Analytical solutions for elastic binary nanotubes of arbitrary chirality

Lai Jiang·Wanlin Guo

Analytical solutions for elastic binary nanotubes of arbitrary chirality

Lai Jiang1·Wanlin Guo1

?The Chinese Society of Theoretical and Applied Mechanics;Institute of Mechanics,Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Analytical solutions for the elastic properties of a variety of binary nanotubes with arbitrary chirality are obtained through the study of systematic molecular mechanics.This molecular mechanics model is first extended to chiral binary nanotubes by introducing an additional out of-plane inversion term into the so-called stick-spiral model, which results from the polar bonds and the buckling of binary graphitic crystals.The closed-form expressions for the longitudinal and circumferential Young’s modulus and Poisson’s ratio of chiral binary nanotubes are derived as functions of the tube diameter.The obtained inversion force constants are negative for all types of binary nanotubes,and the predicted tube stiffness is lower than that by the former stick-spiral model without consideration of the inversion term,reflecting the softening effect of the buckling on the elastic properties of binary nanotubes.The obtained properties are shown to be comparable to available density functional theory calculated results and to be chirality and size sensitive.The developed model and explicit solutions provide a systematic understanding of the mechanical performance of binary nanotubes consisting of III–V and II–VI group elements.

Binary nanotubes·Elastic properties·Molecular mechanics·Chirality·Analytical solutions

DOI 10.1007/s10409-016-0581-3

1 Introduction

Since the first binary nanotube,the hexagonal boron nitride (BN)nanotube,was successfully synthesized in the laboratory[1],explorations for the existence of binary nanotubes with different chemical compositions beyond the carbon element and their size-dependent properties have attracted increasing interest.In particular,the stability of various binary tubular structures for silicon carbide(SiC),beryllium oxide(BeO),zinc oxide(ZnO),boron phosphide (BP),aluminum nitride(AlN),gallium nitride(GaN),gallium phosphide(GaP),and indium phosphide(InP)have been investigated theoretically as well as experimentally[2–10].All these binary nanotubes have hexagonal structures and show exceptional mechanical and physical properties. Most of them are semiconductors independent of geometries,making them suitable for electronic and optoelectronic applications beyond carbon nanomaterials.In contrast to their monoatomic correspondences,binary nanotubes always appear in buckling structures due to their intrinsically polar bonds and out-of-plane deformation when wrapped up in a tubular structure at the nanoscale.Consequently,the two elements in buckled nanotubes are always located on two concentric cylindrical surfaces,resulting in distorted chemical bond structures and distinct mechanical properties.These specific chemical and structural features make binary nanotubes exceptionally sensitive to externally applied strain. However,our mechanical understanding of nanotubes has been limited mainly to monoatomic materials,such as carbon nanotubes(CNTs),and the essential elastic properties of binary nanotubes remain poorly understood,largely because of their complex chemical bond structures.Although the size-and chirality-dependent elastic properties of BN nanotubes have been investigated by numerical methods[11–16],therewas no analytical modeling for any binary nanotube until Jiang and Guo[17]extended the so-called stick-spiral model, which was first established by Chang and Gao[18]based on the molecular mechanics of CNTs,to simplest armchair and zigzag BN nanotubes.The analytical model for single-walled CNTs[18–24]has been proven effective by numerous experimental and numerical investigations,so the governing equations of the stick-spiral model established for CNTs provide us with a useful tool for exploring more complicated binary nanostructures.The extended analytical model for BN nanotubes has proven successful[17],but limited to the two ideal chiralities.Additionally,the prediction of Young’s modulus is relatively higher than the ab initio results[11–15], while the prediction of Poisson’s ratio is lower.Furthermore, many other binary nanotubes remain unexplored analytically in arbitrary chirality.It is extremely time-consuming, even unaffordable,to calculate their mechanical properties of every size and chirality by first-principles approaches.An analytical model is much preferred to cover all these kinds of binary nanotubes since they have analogous molecular and electronic structures.

