Modelling of Nickel Atoms Interacting with Single-Walled...

Research ArticleModelling of Nickel Atoms Interacting withSingle-Walled Nanotubes

Mansoor H Alshehri

Mathematics Department College of Science King Saud University PO Box 2455 Riyadh 11451 Saudi Arabia

Correspondence should be addressed to Mansoor H Alshehri mhalshehriksuedusa

Received 19 December 2018 Accepted 7 February 2019 Published 3 March 2019

Academic Editor Nicola L Rizzi

Copyright copy 2019 MansoorHAlshehriThis is an open access article distributed under theCreativeCommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Herein the encapsulation mechanism of nickel atoms into carbon and boron nitride nanotubes is investigated to determine theinteraction energies between the nickel atomand a nanotubeClassical modelling procedures together with the 6-12 Lennard-Jonespotential function and the hybrid discrete-continuous approach are used to calculate the interaction of a nickel atoms with (119894 119894)armchair and (119894 0) zigzag single-walled nanotubes Analytical expressions for the interaction energies are obtained to determinethe optimal radii of the tubes to enclose the nickel atom by determining the radii that give the minimum interaction energies Wefirst investigate the suction energy of the nickel atom entering the nanotube The atom is assumed to be placed on the axis andnear an open end of a semi-infinite single-walled nanotube Moreover the equilibrium offset positions of the nickel atoms arefound with reference to the cross-section of the nanotubes The results may further the understanding of the encapsulation of Niatoms inside defective nanotubes Furthermore the results may also aid in the design of nanotube-basedmaterials and increase theunderstanding of their nanomagnetic applications and potential uses in other areas of nanotechnology

1 Introduction

Thediscovery of nanotubes such as carbon nanotubes (CNTs)and boron nitride nanotubes (BNNTs) has had an enormousimpact on the study of many technological materials andnanomaterials [1ndash4] CNTs comprise entirely carbon (C)atoms while BNNTs contain an equal number of nitrogen(N) and boron (B) atoms These nanotubes have a one-dimensional tubular structure Their geometric configura-tions can be described by a chiral vector (119894 119895) where 119894 and119895 are a pair of integers known as the chiral vector numbers[5ndash8] The nanotube is called an armchair nanotube when119894 = 119895 a zigzag nanotube when 119895 = 0 and a chiralnanotube when 119894 = 119895 CNT and BNNT structures canbe single-walled or multiwalled [4 5 7 8] Owing to theirgeometries and attractive properties such as resistance tooxidation high absorption energies for somemolecules lightweight mechanical stiffness and high electrical conductanceCNTs and BNNTs have attracted attention in the last twodecades as promising materials for many potential appli-cations including electrochemical actuation field emissionand transistors [5 9ndash13] Although CNTs and BNNT have

structural similarities the electronic structures of BNNT areexpected to be rather different from those of CNTs Forexample the distribution of charge is asymmetric in B-Nbonds in BNNTs as compared to the C-C bonds in CNTswhich might create a significant difference between the twotypes of nanotubes [14] The functionalization of molecularencapsulation inside nanotubes has given rise to a widerange of fundamental applications in areas of electronicsoptics biomechanics thermodynamics biochemistry andmagnetics [13] The encapsulation of molecules inside nan-otubes could be utilized to distinguish metallic CNTs fromsemiconducting CNTs and to increase CNT solubility inorganic media [15] Furthermore one of the most impor-tant biomedical applications of nanotubes is their use asa delivery system for platinum drugs [16ndash18] In additionthe combination of metal atoms and nanotubes is idealfor fabricating novel one-dimensional nanostructures withvarious arrangements and filling nanotubes with magneticelements like Ni can make them potentially applicable for thestorage of magnetic data [19 20] Piskunov et al [20] usedfirst-principles calculations to investigate the encapsulationof nickel atoms into single-walled carbon nanotubes They

HindawiAdvances in Mathematical PhysicsVolume 2019 Article ID 8240587 7 pageshttpsdoiorg10115520198240587

2 Advances in Mathematical Physics

found that the most preferable chiral vector numbers forNi atom insertion into single-walled carbon nanotubes are(55) and (100) Yang et al [21] investigated the interactionof Ni atoms with (55) armchair and (100) zigzag single-walled carbon nanotubes using density functional theorycalculations Their results showed that the energies for Nibinding to CNTs and the structures of CNTs are affected bythe intrinsic defects which include double vacancies singlevacancies and Stone-Wales defects In addition Zhang etal [22] used first-principles calculations to investigate theadsorption and catalytic behaviour of nickel atoms inside(55) (66) and (77) BNNTsThey showed thatNi-BNNTs arepromising low-temperature catalysts for CO oxidation Theencapsulation of various metal nanostructures inside CNTsand BNNTs has been successfully demonstrated by differentgroups using experiments and molecular dynamics simu-lations (see for example [20 23ndash26]) but very few worksin this field have used conventional applied mathematicalmodelling Mathematical modelling together with efforts ofexperimentalists and molecular dynamics (MD) simulationscan provide a deeper understanding of the interactionsbetween molecules In this paper we consider nickel atomsencapsulated inside single-walled armchair (119894 119894) and zigzag(119894 0) single-walled CNTs and BNNTs We use the Lennard-Jones potential function together with the hybrid discrete-continuous approach to determine the optimal nanotube sizeto encapsulate Ni atoms

