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Ab Initio Study of Chirality Effects Onphonon Spectra, Mechanical and Thermal Properties of Nearly Samediameter Single Wall Carbon Nanotubes

H. Tashakori1, B. Khoshnevisan1,2, F. Kanjouri3*,Z.S.Naghavi2

1-PhysicsDepartment,ComputationalPhysicsLab,QomBranch,IslamicAzadUniversity,Qom,I.R.Iran

2-FacultyofPhysics,UniversityofKashan,Kashan,I.R.Iran3-FacultyofPhysics,KharazmiUniversity,Tehran,I.R.Iran

(*)Correspondingauthor:kanjouri@khu.ac.ir(Received: 12 Jan. 2014 and accepted: 17 May 2014)

Abstract:In this paper, we have used density functional perturbation theory (DFPT) and Pseudo-potential method to calculate the phonon spectrum, phonon density of states (DOS), specific heat capacity and mechanical properties of (5,5) armchair and (9,0) zigzag Single Wall Carbon Nanotubes (SWCNTs). Our calculations show that Young’s modulusfor (5,5) and (9,0) nanotubesare higher than 1TPa. We have also shown that the value of compressive Young’s modulus for (5,5) nanotube isgreater thanthat for (9,0) nanotube while thevalue of tensile Young’s modulus for (9,0) nanotubeisgreater than that for(5,5) nanotube. The result of our calculations shows that thespecific heat capacity of (5,5) and (9,0) nanotubes coincides, therefore we may conclude that thespecific heat capacity of nanotubes is independent of their chirality. Furthermore we have found that the atoms in the armchair nanotubes are positioned as close as possible in the direction of the nanotube axis, therefore they could have more resistant against compressive pressure.Keywords: Single-walled carbon nanotubes, Mechanical properties, Young’s modulus.

Int. J. Nanosci. Nanotechnol., Vol. 10, No. 3, Sept. 2014, pp. 163-170

1.INTRODUCTION

During the lastdecades,carbonnanotubes,due tothe ability of understanding the size importancein physical properties and also their applicationsin nanostructure materials, have received muchattentions[1-5].Carbon nanotubes are carbon based structureswhichhavebeenlargelyusedinnewtechnologiesbecauseoftheirspecificelectronicandmechanicalproperties. The physical properties of carbonnanotubeswillchangedependingonthegeometry.Varying thegeometryof carbonnanotubesmakes

it possible to fabricate low dimensional physicalsystems in order to achieve higher advantages intechnology[6-8].Numerous studies have been performed to determine the mechanical properties of thisnano-structured material [9–13]. Theoretical andexperimental investigation indicated an averageYoung’smodulus of around 1 TPa and Poisson’sratio of 0.25–0.28 for single-walled carbonnanotubes (SWNTs), depending on the CNTs’ length,diameter,chirality,samplesynthesis,typeofdefect,measurementtechniques,andcomputationaltheory and parameters.

164

Thevibrationbehaviorofcarbonnanotubes(CNTs)has been extensively investigated due to their importance in nano-electro-mechanical systems(NEMS)andnanosensorapplication.Variousnumericalandexperimentalinvestigationshave been reported on the RBM vibrations of CNTs.S.Basirjafarietalhaveanalyticallystudiedtheradialbreathingmodesofmulti-walledcarbonnanotubes[14].Atheoreticalanalysisoftheradialbreathingmode(RBM)ofcarbonnanotubes(CNTs)subjectedtoaxialpressurehasbeenpresentedbasedonanelasticcontinuummodelbyXiao-WenLeietal[15].Theyhaveinvestigatedtheeffectsofaxialpressure,wavenumbersandnanotubediameterontheRBMfrequency.Carbon nanotubes have the highest tensile strength amongtheknownmaterialsthereforetheymaybeconsidered as the strangest material ever known.There are covalent bonds between carbon atoms.A tensile strength of 63Gpawas found formulti-walled carbon nanotube in 2000 [16] (about athousand times tires).Thehighthermalconductivityinthedirectionofthenanotubeaxis,whichisoneofthemostimportantproperties of carbon nanotubes is in the focusof many researches. Carbon nanotubes will alsobecomeinsulatorsinthedirectionsperpendiculartothe nanotube axis.The specific heat capacity of carbon nanotubes isan important quantity in industrial applicationsand also experimental research. The specific heatcapacity can be determined based on statisticalmechanicscalculations.Coefficientofthermalexpansionofcarbonsingle-wallednanotubeshasbeeninvestigatedanalyticallyand numerically byAskari et al [17]. They haveshownthatthecoefficientofthermalexpansionofcarbon single-walled nanotubes is independent oftheirchirality.In this paper we have theoretically studiedmechanicalandthermalpropertiesofinfinitesinglewall(5,5)and(9,0)nanotubes.Wehavecalculatedthe phonon dispersion, phonon density of states andspecificheatcapacityof thesenanotubes.Ourcalculations indicate that the Youngs’s modulusof CNTs is different in tension and compression.Also the results of our calculations show that the

specific heat capacity of CNTs is independent oftheirchirality.