In this paper,we establish governing equations for a variety of binary nanotubes fabricated in the laboratory and obtain explicit analytical expressions for the longitudinal and circumferential Young’s modulus and Poisson’s ratio of various chiralities as functions of the tube diameter.The obtained negative inversion-force constants reflect the softening effect of the buckling on the elastic properties of binary nanotubes.The solutions are proven to be efficient against available results from first-principles calculations,and the predicted tube stiffness is lower than that by the previous stick-spiral model without consideration of the inversion term.Our work provides a systematic understanding of the mechanical behaviors of binary nanotubes.

2 Stick-spiral model and governing equations for binary nanotubes

We express the total energy of chiral binary nanotubes with three terms:the bond-stretching energy Uρ,the bond angle-bending energy Uθ,and the out-of-plane inversion energy Uω,in the forms of harmonic function.The inversion term Uωis an additional term used to characterize the effects of bond polarity and buckling structures,which was neglected in previous analyses of CNTs[18,19,23]and armchair and zigzag BN nanotubes[17].The total energy is expressed as

where d aiis the bond elongation of bond i,dθjis the variance of the bond angle j,dωkis the variance of the out-of-plane inversion angle k,and Cρ,Cθ,and Cωare the force constants of bond stretching,bond-angle bending,and inversion-angle bending,respectively.An elastic stick with an axial stiffness of Cρa(bǔ)nd an infinite bending stiffness is used to model the force–stretch relationship of a chemical bond.For angle bending,we must employ two spiral springs with different angular stiffness coefficients,Cθ,aand Cθ,c,to model the twisting moments resulting from angle bending on anions and cations,respectively.

A single-walled binary nanotube can be viewed as a binary graphitic sheet rolled into a tube.Similar to the graphene, binary graphitic sheets can be rolled up in different angles, resulting in different chiralities of binary nanotubes.A chiral vector is commonly used to identify the chirality of a nanotube.The chiral vector C can be described as

where n and m are two integers called a chirality index,usually written(n,m)to represent the tube chirality.The absolute value of the chiral vector is the mean circumference of the nanotubes,

where a0is the bond length in the binary graphitic sheet.The chiral angle of the nanotube is given by

Then the chiral angle of each bond(as shown in Fig.1b)is given by

The length of the unit cell is the magnitude of the translation vector,which is defined as

and the absolute value is the length of the unit cell

Fig.1 Schematic illustration of a(10,4)single-walled binary nanotube.a Global structure.b Local structure and top view of bond structures. c Cation(light balls).d Anion(dark balls)

When a chiral binary nanotube is subjected to an external force F and an internal pressure P,we have the relationships between the external forces and the internal force field where R is the average tube radius,which can be obtained by R=C/(2π),piis the internal force contributed on bond i in the plane of inversion angle ωi,and qiis the internal force contributed on bond i perpendicular to the plane of ωi.The geometrical relations of the contribution of the internal forces on an anion–cation bond are shown in Fig.2.

To retain the equilibrium of the unit cell,the internal forces should satisfy the relations

By substituting Eqs.(10)and(11)into Eqs.(8)and(9),we can write the relation between piand qias

Fig.2 Schematic illustration of geometrical relations of internalforces p and q contributed to an anion–cation bond;cations(light balls marked C),anions(dark balls marked A)

That is,for the nanotube is subjected to an external force F,M is written as

for the nanotube is subjected to an internal pressure P,M is written as

The force equilibrium and moment equilibrium equations of the interior force field are

where aiis the bond length of bond i,χijis the torsion angle between the plane through the bond aiparallel to the nanotube axis and the plane of θj,and riis the arm of the internal force qi.The determination of the buckling angle will be discussed in the next section.

