Tuning of Elastic Properties of Nanotubes by Imposing a Transverse

Subscriber access provided by CORNELL UNIVERSITY LIBRARY

Article

Tuning of Elastic Properties of Nanotubes by Imposing Transverse Electric Field: Computational Approach Amin Khorsandi-Langol, Kiana Gholamjani Moghaddam, Seyed Majid Hashemianzadeh, and Zabiollah Mahdavifar J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.6b05703 • Publication Date (Web): 21 Jul 2016 Downloaded from http://pubs.acs.org on July 23, 2016

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry C is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Tuning of Elastic Properties of Nanotubes by Imposing Transverse Electric Field: Computational Approach

Amin Khorsandi-Langola, Kiana Gholamjani Moghaddama, Seyed Majid Hashemianzadeha,*, Zabiollah Mahdavifarb a

Molecular Simulation Research Laboratory, Department of Chemistry, Iran University of Science and Technology, Tehran, Iran b

Computational Chemistry Group, Department of Chemistry, Faculty of Science, Shahid Chamran University, Ahwaz, Iran

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract In this study, we investigated the influence of a transverse electric field on the mechanical properties of carbon (C), boron nitride (BN) and silicon carbide (SiC) nanotubes based on the density functional theory (DFT) and continuum mechanics. In order to evaluate the Young’s modulus of the nanotubes, a compressive distribution loading was implemented in the perpendicular direction to the longitudinal axis of the nanotubes in the presence and absence of the electric field. According to the obtained results from the geometry of the deformed nanotubes, an elliptical function was used to estimate the overall deformation. Based on the Green strain theory, the displacement of each particle is defined by the conversion matrix to transform the initial circular shape of the nanotubes to the final elliptical shape. The classical continuum theory was employed to estimate the relationship between strain energy and strain field of the nanotubes. The evolutionary genetic algorithm was used to achieve an accurate estimation of the fitting of energy function versus strain based on the elasticity theory. In addition, the effect of the electric field on the electronic and mechanical properties of the nanotubes was investigated. The numerical results revealed that the nanotubes exposed to the electric field have more effective stiffness in comparison to the similar case in the off-field conditions.


Page 2 of 36

Page 3 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


1.

Introduction Carbon nanotubes (CNTs) with their unique electronic, optical, magnetic, structural and

mechanical properties are the attractive topics in nanoscience and nanotechnology, although these structures were discovered three decades ago.1 The structural and mechanical properties of CNTs make them useful as the strongest or stiffest elements for a wide range of new applications including nanoscale devices or composite materials.2 Therefore, many efforts have been made to investigate the mechanical properties of the nanotubes using both experimental3,4 and theoretical approaches.5-9 Due to the atomic dimensions of the nanotubes, the computational methods are considered to be one of the most powerful tools to clarify the nanotubes behavior in comparison to expensive experimental methods.10 It is clear that the mechanical properties of the nanotubes depend on the structural properties at the atomic scale.9 Therefore, the prediction of the nanotubes behavior from the atomic perspective has been attracted the interest of the researchers to achieve understanding of the elastic properties of carbon (C),11 boron nitride (BN),12,13 and silicon carbide (SiC)14 nanotubes based on various theoretical approaches including continuum mechanics, molecular dynamics (MD) and density functional theory (DFT). Many studies have been focused on the relationship between size, chirality and mechanical properties of the nanotubes especially the elastic modulus. By employing continuum mechanics and interatomic potential, the elastic behavior of CNTs was investigated. The combination of molecular and solid mechanics indicated that the elastic modulus is inversely proportional to the diameter of nanotubes (almost 1 nm in diameter) and also showed that the nanotubes symmetry affects the mechanical properties.11 However, the simulation results at the atomistic level showed that the structure of CNTs with diameters larger than 1nm has no influence on the elastic modulus.15 Based on the MD simulation, elastic properties of single-walled carbon nanotubes



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(SWCNTs) indicated that the Young’s modulus of SWCNTs is insensitive to the chirality and radius of nanotubes.16-19 However, a low effect of diameter and helicity of CNTs on the Young’s modulus was proved by the molecular mechanics simulation.20 Song et al. reported the structure and size effects on the elastic properties of CNTs using first principle study.21 The Young’s modulus in both axial and transverse directions was calculated for many zigzag and armchair nanotubes.16,17,22,23 It was found that the elastic properties of the nanotubes with diameters almost larger than 1nm are insensitive to the nanotube size. In contrast, when the diameter of the nanotube is almost 1 nm or less than 1 nm, the elastic properties is dependent on the nanotube size. Young’s modulus predictions of BN nanotubes on the basis of the molecular mechanics approach for investigation of their size dependent elastic properties showed that the Young’s modulus of both armchair and zigzag configurations is increased with increasing nanotubes diameter, especially for nanotubes with small radiuses.24 Also, Verma et al. based on the molecular mechanics using Tersoff–Brenner potential, showed that the Young’s modulus of BN nanotubes is changed with nanotubes diameter.25 Therefore, it should be noted that the difference in partial atomic charges between B and N atoms can play a significant role in elastic properties of BN nanotubes.26 Santosh et al. revealed that both the normal and shear modulus are dependent on the magnitude of partial atomic charges of B and N atoms directly.26 In the case of SiC nanotubes, DFT calculations demonstrated that the Young’s modulus are inversely proportional to the Si-C bond length.27 Also, the structural, elastic and electronic properties of other nanotubes, such as BeO nanotubes were reported.28 Furthermore, the effect of structural deformation of the nanotubes on the electronic properties was investigated using two tightbinding and zone-folding approaches.29 The numerical results revealed that the band structure of


