the rupture strength and its ratio vs. boron doping concentration. The results suggest that the doping concentration directly affect the maximum strength in the stress–strain diagram. With increment
1977
in the boron doping value, the rupture stress decreases. Of course, the rate of decrement seems higher in the lower concentrations. The effects of various loading rates on the stress–strain diagram of the pure and 4% boron doped CNTs are investigated in Fig. 6. The figure depicts that higher strain rates leads to the higher elastic modulus and lower failure strains. This effect strengthens with the increment in the strain rates. Fig. 7 shows the effects of temperature variation on the stress– strain diagrams of the pure and 4% boron doped CNTs. As shown in this figure, the increment in the temperature decreases the rupture stresses of both pure and doped CNTs. Fig. 8 presents the rupture stresses of the pure and 4% boron doped CNTs vs. temperature. As shown in this figure, for both cases, the rupture stresses decrease with the increment in the temperature. 3.2. Thermal conductivities As described before and shown in Fig. 2, the CNT is fixed between two hot and cold sources. The variation of the temperature along the
Fig. 6. Effects of various strain rates on the stress–strain diagrams.
1978
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
Fig. 7. Temperature effects on the stress–strain diagrams of pure and doped CNTs.
Fig. 8. Rupture stresses of pure and 4% boron doped CNTs vs. temperature.
Fig. 9. Temperature variation along the CNT length.
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
1979
boron doping with various concentrations on the elastic and rupture of the CNTs were studied. Furthermore, the effects of loading rates and temperature gradients on the mentioned properties were discussed in detail. The results proposed that boron doping can reduce the rupture stress of the CNTs. Moreover, increment in temperature had negative effects on the maximum strength. Also, boron doping affected the thermal conductivities of the CNTs and reduced it drastically.
References
Fig. 10. Effects of boron doping on the thermal conductivities of the CNTs.
CNT length is similar to Fig. 9. The data obtained from molecular dynamics can be interpolated linearly, as shown in this figure. The results corresponding with the thermal conductivities of the boron doped CNTs vs. doping concentration are presented in Fig. 10. The figure reveals that with the increment in the doping concentration, the thermal conductivity decreases. However, this decrement is more dominant in the lower concentrations. As observed in this figure, doping about 5% of boron atoms reduces the thermal conductivity more than 65%.
4. Conclusion The paper investigated the effects of boron doping on the mechanical and thermal properties of the CNTs. The influences of
[1] T. Koretsune, S. Saito, J. Phys. Rev. B 77 (2008) 165417. [2] K. McGuire, N. Gothard, P.L. Gai, M.S. Dresselhaus, G. Sumanasekera, A.M. Rao, J. Carbon 43 (2005) 219–227. [3] P. Wang, C. Zhang, J. Mol. Struct. THEOCHEM. 955 (2010) 84–90. [4] C. Ling-Na, M. Song-Shan, O.Y. Fang-Ping, X. Jin, X. Hui, J. Chin. Phys. B 20 (2011) 017103. [5] L.S. Panchakarla, A. Govindaraj, C.N.R. Rao, J. Inorg. Chim. Acta 363 (2010) 4163–4174. [6] S. Ding, S. Zheng, M. Xie, L. Peng, X. Guo, W. Ding, J. Microporous Mesoporous Mater. 142 (2011) 609–613. [7] S. Ishii, M. Nagao, T. Watanabe, S. Tsuda, T. Yamaguchi, Y. Takano, J. Phys. C 469 (2009) 1002–1004. [8] N.R. Arutyunyan, R. Arenal, E.D. Obraztsova, O. Stephan, A. Loiseau, A.S. Pozharov, V.V. Grebenyukov, J. Carbon 50 (2012) 791–799. [9] Y. Lee, H. Kim, H. Hatori, J. Carbon 42 (2004) 1053–1056. [10] A. Kalijadis, Z. Jovanovic, M. Lausevic, Z. Lausevic, J. Carbon 49 (2011) 2671–2678. [11] L.N. Chen, F.P. OuYang, S.S. Ma, X.Z. Wu, J. Xiao, H. Xu, J. Phys. Lett. A 374 (2010) 4343–4348. [12] M.M.S. Fakhrabadi, A. Allahverdizadeh, V. Norouzifard, B. Dadashzadeh, J. Phys. Lett. A, submitted for publication. [13] B. Mortazavi, S. Ahzi, V. Toniazzo, Y. Remond, J. Phys. Lett. A 376 (2012) 1146–1153. [14] B. Mortazavi, S. Ahzi, J. Solid State Commun. 152 (2012) 1503–1507. [15] B. Mortazavi, A. Rajabpour, S. Ahzi, Y. Remond, S.M. Vaez Allaei, J. Solid State Commun. 152 (2012) 261–264. [16] W.H. Moon, M.S. Son, H.J. Hwang, J. Phys. B 336 (2003) 329–334.