In binary nanotubes,the moment equilibriums for anions and cations should be symmetrical to maintain the continuity of the nanotube structure;thus,we introduce an effective angle-bending force constantThen the moment equilibrium written as a single expression:

The torsion angle χijis calculated by

where

and the bond angle θiis calculated by

The variation of the bond angle θican be obtained by

Then we rewrite Eq.(18)as

where

Similar to the analysis of chiral CNTs[23],we have another constraint condition of deformation

For simplicity,we rewrite Eqs.(23)and(25)as

where the coefficients are defined as

and

Since the change in bond lengths of different tube diameters can be neglected[15],we consider that ai=a0.We choose dφ3as the governing parameter;then from Eqs.(15)–(17), (26),and(27),all other variations of parameters dφi,d ai,anddωiand internal forces piand qican be written as functions of dφ3,

where

and

The axial and circumferential strains when the nanotube is subjected to the external forces can be calculated as

3 Results and discussion

3.1 Buckling

The geometry of binary nanotubes can be characterized by two concentric cylindrical tubes,with anions in the outer cylinder and cations in the inner cylinder.A radial buckling parameter β is defined to describe the buckling of binary nanotubes[11,15],i.e.

where Raand Rcare the radii of the outer(anion)and inner (cation)cylinders,respectively.For different binary nanotubes,the buckling parameter would vary owing to different ionicities of the chemical bonds.For instance,according to some numerical calculations for BN,SiC,and BeO nanotubes [11,15],the buckling parameter is independent of tube chirality.We fit from these numerical results the relationship between the buckling parameter and the tube diameter for arbitrary chirality,

where cβis the buckling factor,with a value of 0.00495 for BN,0.00950 for SiC,and 0.00582 for SiC.For all types of binary nanotubes,we find that an empirical function,

Thus the longitudinal Young’s modulus and Poisson’s ratio are obtained by

And the circumferential Young’s modulus and Poisson’s ratio are obtained by

Table 1 Fundamental parameters used in present model.All data are given by Huber and Hertzberg[25]

3.2 Force constants

The force constants Cρa(bǔ)nd Cθ,effcan be determined using experimental data or ab initio calculation results of Young’s modulus and Poisson’s ratio for planar binary graphitic sheets.Because there is no buckling in planar binary sheets, the inversion term is neglected.We derive the simple expressions of Young’s modulus and Poisson’s ratio for planar binary sheets

For example,substituting the DFT results for planar BN sheets(ES～280 GPa·nm,ν～0.211)[11–15],we obtain Cρ=614.669 nN·nm?1and Cθ,eff=15.11 nN.

An alternative approach to determining the bond-stretching force constant Cρis adopting the Universal force field(UFF) [26].The bond-stretching force constant given by the UFF is

This gives,i.e.,for BN nanotubes Cρ=594.26 nN·nm,in good agreement with our calculation using the DFT results for planar BN sheets.Cθ,effof other binary nanotubes can be determined by a fitting relation when there are no planar data of these binary graphitic crystals.We conclude that there exists a relation between Cθ,effand the effective charges and the bond length by fitting the obtained ones

The values of Cθ,eff(14.94 nN for BN,6.99 nN for ZnO, and 14.41 nN for SiC)obtained using this function are comparable with those obtained using the previously described planar approach(15.91 nN for BN,6.92 nN for ZnO,and 14.26 nN for SiC).BeO is an exception;its value is 11.28 nN,much higher than that using the planar approach(5.67nN).This is mainly due to its highly ionic bonds and the nonnegligible changes in its structure and electronic behaviors when it is rolled up from a planar sheet into a nanotube [15].In particular,a prediction of Cθ,eff=23.90 nN given by this function for CNTs,which,converted into the normal angle-bending force constant,is Cθ=1.75 nN·nm,is comparable with the value of 1.42 nN·nm in Ref.[18].Since the prediction of the bond-stretching force constant Cρfor C-C bonds by our model(537.10 nN·nm?1)is lower than that in their work(742 nN·nm?1),the relatively higher angle bending-force constant predicted by our model is reasonable.

Table 2 Calculated bond-stretching force constant,effective angle-bending force constant,inversion-force constant,and buckling factor calculated by the empirical function

Fig.3 (Color online)a Fitting buckling parameters(solid lines)for three typical binary nanotubes,BN,SiC,and BeO,from DFT results(open symbols),and buckling parameters calculated using buckling factor from empirical function Eq.(43)(dashed lines with solid symbols).b Buckling parameters for other types of binary nanotubes calculated using buckling factor from empirical function

The inversion-force constant Cωmust be determined using available data on the elastic properties of nanotubes.Young’s moduli of(5,5)nanotubes by DFT calculations[15],i.e.,272 GPa·nm for(5,5)BNNTs,162 GPa·nm for(5,5)SiCNTs, and 123 GPa·nm for(5,5)BeONTs,are substituted into Eq.(37),yielding Cω=?1.04 nN·nm for BN,Cω=?0.72 nN·nm for SiC,and Cω=?1.98 nN·nm for BeO.