Page 4 of 36

Page 5 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


carbon nanotubes significantly is changed under uniaxial stress when compared to BN nanotubes. Moreover, the reduction of band gap due to uniaxial stress is mainly considered to justify inducing the semiconductor–metal transition for CNTs while these changes does not take place for BN nanotubes.30-32 Most recently, the effect of an external electric field on nanostructures have been of great interests in promising applications including nano-electromechanical systems and other nanoscale devices because of the variation in the electronic and mechanical properties of the nanostructures. The electric field has a great impact on the electronic properties such as band gap tuning, semiconducting metallic transition, formation of an electric field dipole moment along the direction of applied field and circular cross-section to elliptical cross-section deformation of C,32,33,34 BN,35, 36 SiC and AlN38 nanotubes. However, the influence of the external electric field on the elastic properties of the nanotubes has not been studied extensively. The goal of this study is to reveal the effect of a transverse electric field on the elastic properties of the nanotubes. In this work, six nanotubes C(7,7), C(12,0), BN(7,7), BN(12,0), SiC(6,6) and SiC(10,0) were studied. The deformations of these nanotubes due to the external electric field were investigated via the ab initio approach in the framework of DFT. Furthermore, to obtain the Young’s modulus of the nanotubes, two factors including strain and strain energy were calculated using DFT and Green strain theory, respectively. The classical continuum theory was employed to approximate the deformation of the nanotubes and characterize the energy versus induced strain field. It is noteworthy that the influence of the electric field on the sturcturtal and electronic properties at the atomic scale and consequently variation of the mechanical response of the nanotubes were explored.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 36

2. Methodology 2.1. Computational details In the present work, DFT and classical continuum method were used to study the relationship between the elastic properties of the nanotubes and the transverse electric field. All nanotubes are composed of five rings and terminated with hydrogen atoms. It should be noted that the diameters of all nanotubes are about 1nm. Since the previous theoretical calculations of electronic and structural properties of the nanotubes widely used the DFT method 13,28-30,36, all quantum mechanics calculations were performed based on DFT. All computations were carried out using Gaussian03 package39 at the B3LYP/6-311G(d) level. The band gap and the longitudinal polarizability are strongly dependent on the level of approximation and hybrid functionals like B3LYP as this method provide results with the least deviation from the experimental data40. Furthermore, the Green strain theory was employed to obtain strain for the deformation of the nanotubes resulted from the mechanical force. The classical continuum theory was also used to approximate the deformation of the nanotubes and characterize the energy versus induced strain field. Evolutionary genetic algorithm was applied to achieve an accurate estimation of the fitting of energy function parameters based on elasticity theory versus the obtained strain and strain energy. 2.2. Theory of mechanical properties When external force (stress) is applied to objects made of elastic materials, it can change the shape and the size of the objects (strain). For real materials, stress is proportional to strain when the strain is sufficiently small as follows:

σ = Eε


(1)

Page 7 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Where E is the slope in the linear area (called Young’s modulus), σ and ε is the stress and strain, respectively. For linear elastic solids, there is a strain energy density function,U 0 (ε ) , that can be written as follows:

σ ij =

∂U 0 ∂ε ij

(2)

Where ε ij represents the components of strain tensor and it’s being used to derive the stress– strain relations for the linear elastic solid which begins with the quadratic form of U 0 U 0 = C0 + Cij ε ij + Cijkl ε ij ε kl + Cijklmnε ij ε kl ε mn + ...

