Korea-Australa Rheology Journal Vol. 14, No. 1, March 2002 pp. 25-32 Polyethylene flow predcton wth a dfferental mult-mode Pom-Pom model R.P.G. Rutgers*, N. Clemeur, S. Muke 1 and B. Debbaut 2 Department of Chemcal Engneerng, Unversty of Queensland St Luca, Queensland 4072, Australa 1 Department of Chemcal and Metallurgcal Engneerng, RMIT Unversty, Melbourne Vctora, Australa 3001 2 Polyflow s.a., Louvan-la-Neuve, Belgum (receved Ocober 23, 2001; fnal revson receved February 11, 2002) Abstract We report the frst steps of a collaboratve project between the Unversty of Queensland, Polyflow, Mcheln, SK Chemcals, and RMIT Unversty, on smulaton, valdaton and applcaton of a recently ntroduced consttutve model desgned to descrbe branched polymers. Whereas much progress has been made on predctng the complex flow behavour of many - n partcular lnear - polymers, t sometmes appears dffcult to predct smultaneously shear thnnng and extensonal stran hardenng behavour usng tradtonal consttutve models. Recently a new vscoelastc model based on molecular topology, was proposed by McLesh and Larson (1998). We explore the predctve power of a dfferental mult-mode verson of the pom-pom model for the flow behavour of two commercal polymer melts: a (long-chan branched) low-densty polyethylene (LDPE) and a (lnear) hgh-densty polyethylene (HDPE). The model responses are compared to elongatonal recovery experments publshed by Langouche and Debbaut (1999), and start-up of smple shear flow, stress relaxaton after smple and reverse step stran experments carred out n our laboratory. 1. Introducton The rheologcal propertes of polymers depend sgnfcantly on ther topology. A long-chan branched polymer exhbts a pronounced stran-hardenng behavour n extensonal flow, whle the elongatonal vscosty of a lnear polymer melt hardly rses above the lnear vscoelastc response. In smple shear flow however, both lnear and long-chan-branched polymers show a strong shear-thnnng behavour. Complex combnatons of smple shear and elongatonal flow are often found n ndustral processng geometres, therefore, n some crcumstances, a sutable consttutve equaton should preferably be able to smultaneously predct the materal behavour n each type of flow. Classcal ntegral consttutve models, such as the KBKZ class, do not allow the smultaneous predcton of stran hardenng and shear thnnng n all types of complex flow. And despte progress s currently beng made to adapt the KBKZ model to accommodate ths behavour (e.g. Olley, 2000; Mtsouls, 2001), the numercal treatment of such consttutve equatons remans endowed wth sgnfcant dffcultes. Consttutve theores have focussed on smplfed representatons of the actual polymer archtecture, representng only the most domnant components of the topology. The *Correspondng author: rulander@cheque.uq.edu.au 2002 by The Korean Socety of Rheology very successful tube model of Do and Edwards (1986) has been the bass of much consttutve modellng research. McLesh and Larson (1998) have extended ths theory for branched polymers. The molecules are represented by a smple branched archtecture consstng of two pompoms of q arms each, lnked by a backbone that s confned n a tube. The key feature of ths consttutve model s the presence of separated relaxaton mechansms for the macromolecular orentaton and stretch. To facltate numercal computaton, the ntegral type model has been smplfed to a set of dfferental equatons whch s less demandng n terms of computaton tme. Although n recent papers by Rubo and Wagner (2000) and Wapperom and Keunngs (2001) some dscrepances between the ntegral model and ts dfferental approxmaton have been reported, most current research uses the dfferental verson of the model (Inkson et al., 1999; Bshko et al., 1999, Blackwell et al., 2000, Verbeeten et al., 2001). Inkson et al. (1999) have extended the model to a theoretcal mult-mode model where they show very good fttngs and predctons for varous LPDE melts. It s clamed that the model parameters provde an nsght n the underlyng molecular structure. For example, the stran acheved before the flament break-up n an elongatonal experment s suggested to be a consttutve property. A modfcaton of the orgnal dfferental model was suggested by Blackwell et al. (2000) takng nto account drag-stran couplng of the branch pont. Ths allows the Korea-Australa Rheology Journal March 2002 Vol. 14, No. 1 25