2.CALCULATIONMETHODS

Asiswellknowninharmonicapproximationasolidconsistsofaperiodicarrayofatomsorionsthatwilloscillate around their equilibrium positions. Theenergy needed for dynamical motion is providedby the solid temperature. By calculating the totalenergyofsolidatzerotemperature,wemayobtainstructural properties of carbon nanotubes such aslatticeequilibriumconstantsandbulkmodulus.The vibrational energy of atoms or ions in a dynamicalcrystalatnon-zerotemperaturewillhavesignificant effects on the mechanical and thermalproperties of crystal such as heat capacity andthermal expansion. In this paper Quantum ESPRESSO package hasbeen employed for the DFPT calculations whiletheDFTpart of the calculationsweredoneunderlocaldensityapproximation(LDA)[18].Theeffectoftheinternalelectrons onvalenceelectronicstatesis taken into account by using ultra soft pseudo-potentials[19].Thechosensuper-cellwasahexagonalwith14ÅsidessoanycouplingbetweenneighborSWCNTshasbeeneliminated.Inaddition,theBZintegrationwas carried out via theMonkhorst– Pack scheme[20, 21] with 1×1×12 k-points. The convergencetestwith0.1μeVtolerancehasbeendonefortotalenergyaccordingtotheplanewave’scutoffenergyup to544eV.Moreover,optimizationof the ionicpositions, latticeparametersandcellvolumewereachieved after the relaxation when all forces andstressesbecamelessthan0.25meV/Åand0.05GParespectively.

3.PHONONPROPERTIES

Figures1-2showthephonondispersionsalongthetube’saxis(zdirection)andtheirDOSsforanarmchair(5,5)andanzigzag(9,0)SWCNTs.Considering the lowest energy) acoustic (dispersion curves ,it is deduced that near the Г) k=0)pointthetransverse-acoustic(TA)modebranchesaredoubly

Tashakori, et al.

165

degeneratedandthesecondbranchesarebelongtothelongitudinal-acoustic(LA)mode.

Figure 1: (a) The calculated phonon dispersion relations of an armchair carbon nanotube (5,5) plotted in axial q-vector. (b) Phonon density of states of (5,5) nanotube.

Figure 2: (a) The calculated phonon dispersion relations of an zigzag carbon nanotube (9,0)

plotted in axial q-vector. (b) Phonon density of states of (9,0) nanotube.

Moreover, there is a fourth acoustic mode forCNTs, twisting mode (TW), which shows anionic rotationalwave propagation along the tubes[22]andthelowestthirdbranchshowsthismode,respectively.Slopesoftheacousticbranchesaroundthe Г point (dω /dk)k=0 give their relevant sound velocities.Thesoundvelocitiesof theTAandLAphononsfor(5,5)carbonnanotubeareestimatedasVTA=5.51km/sandVLA=15.127km/s, respectively.

In addition, the velocity of the twisting acousticwave isVTW=14.925km/s for (5,5)nanotube.ThesoundvelocitiesoftheTAandLAphononsfor(9,0)carbonnanotubeareestimatedasVTA=10.0649 km/s and VLA=22.7716km/s,respectively.ThevelocityofthetwistingacousticwaveisVTW=15.428 km/s for (9,0)nanotube.Theacousticvelocitiesarereportedin Table 1.

Table 1: The calculated acoustic velocities of (5,5) and (9,0) carbon nanotubes.

5

velocity of the twisting acoustic wave is 428.15TWV km/s for (9,0) nanotube. The acoustic

velocities are reported in Table 1.

Table 1: The calculated acoustic velocities of (5,5) and (9,0) carbon nanotubes.

skmVTW

skmVLA

skmVTA

Nanotube

14.925 15.127 5.51 (5,5)

15.428 22.7716 10.0649 (9,0)

The velocity of LA mode can be used to calculate the elastic constant along the axis of nanotube

33C via the relation:

33LA

CVr

= (1)

where r is the density of nanotube.