(3)

And in linear elasticity form it can be written as below: U0 = Cijkl ε ij ε kl

(4)

It is noteworthy that the normal strain components e.g. ε x , ε y , ε z are linear functions of normal stress components and by making this assumption, the strain energy density can be rewritten as follows: U 0 (ε x , ε y , ε z ) = Aε x2 + Bε y2 + C ε z2 + F ε x ε y + Gε x ε z + H ε y ε z

(5)

Therefore, the strain energy density is quadratic function of the strain. In addition, due to the relationship between the stress and strain energy functions, the following matrix can be obtained.  σx σ  y  σ z

  2A F G   εx   =  F 2B H   ε     y    G H 2C   ε z 

(6)

The relationship between the stress and the strain tensor for orthotropic materials (that has at least 2 orthogonal planes of symmetry, where material properties are independent of direction within each plane) without shear strain is as follows:



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

 1  E 1 ε  x    ε  = − ϑ12  y   E 1   ε z   ϑ13  −  E1

−

ϑ21 E2 1

E2 −

ϑ23 E2

ϑ31   E3   σx ϑ32   −  σy E3    σ z 1   E3 

Page 8 of 36

−

   

(7)

Where E1, E2, and E3 are the Young’s modulus in 1, 2, and 3 material directions, respectively. ϑij is Poisson’s ratio, defined as the ratio of the transverse strain in the jth direction to the axial strain in the ith direction when the stress was applied in the i-direction.41 As mentioned before, the genetic algorithm was used to estimate fitting parameters in equation (5) with calculations of the strain and strain energy. Finally, the Young’s modulus, per unit volume Y v , can be expressed as:

Y v ,i =

1 Ei , i = x, y and z V0

(8)

Where E i is obtained by equations (6) and (7). V 0 is the volume for a hollow cylinder with length L and thickness r which equals to 2πR × L × r . By assuming that the nanotube is a hollow cylinder, the average bond length (B-N, C-C and Si-C bond) is considered to be the wall thickness of the nanotubes 42 as demonstrated in Figure 1a.

2.2.1. Strain calculation By implementing the mechanical uniaxial force, the cross-section of the nanotubes is changed from the regular circular shape to the elliptical shape. In this case, the tube diameter, which is perpendicular to the force direction, is increased, as shown in Figure 1b. According to the Green strain theory, the relationship between the material line dX, (distance between two points in the neighborhood of each other in the reference configuration) before


Page 9 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


deformation and line dx (distance between previous two points in the deformed configuration) after deformation is defined as: d x = F . dX

(9)

Where F is the deformation gradient and the change in the squared lengths that occurs as a body deformation from the reference to the current configuration can be expressed relative to the original length as below: 2

2

dx − dX = 2dX .ε . dX

(10)

Where ε is called the Green-St. Vanant (Lagrangian) strain tensor or simply the Green strain tensor that can be expressed as: ε =

1 T (F . F − I ) 2

(11)

Where I is the identity tensor.41 According to the Green strain theory, a conversion matrix is defined to estimate the change of the nanotube shape from the circular to elliptical shape. When the longitudinal deformations of the nanotubes become negligible compared to the cross-section deformations, z component of the conversion matrix become equal to 1 as shown in equation 12. af   ai  F = 0   0  

 0   0  1  

0 bf bi 0

(12)

Where ai ,bi and af , bf (a > b) are radiuses of the initial circular cross-section and the final elliptical cross-section, respectively (as shown in figure 1b). Referring to the equation 11, the final strain can be written as follows:



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

2 2 Z − Zi 1 af 1 bf ε x = ( 2 − 1) , ε y = ( 2 − 1) and ε z = f Zi 2 ai 2 bi

Page 10 of 36

(13)

Where Zi and Zf represent the length of the nanotubes before and after deformation, respectively. In this equation, ɛz is assumed to be zero because z component of the conversion matrix for the transformation of the nanotube shape from the circular to elliptical shape is 1.

2.2.2. Strain energy calculation The difference between the energy of the nanotubes before and after deformation is defined as the strain energy,U 0,f (ε x , ε y , ε z ) , such that U 0, f ( ε x , ε y , ε z ) = U f − U i , ( f = 1, 2, 3,... & i = 0)

(14)

Where i is the initial state of the nanotubes and f is the final states of the nanotubes after deformation.

3. Results and discussion In this section, the effect of the electric field on the structural parameters, electronic and elastic properties of the entitled nanotubes will be discussed.

3.1. Structural and Electronic Properties 3.1.1 Geometry By applying a transverse electric field, the structure of the nanotubes are deformed. In fact, the nanotubes are stretched in the direction of the implemented electric field. Under the influence of the electric field, the initial circular cross-section of the nanotubes is deformed to the elliptical form, so that the longer and smaller diameters of the nanotubes are placed in the y and x directions, respectively as displayed in Figure 1c. The bond length of the nanotubes is changed according to the orientation of the nanotubes bonds in the presence of the electric field. For quantitative analysis of this variation, the approximate location of bonds in the nanotubes toward 10 ACS Paragon Plus Environment