R.P.G. Rutgers, N. Clemeur, S. Muke and B. Debbaut arms to retract nto the tube before the maxmum stretch s acheved and leads therefore to a smoother transton for the elongatonal vscosty when the pom-pom molecule s fully stretched. Öttnger (2001) analysed the thermodynamc admssblty of the pom-pom model n the framework of nonequlbrum thermodynamcs. He found the pom-pom model thermodynamcally admssble, but suggested a modfcaton of the orentaton relaxaton that produces a non-zero second normal-stress dfference. Verbeeten et al. (2001) have very recently proposed alteratons to the model, removng the fnte extensblty condton and addng a non-zero second normal stress dfference. Predctons for a low and hgh densty polyethylene were shown n close agreements wth expermental data. To date, complex flow smulaton of the pom-pom model s only just startng to be nvestgated. Bshko et al. (1999) and Wapperom and Keunngs (2001) performed numercal smulatons of contracton and contracton/expanson flows wth a sngle-mode model usng ether the dfferental or the ntegral versons of the model. Usng a sgnfcant solvent vscosty (possbly to avod numercal nstabltes) Bshko et al. (1999) showed smlartes wth expermental brefrngence patterns obtaned for a branched polymer, although no drect comparson s made. Very recently, Lee et al. (2001) reported two-dmensonal transent smulatons of the flow n a multpass rheometer. They compared the smulated stresses wth expermental brefrngence data for branched and lnear polymers. Although the stress pattern s found to be dfferent for the lnear and branched polymer, the agreement s only qualtatve. In ther paper, the authors also ntroduce a second modfcaton to the orgnal pom-pom model, by ntroducng a varable orentaton relaxaton tme for deceleratng flows. In the present paper, the modfed, although not ncludng ths latest modfcaton, approxmate dfferental verson of the pom-pom model s used to characterse a commercal (long-chan branched) Low Densty and (lnear) Hgh Densty Polyethylene (LDPE and HDPE respectvely). The HDPE studed here, although assumed to have a lnear archtecture, shows marked stran-hardenng n extensonal flow, justfyng the use of a model prmarly desgned for long-chan branched materals. The lnear parameters of the model are selected usng the lnear relaxaton spectrum whle the non-lnear parameters are dentfed on the bass of transent unaxal extensonal vscosty curves only. The resultng model s then utlsed to predct the behavour for rheometrc experments. Stress relaxaton after sngle and mult-step stran deformatons are predcted and compared to expermental data for the LDPE melt. Transent recovery experments after unaxal extenson are smulated for the HDPE. For all flow stuatons, the pom-pom model results are compared to the predctons obtaned wth a Phan Then- Tanner model (PTT) (Phan-Then and Tanner, 1977) whose parameters are ftted usng the same expermental data as the pom-pom model as well as steady shear vscosty obtaned usng the Cox-Merz rule. 2. Consttutve equatons The pom-pom model, proposed by McLesh and Larson (1998) s derved from macromolecular concepts that stem from the Do-Edwards tube model (Do and Edwards, 1986). The molecules are represented by the smplest branched archtecture, consstng of a backbone lnkng two pom-poms of q arms each. The topologcal