Drawing of Hollow Multilayered All-Polymer Fibers

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1 Mate. Res. Sc. Symp. Pc. Vl Mateals Reseach Scety 090-S01-05 Dawng f Hllw Multlayeed All-Plyme Fbes El Pne, Chales Dubs, Nng Gu, Yan Ga, Alexande Dupus, Suanne Lacx, and Maksm Skbgaty Écle Plytechnque de Mntéal, Mntéal, Québec, H3C 3A7, Canada ABSTRACT We pesent a flud dynamcs mdel f the dawng f hllw multlaye plyme ptcal fbe. A newtnan mdel s cnsdeed assumng slende gemetes. Hllw ce cllapse dung dawng and laye thckness nn-unfmty ae nvestgated as a functn f daw tempeatue, daw at, feedng speed, ce pessuatn and msmatch f mateal ppetes n a multlaye. INTRODUCTION Hllw ce multlaye and mcstuctued ptcal fbes (MOF) f adatn gudng n the nea and md-nfaed (IR) [1-11] have ecently eceved clse attentn as they pmse cnsdeable advantage ve the sld ce cuntepats n applcatns elated t pwe gudance at almst any IR wavelength f mltay, ndusty and medcal applcatns, IR magng and sensng, and even TH tansmssn. Due t ts cmplexty, fabcatn f such wavegudes s an actve feld f eseach. Fu man methds have been dentfed f hllw ce fbe manufactung, each ffeng ts wn advantages and challenges. Fst methd s a depstn f metall-delectc flms n the nsde f a dawn capllay by lqud-phase catng [5,6]; techncal challenges n enfcng thckness unfmty n the esultant catngs lmt fbe length t the dstances f 10m. Secnd methd s a capllay stackng methd [7-9] whee glass capllaes ae aanged n a pedc manne and then dawn; s fa such fbes have been mstly demnstated t gude belw 3µ m due t the nn-tanspaency f slca and plyme mateals used n the fabcatn. Thd methd s a depstn f adally unfm thn flms n a dawn substate fbe by means f physcal chemcal vap depstn methds [10]; man challenge f ths technlgy s pesumably unfmty f the esultant catngs, and a thughput due t a elatvely slw depstn pcess. Fnally, flm llng pcess [11] stats wth a depstn f a glass (chalcgende) flm n tp f a plyme flm wth cnsecutve llng aund a mandel tube, tube etchng and dawng; ptental challenges nclude fbe pfle ptmatn whch s smewhat nntval due t a stctly pedc eflect gemety mpsed by the fabcatn methd, anthe ptental challenge s cntllng b-cmpatblty f a esultant fbe. In u eseach gup we study fabcatn f all-plyme hllw multlaye fbes. Althugh efactve ndex cntast between layes n an all-plyme Bagg fbe s elatvely small (at mst 1317./. ), as demnstated n [1] lqud ce all-plyme Bagg fbes can be desgned t gude vey well bth TE and TM lke mdes, whle gas flled all-plyme Bagg fbes can gude effectvely a TE plaed mde. We beleve that fabcatn smplcty, and ptental bcmpatblty f such fbes can be attactve f applcatns n b-medcal sect. Recently, u eseach gup has succeeded n develpng tw methdlges f fabcatn f multlayeed