Page 11 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


the y-axis is shown in Figure 2a and 3a for BN(12,0) and BN(7,7) nanotubes, respectively. It should be noted that y-axis is oriented perpendicular to the longitudinal axis of the nanotubes. However, according to Figures 2a and 3a, the bonds located in a and b regions are parallel to the electric field direction, while the bonds located in the c and d regions are perpendicular to the electric field direction. The results showed that the length of parallel (perpendicular) bonds to the direction of the electric field have the maximum (minimum) changes as presented in Figures 2b and 3b for BN(12,0) and BN(7,7) nanotubes, respectively. The corresponding figures for other nanotubes are reported in the supporting information (S1-S4). It is clear that in the cases of BN and SiC nanotubes, the dipole moment rotation occurs in the presence of the electric field which is proportional to the partial charges of atoms. The required energy for rotation of the dipole moment along the direction of field is expressed by: → →

U = − E .q .d cos θ = − E . p

(15)

Where E, q, d and p are the electric field vector, electrical charge, distance between two opposite charges and dipole moment, respectively. The most stable and unstable states are occurred when θ = 0,θ = 180 , respectively. According to Equation 15, the variation of bond length is the most effective parameter for minimizing the potential energy of the dipole moment. In the case of θ < 90 , with increasing bond length under transverse electric field, the potential energy of dipole moment would be more stable than the initial state (in the off-field condition). On the contrary, for θ > 90 , the potential energy of dipole moment would be more stable with decreasing bond length. The data are summarized in the supporting information, Table S1. Based on Equation 15, the electric field can minimize the potential energy of dipole moment via three variables. Since the nanotubes bonds are restricted in the geometry, the changes of bond



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 36

angles of nanotubes are insignificant in the presence of the electric field. Therefore, the bond length and charge are the parameters to minimize potential energy of dipole moment. To verify these results, various analyses were performed as discussed in the following sections.

3.1.2 NBO (Natural Bond Orbital) analysis The NBO analysis showed that the charge distribution on the nanotube atoms is changed when the electric field is applied. In other words, the uniform distribution of the electrical charges, which is proportional to the external electric field direction, will break up as demonstrated in Figure 4 for BN(12,0) nanotube. As shown in this figure, the uniform charge distribution of the nanotube (Figure 4b) is dependent on the position of atoms in direction of the applied field (Figure 4a). In fact, the bonds aligned along the electric field direction present the maximum changes in the charge distribution. Moreover, the constant external electric field breaks up the symmetry of the molecular orbitals, so that the highest occupied molecular orbital (HOMO) is localized at the negative source charge, while the lowest unoccupied molecular orbital (LUMO) is localized at the positive source charge. Figure 5 shows redistribution of the LUMO and HOMO of BN (12,0) nanotube along the direction of the electric field. The applied electric field induces positive and negative charges at the top and bottom of the nanotubes, respectively. Hence the energy of the orbitals near the negative source charge of the electric field shifts to the more positive values due to the induction of positive charges from electric field to the nanotubes. This is in contrast to the orbitals near the positive source charge of the electric field, in which the amount of energy becomes more negative because of the induction of negative charges from the electric filed to the nanotubes (Figure 6).


Page 13 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Based on the NBO analysis, π and π* orbitals have the most contribution in the formation of HOMO and LUMO, respectively. Therefore, the energy change of these orbitals is considered as HOMO and LUMO energy level change. The calculations of energy levels of molecular orbitals in the presence of the external electric field showed that the molecular orbitals energy is changed in such a way that the populations of HOMO in the positive direction of the electric filed and LUMO in the negative direction are increased due to induced electrical charge from the electric filed to the nanotubes. This result is shown in Figure 7 for BN(12,0) nanotube. Similar results obtained for other nanotubes are given in the supporting information (Figure S10-S14).

3.1.3. Stability of the nanotubes in the presence of the electric field As shown in Table 1, the DFT calculations indicated that, in spite of the exerting change in the configuration of the nanotubes, the external electric field leads to the most stable structures. In the other words, the delocalization energy (Edeloce) of the nanotubes becomes more negative under the influence of the electric field. According to Hellmann–Feynman theorem43 and Taylor expansion, the energy of the molecule in the presence of the electric field can be expressed as follows.

1 U =U 0 − µ y 0 E y − α yy E y2 + ... 2

(16)

Where U ,U 0 , µ y 0 , E y and α yy are the energy of the molecule in the on-field condition, energy of the molecule in the off-field condition, electric dipole moment, electric field in the ydirection and polarizability in the y-direction, respectively. Therefore, the polarizability of the nanotubes in the presence of electric field is the main factor for stability of the nanotubes, despite the structural deformations. The calculated polarizability for all nanotubes is listed in the Table 2.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Comparing the DOS spectra of the nanotubes in the absence and presence of the external electric field is deduced that the electric field lead to the break-up of the energy degeneracy as shown in Figure 8 for BN(12,0) nanotube. This phenomenon causes to increase charge transfer to higher energy level because of a decrease in the band gap. By reducing the band gap, the probability of the charge transfer is increased, resulting in the delocalization energy change. It can be concluded that in addition to polarizability, the delocalization energy due to the break-up of degeneracy of the orbitals is the other factor that provides more stability to the nanotubes under applied electric field. It should be noted that, in the case of SiC (10,0) nanotube, the main reason of decreasing the delocalization energy is related to metal-like properties of SiC(10,0) unlike SiC(6,6) and other nanotubes (Figures S5-S9 in the supporting information).