constrants caused by neghbourng chans are represented by a confnng tube. The chan s not allowed to move more than the tube dameter perpendcular to the tube axs, whle along the tube, the pom-poms wll not allow the backbone to reptate or retract as freely as a lnear polymer. Ths last feature leads to a strong stran-hardenng behavour. In flow, the tube wll be stretched and wll drag the backbone wth t. When a certan stretch s attaned, t becomes more favorable to lose entropy by wthdrawng the free ends nto the tube than to contnue to stretch the backbone. The requrement that Gaussan chan statstcs are mantaned n equlbrum mples that the maxmum allowed stretch sustanable by the backbone s equal to the number of arms, q. Beyond ths lmt, the tenson n the backbone s suffcent to wthdraw the free ends nto the tube. The dynamc varables nvolved n the descrpton of the molecule are the dmensonless stretch rato of the backbone λ(t), the orentaton tensor S(t) whch measures the dstrbuton of unt vectors along the backbone and s c (t), the dmensonless dstance of arms wthdrawal nto the tube. McLesh and Larson (1998) have shown that the arms wthdrawal nto the tube only has an nsgnfcant contrbuton to the stress and can therefore be gnored (s c (t)=0). Thus, the man dfference wth other consttutve models s the separaton of orentaton and stretch relaxaton mechansms n the mathematcal descrpton of the model. In order to smulate a polyethylene wth molecular weght and branchng dstrbuton, a mult-mode approxmate dfferental model proposed by Inkson et al. (1999) s used. Ths mult-modal approach descrbes polydsperse long chan branched molecules by a theoretcal blend of pom-pom molecules wth dfferent number of arms and by assgnng orentatonal relaxaton tmes from the lnear relaxaton spectrum. The relaxatons of the dfferent segments of the long chan branched molecules are decoupled from each other. A correcton suggested by Rubo and Wagner (2000) s adopted as well as a local branch-pont dsplacement factor ν *, proposed by Blackwell et al. (2000). The followng set of equatons s thus used: σ() t = 3g λ 2 ()S t (), t (1) 26 Korea-Australa Rheology Journal
Polyethylene flow predcton wth a dfferental mult-mode Pom-Pom model A 1 () t + ---- ( A () t I) = 0, τ b S () t = -------------------------- A () t trace( A () t ), D -----λ Dt () t = λ () v:s t ( () t ) ---- 1 ( λ τ () t 1)e v * ( ( λ () t 1) ) s strctly for λ () t < q, λ () t = q, otherwse (2) (3) (4) Equaton (1) relates the total extra-stress tensor σ to the orentaton tensor S and the backbone stretch λ, orgnatng from the ndvdual modes. The subscrpt dentfes the mode number. A s an auxlary tensor used to obtan the tensor S through equaton (3). Equatons (2) and (4) descrbe the evoluton of the orentaton tensor and the backbone stretch respectvely. The plateau modul g and the backbone orentaton relaxaton tme τ b are ftted usng lnear vscoelastc measurements. The non-lnear parameters, determned from the transent elongatonal vscosty curves only, are the relaxaton tme for stretch τ s and the * number of arms q. The parameter v was chosen equal to 2/q as recommended for LDPE melts by Blackwell et al. (2000). The pom-pom predctons are compared to a mult-mode Phan-Then-Tanner (PTT) model (Phan-Then and Tanner, 1977). In ths model, the total extra-stress tensor σ also results from ndvdual contrbutons σ, whch obey the followng consttutve relatonshp: ε exp ---tr( σ (5) g ) σ + τ 1 ξ --- 2 σ + ξ ---σ 2 = g τ ( v + v T ) where τ are the relaxaton tmes and ε, ξ are the non-lnear parameters. The symbols and stand for the upperand lower-convected tme dervatve operators, respectvely. 