It would seem to contradict the conventional force field theories that the inversion-force constants in our model have negative values,but this is reasonable if we consider the angle-bending term and the inversion term in Eq.(1)together. The value of the angle-bending force constant Cθ,effused in our model is determined from the case of planar binary graphitic sheets,and it is valid only for small variances of the bond angle.For large variances of the bond angle,Cθ,effwould no longer be constant and would change with the bond angle[26].The increased charge transfer from the inner surface to the outer surface of the binary nanotubes as the decreasing tube diameter[15]would also reduce the anglebending force constant Cθ,eff.Thus,for the binary nanotubes of small tube diameter,the real Young’s modulus is much lower than predicted by the conventional stick-spiral model, which neglects the inversion term[17],as shown in Figs.4a, 5a,and 6a.A negative inversion-force constant represents a good modification when the angle-bending force constant for planar graphitic sheets is adopted for the nanotube.Though the inversion energy term is comparatively small in terms of the total energy,the total stiffness of the nanotubes would be lowered by including the inversion-force constant Cω.

On the other hand,the inversion energy term represents the tendency for the bond configuration of anions to form a sp3configuration[1].The inversion energy is relatively small compared with the bond-stretching and angle-bending energy.Although the total energy is minimized for the current equilibrium structure,the inversion energy is far from minimized.In fact,in this case,the inversion term has a monotonic function form.Increasing the inversion angle would decrease the inversion energy,whereas decreasing the inversion angle would increase the inversion energy.In the present molecular mechanics model framework,the inversion energy is confined to the harmonic function form;thus,it is better to use a negative force constant to describe the inversion term.To describe the inversion energy more precisely,the static electric interaction form should be introduced and a completely different force field established based on this form.

For other binary nanotube types,by analyzing the relation between the previously obtained Cωand the corresponding effective charges and bond lengths,we propose that Cωhas the following expression

Fig.4 a Predicted longitudinal Young’s modulus of BN nanotubes as functions of tube diameter.The prediction of the present model with the inversion term(solid lines with solid symbols)and without an inversion term(dashed lines with solid symbols)are compared with the DFT calculation results[15](discrete open symbols).b Longitudinal Young’s modulus of BN nanotubes in various chiralities as functions of tube diameter

Fig.5 a Predicted longitudinal Young’s modulus of SiC nanotubes as functions of tube diameter.The prediction of the present model with the inversion term(solid lines with solid symbols)and without an inversion term(dashed lines with solid symbols)are compared with the DFT calculation results[15](discrete open symbols).b Longitudinal Young’s modulus of SiC nanotubes in various chiralities as functions of tube diameter

The calculated force constants for various binary nanotubes are listed in Table 2.

3.3 Elastic properties

The longitudinal Young’s modulus and Poisson’s ratio of binary nanotubes are calculated using Eqs.(37)and(38). The results of the calculated Young’s modulus of BN,SiC, and BeO nanotubes are compared with the available DFT calculation results[15]shown in Figs.4–6.Meanwhile,we also calculated the Young’s modulus using the model excluding the inversion term for comparison.It is shown that the predictions of the Young’s modulus of various binary nanotubes are all in good agreement with the DFT results,while the predictions given by the model excluding the inversion term are relatively higher.This indicates that binary nanotubes are softened and their stiffness reduced by buckling,and the introduction of the inversion term is a good modification of the stick-spiral model.