3.2. Elastic properties In this study, the effect of the transverse electric field on the mechanical properties of the nanotubes such as Young’s modulus and Poison’s ratio was investigated. The mechanical loading was applied in the absence and presence of the transverse electric field. For off-field condition, the mechanical force was applied in the direction of perpendicular to the longitudinal axis (z-axis). It should be noted that the implemented mechanical loading was applied by several incremental load steps. The results showed that the initial circular cross-section of the nanotubes are deformed to the final elliptical ones under mechanical force, in which the longer diameter of ellipse is aligned in the x-axis direction and its smaller diameter is located in the direction of the y-axis. The loading condition of the nanotubes is illustrated in Figure 9a. The energy for each step of the deformation was calculated using the single point energy by DFT method. As mentioned before, the strain was calculated by Green strain tensor. The Young’s modulus and Poisson’s ratios of the nanotubes after exposure to the mechanical loading are reported in Table


Page 14 of 36

Page 15 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


2. The results illustrated that the polarizability is inversely proportional to the Young’s modulus. This trend can be observed not only among different nanotubes but also in the same family of the nanotubes. The electric field of 0.02 a.u. was applied in the perpendicular to the nanotubes axis. Geometry optimization was carried out for the nanotubes in the presence of the transverse electric field. It was found that the nanotubes respond to the electric field, so that they are stretched in the direction of the implemented electric field. The initial circular cross-section of the nanotubes is changed to the elliptical form. Furthermore, in order to investigate the relationship between the transverse electric field and elastic properties, mechanical force was applied in the direction of perpendicular to the longitudinal axis (z-axis), aligned in the direction of electric field. In order to compare the Young’s modulus in the presence and absence of the electric field, the deformation of the nanotubes in the presence of the electric field was considered as a mechanical loading. This strategy for investigation of the Young’s modulus in the presence of the electric field is shown in Figure 9b. The obtained Young’s modulus of the nanotubes exposed to electric field is reported in Table 3. By comparing two series of the Young’s modulus, regardless of the nanotube type, the elastic modulus follows the increasing trend by applying the transverse electric field. The DFT results showed that the most stable configuration of the nanotubes in the transverse electric field is the elliptical cross-section form. The orientation of bonds provides the minimum potential energy of dipole moment of bonds under electric field. By applying mechanical force in the presence of the electric field, the conformation of the nanotubes is changed and consequently the smaller diameter is placed along the direction of the electric field and the larger diameter is placed in the perpendicular direction to the field as shown in



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 16 of 36

Figure 9b. In the presence of the electric field, new orientation of bond resulting mechanical force induces more positive potential than initial orientation of bond. The mechanical uniaxial force inclines to maximize the potential energy of the dipole moment unlike the electric filed. As a result, under transverse electric field, more force is required for the same deformation compared to the absence of the electric field. For this reason, it can be concluded that the stiffness of the nanotubes under transverse electric field follows the increasing trend, regardless of the nanotube type.

4. Conclusions In this study, the influence of the transverse electric field on the electronic and mechanical properties of six nanotubes such as C(7,7), C(12,0), BN(7,7), BN(12,0), SiC(6,6) and SiC(10,0) were investigated using density functional theory and continuum mechanics. In the presence of the transverse electric field, the obtained results showed that the circular shape of the nanotubes is deformed to the elliptical shape, in which the tube diameter along the field direction is increased whereas the diameter perpendicular to the field direction is reduced. It should be noted that the structural deformations of heteroatomic nanotubes are increased compared to other nanotubes. The NBO analysis clarified that the major reason for deformation of the nanotubes is potential energy stabilization of dipole moment of bonds. Moreover, the electric field makes the structures more stable despite the structural deformations. Based upon the results, the polarizability of the nanotubes in the presence of the electric field and delocalization energy lead to stabilization of the nanotubes. The application of the electric field redistributes the population of the HOMO and LUMO along the direction of the electric field. It should be noted that the electric field changes the distribution of the electrical charge over the nanotubes by removing degeneracy of energy levels, so that the change in the charge distribution is correspond to the


Page 17 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


change in the orientation of bonds in response to the electric field. These results are well in accordance with the DOS spectra of the nanotubes. The results of the effect of the external electric field on the mechanical properties of the nanotubes showed that by applying only an electric field without any chemical changes, the nanotubes resistance can change against external stress. The results indicated that by applying a transverse electric field, the nanotubes are stretched along the electric field direction (y-axis). When a mechanical force is applied in the perpendicular direction to the electric field in such a way that the nanotubes are more stretched along the y-axis, the Young’s modulus will be decreased. In contrast, when a mechanical force is applied along the direction of the applied field in such a way that the deformation of nanotubes is decreased, the Young’s modulus will be increased.