2 Potential Energy

The van der Waals interaction energy between two atoms ata distance 119903 apart is derived from the classical Lennard-Jonespotential function Φ(119903) which is given by

Φ(119903) = minus1198601199036 + 11986111990312 (1)

where 119860 and 119861 are the attractive and repulsive constantsrespectively The interaction energy between two nonbondedmolecular structures can be obtained as a summation for eachatomic pair namely

119864 = sum119901

sum119902

Φ(119903119901119902) (2)

whereΦ(119903119901119902) is a potential function for atoms 119901 and 119902 that areseparated by a distance 119903119901119902 on two distinct molecular struc-tures Using the continuous approach (atoms are assumedto be uniformly distributed over the entire surfaces of themolecules) the double summation in (2) is replaced by thefollowing double-surface integral for each molecule

119864 = 12057811205782 int1198721

int1198722

Φ (119903) 1198891198721 1198891198722 (3)

where 1205781 and 1205782 are the mean atomic surface densities ofthe first and the second molecule respectively and 119903 is thedistance between the two typical surface elements 1198891198721 and1198891198722 Note that the mean atomic surface density can becalculated by dividing the number of atoms which makeup the molecule by the surface area of the molecule (ie

Table 1 Numerical values of the attractive and repulsive constants

Interaction A (eV A6) B (eV A12)C-Ni 4797 3344917BN-Ni 107947 7666393

120578 = number of the atomssurface area of molecule) Inthis paper we use the hybrid discrete-continuous approach todetermine the interaction energyThe hybrid model which isintroduced by elements of both (2) and (3) is given by

119864 = sum119896

120578intΦ (119903119896) 119889119872 (4)

where Φ(119903119896) is the potential function 120578 denotes the surfacedensity of atoms on the molecule which is consideredcontinuous and 119903 is the distance between an atom 119896 in themolecule which is modelled as a summation over each of itsatoms and a typical surface element 119889119872 on the continuouslymodelledmoleculeThe values of the vanderWaals diametersand well depths given in [27] are used here and we note thatthese values have been widely used in many research areasWe find the attractive (A) and repulsive (B) constants forthe interaction between different types of atoms 119898 and 119899as shown in Table 1 utilizing the van der Waals diameters120590 and well-depths 120576 from Rappi et al [27] for each atomtype We then used the empirical combining rules 120590119898119899 =(120590119898 + 120590119899)2 and 120576119898119899 = radic120576119898120576119899 where 119860 = 41205761205906 and119861 = 412057612059012 In addition the surface density of atoms oncarbon and boron nitride nanotubes is assumed to be thesame as those of their parent materials namely graphene andhexagonal boron nitride The surface density of atoms on asheet which comprises a tessellation of hexagonal rings can beeasily calculated using the relation 120578 = 4radic399848582 where 984858 =142 A and 984858 = 145 A for lengths of one carbonndashcarbon andboronndashnitride bonds respectively thus the atomic surfacedensities of carbon and boron nitride nanotubes are 120578119888 =03821 Aminus2 and 120578119887119899 = 03682 Aminus2 respectively Moreover thenanotube radii can be calculated from 119886 = 120572radic1198942 + 1198952 + 1198941198952120587where (119894 119895) denotes the chiral vector and 120572 = radic3984858 A3 Suction Energy of Ni Atom EnteringSingle-Walled Nanotube

In this section we consider the energy of a nickel atom as itenters single-walled carbon and boron nitride nanotubes Inthis case the nanotube is assumed to be a semi-infinite cylin-der without taking into account the effects of the other endand the cylinder length Thus we only need to consider theshort range nature of the van derWaals forcesTheNi atom isassumed to have the coordinates (0 0 119885) and the nanotube ofradius 119886 is given parametrically by (119886 cos 120601 119886 sin 120601 119911) whereminus120587 le 120601 le 120587 and minusinfin lt 119911 lt infin as shown in Figure 1 Inthis configuration the distance 119903 between the Ni atom and atypical point on the tube is given by

1199032 = 1198862cos2 120601 + 1198862sin2 120601 + (119911 minus 119885)2 = 1198862 + (119911 minus 119885)2 (5)

Advances in Mathematical Physics 3

a

x

r

z

Z

Figure 1 Schematic illustration of a Ni atom entering a single-walled nanotube

Therefore the total interaction energy of the Ni entering thetube is given by

119864 = 119886120578119905 int120587minus120587

intinfin0

(minus1198601199036 + 11986111990312 ) 119889119911 119889120601= 119886120578119905 (minus1198601198673 + 1198611198676)

(6)

where 120578119905 is the mean atomic surface density of the nanotube(CNT or BNNT) We evaluate the integral 119867119899 (119899 = 3 6) asfollows

119867119899 = int120587minus120587

intinfin0

1[1198862 + (119911 minus 119885)2]119899 119889119911 119889120601 (7)

The integral over 120601 is trivial and therefore

119867119899 = 2120587intinfin0

1[1198862 + (119911 minus 119885)2]119899 119889119911 (8)

since the integrand is independent of 120601 The integral 119867119899 isevaluated over 119911 by making the variable change 119904 = 119911 minus 119885and we obtain