3. Materals and experments 3.1. Charactersaton of lnear and non-lnear model parameters Two dfferent materals were charactersed and smulated n ths study. The LDPE melt (referred to as LD20) of BP has a molecular weght M w of 82995 (from GPC data) and a hgh polydspersty of Mw/Mn 4.94 (from GPC data). The precse molecular structure s unknown, but t s expected that a hgh proporton of long chan branches exsts, as s typcal for LDPE. The shear behavour of the materal at 180 o C was charactersed at The Unversty of Queensland (UQ) and the extensonal behavour at 180 o C was determned at the Royal Melbourne Insttute of Technology Unversty (RMIT). Dynamc rheometry n the lnear vscoelastc doman was performed under Ntrogen on an ARES rheometer (Rheometrcs Scentfc) yeldng the relaxaton spectrum for a frequency range of 10 2 to 10 2 s 1. Fg. 1. Transent unaxal extensonal vscosty for the LDPE at 180 C. The symbols represent the expermental data, the contnuous curves refer to the pom-pom fttng and the dotted curves to the PTT fttng. For the LDPE the relaxaton spectrum (τ b, g ) was obtaned through a downhll smplex method where the tme constants are spaced evenly on the logarthmc scale. The elongatonal behavour was charactersed wth an RME (Elongatonal Rheometer for Melts - Rheometrc Scentfc) for stran rates of 0.01, 0.1 and 0.86 s 1 (Fgure 1) and used for the fttng of the non-lnear parameters. In ths Messner-type rheometer two metal conveyor belts are used to hold the rectangular polymer sample by ts extremtes (Messner, 1994). Between the conveyor belts, the sample s supported by a cushon of nert gas. The belts rotate n opposte drectons and, as a result, the melt s elongated homogeneously at a constant stran rate. As observed by (Schwezer, 2000), the real stran rate appled must sometmes be corrected, n partcular for hgh stran rate values. The second materal, a HDPE melt, s a general-purpose blow mouldng resn produced by Solvay. The expermental data publshed and descrbed by Langouche and Debbaut (1999) are used here and were obtaned n the same manner as for the LDPE. 3.2. Charactersaton of non-lnear behavour n reverse flows: reverse step-stran of LDPE The purpose of the correspondng mathematcal exercse here s to assess the predctve capablty, and hence the sutablty of the selected consttutve models and concepts n flow stuatons charactersed by relatvely smple and well-controlled flow knematcs, whch nvolve reversal mechansms. In order to evaluate the predctve capabltes of both PTT and pom-pom models for these melts, a comparson between experment and predcton must be carred out for other flow types than those used to ft the parameters of the model. We consder a sngle step stran experment (referred to as type A, analogous to Wagner, 1998) wth varous stran ampltudes, as well as several double Korea-Australa Rheology Journal March 2002 Vol. 14, No. 1 27
R.P.G. Rutgers, N. Clemeur, S. Muke and B. Debbaut step stran experments, as suggested by Wagner (1998). In the double step experments, three condtons are consdered. These experments, referred to as type-b, -C and -D, start from rest state and nvolve a step stran of assgned ampltude γ 1 (here γ 1 =4). After a specfed tme nterval (here t=0.2 s), a reverse step stran γ 2 of -γ 1 /2, -γ 1 and -2γ 1 s appled for the type-b, -C and -D experments respectvely. Not only do these experments represent a severe test for the consttutve models, they are furthermore hghly representatve for the complex flow knematcs undergone by melts n ndustral processes, where sequences of contracton and expanson are often met. 3.3. Charactersaton of non-lnear behavour n reverse flows: transent extensonal recovery of HDPE Varous types of experments are recommended to test consttutve equatons (Tanner, 2000). Amongst these, transent extensonal recovery has some smlarty wth the reverse-step-stran experment descrbed above but s also a so-called strong flow, n vew of ts elongatonal character. In ths type of experment, the sample s allowed to recover freely after beng stretched under well-controlled condtons. The stress relaxaton s observed, as well as the transent deformaton of the sample, whch s no longer mposed. Transent recovery experments were performed by Langouche and Debbaut (1999) for the HDPE melt and compared to the mult-mode PTT model predctons. In the present paper, a smlar analyss s performed usng the pom-pom model and the results are compared to the expermental data and the PTT model predctons. 