AN ATOMISTIC-BASED CONTINUUM ANALYSIS FOR NONELASTIC BEHAVIORS OF CARBON NANOTUBES

Transcription

1 16 TH INTRNATIONAL CONFRNC ON COMPOSIT MATRIALS AN ATOMISTIC-BASD CONTINUUM ANALYSIS FOR NONLASTIC BHAVIORS OF CARBON NANOTUBS Tohiaki Natuki# and Qing-Qing NI -mail Faculty of Txtil Scinc & Tchnology, Shinhu Univrity , Tokida, Uda, Nagano Japan Kyword: Carbon notub, Mchanical proprti, Failur, Simulation, Prdiction Abtract Bad on an atomitic-bad continuum alyi, a tructural mchanic approach i prntd to imulat tr-train bhavior of carbon notub (CNT Th intratomic potntial i dirctly incorporatd into th continuum alyi through a pring modl According to th prnt modl, th nonlir latic proprti of both igag and armchair tub ar invtigatd Thi rult how that th atomic tructur of CNT hav a ignificant influnc on th tr-train bhavior Th armchair igag tub xhibit largr trtrain rpon than th igag notub undr tnil loading Th prnt thortical approach giv om vry impl formula and can b ud to prdict th mchanical proprti for CNT 1 Introduction Carbon notub (CNT hav bn th ubjct of intn rarch du to thir low dnity, rmarkabl mchanical, thrmal and lctrical proprti [1-3] In particular, du to th high pcific tiffn and trngth, CNT can b conidrd to b vry promiing matrial a rinforcmnt in compoit matrial A potntial application of CNT i th CNT-bad compoit inc th incluion of CNT within variou matric can obviouly improv not only th mchanical proprty but alo th phyical proprty Th addition of jut 1 wt% CNT to polytyrn rult in an incra of latic modulu of CNT compoit by 35-%, and trngth by approximatly 5% [] Th xtrmly mall dimnion of CNT with diamtr of a fw momtr and lngth of a fw micron impo a trmndou challng for xprimntal tudy of mchanical proprti xprimntal mthod to maur th mchanical proprti of CNT ar bad mainly on th tchniqu of tranmiion lctron microcopy (TM and atomic forc microcopy (AFM [5, 6] A largr variation of th latic modulu wa rportd to b TPa for multi-walld carbon notub (MWNT by Wong t al [5] Th mchanical maurmnt uch a th latic modulu and th tnil tr hav contributd to confirm that CNT hav xcptiol mchanical proprti Sinc th CNT ar vry mall in i and hardly controlld, thir mchanical proprti wr not maurd xactly pcially, it i rathr difficult to dirctly tat th ffct of tructur of CNT on th latic modulu and trngth du to ytmatic xprimntal rror Computatiol imulation for prdicting mchanical and phyical proprti of CNT ha bn rgardd a a powrful tool rlativ to th xprimntal difficulty Thr ar two major catrgori of molcular dymic (MD and olid mchanic for CNT imulation Yakobon and coworkr [7-10] hav ud th MD mthod for imulating th laticity and platicity proprti, mchanim of train rla, and intabiliti byond lir rpon Although th claical MD and ab initio mthod hav bn ud quit xtnivly to tudy and prdict th mchanical proprti and failur of CNT, an availabl computatiol powr i dmandd for th MD mthod From th crytal laticity approach, on might th poibility of applying th olid mchanic to th computatiol mchanic of notub inc th continuum concpt of tr can b xtractd from a molcular modl [11] In thi work, w prnt a thortical alyi on th nonlatic bhavior of th CNT bad on an atomitic-bad continuum alyi incorporating intratomic potntial Th tr-train rlationhip of th CNT having diffrnt tructur (th igag and armchair notub i invtigatd 1