119867119899 = 2120587intinfinminus119885

1(1198862 + 1199042)119899 119889119904 (9)

Upon making the substitution 119904 = 119886 tan 120593 the integral 119867119899becomes

119867119899 = 21205871198861minus2119899 int1205872120594

sec2minus2119899120593119889120593= 21205871198861minus2119899 int1205872

120594cos2119899minus2120593119889120593

(10)

where 120594 = tanminus1(minus119885119886) The above integral can be evaluatedusing the formula from [28] (∮ 2512(2))int1205872120594

cos2119899minus2120593119889120593 = [ sin 1205932119899 minus 2 (cos(2119899minus3)120593

+ 119899minus2sum119896=1

((2119899 minus 3) (2119899 minus 5) sdot sdot sdot (2119899 minus 2119896 minus 1) cos2119899minus2119896minus31205932119896 (119899 minus 2) (119899 minus 3) sdot sdot sdot (119899 minus 1 minus 119896) ) + (2119899 minus 3) 1205932119899minus1 (119899 minus 1))]

1205872

120594

(11)

where sin 120594 = minus119885(1198862 + 1198852)minus12 and thus cos 120594 = 119886(1198862 +1198852)minus12 For the values of 119899 = 3 and 6 the integral 119867119899 can beexpanded and evaluated to yield

1198673 = 120587119886minus5 [31205878 + 34 tanminus1 (119885119886 ) + 31198851198864 (1198862 + 1198852)+ 11988511988632 (1198862 + 1198852)2]

(12)

and

1198676 = 120587119886minus11 [ 91205873840 + 3640 tanminus1 (119885119886 ) + 11988511988695 (1198862 + 1198852)5

+ 9119885119886740 (1198862 + 1198852)4 +

7119885119886560 (1198862 + 1198852)3 +

119885119886316 (1198862 + 1198852)2

+ 311988511988680 (1198862 + 1198852)]

(13)

31 Results and Discussion Using the aforementioned valuesof constants and the algebraic computer package MAPLEthe numerical solutions of the interaction energies of the Niatom entering various carbon and boron nitride nanotubesare obtained As shown in Figure 2 the results indicate thatthe Ni atom is accepted for all nanotubes with different radiiand the energies are more negative inside the tube (positive119885) than outside the tube (negative119885) Table 2 summarizes theresults of the interaction energies between Ni atom and CNTsand BNNTs Figure 2 and Table 2 show that the interactionenergy is strongly dependent on the radius 119886 of the nanotubeWe observe that the nanotubes with radii bigger than asymp 2A are accepted the Ni atom for both CNTs and BNNTs andthe values of interaction energy of Ni atom with nanotubeshave been found increasing with radius of the nanotubegrowth Also by minimizing the energy the results predictthat the optimal of the values of the radius to enclose theNi atom are 20344 A and 2072 A for CNT and BNNTcorresponding (3 3) armchair nanotube and 2741 A and2790 A for CNT and BNNT corresponding (7 0) zigzagnanotube respectively

4 An Offset Ni Atom Insidea Single-Walled Nanotube

In this section we study the interaction of the Ni atomsituated inside a carbon and boron nitride nanotube Wecalculate the potential energy of the interaction between theNi atom and the nanotube where the Ni atom is assumedto be located at a distance 120575 away from the tube axis Todetermine the preferred position of a Ni atom inside a CNTor BNNT Ni is assumed to be located at (120575 0 0) insideaxially symmetric cylindrical polar coordinates of radius 119886as shown in Figure 3 and the nanotubes are assumed toextend infinitely with coordinates (119886 cos120601 119886 sin 120601 119911) where


minus15 minus10 minus5 0 5 10 15

Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

minus05

(33) (44) (55) (66)

(a) Armchair CNTs (119894 119894)


Z Distance (A)

Inte

ract

ion

E (e

V)

minus005

minus010

minus015

minus020

(70) (80) (90) (100) (110)

(b) Zigzag CNTs (119894 0)


Z Distance (A)

Inte

ract

ion

E (e

V)

minus02

minus04

minus06

minus08

minus1

(33) (44) (55) (66)

(c) Armchair BNNTs (119894 119894)


Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

(70) (80) (90) (100) (110)

(d) Zigzag BNNTs (119894 0)

Figure 2 Typical interaction energies for a Ni atom entering nanotubes with different radii 119886

minus120587 le 120601 le 120587 and minusinfin lt 119911 lt infin Thus the distance 119903 from theNi atom to the wall of the nanotube is given by

1199032 = 1198862 minus 2119886120575 cos 120601 + 1205752 + 1199112= (119886 minus 120575)2 + 4119886120575 sin2 (1206012) + 1199112 (14)

and the total potential energy of the interaction of theNi atominside the nanotube is obtained by

119864 = 119886120578119905 int120587minus120587

intinfinminusinfin

(minus1198601199036 + 11986111990312 ) 119889119911 119889120601= 120578119905 (minus1198601198703 + 1198611198706)

(15)

Now we evaluate the integral 119870119899 (119899 = 3 6) as follows119870119899= 119886int120587minus120587

intinfinminusinfin

1[(119886 minus 120575)2 + 4119886120575 sin2 (1206012) + 1199112]119899 119889119911 119889120601

(16)