ASSOCIATE CONTENT Supporting Information The material includes figures of bond location and bond length variation of SiC(6,6), SiC(10,0), C(12,0), C(7,7) nanotubes, DOS spectra and energy change of π and π* orbitals of BN(7,7), SiC(6,6), SiC(10,0), C(7,7), C(12,0) nanotubes, curve fitting between energy and the electric field of C(7,7), C(12,0), BN(7,7), BN(12,0), SiC(6,6), SiC(10,0) and a table of variation of bond lengths for BN (12,0) nanotube in the presence of the electric field . This information is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Authors *E-mail: [email protected] Tel: +982177240287



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

ACKNOWLEDGMENT The authors gratefully appreciate the Iran University of Science and Technology (IUST) and Iranian Nanotechnology Initiative Council for the partial financial supports.


Page 18 of 36

Page 19 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


REFERENCES (1) Iijima S. Helical microtubules of graphitic carbon. Nature 1991, 354, 56–58. (2) Yakobson, B. I.; Avouris, P. Mechanical properties of carbon nanotubes. Appl. phys. 2001, 80, 287–327. (3) Dettlaff-Weglikowska, U.; Skakalova, V.; Graupner, R.; Jhang, S. H.; Kim, B. H.; Lee, H. J.; Ley, L.; Park, Y. W.; Berber, S.; Tomanek D.; et al. Effect of SOCl2 Treatment on Electrical and mechanical properties of single-wall carbon nanotube networks. J. Am. Chem. Soc. 2005, 127, 5125–5131 (4) Ruoff, R. S.; Qian, D.; Liu, W. K. Mechanical properties of carbon nanotubes: theoretical predictions and experimental measurements. C. R. Physique. 2003, 4, 993–1008. (5) Borders, T. L.; Fonseca, A. F.; Zhang, H.; Cho, K.; Rusinko, A. Developing descriptors to predict mechanical properties of nanotubes. J. Chem. Inf. Model. 2013, 53, 773−782. (6) Lourenço, M. P.; Oliveira, C. D.; Oliveira, A. F.; Guimaraes, L.; Duarte, H. A. Structural, electronic, and mechanical properties of single-walled chrysotile nanotube models. J. Phys. Chem. C 2012, 116, 9405−9411. (7) Reich, S.; Thomsen, C.; Ordejón, P. Elastic properties of carbon nanotubes under hydrostatic pressure. Physical Review B, 2002, 65, 153407-153411. (8) Cherian, R.; Mahadevan, P. Elastic properties of carbon nanotubes: an atomistic

approach. J. Nanosci. Nanotechnol. 2007, 7, 1779-1782. (9) Lei, X.; Natsuki, T.; Shi, J.; Ni, Q.Q. Analysis of Carbon Nanotubes on the Mechanical Properties at Atomic Scale. J. Nanomater. 2010, 2011, 1-10.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 36

(10) Cheng, H. C.; Liu, Y.-L.; Hsu, Y.-C.; Chen, H.-W. Atomistic-continuum modeling for mechanical properties of single-walled carbon nanotubes. Inter. J. Solids and Struct. 2009, 46, 1695−1704. (11) Natsuki, T.; Tantrakarn, K.; Endo, M. Effects of carbon nanotube structures on mechanical properties. Appl. Phys. A. 2004, 79, 117−124. (12) Peng, Y.-J.; Zhang, L.-Y.; Jin, Q.-H.; Li, B.-H.; Din, D.-T. Ab initio studies of elastic properties and electronic structures of C and BN nanotubes. Physica E 2006, 33, 155−159. (13) Kudin, K. N.; Scuseria, G. E.; Yakibson, B. I. C2F, BN, and C nanoshell elasticity from ab initio computations. Phys. Rev. B 2001, 64, 235406−10. (14) Zhou, R. L.; Wang, L.; Pan. B. C. Elastic and melting properties of crystalline SiC nanotubes. J. Phys. Chem. C 2010, 114, 8199–8205. (15) Wernik, J. M.; Meguid, S. A. Atomistic-based continuum modeling of the nonlinear behavior of carbon nanotubes. Acta. Mech. 2009, 212, 167−179. (16) Rafiee, R.; Maleki Moghadam, R. On the modeling of carbon nanotubes: A critical review. Composites: Part B 2014, 56, 435–449. (17) Cao, G.; Chen, X. The effects of chirality and boundary conditions on the mechanical properties of single-walled carbon nanotubes. Inter. J. Solids and Struct. 2007, 44, 5447– 5465. (18) Jin, Y.; Yuan, F. G. Simulation of elastic properties of single-walled carbon nanotubes. Composites Sci. Tech. 2003, 63, 1507−1515. (19) Wenxing, B.; ChangChun, Z.; WanZhao, C. Simulation of Young’s modulus of singlewalled carbon nanotubes by molecular dynamics. Physica B 2004, 352, 156−163. 20 ACS Paragon Plus Environment