2 AUTHOR 1,! Thortical Approach 1 Molcular tructural mchanic of carbon notub A ingl-walld carbon notub (SWNT can b rgardd a a hollow cylindr rolld from a graphn ht SWNT compod of carbon hxagon ar uually indxd by th chiral vctor intgr (n,m Th diamtr of notub can b calculatd a ( n + m mn n + (1 π whr b i th C-C bond lngth, which i 01 nm CNT ar claifid into thr catgori of igag, armchair and chiral in trm of th chiral vctor intgr (n,m a follow: 1 Zigag notub ( m0 Armchair notub ( nm 3 Chiral notub ( m n Conidring th layr thickn t, th ffctiv diamtr of CNT,, i givn by ( n + m + mn t + ( π Sytm potntial nrgy for carbon Th mchanical proprti of olid matrial mut ultimatly dpnd on th trngth and thir intraction bond Th bondd and non-bondd intraction of th atom in a molcular tructur can b dcribd by uing molcular mchanic Th forc that xit for ach bond ar dcribd by th forc fild o that th forc contribut to th molcular potntial nrgy of th forc fild ( Fig1 can b xprd a m τ ω νdw L (3 τ ω whr,, and ar bonding nrgy, dfind a bond trtching, angl variation, torion and invrion, rpctivly Th nonbonding nrgy dw conit of van dr Waal forc ν and l lctrotatic intraction that ar quit wak For CNT ubjctd to th axial loading, torion, invrion and nonbonding intraction ar vry mall and can b ngligibl Thrfor, th ytm potntial nrgy of th notub with carbon-tocarbon bond in q (3 can b implifid a m + ( In th prnt tudy, w u a modifid Mor potntial function in which a bond-angl-bnding potntial i addd, givn a [( 1 1] D [ ] (5 1 k ( 1 + k ( (6 whr b and ar th variation of th bond lngth and angl btwn two nighbor bond, rpctivly Torion Non-bondd Bond-angl variation Bond trtching Carbon atom Invrion Fig 1 Atomic tructur modl of a hxagol unit cll Th nrgy paramtr of carbon-to-carbon bond ar givn by th contant of th modifid Mor modl [1] D 1807V, k 5617 V rad, 1 β 383nm, k 075rad Diffrntiating q (5 and (6, w obtain th trtching and bnding forc of atom bond in th molcular fild a ( F 1 (7 [ + ( ] M k 1 (8 Th molcular forc fild dfind by q (7 and (8 ar lir in low loadd CNT, and ar givn a F β b (9 M k (10

3 PAPR TITL Figur how th molcular mchanic modl ubtitutd with a pring modl K and K ar th trtching and bnding contant, rpctivly whr γ π n Form q (11 and (1, th variation of bond lngth yild ln β P co ( (16 P ln, P (17 β Fig lmnt of a pring modl in CNT 3 Solution of nonlir latic proprti W can obtain th tr-train olution bad on author prviou work [13, 1] Figur 3 how th thr-dimniol chmatic illutration of th forc and momnt acting on a bond of th igag notub, and th gomtrical rlationhip among th atom Lt u conidr th forc P and th momnt M acting on a carbon-to-carbon bond Th trtch and angular dformation of bond ar caud by th axial forc and th bnding momnt, rpctivly For a igag notub ubjctd to an axial load, forc quilibrium to bond xtnion of tick i givn by β β P co 1 (11 ( ( In th imulation modl, th wall thickn of SWNT i only givn a a continuum aumption W can rgard th wall thickn of CNT a ro For a igag notub with th diamtr of D, th tr and train can b obtaind from P σ (18 Db in b in ε ( ( + co( b( 1+ co + (19 q (11-(19 giv a t of quation to imulat th tr-train rpon for th igag notub β β ( P 1 (1 whr and ar th dformation of and bond with diffrnt alignmnt i dfind a a half bond angl Th momnt quilibrium to bond i givn by M j k + k ( ] 1+ ( co Pb ( [ ] ϕ ( j A, B (13 M M A + M B in (1 According to th gomtrical rlation of angl hown in Fig 3, w obtain coγ 3in γ (15 Fig 3 Alyi of forc and gomtrical rlation for th igag notub For th tr-train rpon of armchair notub, w hav th am alytical tp a tho of th igag notub According to th alyi of forc and gomtrical rlation hown in Fig, th tr-train rpon for th armchair notub can b obtaind from quation a follow: 3