Table 2 Main results of the interaction of a Ni atom entering nanotubes

Tube Type Tube Radius (A) Interaction (eV)CNT BNNT CNTNi BNNTNi

(33) 20344 2072 -0554 -1145(44) 2713 2761 -0246 -0518(55) 3391 3451 -0102 -0214(66) 4069 4142 -0049 -0103(70) 2741 2790 -0237 -0497(80) 3132 3188 -0139 -0293(90) 3523 3587 -0087 -0183(100) 3915 3985 -0057 -0120(110) 4307 4384 -0039 -0082

a

x

r

z

Figure 3 Schematic illustration of an offset Ni atom inside a single-walled nanotube

Webegin by defining1205962 = (119886minus120575)2+4119886120575 sin2(1206012) andmakingthe substitution 119911 = 120596 tan120595 which gives

119870119899 = 119886int1205872minus1205872

cos2119899minus2 120595119889120595int120587minus120587

11205962119899minus1119889120601 (17)

Thefirst integral can be evaluated using the beta function andthus 119870 becomes

119870119899 = 119886B(119899 minus 12 12)int120587

minus120587

11205962119899minus1119889120601 (18)

where B(119901lowast 119902lowast) denotes the beta function To evaluate thesecond integral of119870 the use of 120582 = sin2(1206012) yields

119870119899 = 2119886(119886 minus 120575)2119899minus1B(119899 minus 12 12)times int10120582minus12 (1 minus 120582)minus12 (1 minus 120603120582)12minus119899 119889120582

(19)

where 120603 = minus4119886120575(119886 minus 120575)2 In the Euler form the integral isgiven by

119870119899 = 2120587119886(119886 minus 120575)2119899minus1B (119899 minus 12 12)times 119865(119899 minus 12 12 12 1 120603)

(20)

where 119865(119886lowast 119887lowast 119888lowast 119911lowast) denotes the usual hypergeometricfunction Since 119888lowast = 2119887lowast we may use the quadratictransformation from [29] which gives

119870119899 = 21205871198862119899minus2B (119899 minus 12 12)times 119865(119899 minus 12 119899 minus 12 12 1 120575

2

1198862) (21)

41 Results and Discussion By utilizing the aforementionedconstant values together with the algebraic computer pack-age MAPLE we evaluated (15) to determine the preferredposition of a Ni atom inside (5 5) (6 6) (10 0) and (11 0)carbon and boron nitride nanotubes By minimizing the totalpotential energy of the Ni atom inside CNTs and BNNTsFigure 4 shows that the preferred location of the Ni atominside (5 5) for both CNT and BNNT is on the tube axis (ie120575 = 0) In addition for (6 6) (10 0) and (11 0) nanotubeswe find that for the case of CNT 120575 = 0952 A 120575 = 0756A and 120575 = 1230 A respectively For the case of BNNT120575 = 1001 A 120575 = 0804 A and 120575 = 1280 A respectivelyThese results indicate that as the radius of the tube increasesthe location where the minimum energy occurs tends to becloser to the nanotube wall

5 Conclusions

This study investigated the interaction between nickel atomsand carbon and boron nitride nanotubes using the 6-12Lennard-Jones potential function and the hybrid discrete-continuous approach to determine the preferred radii of thenanotubes for encapsulating the nickel atoms We observedthat the encapsulation of Ni atoms inside both CNTs andBNNTs depended strongly on the nanotube radius In addi-tion we used the algebraic computer package MAPLE tonumerically evaluate the interaction energies between a Niatom and the nanotubes The results indicated that theencapsulation of the Ni atom into a single-walled CNTs andBNNTs nanotubes may occur for nanotubes with greaterthan 2 A Our results are in a good agreement with othersworks [20 22] Moreover these results indicate that theinteraction between BNNTs and Ni atom is stronger than


(55) (66) (100) (110)

0

05 1 15 2

offset from tube axis (A)

Inte

ract

ion

E (e

V)

minus005

005

010

minus010

(a) CNTs

(55) (66) (100) (110)

05 1 15 2


0

Inte

ract

ion

E (e

V)

minus01

01

minus02

(b) BNNTs

Figure 4 Total potential energy of an offset Ni atom inside CNTs and BNNTs with respect to the radial distance 120575

the interaction between CNTs and Ni atom since the formershowed the lowest minimum energy The results of thisstudy may stimulate further studies of other metal atominteractions inside nanotubes

Data Availability

No data have been used to support this study

Conflicts of Interest

The author declares that they have no conflicts of interest

Acknowledgments

This project was supported by King Saud University Dean-ship of Scientific Research College of Science ResearchCenter

References

[1] A Rubio J L Corkill and M L Cohen ldquoTheory of graphiticboron nitride nanotubesrdquo Physical Review B vol 49 no 7 pp5081ndash5084 1994

[2] X Blase A Rubio S G Louie and M L Cohen ldquoStability andband gap constancy of boron-nitride nanotubesrdquo EPL vol 28pp 335ndash340 1994

[3] S Iijima and T Ichihashi ldquoSingle-shell carbon nanotubes of 1-nm diameterrdquo Nature vol 363 pp 603ndash605 1993

[4] S Iijima ldquoHelicalmicrotubules of graphitic carbonrdquoNature vol354 pp 56ndash58 1991