Contents lists available at SciVerse ScienceDirect
Solid State Communications journal homepage: www.elsevier.com/locate/ssc
Effects of boron doping on mechanical properties and thermal conductivities of carbon nanotubes Mir Masoud Seyyed Fakhrabadi n, Akbar Allahverdizadeh, Vahid Norouzifard, Behnam Dadashzadeh Karaj Branch, Islamic Azad University, Karaj, Iran
a r t i c l e i n f o
abstract
Article history: Received 21 July 2012 Accepted 5 August 2012 by E.L. Ivchenko Available online 15 August 2012
The paper presents the elastic and failure behaviors as well as the thermal conductivities of pure and boron doped carbon nanotubes (CNTs). The molecular dynamic simulation technique is applied to investigate the effects of various doping concentrations, loading rates and temperatures on the mechanical characteristics of the mentioned nanostructures. The results show the decrement in the maximum strengths and thermal conductivities when the CNTs are doped with boron. For example, only 5% boron doping reduces the thermal conductivity more than 65%. & 2012 Elsevier Ltd. All rights reserved.
Keywords: A. Carbon nanotubes C. Mechanical behaviors D. Thermal conductivities D. Boron doping
1. Introduction Carbon nanotubes (CNTs) have demonstrated excellent properties in many fields including electrical, mechanical, civil and material engineering as well as medical and biomedical applications. Hence, there has been tremendous growth in the researches conducted on these nanomaterials in the recent years. From mechanical and physical points of views, CNTs, in comparison with many nanomaterials, show higher strength to weight ratios and elastic modulus as well as higher electrical and thermal conductivities. Thus, they are considered exceptional choices in nanoengineering. Although, pure CNTs have remarkable characteristics, doping them with other materials leads to increase in their abilities for different applications. For example, electronic and optical properties of the boron and nitrogen doped CNTs dominate to those for the pure CNTs. Therefore, this is considered an effective technique to make the properties better in the recent years. Koretsune and Saito investigated the electronic properties of boron doped single-walled CNTs via application of the first principles technique based on the density functional theory [1]. In their research, the total energy, band structure, and density of states were evaluated. They proved that a narrower CNT required less energy cost to substitute a carbon atom with a boron one. Synthesis and Raman characterization of boron doped singlewalled CNTs were conducted by McGuire et al. [2]. In this paper,
n
Corresponding author. Tel./fax: þ 98 935 5928477. E-mail addresses: [email protected], [email protected] (M.M.S. Fakhrabadi). 0038-1098/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ssc.2012.08.003
a research was conducted on the doped single-walled CNT bundles with varying amounts of boron via application of the pulsed laser vaporization method. The CNTs with 0.5–10% boron were fabricated by mixing elemental boron with carbon paste and the Co/Ni catalysts. Wang and Zhang investigated theoretically the different doping techniques of boron and nitrogen doped CNTs [3]. In their article, the boron and nitrogen doped (4,4)i and (8,0)i (i¼3–5) CNTs, BCyN (y¼1–4, 6 and 8), B3CyN3 (y¼2 and 4), B2CN2, as well as BN nanotubes were studied by B3LYP/6-31G * approach. The lowest energy structures of BxCyNx nanotubes of doped (8,0)i CNTs revealed that the boron and nitrogen atoms prefer to dope (8,0)i CNTs from bottom layers to top layers one by one. Ling-Na et al. applied the first-principles method to analyze the electronic band structure and the quantum transport properties of metallic CNTs with boron/nitrogen co-doping [4]. They discovered that both of the mentioned properties were very sensitive to the doping