2 all-plyme hllw pefms. One appach uses cnsecutve depstn f layes f tw dffeent plymes by slvent evapatn n the nsde f a tatng plyme claddng tube [13]. Othgnal slvents wee fund, and slvent evapatn pcess was develped f bth PMMA(Plymethyl methylacylate)/ps(plystyene) and PVDF(Plyvnyldene flude)/pc(plycabnate) mateal cmbnatns. In Fgue 1(a), a 30cm lng all-plyme pefm f 10 cnsecutve PMMA/PS layes depsted n the nsde f a PMMA claddng tube s pesented, whle n Fgue 1(b) pefm cssectn s shwn. Altenatve methd uses a cllng f tw dssmla plyme flms smlaly t [11]; n Fgue 1(c), a 0cm lng all-plyme pefm f 19 cnsecutve plystyene PVDF/PC s pesented, whle n Fgue 1(d) cssectn f a dawn fbe s shwn. Fgue 1. a) 30cm lng all-plyme pefm f 10 cnsecutve PMMA/PS layes depsted n the nsde f a PMMA claddng tube b) PMMA/PS pefm cssectn c) 0cm lng pefm f 19 cnsecutve PVDF/PC layes d) cssectn f a dawn PVDF/PC fbe. Afte pefm s fabcated, hllw MOFs ae manufactued by pefm heatng and dawng. Gemety f the fnal fbe can be sgnfcantly nfluenced by cntllng vaus paametes n the dawng pcess such as tempeatue dstbutn n a funace, fbe dawng and pefm feed velctes, as well as pessuatn f the hllw ce. Meve, f seveal mateals ae used n a sngle pefm, dawng pcess can be nfluenced geatly by the msmatch n the vscstes f the cnsttutve mateals. Pevus studes n fbe dawng have fcused manly n spnnng mlten theadlnes [14,15] dawng cnventnal sld ptcal fbes [16,17]. Dawng f hllw fbes was fst studed n [18] whee the asympttc thn-flament equatns wee btaned but the effects f suface tensn wee neglected. A me cmplete analyss s gven n [19-1]. When elatvely hgh tempeatues lw speeds ae used, the suface tensn fce can be f mptance and t can cause an patal even cmplete cllapse f the hllw ce. Ths hle cllapse affects nt nly the ntal at between the nne and ute dametes f the pefm but als the unfmty f layes thckness. The pupse f ths pape s t chaactee such a hle cllapse dung the dawng f hllw fbes when the suface tensn effects ae nn neglgble. In patcula, we nvestgate hw the hle cllapse s affected by standad cntl paametes such as daw tempeatue, daw at, feedng speed, ce pessuatn and msmatch f mateal ppetes n a multlaye.

3 BASIC EQUATIONS Schematc f a hllw multlaye pefm pfle dung dawng s shwn n Fgue. Fgue. Schematc f a hllw multlaye pefm dung dawng. Dffeent cls cespnd t dffeent mateals n a multlaye. F an ncmpessble axsymmetc steady flw the equatns f cnsevatn f mass and mmentum n cylndcal cdnates ae as fllws: 1 ( v ) v + = 0 ( ) v v p 1 τ τθθ τ ρ v + v = + + v v p 1 ( τ ) τ ρ v + v = ρg (1) () whee and ae the adal and axal cdnates, v and v ae the and cmpnents f the velcty vect v, ρ s a cnstant densty, p s a pessue, τ j s an exta-stess and g s a gavtatnal acceleatn. The cmpnents f a ttal stess tens σ ae σ = pδ + τ (3) j j j

4 The defntn f τ j depends upn the plyme mdel, and s dscussed n detals late. F these equatns, we need t specfy the bunday cndtns. At the ntefaces between dffeent layes the knematc cndtns ae v = R jv at = R (4) j whee Rj = Rj( ) dente the ntefaces between layes and the ndex j = 1, N s used t numbe them statng fm the nne ne. The pmes dente the devatve wth espect t. Snce the fst and the N-th ntefaces ae extenal ntefaces, we wll dstngush them by dentng R R1 and R RN f the nne and ute bundaes espectvely. Hllw ce can be pessued n de t cntl ts cllapse unde the actn f a suface tensn. In ths case, at the nne nteface the dynamc bunday cndtns ae σ n = ( γκ P ) n σ t= 0 whee γ dentes the suface tensn ceffcent, and 1 / 3 / κ = 1 R(1 + R ) R (1 + R ) (6) s the cuvatue, whle P s the hle vepessue (cnstant ambent pessue has n effect n the flw). Outwad-pntng nmal at the nne bunday n s defned as: T 1 ( ) 0 R n n n n =, θ, =,,, (7) 1+ R 1+ R whle T t = ( n, 0, n ) (8) s the unt tangent vect. In a smla way, the dynamc bunday cndtns at the ute bunday ae σ n = γκ n (9) σ t= 0 whee n, t and κ satsfy the same equatns as n, t and κ espectvely wth R eplaced by R. At the ntefaces between the ntenal layes, we wll cnsde a cntnuus stess and velcty. In the axal dectn the bunday cndtns ae the knwn values f the dawng ( V d ) and the feedng ( V f ) velctes. Futheme, as an ntal cndtn, the values R j (0) ae knwn. THIN FILAMENT EQUATIONS One f the basc dmensnless paametes n the pblem s the at between the pefm adus and the length f the neck dwn egn defned as ε. In the case when ε <<1 a s called thn flament appxmatn can be used. Thee ae tw dffeent appaches f smplfyng the equatns n ths case. In the fst appach [19,], the vaables ae expanded as pwe sees n ε and nly the dmnant tems ae etaned n the equatns. In the secnd appach [18], whch we als fllw n ths pape, the equatns ae aveaged ve the csssectn at each value f. (5)