[5] M S Dresselhaus G Dresselhaus and P Avouris Carbon Nan-otubes Advanced Topics in the Synthesis Structure Propertiesand Applications vol 80 Springer Berlin Germany 2001

[6] B J Cox and J M Hill ldquoGeometric polyhedral models for nan-otubes comprising hexagonal latticesrdquo Journal of Computationaland Applied Mathematics vol 235 no 13 pp 3943ndash3952 2011

[7] C Zhi Y Bando C Tang and D Golberg ldquoBoron nitridenanotubesrdquo Materials Science and Engineering R Reports vol70 no 3ndash6 pp 92ndash111 2010

[8] A Nigues A Siria P Vincent P Poncharal and L BocquetldquoUltrahigh interlayer friction in multiwalled boron nitridenanotubesrdquoNature Materials vol 13 pp 688ndash693 2014

[9] G Mpourmpakis and G E Froudakis ldquoWhy boron nitridenanotubes are preferable to carbon nanotubes for hydrogenstorage an ab initio theoretical studyrdquoCatalysis Today vol 120no 3-4 pp 341ndash345 2007

[10] Y Chen J Zou S J Campbell and G L Caer ldquoBoron nitridenanotubes pronounced resistance to oxidationrdquoApplied PhysicsLetters vol 84 no 13 pp 2430ndash2432 2004

[11] M Ishigami S Aloni and A Zettl ldquoProperties of boron nitridenanotubesrdquo AIP Conference Proceedings vol 696 pp 94ndash992003

[12] D Hongjie ldquoCarbon nanotubes synthesis integration andpropertiesrdquo Accounts of Chemical Research vol 35 Article ID10351044 pp 1035ndash1044 2002

[13] D Cui ldquoBiomolecules functionalized carbon nanotubes andtheir applicationsrdquo inMedicinal Chemistry and PharmacologicalPotential of Fullerenes and Carbon Nanotubes vol 1 of CarbonMaterials Chemistry and Physics pp 181ndash221 Springer Nether-lands Dordrecht Netherlands 2008

[14] M L Cohen and A Zettl ldquoThe physics of boron nitridenanotubesrdquo Physics Today vol 63 no 11 pp 34ndash38 2010

[15] J Li J KoehneH Chen et al ldquoCarbon nanotubenanoelectrodearray for ultrasensitive DNA detectionrdquoNano Letters vol 3 no5 pp 597ndash602 2003

[16] H Hillebrenner F Buyukserin J D Stewart and C R MartinldquoTemplate synthesized nanotubes for biomedical delivery appli-cationsrdquo Nanomedicine vol 1 no 1 pp 39ndash50 2006


[17] Z Khatti and S M Hashemianzadeh ldquoBoron nitride nanotubeas a delivery system for platinumdrugs drug encapsulation anddiffusion coefficient predictionrdquo European Journal of Pharma-ceutical Sciences vol 88 pp 291ndash297 2016

[18] C R Martin and P Kohli ldquoThe emerging field of nanotubebiotechnologyrdquoNature Reviews Drug Discovery vol 2 no 1 pp29ndash37 2003

[19] P Roy S Berger and P Schmuki ldquoTiO2 nanotubes synthesisand applicationsrdquo Angewandte Chemie International Editionvol 50 no 13 pp 2904ndash2939 2011

[20] S Piskunov J Kazerovskis Y F Zhukovskii PNDyachkov andS Bellucci ldquoPhysics chemistry and application of nanostruc-turesrdquo in Proceedings of the International Conference Nanomeet-ing vol 80 pp 139ndash142 World Scientific Publishing Co PteLtd Minsk Belarus 2013

[21] S H Yang W H Shin J W Lee S Y Kim S I Woo and JK Kang ldquoInteraction of a transition metal atom with intrinsicdefects in single-walled carbon nanotubesrdquo Journal of PhysicalChemistry B vol 110 no 28 pp 13941ndash13946 2006

[22] Y Zhang Y Liu Z Meng et al ldquoConfinement boosts COoxidation on an Ni atom embedded inside boron nitridenanotubesrdquo Physical Chemistry Chemical Physics vol 20 pp17599ndash17605 2018

[23] D Golberg F Xu and Y Bando ldquoFilling boron nitride nan-otubes with metalsrdquo Applied Physics A vol 76 pp 479ndash4852003

[24] C Tang Y Bando D Golberg X Ding and S Qi ldquoBoronnitride nanotubes filled with Ni and NiSi2 nanowires in siturdquoJournal of Physical Chemistry B vol 107 no 27 pp 6539ndash65432003

[25] T Oku N Koi I Narita K Suganuma and M NishijimaldquoFormation and atomic structures of boron nitride nanotubeswith cup-stacked and Fe nanowire encapsulated structuresrdquoMaterials Transactions vol 48 no 4 pp 722ndash729 2007

[26] L Jin X Zhao X Qian and M Dong ldquoNickel nanoparticlesencapsulated in porous carbon and carbon nanotube hybridsfrom bimetallic metal-organic-frameworks for highly efficientadsorption of dyesrdquo Journal of Colloid and Interface Science vol509 pp 245ndash253 2018