4. Predcton of reverse step stran behavour of LDPE For the LDPE smulatons, both pom-pom and PTT models were coded usng Matlab usng an algorthm adapted for stff problems. The resultng ODE equatons were solved usng a thrd-order Runge-Kutta scheme wth adaptve stepsze control. In transent extensonal deformaton, pronounced stran hardenng was observed for both materals. The model parameters ftted to the lnear vscoelastc behavour and to the transent elongatonal data are lsted n Table 1. The expermental extensonal data as well as the ftted model response are shown n Fgure 1. The agreement s very good for the stran rate of 0.86 s 1 and 0.1 s 1 but not for the stran rate of 0.01 s 1 probably because of the lack of lnear vscoelastc data at low frequences. A smlar problem was already reported by Inkson et al. (1999). Snce most of the experments dscussed below nvolve characterstc tmes well below the longest relaxaton tme, t s beleved that ths dscrepancy at low stran rates wll not sgnfcantly affect the model predctons. To start wth, we evaluate the predctve capablty of the Table 1. Parameters for the seven-mode pom-pom and Phan- Then-Tanner (PTT) models g [Pa] τ b (τ for PTT) pom-pom (ν *=2/q I ) PTT [s] τ b /τ s q ξ ε 1 6.282 10 4 1.000 10 2 2 2 4.0 10 3 0.129 2 1.113 10 4 4.641 10 2 2 2 4.0 10 3 0.232 3 8.760 10 3 2.154 10 1 2 2 4.0 10 3 0.114 4 2.405 10 3 1.000 10 0 2 2 1.2 10 2 0.124 5 4.968 10 2 4.641 10 0 2 16 1.2 10 2 0.468 6 4.662 10 1 2.154 10 1 2 20 4.0 10 3 0.352 7 3.156 10 0 1.000 10 2 1.5 32 4.0 10 3 0.438 Fg. 2. Expermental stress profles (a) and stran profles (b) for type-b, (open crcles) -C (open trangles) and -D (closed damonds) experments. Insert shows on a lnear tme scale the detal of the reverse step stran. selected models for the LDPE melt response n sngle stepstran. Experments were performed on the ARES rheometer wth ar-bearng actuator, wth stran ampltudes of 0.1, 2 and 4. The actual mposed stran as recorded by the rheometer (rather than the deal step-stran) was smulated to compute the stress response for both the PTT and pompom consttutve models. A very good agreement was found between the data and the predctons. The PTT model exhbts oscllatons when approachng full relaxaton, leadng to negatve values n the stress response wthn a relatvely narrow nterval around zero. Ths mechansm s a model artefact smlar to the one reported by Georgou and Vlassopoulos (1998) and Tanner (2000) for a Johnson-Segalman flud model. The varous double-step stran experments provde a deeper nsght n the predctve performance of the model. In Fgure 2(b) the stran as expermentally recorded s shown. In all cases, there s a dscrepancy between the deal step stran and the actual mposed stran: A stran growth develops durng a tme nterval of about 0.06 s after the start of the experment. A smlar transent s found when the reverse stran s appled. Moreover, for type-c and -D experments, the stran reversal exhbts an under- 28 Korea-Australa Rheology Journal