4 AUTHOR 1,! Q σ a (0 Db co( + ] ( + + in( + b co ε a (1 bin k Q ( ] + k 1+ ( b co( + ln β [ ] ( + Q in Q Q in ( + coϕ ( (3 Th angl rlation of th armchair tub ( n, n btwn and α hav whr γ π n 3 coγ co γ ( Fig Alyi of forc and gomtrical rlation for th armchair notub According to th quilibrium quation on th forc and th momnt, th tnil modulu of CNT can b aily calculatd from blow formula [13]: whr λ K K K + 9λ K 8 3 n (5 ( π n ( π n 3 + co λa for armchair tub (6 7 co ( π n ( π n 1 co λ for igag tub (7 5 3co 3 Numrical Simulation and Dicuion In thi computatiol imulation, th two main typ of CNT, that i th igag and armchair tub ar dicud In ordr to undrtand th tructural dpndnc of th mchanical proprti for CNT, w prformd th computr imulation for th (17, 0 igag, th (6, 0 igag and th (15, 15 armchair notubr Th (6, 0 igag tub and th (15, 15 armchair tub hav a rly qual diamtr, which ar about 0 nm Th (17, 0 igag tub i 13 nm in diamtr Figur 5 how th tr-train rpon of CNT with diffrnt tructur In mall train rang, lop of th tr-train curv ar dfind a th latic modulu For th am diamtr, th latic modulu of a igag tub i th am with that of an armchair tub Thi man that th latic modulu of CNT i indpndnt of th notub tructur It can b n from Fig 6 that th latic modulu of th armchair tub i lightly largr than that of th igag tub only whn th notub diamtr i vry mall (about 07 nm A th notub diamtr incra, th latic modulu of th igag and armchair tub bgin to hav th am valu A SWNT diamtr i uually in th rang of nm Bad on th prnt imulation, w obtain th tnil modulu of about 10 TPa ( Fig 6 A hown in Fig 5, th axial tnil tr a a function of train hav a nonlir dpndnc It can b found that th notub tructur hav a ignificant influnc on th tr-train bhavior For th igag and armchair notub with an idntical diamtr, th tr-train curv ovrlap at th initial loading tag (mall train Howvr, th tnil tr of th armchair tub i largr than tho of th igag tub in larg train Morovr, th armchair notub how th highr train bhavior than th igag tub For idntical tructur, uch a th igag or th armchair tub, thir fractur train (at th maximum tr i indpndnt of th diamtr of CNT In thi imulation, w prdict th maximum tr of th (17, 0 igag tub to b about 69 GPa around a train of 1% Th (6, 0 igag tub hav th am 1% train but th maximum tr of

5 PAPR TITL about 5 GPa For th (15, 15 armchair tub, th maximum tr i approximatly 60 GPa around 15% train tructur Compard with th igag nnotu, th armchair notub can undrgo not only largr tnil tr but alo largr fractur train Tnil tr (GPa Modulu (TPa Diamtr (nm Fig 6 Variation of latic modulu with notub diamtr for igag and armchair notub Concluion A nocal continuum thory i tablihd bad on a link btwn molcular and olid mchanic Uing th atomitic-bad continuum approach, th tr-train rpon can b aily obtaind to dcrib th nonlir bhavior of CNT Th latic modulu of CNT i indpndnt of thir Rfrnc [1] Lau KT, Hui D Th rvolutiory cration of nw advancd matrial: carbon notub compoit Zigag (17 0 Compoit: Part B Vol33, pp63-77, 00 Armchair (15,15 [] Thotnon T, Rn Z, Chou TW Advanc in th cinc and tchnology of carbon notub and thir compoit: a rviw Comp Sci Tch Vol 61, pp , 001 Zigag (6 0 [3] Schadlr LS, Gianri SC, Ajayan PM Load tranfr in carbon notub poxy compoit Appl Phy Ltt Vol 73 (6, pp 38-38, 1998 [] Qian D, Dicky C, Andrw, R, Rantll T Load tranfr and dformation mchanim in carbon Strain, ε (% notub-polytyrn compoit Appl Phy Ltt 000; 76(0: [5] Wong W, Shhan P, Libr CM Nanobam Fig 5 Str-train rpon of CNT mchanic: laticity, trngth, and toughn of norod and notub Scinc Vol 77(6: undr tnil loading 1975, 1997 [6] Tracy MMJ, bbn TW, Gibon JM xcptiolly high young modulu obrvd for individual notub Natur Vol 381(658, pp , 1996 Zigag [7]Yakobon BI, Brabc CJ, Brnholc J Armchair Nanomchanica of carbon tub: Intabiliti byond lir rpon Phy Rv Ltt Vol 76(1, pp , 1996 [8]Yakobon BI Mchanical rlaxation and intramolcular platicity in carbon notub Appl Phy Ltt Vol 7(8, pp , 1998 [9] Nardlli MB, Yakobon BI, Brnholc J Brittl and ductil bhavior in carbon notub Phy Rv Ltt Vol 81(1, pp , 1998 [10] Kudin KN, Scuria G, Yakobon BI C F, and C nohll laticity from ab inito computation Phy Rv B Vol 6(3, pp , 001 [11] Qian D, Wagnr GJ, Liu WK, Yu MF, Ruoff RS Mchanic of carbon notub Appl Mch Rv Vol 55(6, pp , 00 [1] Blytchko T, Xiao SP, Schat GC, Ruff RS Atomitic imulation of notub fractur Phy Rv B Vol 65(3, pp , 00 [13] Natuki T, Tantrakarn K and ndo M ffct of carbon notub tructur in mchanical proprti Appl Phy A, Vol 79, pp 117-1, 00 [1] Natuki T and ndo M Str imulation of carbon notub in tnion and comprion Carbon, Vol, pp , 006 5