Although the density is an ill-defined concept in the case of CNTs [23, 24], here a nanotube is

assumed to be a rolled graphene with 3.4Å thickness (the separation between Carbon layers in

graphite). Table 2 shows the longitudinal elastic constants and also the number of ions per unit-

cell of the considered SWCNTs.

Table 2 Radius, number of C atoms per unit-cell and elastic constant (C33) for (5,5) and (9,0) SWCNTs.

C33 (GPa) N/unit-cell Radius (Å) SWCNT

515.9 20 3.4017 (5,5)

1168.6 36 3.5302 (9,0)

13

224( 3cm 665)RBM D Dn - = - (2)

The velocity of LA mode can be used to calculatethe elastic constant along the axis of nanotube 33C via the relation:

33LA

CV

r= (1)

where r is the density of nanotube. Although the density is an ill-defined concept in the case of CNTs ,[24 ,23] here a nanotube is assumed to be a rolled graphene with3.4 Å thickness) the separation between Carbon layers in graphite .(Table 2shows the longitudinal elastic constants and also the number of ions per unit-cell of the consideredSWCNTs.

Table 2: Radius, number of C atoms per unit-cell and elastic constant (C33) for (5,5) and (9,0)

SWCNTs.

5

velocity of the twisting acoustic wave is 428.15TWV km/s for (9,0) nanotube. The acoustic

velocities are reported in Table 1.

Table 1: The calculated acoustic velocities of (5,5) and (9,0) carbon nanotubes.

skmVTW

skmVLA

skmVTA

Nanotube

14.925 15.127 5.51 (5,5)

15.428 22.7716 10.0649 (9,0)

The velocity of LA mode can be used to calculate the elastic constant along the axis of nanotube

33C via the relation:

33LA

CVr

= (1)

where r is the density of nanotube.

Although the density is an ill-defined concept in the case of CNTs [23, 24], here a nanotube is

assumed to be a rolled graphene with 3.4Å thickness (the separation between Carbon layers in

graphite). Table 2 shows the longitudinal elastic constants and also the number of ions per unit-

cell of the considered SWCNTs.

Table 2 Radius, number of C atoms per unit-cell and elastic constant (C33) for (5,5) and (9,0) SWCNTs.

C33 (GPa) N/unit-cell Radius (Å) SWCNT

515.9 20 3.4017 (5,5)

1168.6 36 3.5302 (9,0)

13

224( 3cm 665)RBM D Dn - = - (2)

1

33CVLA (1)

31 6652243)cm(

DDRBM (2)

02

21

EV

Yeq

(3)

0

0

0

0

0

0 ,/

/L

YALFLA

FLLLAFY

(4)

0

0

//LLRR

(5)

d

e

eTk

hgTCTk

h

Tkh

Bv

B

B

2

2

01

)()(max

(6)

(2)

The other vibrational characteristic feature of theSWCNTs is their radial breathing mode (RBM) frequencies (the most important low-frequency

International Journal of Nanoscience and Nanotechnology

166

Raman active mode). It involves a collectivemovement of atoms towards and away from thecentralaxisanditsvalueisdifferentforeachtube.The RBM frequency dependence on diameter ofSWCNTs is commonly accepted via an empiricalformula [23]: Disthediameterofthenanotube(Å).Radial breathing modes calculated and radialbreathing modes obtained from the empiricalequation(2)arecomparedintable3.

4.MECHANICALPROPERTIES

The main mechanical characteristics of theSWCNTsareYoungmodulus,Y,andPoissonratio.TheYvaluedirectionisdefinedas:

(3)

where E is the formation energy, Veq and ε arethe equilibrium volume and the subjected strainrespectively. Here we have used two methods todeterminetheYvalues;i-usingtheDFTapproachfor minimizing the formation energy versus thesubjected strain along the tube’s axis (Murnaghanmethod[26]),ii-usinglinearelasticapproximation:

1

33CVLA (1)

31 6652243)cm(

DDRBM (2)

02

21

EV

Yeq

(3)

0

0

0

0

0

0 ,/

/L

YALFLA

FLLLAFY

(4)

0

0

//LLRR

(5)

d

e

eTk

hgTCTk

h

Tkh

Bv

B

B

2

2

01

)()(max

(6)

(4)

Where e is the inserted strain, 0L is the equilibriumlength, 0A isthecross-sectionandκistheequivalentelasticconstantforlongitudinal

expansionor contractionofnanotube.Tofindthe κ value, we plot the energy variations vs

2( ) 2S L= ∆ where L∆ is the length variation oftubes.Theκvaluecanbecalculatedviatherelation: E Sκ∆ = (elasticapproximation)Figures 3 and 4 show the energy variation ofCNT (5,5) and (9,0) versus S in tension andcompression.As it canbeseen from thesefiguresthe energy linearly increases by increasing S. Bycalculating the slope of the curve we can obtainthevalueofYoung’smodulusofCNTs in tensionand compression.Howeverwe canfind the valueofYoung’smodulusMurnaghanmethod.Table. 4indicatestheresultsofourcalculations.