[27] A K Rappi C J Casewit K S Colwell W A Goddard IIIW M Skid and J Am ldquoUFF a full periodic table force fieldfor molecularmechanics andmolecular dynamics simulationsrdquoJournal of the American Chemical Society vol 114 no 25 pp10024ndash10035 1992

[28] I S Gradshteyn and I M Ryzhik Table of Integrals Series andProducts Academic Press San Diego Calif USA 6th edition2000

[29] A Erdelyi W Magnus F Oberhettinger and F G TricomiHigher Transcendental Functions vol 1 McGraw-Hill NewYork NY USA 1953

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found that the most preferable chiral vector numbers forNi atom insertion into single-walled carbon nanotubes are(55) and (100) Yang et al [21] investigated the interactionof Ni atoms with (55) armchair and (100) zigzag single-walled carbon nanotubes using density functional theorycalculations Their results showed that the energies for Nibinding to CNTs and the structures of CNTs are affected bythe intrinsic defects which include double vacancies singlevacancies and Stone-Wales defects In addition Zhang etal [22] used first-principles calculations to investigate theadsorption and catalytic behaviour of nickel atoms inside(55) (66) and (77) BNNTsThey showed thatNi-BNNTs arepromising low-temperature catalysts for CO oxidation Theencapsulation of various metal nanostructures inside CNTsand BNNTs has been successfully demonstrated by differentgroups using experiments and molecular dynamics simu-lations (see for example [20 23ndash26]) but very few worksin this field have used conventional applied mathematicalmodelling Mathematical modelling together with efforts ofexperimentalists and molecular dynamics (MD) simulationscan provide a deeper understanding of the interactionsbetween molecules In this paper we consider nickel atomsencapsulated inside single-walled armchair (119894 119894) and zigzag(119894 0) single-walled CNTs and BNNTs We use the Lennard-Jones potential function together with the hybrid discrete-continuous approach to determine the optimal nanotube sizeto encapsulate Ni atoms

2 Potential Energy

The van der Waals interaction energy between two atoms ata distance 119903 apart is derived from the classical Lennard-Jonespotential function Φ(119903) which is given by

Φ(119903) = minus1198601199036 + 11986111990312 (1)

where 119860 and 119861 are the attractive and repulsive constantsrespectively The interaction energy between two nonbondedmolecular structures can be obtained as a summation for eachatomic pair namely

119864 = sum119901

sum119902

Φ(119903119901119902) (2)

whereΦ(119903119901119902) is a potential function for atoms 119901 and 119902 that areseparated by a distance 119903119901119902 on two distinct molecular struc-tures Using the continuous approach (atoms are assumedto be uniformly distributed over the entire surfaces of themolecules) the double summation in (2) is replaced by thefollowing double-surface integral for each molecule

119864 = 12057811205782 int1198721

int1198722

Φ (119903) 1198891198721 1198891198722 (3)

where 1205781 and 1205782 are the mean atomic surface densities ofthe first and the second molecule respectively and 119903 is thedistance between the two typical surface elements 1198891198721 and1198891198722 Note that the mean atomic surface density can becalculated by dividing the number of atoms which makeup the molecule by the surface area of the molecule (ie

Table 1 Numerical values of the attractive and repulsive constants

Interaction A (eV A6) B (eV A12)C-Ni 4797 3344917BN-Ni 107947 7666393

120578 = number of the atomssurface area of molecule) Inthis paper we use the hybrid discrete-continuous approach todetermine the interaction energyThe hybrid model which isintroduced by elements of both (2) and (3) is given by

119864 = sum119896

120578intΦ (119903119896) 119889119872 (4)

where Φ(119903119896) is the potential function 120578 denotes the surfacedensity of atoms on the molecule which is consideredcontinuous and 119903 is the distance between an atom 119896 in themolecule which is modelled as a summation over each of itsatoms and a typical surface element 119889119872 on the continuouslymodelledmoleculeThe values of the vanderWaals diametersand well depths given in [27] are used here and we note thatthese values have been widely used in many research areasWe find the attractive (A) and repulsive (B) constants forthe interaction between different types of atoms 119898 and 119899as shown in Table 1 utilizing the van der Waals diameters120590 and well-depths 120576 from Rappi et al [27] for each atomtype We then used the empirical combining rules 120590119898119899 =(120590119898 + 120590119899)2 and 120576119898119899 = radic120576119898120576119899 where 119860 = 41205761205906 and119861 = 412057612059012 In addition the surface density of atoms oncarbon and boron nitride nanotubes is assumed to be thesame as those of their parent materials namely graphene andhexagonal boron nitride The surface density of atoms on asheet which comprises a tessellation of hexagonal rings can beeasily calculated using the relation 120578 = 4radic399848582 where 984858 =142 A and 984858 = 145 A for lengths of one carbonndashcarbon andboronndashnitride bonds respectively thus the atomic surfacedensities of carbon and boron nitride nanotubes are 120578119888 =03821 Aminus2 and 120578119887119899 = 03682 Aminus2 respectively Moreover thenanotube radii can be calculated from 119886 = 120572radic1198942 + 1198952 + 1198941198952120587where (119894 119895) denotes the chiral vector and 120572 = radic3984858 A3 Suction Energy of Ni Atom EnteringSingle-Walled Nanotube