concentration of the boron and nitrogen pairs. They contended that the energy gaps increased with doping concentration increasing both along the axis and around of the CNT, because the mirror symmetry of the nanostructure was broken via application of boron and nitrogen doping. Panchakarla et al. investigated the boron and nitrogen doped CNTs and graphene sheets [5]. They claimed that doping with boron and nitrogen caused remarkable changes in the Raman spectra of the CNTs. In their article, they presented the synthesis, characterization and properties of the mentioned nanostructures. In another research, Ding et al. suggested a one-pot synthesis technique with the combination of boron doping and fabrication of mesoporous carbon via application of sol–gel approach using boric acid as the catalyst, dopant and pore-forming agent [6].
1974
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
In this paper, the resultant materials were analyzed using TEM, small-angle and wide-angle XRD, Raman, nitrogen sorption, SEM, XPS and B MAS NMR measurement methods. Ishii et al. synthesized a large amount of boron doped multiwalled CNTs using hot-filament chemical vapor deposition (CVD) [7]. The synthesis was performed in a flask using a methanol solution of boric acid as a source material. The SEM, TEM and micro-Raman spectroscopy were applied to evaluate the structural properties of the obtained nanomaterials. Arutyunyan et al. investigated the boron and nitrogen doping of the CNTs
and carbon flakes using high resolution TEM and electron energy loss spectroscopy [8]. The results obtained from optical absorption spectroscopy measurements showed a monotonous increment in the bandgap value of the synthesized CNTs with the increment of the content of boron and nitrogen phases in the initial mixture of the synthesis compounds. Lee et al. studied the effects of substitutional boron on the oxidation of CNTs in air and oxygen plasma [9]. They doped multi-walled CNTs with boron at the initial loading of 1 and 2.5 wt% and investigated their oxidation properties in both air and oxygen plasma to evaluate the effects of substitutional boron in carbon oxidation and assess the theory that the preferential attack by atomic oxygen on basal plane was due to delocalized pelectrons. Kalijadis et al. doped glassy carbon with boron using chemical modification and irradiating the precursor polymer with boron ions [10]. A polymer with monotonous distribution of boron in the bulk was fabricated by chemical modification and the polymer was situated in a narrow region under the surface via irradiation with B3 þ. The obtained materials were carbonized at high temperatures and examined using X-ray diffraction, Raman spectroscopy, temperature programmed desorption (TPD) and hardness measurements. Chen et al. applied the first principles techniques to estimate the electronic band structure of metallic CNTs with boron and nitrogen co-doping [11]. They revealed the formation energies proposing that the boron and nitrogen co-doping configuration was the energetically stable structure. They also showed that the electronic structure properties depended on the doping concentration of the CNTs and doping positions. Fakhrabadi et al. investigated the mechanical properties and thermal conductivities of the nitrogen doped CNTs using molecular dynamics [12]. In their research, the elastic and failure properties as well as thermal conductivities of the mentioned nanostructures for different doping concentrations were studied. Mortazavi et al. employed the molecular dynamics simulations to study the tensile behavior of the graphene [13]. The accuracy of the obtained results
Fig. 2. Schematic view of the model to determine the thermal conductivities of the CNTs.
Fig. 1. Schematic views of the pure and boron doped CNTs.
Fig. 3. Stress–strain diagram of the pure and 4% boron doped CNTs.