Polyethylene flow predcton wth a dfferental mult-mode Pom-Pom model shoot below the assgned value durng a short tme nterval. Most probably, these expermental artefacts are n connecton wth the nerta of the devce, and hence cannot easly be crcumvented. The actual flow knematcs have therefore been consdered to predct the PTT and PomPom model responses. Fgure 2(a) shows the development of the shear stress for all three double-step stran experments. Frst of all, we fnd n all three cases an dentcal stress response to the frst step stran: ths ndcates excellent repeatablty of the experments. The stress development exhbts a rapd growth n response to the appled step stran. Once the nomnal stran s reached, the stress starts ts decay untl, at tme t=0.2 s, the reverse stran s appled, and from there, dfferent responses are expected, correspondng to the dfferent double-step stran experments. All three experments are dscussed below, but the model predctons are only shown graphcally n Fgure 3, for the most severe type-d experment. For the type-b experment (γ 2 = -½γ 1 ), once the reversal stran s appled, the stress almost nstantaneously changes sgn. Durng decay, ths stress remans negatve for a short tme nterval of less than 0.2 s. After that tme, the stress becomes postve agan, presumably because the stress from the orgnal larger postve stran s stll decayng n the materal. A very good agreement s found between the predctons obtaned wth both the PTT and pom-pom consttutve models and the correspondng expermental data for the total tme nterval. Although both models appear to slghtly over-predct the peak n the ntal stress response ths s attrbuted to the sparse expermental data due to the extremely short tme nterval. In the type-c experment (γ 2 = γ 1 ), upon reversal of the stran a negatve stress peak s obtaned of almost equal magntude to the orgnal stress peak response to the frst step stran. Ths stress subsequently decays monotoncally to zero, wthout becomng postve. Agan a very good agreement s found between predctons and expermental data for the total tme nterval for both models. The predctons wth the PTT model are essentally the same as those obtaned wth the pom-pom model. However, the prevously mentoned oscllatng response of the PTT model develops after the second stran s appled, although ths s actually apparent on a logarthmc plot only,.e. the correspondng values are close to zero. Fnally, as shown n Fgure 3, when the second stran s appled n the type-d experment (γ 2 = 2γ 1 ), the stress rapdly becomes negatve and exhbts smlar behavour as durng the frst stran, wth an opposte sgn. The stress remans negatve untl complete relaxaton. A smlar behavour s predcted wth the pom-pom model, whch s also able to accurately render the somewhat sophstcated stress response at the nstant of the stran change. At ths pont the PTT model performs less well n predctng the Fg. 3. Stran hstory and shear stress response for type-d experments. Symbols represent expermental data, contnuous curve Pom-Pom model response, dotted curve PTT response. overshoot, and the predcted response s anew affected by the oscllaton mechansm already mentoned above. 5. Rheometrc predctons for HDPE For the HDPE melt, a 9-mode pom-pom model was ftted to the publshed lnear vscoelastc and unaxal extenson data. (Langouche and Debbaut, 1999). Fgure 4 shows that good fts of the transent unaxal extenson data are obtaned for all stran rates; thanks to the wder relaxaton spectrum, a better ft s obtaned at 0.01 s 1 than achevable for the LDPE (Fgure 1). In Table 2, we report the lnear and non-lnear parameters that were dentfed for the pompom model. The branchng parameter q dsplays a narrow dstrbuton of low values, as expected for ths predomnantly lnear molecular archtecture. To obtan a good fttng of the unaxal elongatonal propertes, the recommendatons gven by Inkson et al. (1999) where not strctly followed n the selecton of the other non-lnear parameters Fg. 4. Transent unaxal extensonal vscosty for the HDPE at 180 C. The symbols represent the expermental data, the contnuous curves refer to the ftted pom-pom response. Korea-Australa Rheology Journal March 2002 Vol. 14, No. 1 29