Page 21 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(20) Sears. A.; Batra, R. C. Macroscopic properties of carbon nanotubes from molecularmechanics simulations. Phy. Rev. B. 2004, 69, 235−406. (21) Song, X.; Ge, Q.; Yen, S-C. A first-principles study on the elastic properties of singlewalled carbon nanotubes. Nano Engin. Nanosys. 2010, 223, 163−168. (22) Lu, X.; Hu, Z. Mechanical property evaluation of single-walled carbon nanotubes by finite element modeling. Composites: Part B 2012, 43, 1902–1913. (23) Gupta, S.; Dharamvir, K.; Jindal, V. K. Elastic modulus of single-walled carbon nanotubes and their ropes. Phys. Rev. B 2005, 72, 165428–16. (24) Jiang, L.; Guo, W. A Molecular mechanics study on size-dependent elastic properties of single-walled boron nitride nanotubes. J. Mechanic. Phys. Solid. 2011, 59, 1204–1213. (25) Verma, V.; Jindal, V K.; Dharamvir, K. Elastic modulus of a boron nitride nanotube. Nanotech. 2007, 18, 435711–6. (26) Santosh, M.; Maiti, P. K.; Sood, A. K. Elastic properties of boron nitride nanotubes and their comparison with carbon nanotubes. J. Nanoscience and Nanotech. 2009, 9, 1–6. (27) Zhang, A.; Gu, X.; Liu, F.; Xie, Y.; Ye, X.; Shi, W. A Study of the size-dependent elastic properties of silicon carbide nanotubes: First-principles calculations. Phys. Lett. A 2012, 376, 1631–1635. (28) Baumeier, B.; Krüger, P.; Pollmann, J. Structural, elastic, and electronic properties of SiC, BN, and BeO nanotubes. Phys. Rev. B 2007, 76, 085407–10. (29) Li-Jun, P.; Yu, J.; Qiang, S.; Xing, H. Electronic properties of boron nanotubes under uniaxial strain: a DFT study. Chin. Phys. Lett. 2011, 28, 087103–4.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 36

(30) Ju, S. P.; Wang, Y. C.; Lien, T. W. Tuning the electronic properties of boron nitride nanotube by mechanical uniaxial deformation: A DFT study. Nanoscale Res. Lett. 2011, 6, 160–170. (31) Wang, Z. G.; Li, Z.; Cheng, D. M. Effects of uniaxial strain on the band structure of boron nitride nanotubes: A first principles study. Eur. Phys. J. Appl. Phys. 2009, 46, 20601–5. (32) Lee, C. H.; Chen, R. B.; Li, T. S.; Chang, C. P.; Lin, M. F. Electronic structures of finite carbon nanotubes under external fields. J. Phys. Condens. Matter 2006, 18, 9427–9434. (33) Pacheco, M.; Barticevic, Z.; Rocha, C. G.; Latge, A. Electric field effects on the energy spectrum of carbon nanotubes. J. Phys. Condens. Matter 2005, 17, 5839–5847. (34) Roknabadi, M. R.; Koohjani, F.; Shahtahmasbi, N.; Modarresi, M.; Ghodrati, M. Estimation of the Young’s modulus of single-walled carbon nanotubes under electric field using tight-binding method. Superlattice Microst. 2013, 59, 178–186. (35) Ghassemi, H. M.; Lee, C. H.; Yap, Y. K.; Yassar, R. S. Field emission and strain engineering of electronic properties in boron nitride nanotubes. Nanotech. 2012, 23, 105702–6. (36) Dai, Y.; Guo, W.; Zhang, Z.; Zhou, B.; Tang, C. Electric-field-induced deformation in boron nitride nanotubes. Appl. Phys. 2009, 42, 085403–4. (37) Alfieri, G.; Kimoto, T. Engineering the band gap of SiC nanotubes with a Transverse electric field. Appl. Phys. Lett. 2010, 97, 043108–3. (38) Machado, M.; Azevedo, S. Stability and electronic properties of AlN nanotubes under the influence of external electric field. Eur. Phys. J. B 2011, 81, 121–125. 22 ACS Paragon Plus Environment