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
was validated by comparing the simulation outcomes with the available experimental data. By performing uniaxial tension simulations, they studied the effects of different strain rates, chiralities, nanoribbon widths and number of atomic planes on the elastic and failure properties of the graphene. In another paper, Mortazavi and Azhi analyzed the effects of boron atom substitution on the thermal conductivity and mechanical properties of the single-layer graphene via application of the molecular dynamics technique [14]. They applied the uniaxial tension condition and revealed that this substitution slightly decreased the elastic modulus and tensile strength of the graphene. Mortazavi et al. studied the effects of nitrogen doping and curvature on the thermal conductivity of the graphene sheets using molecular dynamics simulations [15]. They applied the Tersoff potential in their research. It was revealed that just 1% concentration of nitrogen doping in the graphene decreased its thermal conductivity more than 50%. Some of the papers reviewed above proposed that doping may affect the characteristics of the CNTs such as their mechanical and thermal behaviors that should be investigated in detail. The mentioned influences may be considered advantages or disadvantages of this technique. In this paper, we are going to study them. The effects of various boron doping rates, temperatures and loading rates on the elastic and failure characteristics of the CNTs as well as the thermal conductivities are discussed in the following lines.
2. Atomistic simulation techniques In this section, the concepts of the applied atomistic simulation method are presented. The simulations are performed using LAMMPS
1975
(Large-scale Atomic/Molecular Massively Parallel Simulator) molecular dynamics software. The well-known Tersoff potential [16] in Eq. (1) is applied to model the covalent bonds between atoms either C–C or C–B. The constants corresponding with carbon and boron atoms to apply in the Tersoff potential are adopted from [16] E¼
1XX V 2 i j a i ij
ð1aÞ
V ij ¼ f C ðr ij Þ½f R ðr ij Þ þ bij f A ðr ij Þ
ð1bÞ
8 > < 1, f C ðrÞ ¼ 12 12 sin p2 rR D , > : 0,
ð1cÞ
r o RD RD o r oR þ D r 4 R þD
f R ðrÞ ¼ A expðl1 rÞ
ð1dÞ
f A ðrÞ ¼ B expðl2 rÞ
ð1eÞ
n n
bij ¼ ð1 þ b zij Þð1=2nÞ
zij ¼
X
ð1fÞ m
f c ðr ik Þgðyijk Þexp½l3 ðr ij r ik Þm
ð1gÞ
k a i,j
gðyÞ ¼ gijk 1 þ
c2 d
2
c2 2
!
½d þ ðcos ycosy0 Þ2
ð1hÞ
where fR denotes a two-body term and fA represents the three-body interactions. The summations in the formula are over all neighbors j and k of atom i within a cutoff distance¼RþD. The paper includes uniaxial tensile loading of the pure and boron doped CNTs and their thermal conductivities. The length and chirality of the CNTs considered in this paper are, respectively,
Fig. 4. Elongation and failure of the 4% doped CNT under a tensile load.
1976
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
10 nm and (10,10). In addition, periodic boundary conditions are assumed for the molecular dynamics simulations via application of LAMMPS software. Fig. 1 illustrates schematically both pure and boron doped CNTs. In this figure, the brown spheres represent boron atoms doped in the CNT structure. In uniaxial tensile loading, the lower sides of the CNTs are fixed while the upper sides are applied different strain rates in various simulations representing the applied load. It is worth noting that the upper side is only allowed to move in the vertical direction. The effects of various loading rates and temperatures are investigated on the elastic and failure properties of the pure and boron doped CNTs. The boron atoms are considered with different concentrations to comprehend various results. More results help us to predict the behaviors better and more accurate. To investigate the thermal conductivities of the CNTs and effects of various boron concentrations on them, Fourier conduction relation (Eq. (2)) is used. In this equation, qz is the heat flux, k is the thermal conductivity, A is the cross sectional area and dT/dz is the temperature gradient along the longitudinal axis
(in this paper, z) qz ¼ kA
dT dz
ð2Þ
The unknown values in Eq. (2) (qz and dT/dz) are computed from MD simulation and then the thermal conductivities can be evaluated. For the models studied in this paper, two sides of the CNTs are mechanically fixed and the two hot and cold regions are considered in two ends (Fig. 2). The temperatures of the two regions do not vary during the simulation. The required parameters in Eq. (2) are obtained from the middle atoms. The effects of various boron doping concentrations on the thermal conductivities are studied in detail.