In this section we consider the energy of a nickel atom as itenters single-walled carbon and boron nitride nanotubes Inthis case the nanotube is assumed to be a semi-infinite cylin-der without taking into account the effects of the other endand the cylinder length Thus we only need to consider theshort range nature of the van derWaals forcesTheNi atom isassumed to have the coordinates (0 0 119885) and the nanotube ofradius 119886 is given parametrically by (119886 cos 120601 119886 sin 120601 119911) whereminus120587 le 120601 le 120587 and minusinfin lt 119911 lt infin as shown in Figure 1 Inthis configuration the distance 119903 between the Ni atom and atypical point on the tube is given by

1199032 = 1198862cos2 120601 + 1198862sin2 120601 + (119911 minus 119885)2 = 1198862 + (119911 minus 119885)2 (5)


a

x

r

z

Z



119864 = 119886120578119905 int120587minus120587

intinfin0

(minus1198601199036 + 11986111990312 ) 119889119911 119889120601= 119886120578119905 (minus1198601198673 + 1198611198676)

(6)


119867119899 = int120587minus120587

intinfin0

1[1198862 + (119911 minus 119885)2]119899 119889119911 119889120601 (7)


119867119899 = 2120587intinfin0

1[1198862 + (119911 minus 119885)2]119899 119889119911 (8)


119867119899 = 2120587intinfinminus119885

1(1198862 + 1199042)119899 119889119904 (9)


119867119899 = 21205871198861minus2119899 int1205872120594


120594cos2119899minus2120593119889120593

(10)



+ 119899minus2sum119896=1


1205872

120594

(11)


1198673 = 120587119886minus5 [31205878 + 34 tanminus1 (119885119886 ) + 31198851198864 (1198862 + 1198852)+ 11988511988632 (1198862 + 1198852)2]

(12)

and

1198676 = 120587119886minus11 [ 91205873840 + 3640 tanminus1 (119885119886 ) + 11988511988695 (1198862 + 1198852)5

+ 9119885119886740 (1198862 + 1198852)4 +

7119885119886560 (1198862 + 1198852)3 +

119885119886316 (1198862 + 1198852)2

+ 311988511988680 (1198862 + 1198852)]

(13)






Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

minus05

(33) (44) (55) (66)

(a) Armchair CNTs (119894 119894)


Z Distance (A)

Inte

ract

ion

E (e

V)

minus005

minus010

minus015

minus020

(70) (80) (90) (100) (110)



Z Distance (A)

Inte

ract

ion

E (e

V)

minus02

minus04

minus06

minus08

minus1

(33) (44) (55) (66)



Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

(70) (80) (90) (100) (110)






119864 = 119886120578119905 int120587minus120587

intinfinminusinfin

(minus1198601199036 + 11986111990312 ) 119889119911 119889120601= 120578119905 (minus1198601198703 + 1198611198706)

(15)


intinfinminusinfin

1[(119886 minus 120575)2 + 4119886120575 sin2 (1206012) + 1199112]119899 119889119911 119889120601

(16)




(33) 20344 2072 -0554 -1145(44) 2713 2761 -0246 -0518(55) 3391 3451 -0102 -0214(66) 4069 4142 -0049 -0103(70) 2741 2790 -0237 -0497(80) 3132 3188 -0139 -0293(90) 3523 3587 -0087 -0183(100) 3915 3985 -0057 -0120(110) 4307 4384 -0039 -0082

a

x

r

z



119870119899 = 119886int1205872minus1205872


11205962119899minus1119889120601 (17)


119870119899 = 119886B(119899 minus 12 12)int120587

minus120587

11205962119899minus1119889120601 (18)



(19)



(20)



2

1198862) (21)


5 Conclusions



(55) (66) (100) (110)

0

05 1 15 2


Inte

ract

ion

E (e

V)

minus005

005

010

minus010

(a) CNTs

(55) (66) (100) (110)

05 1 15 2


0

Inte

ract

ion

E (e

V)

minus01

01

minus02

(b) BNNTs



Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









a

x

r

z

Z



119864 = 119886120578119905 int120587minus120587

intinfin0

(minus1198601199036 + 11986111990312 ) 119889119911 119889120601= 119886120578119905 (minus1198601198673 + 1198611198676)

(6)


119867119899 = int120587minus120587

intinfin0

1[1198862 + (119911 minus 119885)2]119899 119889119911 119889120601 (7)


119867119899 = 2120587intinfin0

1[1198862 + (119911 minus 119885)2]119899 119889119911 (8)


119867119899 = 2120587intinfinminus119885

1(1198862 + 1199042)119899 119889119904 (9)


119867119899 = 21205871198861minus2119899 int1205872120594


120594cos2119899minus2120593119889120593

(10)



+ 119899minus2sum119896=1


1205872

120594

(11)


1198673 = 120587119886minus5 [31205878 + 34 tanminus1 (119885119886 ) + 31198851198864 (1198862 + 1198852)+ 11988511988632 (1198862 + 1198852)2]

(12)

and

1198676 = 120587119886minus11 [ 91205873840 + 3640 tanminus1 (119885119886 ) + 11988511988695 (1198862 + 1198852)5

+ 9119885119886740 (1198862 + 1198852)4 +

7119885119886560 (1198862 + 1198852)3 +

119885119886316 (1198862 + 1198852)2

+ 311988511988680 (1198862 + 1198852)]