Page 23 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(39) Frisch, M. J.; et al. Gaussian03, Revision B. 03, Gaussian, Inc., Pittsburgh, PA, 2003. (40) Demichelis, R.; Noël, Y.; D’Arco, P.; Rérat, M; Zicovich-Wilson, C.; Dovesi, R. Properties of Carbon Nanotubes: An ab Initio Study Using Large Gaussian Basis Sets and Various DFT Functionals. J. Phys. Chem. C 2011, 115, 8876–8885 (41) Reddy, J. N. An introduction to continuum mechanics; Cambridge University Press: New York, 2008. (42) Gupta, S.S.; Bosco, F.G.; Batra, R.C. Wall thickness and elastic modulus of singlewalled carbon nanotubes from frequencies of axial, torsional and in extensional modes of vibration. Comput. Mater. Sci. 2010 , 47, 1049-1059 (43) P. W. Atkins, R. S. Friedman, Molecular Quantum Mechanics; Oxford University Press: New York, 2005.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 36

Table 1. Total energy (Etotal) and delocalization energy (EDeloc) of the nanotubes in the absence and presence of the electric field. Nanotube

Etotal(a.u), E=0.0 a.u

Edeloc (kcal/mol), E=0.0 a.u

Etotal(a.u), E=0.02 a.u

Edeloc (kcal/mol), E=0.02 a.u

C(7,7)

-5887. 12

-29.95

-5887.50

-53.38

C(12,0)

-5503.30

-77.62

-5503.77

-82.13

BN(7,7)

-6156.69

-50.25

-6156.99

-129.55

BN(12,0)

-5755.58

-52.27

-5755.86

-55.92

SiC(6,6)

-21638.45

-85.40

-21639.00

-375.64

SiC(10,0)

-19669.97

-72.07

-19670.72

-65.49


Page 25 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2. Calculated Young’s modulus and Poisson’s ratios of the nanotubes exposed mechanical loading.

Nanotube

Ey(GPa)

Ex(GPa)

ϑ xy

ϑyx

Polarizability (a.u)

C(7,7)

144.21

98.70

0.82

1.20

2003.20

C(12,0)

134.39

103.89

0.88

1.14

2363.40

BN(7,7)

154.19

140.50

0.95

1.05

1513.24

BN(12,0)

184.39

162.89

0.94

1.06

1382.16

SiC(6,6)

32.30

29.42

0.95

1.05

3108.00

SiC(10,0)

29.91

24.10

0.90

1.11

3236.20



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 26 of 36

Table 3. Calculated Young’s modulus of the nanotubes exposed to electric field Nanotube

Ey(GPa), E = 0.02a.u

Ex(GPa), E = 0.02a.u

ϑ xy

ϑyx

C(7,7)

236.09

175.75

0.87

1.17

C(12,0)

206.50

172.10

0.91

1.20

BN(7,7)

252.07

229.88

0.95

1.05

BN(12,0)

280.27

247.91

0.94

1.06

SiC(6,6)

64.79

57.67

0.94

1.06

SiC(10,0)

65.33

52.80

0.90

1.11


Page 27 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1. (a) Single wall carbon nanotube and an equivalent cylindrical tube, (b) cross-section of BN(12,0) nanotube before and after deformation (initial and final elliptical form) left to right, respectively, and (c) deformation of the BN(12,0) nanotube under transverse electric field (the dark arrow indicates the direction of the applied field).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 2. (a) Bonds location according to their Y component and (b) bond length variation in the absence (circle shape) and presence (triangular shape) of the external electric field for BN(12,0) nanotube.


Page 28 of 36

Page 29 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3. (a) Bonds location according to their Y component and (b) bond length variation in the absence (circle shape) and presence (triangular shape) of the external electric field for BN(7,7) nanotube.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 4. (a) The location of N atoms according to their Y component and (b) charge distribution of N atoms in the presence of the electric field for BN(12,0) nanotube.


Page 30 of 36

Page 31 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 5. Redistribution of the LUMO and HOMO along the direction of the electric field for BN (12,0) nanotube.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 6. The schematic shifting of the HOMO and LUMO by applying the external electric field for BN (12,0) nanotube.


Page 32 of 36

Page 33 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 7. The energy changes of π and π* orbitals in the electric field for BN (12,0) nanotube.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 8. The calculated DOS spectra of the BN(12,0) nanotube. Filled area is related to DOS before applying electric field and line is related to DOS after applying electric field.


Page 34 of 36

Page 35 of 36

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 9. (a) Deformation of BN (12,0) nanotube under loading in the direction of perpendicular to the longitudinal axis (z axis) in the absence of the electric field and (b) the obtained Young’s modulus affected by electric field for BN(12,0) nanotube.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

1761x908mm (96 x 96 DPI)

ACS Paragon Plus Environment

Page 36 of 36

Tuning of Elastic Properties of Nanotubes by Imposing a Transverse

Recommend Documents