3. Results and discussion This section includes the results of molecular dynamics simulation described in the previous section. It is divided into
Fig. 5. (a) Rupture stress vs. boron doping concentration and (b) rupture stress ratio vs. concentration.
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
two sub-sections; one for the mechanical properties and another for the thermal conductivities.
3.1. Mechanical properties As shown in Fig. 1, the lower layers of the CNTs are fixed and the upper layers are allowed to move only in the vertical direction. Fig. 3 illustrates the stress–strain diagrams of the pure and 4% boron doped CNTs under uniaxial loading conditions. The figure proposes that boron doping decreases the ultimate strength of the CNT. Fig. 4 displays the elongation and failur of the 4% boron doped CNT under the uniaxial tensile loading condition. The figure shows that the nanostructure tolerates the load up to maximum value and then fails due to the crack propogation. Fig. 5(a) and (b) depicts, respectively, the variation of the rupture strength and its ratio vs. boron doping concentration. The results suggest that the doping concentration directly affect the maximum strength in the stress–strain diagram. With increment
1977
in the boron doping value, the rupture stress decreases. Of course, the rate of decrement seems higher in the lower concentrations. The effects of various loading rates on the stress–strain diagram of the pure and 4% boron doped CNTs are investigated in Fig. 6. The figure depicts that higher strain rates leads to the higher elastic modulus and lower failure strains. This effect strengthens with the increment in the strain rates. Fig. 7 shows the effects of temperature variation on the stress– strain diagrams of the pure and 4% boron doped CNTs. As shown in this figure, the increment in the temperature decreases the rupture stresses of both pure and doped CNTs. Fig. 8 presents the rupture stresses of the pure and 4% boron doped CNTs vs. temperature. As shown in this figure, for both cases, the rupture stresses decrease with the increment in the temperature. 3.2. Thermal conductivities As described before and shown in Fig. 2, the CNT is fixed between two hot and cold sources. The variation of the temperature along the
Fig. 6. Effects of various strain rates on the stress–strain diagrams.
1978
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
Fig. 7. Temperature effects on the stress–strain diagrams of pure and doped CNTs.
Fig. 8. Rupture stresses of pure and 4% boron doped CNTs vs. temperature.
Fig. 9. Temperature variation along the CNT length.
M.M.S. Fakhrabadi et al. / Solid State Communications 152 (2012) 1973–1979
1979
boron doping with various concentrations on the elastic and rupture of the CNTs were studied. Furthermore, the effects of loading rates and temperature gradients on the mentioned properties were discussed in detail. The results proposed that boron doping can reduce the rupture stress of the CNTs. Moreover, increment in temperature had negative effects on the maximum strength. Also, boron doping affected the thermal conductivities of the CNTs and reduced it drastically.
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Fig. 10. Effects of boron doping on the thermal conductivities of the CNTs.
CNT length is similar to Fig. 9. The data obtained from molecular dynamics can be interpolated linearly, as shown in this figure. The results corresponding with the thermal conductivities of the boron doped CNTs vs. doping concentration are presented in Fig. 10. The figure reveals that with the increment in the doping concentration, the thermal conductivity decreases. However, this decrement is more dominant in the lower concentrations. As observed in this figure, doping about 5% of boron atoms reduces the thermal conductivity more than 65%.
4. Conclusion The paper investigated the effects of boron doping on the mechanical and thermal properties of the CNTs. The influences of
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