5 The aveage ϕ ( ) f a vaable ϕ ( ) s defned as 1 R ϕ( ) = ( ) d π ϕ, (10) R π R R F the axal velcty, the assumptn v = v s made explctly. We nte fst that f a slw vayng thn flament; R j <<1, and by neglectng tems f the de R j the bunday cndtns (5) and (9) take the fllwng fm: γ γ σ = P σ = R R γ γ σ= + P R at = R σ = R R R at = R (11) γ γ σ R = P σ R = R R Multplyng the -cmpnent f the mmentum equatn () by R, cnsdeng d d R π τd 0 R π, ntegatng fm R t, neglectng the netal tem because f the small value f the adal velcty and usng the bunday values f σ gven by (11), we btan ( R R ) R P p τ + τ γ + θθ = + R R Multplyng the -cmpnent f the mmentum equatn () by π, ntegatng fm R t R, usng the bunday values f τ = σ gven by (11) as well as the equatn (1) and neglectng tems f elatve de R j, we btan Q Q Q τ + τθθ ρ v = τ γ ( R R ) ρg v (13) v whee πq = π R R v s the cnstant vlumetc flw ate. Ths s the axal fce balance equatn. NEWTONIAN FLOW In geneal plymes ae vscelastc and the vscsty depends n the knematcs f the flw. But at hgh tempeatues, f sme plymes such as PMMA and PS, we can stll cnsde a Newtnan mdel. The cnsttutve equatn f the Newtnan flud s T τ = η(, )( v + v ) (14) whee η s the vscsty whch n u case depends n and. In ths pape we cnsde a unfm tempeatue n any gven css-sectn, thus T s nly a functn f. We als assume an axal vaatn f the tempeatue dependent vscsty f each cnsttuent mateal. Fm the cntnuty equatn ne fnds: ' (1)

6 v A v = + (15) whee A= A( ) s a functn t be detemned late. The exta-stess tens takes the fm: A η v A τ = 0 η v 0 (16) 0 0 η v v whee the cmpnent τ = η has been neglected. Fm (16) the fllwng elatn hlds f the aveaged cmpnents f an exta-stess tens τ + τθθ τ = 3η v (17) and the axal fce balance equatn (13) can be wtten n the fm ' 3 v Q ρqv = ηq + γ ( R + R ) + ρg (18) v v The expessn f A s btaned by dectly ntegatng the -cmpnent f the mmentum equatn () fm R t R and usng the bunday values f σ gven by (11) P γ ( 1/ R + 1/ R ) A = (19) R η ( ) 4 R 3 d Cmbnng the knematc bunday cndtns (4) wth elatn (15), we btan NUMERICAL SOLUTION ' j Rv = A (0) The last thee equatns may be cnsdeed as a system f cupled dffeental equatns f v ( ), A ( ) and Rj ( ). Ths system f equatns can be slved wth an teatve methd. Statng fm an abtay ntal dstbutn f v ( ), say a lnea dstbutn between the feedng and dawng speed, the ntal value equatns (0) can be ntegated n de t btan Rj ( ) wth the value f A gven by (19). Ths functns ae then used t slve the bunday value pblem (18) t btan a new functn v ( ) passng s at the next teatn. F the examples gven late n ths pape we have tested ths pcedue and t cnveges vey fast (less than 00 teatns). In mst cases f pactcal mptance, netal, gavtatnal and capllay tems n (18) can be neglected and t takes a smple fm v η = C (1) v