Figure 3: Energy variation versus 2( ) 2S L= ∆ for (5,5) SWCNT: a-tensile state, b-compressive state

TheYvaluesinthirdcolumnoftable4havebeenobtainedvia theMurnaghanmethod;whereas, fordetermining the compressive and tensileY values(fourthandfifthcolumnsofthetable)wehaveusedthe elastic approximation (Figures 3-4). It must

6

D is the diameter of the nanotube (Å).

Radial breathing modes calculated and radial breathing modes obtained from the empirical

equation (2) are compared in table 3.

Table 3: The radial breathing mode (RBM) of (5,5) and (9,0) carbon nanotubes. Number of C atoms in unit cell (N).

Radius of the nanotube (R).

Nanotube N R(Å) 1(cm )RBMn

- empirical Eq.(2)

1(cm )RBMn-

Calculated Other

calculation

(5,5) 20 3.4017 327.58 340.88 332.881 [25] (9,0) 36 3.5301 315.81 317.21 -

Mechanical properties

The main mechanical characteristics of the SWCNTs are Young modulus, Y, and Poisson ratio.

The Y value direction is defined as:

2

20

1 eq

EYV ee =

¶ِو ÷ç ÷= ç ÷ç ÷ç ¶è ّ (3)

where E is the formation energy, Veq and ε are the equilibrium volume and the subjected strain

respectively. Here we have used two methods to determine the Y values; i- using the DFT

approach for minimizing the formation energy versus the subjected strain along the tube’s axis

(Murnaghan method [26]), ii- using linear elastic approximation:

0 0 0

0 0 0

/,

/F A FL YAY F L

L L A L Ls k ke

= = = =ق D =D D

(4)

Where e is the inserted strain, 0L is the equilibrium length, 0A is the cross-section and κ is the equivalent elastic constant for longitudinal expansion or contraction of nanotube. To find the κ

Table 3: The radial breathing mode (RBM) of (5,5) and (9,0) carbon nanotubes. Number of C atoms in unit cell (N). Radius of the nanotube (R).

Tashakori, et al.

167

be pointed out that there is a very good agreement betweenaveragevaluesof the4thand5thcolumnsandthevaluesofthethirdcolumn.Asitisexpected,thecompressivemodulusvaluesarebiggerthanthetensileonesandthe6thcolumnofthetableshowstheirdifferences.

Figure 4: Energy variation versus 2( ) 2S L= ∆ for(9,0) SWCNT: a-tensile state, b-compressive state

Another important mechanical property of amaterial in the linear regime is the Poisson ratio, which has been defined in amacroscopic contextasthenegativeratiooftherelativechangeinradiusover the relative elongation:

(5)

1

33CVLA (1)

31 6652243)cm(

DDRBM (2)

02

21

EV

Yeq

(3)

0

0

0

0

0

0 ,/

/L

YALFLA

FLLLAFY

(4)

0

0

//LLRR

(5)

d

e

eTk

hgTCTk

h

Tkh

Bv

B

B

2

2

01

)()(max

(6)

Where, R0 andL0 are relevant unperturbed radius and length of the sample.Inthisregard,byconsiderationofanisolatedtubeand finding out the slope of its radius variationversusitsperturbingunit-cellelongation(Figure5)andmultiplyitby(-L0/R0),itwouldbepossibletocalculatethePoissonratio.Theresultsofcalculationfor thenarrowSWCNTsareshown in table5.Asthis table shows the Poisson ratios for the zigzag(9,0)nanotubeishigherthanthatforarmchair(5,5)nanotube.

8

Fig. 4: Energy variation versus 2( ) 2S L for (9,0) SWCNT: a-tensile state, b-compressive state

Table.4 Young modulus, Y, for (5,5) and (9,0) SWCNTs. The last column shows ∆Y=Ycomperessive-Ytensile

∆Y(GPa) Y(Tensile)(TPa) Y(compressive) (TPa) Y(TPa) R(Å) nanotube

278.7 0.9557 1.2344 1.0885 3.4017 (5,5) 39.9 1.0280 1.0679 1.0458 3.5302 (9,0)

Another important mechanical property of a material in the linear regime is the Poisson ratio,

which has been defined in a macroscopic context as the negative ratio of the relative change in

radius over the relative elongation:

ν0 0

R LR L

é ùD Dê ْ= - ê ْë û (5)

Where, R0 and L0 are relevant unperturbed radius and length of the sample.