(13)






Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

minus05

(33) (44) (55) (66)

(a) Armchair CNTs (119894 119894)


Z Distance (A)

Inte

ract

ion

E (e

V)

minus005

minus010

minus015

minus020

(70) (80) (90) (100) (110)



Z Distance (A)

Inte

ract

ion

E (e

V)

minus02

minus04

minus06

minus08

minus1

(33) (44) (55) (66)



Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

(70) (80) (90) (100) (110)






119864 = 119886120578119905 int120587minus120587

intinfinminusinfin

(minus1198601199036 + 11986111990312 ) 119889119911 119889120601= 120578119905 (minus1198601198703 + 1198611198706)

(15)


intinfinminusinfin

1[(119886 minus 120575)2 + 4119886120575 sin2 (1206012) + 1199112]119899 119889119911 119889120601

(16)




(33) 20344 2072 -0554 -1145(44) 2713 2761 -0246 -0518(55) 3391 3451 -0102 -0214(66) 4069 4142 -0049 -0103(70) 2741 2790 -0237 -0497(80) 3132 3188 -0139 -0293(90) 3523 3587 -0087 -0183(100) 3915 3985 -0057 -0120(110) 4307 4384 -0039 -0082

a

x

r

z



119870119899 = 119886int1205872minus1205872


11205962119899minus1119889120601 (17)


119870119899 = 119886B(119899 minus 12 12)int120587

minus120587

11205962119899minus1119889120601 (18)



(19)



(20)



2

1198862) (21)


5 Conclusions



(55) (66) (100) (110)

0

05 1 15 2


Inte

ract

ion

E (e

V)

minus005

005

010

minus010

(a) CNTs

(55) (66) (100) (110)

05 1 15 2


0

Inte

ract

ion

E (e

V)

minus01

01

minus02

(b) BNNTs



Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018










Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

minus05

(33) (44) (55) (66)

(a) Armchair CNTs (119894 119894)


Z Distance (A)

Inte

ract

ion

E (e

V)

minus005

minus010

minus015

minus020

(70) (80) (90) (100) (110)



Z Distance (A)

Inte

ract

ion

E (e

V)

minus02

minus04

minus06

minus08

minus1

(33) (44) (55) (66)



Z Distance (A)

Inte

ract

ion

E (e

V)

minus01

minus02

minus03

minus04

(70) (80) (90) (100) (110)






119864 = 119886120578119905 int120587minus120587

intinfinminusinfin

(minus1198601199036 + 11986111990312 ) 119889119911 119889120601= 120578119905 (minus1198601198703 + 1198611198706)

(15)


intinfinminusinfin

1[(119886 minus 120575)2 + 4119886120575 sin2 (1206012) + 1199112]119899 119889119911 119889120601

(16)




(33) 20344 2072 -0554 -1145(44) 2713 2761 -0246 -0518(55) 3391 3451 -0102 -0214(66) 4069 4142 -0049 -0103(70) 2741 2790 -0237 -0497(80) 3132 3188 -0139 -0293(90) 3523 3587 -0087 -0183(100) 3915 3985 -0057 -0120(110) 4307 4384 -0039 -0082

a

x

r

z



119870119899 = 119886int1205872minus1205872


11205962119899minus1119889120601 (17)


119870119899 = 119886B(119899 minus 12 12)int120587

minus120587

11205962119899minus1119889120601 (18)



(19)



(20)



2

1198862) (21)


5 Conclusions



(55) (66) (100) (110)

0

05 1 15 2


Inte

ract

ion

E (e

V)

minus005

005

010

minus010

(a) CNTs

(55) (66) (100) (110)

05 1 15 2


0

Inte

ract

ion

E (e

V)

minus01

01

minus02

(b) BNNTs



Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018











(33) 20344 2072 -0554 -1145(44) 2713 2761 -0246 -0518(55) 3391 3451 -0102 -0214(66) 4069 4142 -0049 -0103(70) 2741 2790 -0237 -0497(80) 3132 3188 -0139 -0293(90) 3523 3587 -0087 -0183(100) 3915 3985 -0057 -0120(110) 4307 4384 -0039 -0082

a

x

r

z



119870119899 = 119886int1205872minus1205872


11205962119899minus1119889120601 (17)


119870119899 = 119886B(119899 minus 12 12)int120587

minus120587

11205962119899minus1119889120601 (18)



(19)



(20)



2

1198862) (21)


5 Conclusions



(55) (66) (100) (110)

0

05 1 15 2


Inte

ract

ion

E (e

V)

minus005

005

010

minus010

(a) CNTs

(55) (66) (100) (110)

05 1 15 2


0

Inte

ract

ion

E (e

V)

minus01

01

minus02

(b) BNNTs



Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018









(55) (66) (100) (110)

0

05 1 15 2


Inte

ract

ion

E (e

V)

minus005

005

010

minus010

(a) CNTs

(55) (66) (100) (110)

05 1 15 2


0

Inte

ract

ion

E (e

V)

minus01

01

minus02

(b) BNNTs



Data Availability




Acknowledgments


References






































Journal of












Journal of







Volume 2018






Volume 2018





























Journal of












Journal of







Volume 2018






Volume 2018








Modelling of Nickel Atoms Interacting with Single-Walled...

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