7 whee C s a cnstant. Ths equatn, whch s nw uncupled fm the the tw, can be easly ntegated t gve d 0 η( ) V d v( ) = exp lnvf + ln () L d Vf 0 η( ) whee L s a funace length. Once the axal velcty s knwn, the ntal value equatns (0) can be easly ntegated t btan the pfle f a dawn stuctue. One f the key aspects f hllw pefm dawng s the patal even cmplete cllapse f a cmpessble ce as a esult f the suface tensn fces actng at the fee bundaes. In what f p R R fllws we chaactee the hle cllapse by the at f / p, whee ndex f defnes the dawn R R fbe, whle ndex p defnes the pefm. Hle cllapse typcally esults n a faste eductn f a smalle ce adus cmpaed t the lage ute adus. Thus, statng wth dentcal thcknesses f the same mateal layes n a pefm, n a dawn fbe the nne layes wll becme thcke than the ute nes. We wll h chaactee the thckness nn-unfmty by the at h between the thckness f the ute laye and the thckness f the nne ne, assumng they wee equal n a pefm. In the fllwng we nvestgate hw the hle cllapse and laye thckness nn-unfmty s nfluenced by vaus cntl paametes. As an example, we cnsde dawng f a multlaye hllw Bagg fbe pefm whee claddng tube and ne f the tw mateals f a multlaye s PMMA, whle the the mateal s a dffeent plyme. Mateals n cnsecutve plyme layes ae altenated t ceate a pedc multlaye stuctue. Effects f daw at, tempeatue and vscsty msmatch In u calculatns we assume a unaxal tempeatue dstbutn wth a maxmum at the funace cente (Fgue 3). In the fllwng when we vay the maxmum value f the tempeatue (T) we smply escale the whle pfle. We assume that Newtnan vscsty f PMMA beys an Ahenus type dependency [3] η( T ) = exp 935 N. s/ m T 170 (3)

8 Fgue 3. Tempeatue dstbutn n the funace. In u fst calculatn vscsty f the the plyme s assumed t be smply tw tmes hghe than that f PMMA. Althugh we ecgne that t mdel cectly the flw f a patcula plyme we need t use ts ppe tempeatue dependent vscsty, n u fst smulatn we athe want t hghlght the maj effect f addng anthe mateal nt a pefm. Patculaly, we want t nvestgate hw the ce cllapse s affected by the vscsty f the secnd mateal despte f ts small vlume factn n the pefm. We cnsde dawng f a pefm wth extenal and ntenal damete and 5.4mm espectvely. PMMA tube s cated n the nsde wth 5 altenated layes f PMMA and anthe plyme wth a vscsty tw tmes hghe, each ne f them havng a thckness f 50 µ m. The value f suface tensn ceffcent s cnsdeed cnstant f exte ntefaces γ = 003. N/m [4] and bth denstes ae cnsdeed t be 1195kg/m 3 [3]. We als assume a funace length L = 30 cm, a cnstant pefm feedng velcty V f = 5. µ m/s and a e pessuatn P = 0. Fst, we cnsde the effects f vayng the dawng at defned as D = Vd/ Vf and the maxmum tempeatue n a funace. In Fgue 4(a), sld lnes epesent the paamete f a ce f p R R cllapse f / p as a functn f a daw at D R R = Vd/ Vf f dffeent values f the maxmal tempeatue. Dashed lnes epesent the paamete cuves esultng n a cnstant utsde damete D R = 15µ m and D = 50µ m afte the daw. F cmpasn, n dtted cuves we pesent the ce cllapse f n the plyme s pesent n the pefm (dawng f a smple PMMA tube f the same nne and ute ad as a multlaye pefm). We see that t pevent hle cllapse hghe daw ats and lwe tempeatues have t be used. Bth cases demand hghe daw fce whch mght lead t fbe beakage. We als bseve that the cllapse f the tube wth the same thckness as the multlaye pefm s me pnunced because f a lwe aveage vscsty. Effect f the tempeatue n a ce cllapse s easy t undestand as vscsty deceases apdly wth eductn f the tempeatue, thus hndeng the hle cllapse.

9 (a) (b) Fgue 4. (a) Nmaled at RR / as a functn f a daw at D f dffeent tempeatues. Sld lnes descbe dawng f a multlaye pefm. Dtted lnes descbe dawng f a smple tube havng exactly the same dmensns as a multlaye pefm. Dashed lnes epesent the cuves f a cnstant utsde damete. (b) Rat h / h between the nne and ute laye thcknesses as a functn f the daw at f dffeent tempeatues. Effect f the daw at s me subtle. Statng wth pefms f the same damete, daw at ncease leads t the eductn f a esultant fbe css-sectn. As a cnsequence, the fces f suface tensn becme me pnunced, thus favng the hle cllapse. Ths s vecmpensated by the fact that nceasng the daw at leads t the hghe axal velctes, thus the tme a css-sectn spends n a melted ne dmnshes whch wks aganst the hle cllapse.