In this regard, by consideration of an isolated tube and finding out the slope of its radius

variation versus its perturbing unit- cell elongation (Fig.5) and multiply it by (-L0/R0), it would

be possible to calculate the Poisson ratio. The results of calculation for the narrow SWCNTs are

shown in table 5. As this table shows the Poisson ratios for the zigzag (9,0) nanotube is higher

than that for armchair (5,5) nanotube.

Table 4: Young modulus, Y, for (5,5) and (9,0) SWCNTs. The last column shows ∆Y=Ycomperessive-Ytensile

Figure 5: Least square fitting for variation of the narrow (5,5) and (9,0) SWCNTs versus their unit-cell elongation

International Journal of Nanoscience and Nanotechnology

168

Table 5: The poisson is calculated for carbon nanotubes

PoissonR(Å)nanotube0.1563037353.4017(5,5)0.2171322763.5302(9,0)

4. SPECIFIC HEAT

Inprinciple, except in very low temperature region the phononcontributiontothespecificheat,Cv(T) of materials is significant. After calculating thephononDOSforthenarrowSWCNTs,wecanfindout the temperature dependence of their specificheataccordingtothefollowingexpression[27]:

(6)

Whereνisphonon’sfrequency, ( )g n is phonon DOSand maxn isthehighestphononfrequencyof the material and is related to the Debyetemperature, DQ ( maxB D hk nQ = ).The temperature dependence of specific heatfor(5,5)(armchair)and(9,0)(zigzag)SWCNTisshowninfigure6.Itisobservedthatspecificheat exhibits almost the same behavior withtemperature for both tube types.

Figure 6: Temperature dependence of specific heat of (5,5) and (9,0) carbon nanotube.

5.CONCLUSION

Our results for radialbreathingmodes (RBM)arein good consistentwith the experimental ones. Inequation (2), nanotubes radii are the parametersconsideredtocalculatetheRBMwhilethechiralitydependence of these modes is ignored in thisequation. Our theoretical results predict that theRBMdependsonthechirality.ThemostimportantapplicationofRBMisinnanotubespurification.TheYoung’smodulusfor(5,5)and(9,0)nanotubesare greater than 1Tpa, based on our calculations.The value of compressive Young’s modulus for(5,5)nanotubeisgreaterthanthatfor(9,0)nanotubewhilethevalueoftensileYoung’smodulusfor(9,0)nanotubeisgreaterthanthatfor(5,5)nanotube.We have also found that the atoms in armchairnanotubes arepositioned as close aspossible inthedirectionofthenanotubeaxis,thereforetheycould have more resistant against compressivepressure. Furthermore we have shown that theatoms in zigzag nanotubes are positioned as faraspossibleinthedirectionofthenanotubeaxis,therefore theycouldhavemore resistantagainsttensile pressure. The value of elastic constant coefficient for (5,5)nanotubesinthedirectionofnanotubeaxis(C33)iscompletelydifferent incomparisonwith thevalueofYoung’smodulus. Due to the concentration ofatomsinthedirectionofnanotubeaxis,theelasticcoefficientperpendiculartonanotubeaxisbecomesimportant while this cannot be observed in (9,0)nanotubes.FinallywehavefoundthatthePoissonratios for thezigzag (9,0)nanotube ishigher thanthatforthearmchair(5,5)nanotube.In room temperature (300 K), the specific heatcapacity per mole is equal to 1R (universalconstant)which isdue to theassumptionofquasione dimensionality of nanotubes.The outcome of our calculations shows that thespecificheatcapacityoftwo(5,5)nanotubeand(9,0)nanotubecoincides,thereforewemayconcludethatthespecificheatcapacityofnanotubesisindependentof their chirality.This result is in agreementwiththe result of analytical andnumerical calculationsin Ref. [17]

Tashakori, et al.

1

33CVLA (1)

31 6652243)cm(

DDRBM (2)

02

21

EV

Yeq

(3)

0

0

0

0

0

0 ,/

/L

YALFLA

FLLLAFY

(4)

0

0

//LLRR

(5)

d

e

eTk

hgTCTk

h

Tkh

Bv

B

B

2

2

01

)()(max

(6)

169

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