Fig.6 (a)Predicted longitudinal Young’s modulus of BeO nanotubes as functions of tube diameter.The prediction of the present model with the inversion term(solid lines with solid symbols)and without an inversion term(dashed lines with solid symbols)are compared with the DFT calculation results[15](discrete open symbols).b Longitudinal Young’s modulus of BeO nanotubes in various chiralities as functions of tube diameter

Fig.7 Predicted longitudinal Young’s modulus.a ZnO nanotubes.b Other III–V group binary nanotubes as functions of tube diameter

The longitudinal Young’s modulus and Poisson’s ratio of binary nanotubes in different chiralities are calculated and shown in Figs.4–8.All types of binary nanotubes show the same progression with that of BN nanotubes with increasing tube diameter since they have geometrical structures similar to those of BN nanotubes.The Young’s modulus increases with an increasing tube diameter,whereas the Poisson’s ratio decreases.For the same diameter,the smaller the tube chiral angle,the smaller the Young’s modulus and the larger Poisson’s ratio are.This is due to the larger distortion of the bond angles in the nanotubes with small chiral angles.It indicates that the elastic properties of binary nanotubes are size and chirality dependent.

In particular,our model also applies to CNTs.Since there is no buckling in CNTs,the inversion angle ωiis set to zero. According to Eq.(16),the variance of the inversion angle is zero,so the out-of-plane inversion energy Uωin Eq.(1) vanishes for CNTs.Then the expressions of the Young’s modulus and Poisson’s ratio evolve into those in Ref.23. The force constants Cρa(bǔ)nd Cθ,effare determined using the experimental data of the Young’s modulus and Poisson’s ratio(ES～360 GPa·nm,ν～0.16)for planar graphene sheets,as mentioned earlier.The predictions of the longitudinal Young’s modulus and Poisson’s ratio are shown in Fig.9, and the result is converged to that of the stick-spiral model for CNTs[23].

The circumferential Young’s modulus and Poisson’s ratio of binary nanotubes are calculated using Eqs.(39)and(40). We find that the predicted circumferential Young’s modulus and Poisson’s ratio of all binary nanotubes are identical with the longitudinal Young’s modulus and Poisson’s ratio.The circumferential Young’s modulus and Poisson’s ratio of BNnanotubes,for example,are shown in Fig.10.This indicates that the in-plane elastic properties of binary nanotubes are isotropic.

Fig.8 Predicted longitudinal Poisson’s ratio of various binary nanotubes as functions of tube diameter.a BNNTs.b SiCNTs,c BeONTs.d Other binary nanotubes

Fig.9 Predicted longitudinal.a Young’s modulus.b Poisson’s ratio of CNTs as functions of tube diameter

Fig.10 Predicted circumferential.a Young’s modulus.b Poisson’s ratio of BN nanotubes as functions of tube diameter

4 Concluding remarks

We have developed a set of molecular mechanics solutions for the chirality-and size-dependent elastic properties of single walled binary nanotubes by modifying the stick-spiral model. An out-of-plane inversion term is introduced to characterize the effects of bond polarity and buckling structures on the elastic properties of binary nanotubes.Parameters and analytical solutions were obtained for all binary nanotubes synthesized in the laboratory.The surface elastic properties,i.e.,longitudinal and circumferential Young’s modulus and Poisson’s ratio,are explicitly presented as functions of the tube chirality and diameter.The obtained inversion-force constants are negative for all types of binary nanotubes,leading to a stiffness of the nanotubes that is lower than that obtained using our former stick-spiral model without considering the inversion term.For different binary nanotubes, the force constants are very sensitive to the bond length and the effective charges,leading to a significant dependence of elastic properties on the elements constituting the nanotubes.For all kinds of binary nanotubes,the Young’s modulus increases with increasing tube diameter,while the Poisson’s ratio decreases,in all chiralities.The predictions are in good agreement with available numerical calculation results.

Acknowledgments This work was supported by the 973 Program (Grants 2013CB932604,2012CB933403),a project funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions,and Jiangsu Innovation Program for Graduate Education (Grant CXZZ12_0140).

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? Wanlin Guo wlguo@nuaa.edu.cn

1State Key Laboratory of Mechanics and Control for Mechanical Structures and Key Laboratory for Intelligent Nano Materials and Devices(MOE),Nanjing University of Aeronautics and Astronautics,Nanjing 210016,China

28 January 2016/Revised:1 March 2016/Accepted:21 April 2016/Published online:8 September 2016