10 In Fgue 4(b), thckness nn-unfmty paamete s pesented as a functn f the daw at f dffeent values f the maxmum tempeatue. The cuves ae smla t thse f the hle cllapse n Fgue 4(a). Fm mass cnsevatn, the at between the css-sectn aeas f dffeent layes emans cnstant fm whch t fllws that the thckness nn-unfmty paamete s pptnal t the hle cllapse. We nw descbe n me detals the effect f msmatch n the plyme vscstes when tw dffeent mateals ae used n the same pefm. As seen n Fgue 4(a), the hle cllapse s cnsdeably less pnunced f the multlaye stuctue despte the fact that the hghe vscsty mateal ccupes nly a vey small factn f the ttal vlume and ts vscsty s nly tw tmes hghe than that f PMMA. In what fllws we assume that multlaye pefm s made f PMMA and anthe plyme. F the vscsty f a secnd mateal smla Ahenus law as f PMMA s assumed (3) 1 1 η( T ) = η0exp α (4) T T0 whee η 0 and α ae cnstant ceffcents and T 0 s a efeence tempeatue. Fst, we nvestgate the effects f changng η 0 whle keepng the the paametes unchanged, whch cespnds t the case f usng the same plyme, but wth a dffeent mlecula mass. Secnd, we nvestgate the effects f changng T 0 whch cespnds t the case f vayng the plyme mateal. In Fgue 5, we cnsde dawng f pefms f vaus cmpstns at a fxed maxmum funace tempeatue f 190 C and a daw ate D = Multlaye pefm gemety s the same as descbed abve; msmatch n the plyme vscstes s descbed n tems f the ats f the mateal paametes η0/ η 0, PMMA and ( T0 T0, PMMA) / T0, PMMA. Fm Fgue 5, we see that the hle cllapse depends sgnfcatvely n the vscsty f a secnd mateal and can be pevented by chsng a plyme wth an apppate vscsty. (a)

11 (b) Fgue 5. Effect f msmatch n the vscstes f mateals n a multlaye n ce cllapse (sld lnes), and laye nn-unfmty (dtted lnes).maxmum funace tempeatue s T = 190 C, daw ate D = Effects f the pessuatn and pefm feedng velcty Othe paametes that nfluence ce cllapse ae the hle vepessue and the feedng speed. By nceasng the hle pessue we expect t educe the hle cllapse. Als, f a gven daw at, by nceasng the pefm feedng speed we expect eductn f the ce cllapse as fbe css-sectn wuld spend less tme n the melted ne. (a)

12 (a) Fgue 6. The hle cllapse and thckness nn-unfmty as a functn f the hle vepessue (a) and feedng speed (b). Maxmal funace tempeatue s T = 190 C, daw ate D = We cnsde dawng f the same pefm as n a pevus sectn usng the daw at D = 5000, whle the the daw paametes eman unchanged. In Fgue 6, ce cllapse and laye thckness nn-unfmty ae pesented as functns f the hle vepessue P and feedng speed V f. As expected, the hle cllapse s vey senstve t pessuatn, and n pncple, can be educed by nceasng the pessue. Tme evlvng tansent daw smulatns, nt dscussed n ths pape, als shw that abve a cetan ctcal value f an vepessue, whch n u case s less than 7Pa, even f the fbe des nt blw up mmedately, the dawng pcess neve eaches ts steady state. A me subtle way f cntllng the ce cllapse s by changng the pefm feedng speed, althugh, f a gven daw at, ths culd be lmted by the maxmal daw velcty. CONCLUSION Dawng f multlaye hllw plyme fbes was studed usng the thn flament appxmatn. A Newtnan mdel f the flw was cnsdeed. We have chaacteed suface tensn mtgated ce cllapse and clsely elated laye thckness nn-unfmty. We have demnstated that by vayng vaus cntl paametes such as funace tempeatue, feedng speed and pessuatn t s pssble t educe hle cllapse. Whle hle pessuatn pvdes a vey effectve way f cmpensatng f the hllw ce cllapse, t was fund that the fnal fbe dmensns ae vey senstve t the value f an vepessue. Meve, the daw pcess culd newe each a steady state f vepessue lage than a cetan ctcal value wee used.

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