R.P.G. Rutgers, N. Clemeur, S. Muke and B. Debbaut Table 2. Parameters for the nne-mode pom-pom and Phan- Then-Tanner (PTT) models g [Pa] τ b (τ for PTT) [s] Pom-pom PTT τ b /τ s q ν * ξ ε 1 1.803 10 5 4.495 10 3 3.02 1 0.0 0.067 0.093 2 7.561 10 4 1.976 10 2 3.00 1 0.0 0.065 0.565 3 3.216 10 4 8.684 10 2 3.00 1 0.0 0.113 0.919 4 1.307 10 4 3.817 10 1 11.7 1 0.0 0.074 0.920 5 4.876 10 3 1.678 10 0 28.4 1 0.0 0.142 0.271 6 1.557 10 3 7.375 10 0 49.2 2 1.0 0.206 0.543 7 4.822 10 2 3.241 10 1 10.8 4 1.0 0.246 0.146 8 7.868 10 1 1.425 10 2 2.28 6 1.0 0.310 0.039 9 3.573 10 1 6.262 10 2 3.01 6 0.3 0.531 10 5 (τ s and ν *). The parameters for the PTT model were adopted from Langouche and Debbaut (1999). The pom-pom fttng of the lnear vscoelastc data and the predcton of the steady shear behavour s gven n Fgure 5. Assumng valdty of the Cox-Merz rule for the expermental data, the pom-pom predcton slghtly overpredcts the shear thnnng behavour of the materal. In transent elongatonal recovery experments usng an RME devce, an ntal stress state s generated by stretchng the sample at a gven stran rate up to a gven Hencky stran. Once the desred stran s acheved, the sample s cut at one end, and ts recovery s measured by recordng the length L(t) usng vdeo equpment. In order to predct the response n terms of the stran, an explct pom-pom equaton n terms of stran as a functon of stress and stran hstory must be formulated. The recovery after elongaton occurs n two stages: At tme t 0 the materal undergoes an nstantaneous recovery resultng n an nstantaneous change n stran E. Fg. 5. Lnear vscoelastc and steady shear behavour of HDPE. Symbols represent expermental data; contnuous curves the ftted Pom-Pom lnear vscoelastc behavour; dashed curve the predcted steady shear pom-pom response. To deal wth the large number of tme steps nvolved n the recovery smulaton, a C ++ program was developed. The recovery factor (.e. the rato of the current length L(t) over the ntal sample length L 0 after stretch) s plotted n Fgure 6 for varous extenson hstores. Predctons of the pom-pom model are n good agreement wth expermental data for moderate stran rate. The pom-pom predcton slghtly overestmates the recovery after extenson wth a Hencky stran of 2 and a stran rate of 1. The usual expermental sources of error may explan ths. In addton, as suggested by Langouche and Debbaut (1999), an rrevers- At t 0 the stran rate and stran become ε = Eδ( t t 0 ) and ε = ε 0 + E respectvely, whch yelds for the components of the auxlary tensor A (the subscrpt s momentarly omtted for sake of clarty): A 11 ( t + 0 ) = A 11 ( t 0 )e E and A 33 ( t + 0 ) = A 33 ( t 0 )e 2E (6) Assumng the affne stretch hypothess for all modes, one can wrte for the backbone stretch: λ( t + 0 ) = λ( t 0 ) 2A 11 ( t 0 )e E + A ( t )e 2E --------------------------------------------------------- 33 0 2A 11 ( t 0 ) + A 33 ( t 0 ) Assumng that the stresses become sotropc (.e. σ 11 ( t + 0 ) = σ 33 ( t + 0 )) as a result of the nstantaneous recovery, t s possble to solve for the nstantaneous change n stran E usng the known state obtaned at the end of the elongaton. S + + + 11, S 33 and λ are determned usng (6) and (7). Afterwards, a natural recovery occurs where the materal tends to slowly recover towards ts orgnal shape. Equaton (2) and equaton (4) become, respectvely: da --------- 11 + 1 ε + --- dt τ b A11 = da 33 --------- 1 2ε --- dt τ b A 33 = --- 1 τ b (7) and A 22 =A 11 (8) ----- dλ = λε ( S (9) dt 33 S 11 ) --- 1 ( λ 1)e ν* ( λ 1) τ s Durng ths second phase a vanshng total elongatonal stress dfference s assumed: ---- d ( σ (10) dt 33 σ 11 ) = ---- d 3g λ 2 ( A 33 A 11 ) -------------------------------------- dt A 33 + 2A 11 = 0 These equatons can be rewrtten nto an explct expresson for the stran rate: ε () t = 2λ Σ g ( A 33 A 11 )( λ 1)exp( ν * ( λ 1) ) 3λ 2 --------------------------------------------------------------------------------------- g ( A 33 A 11 ) + ------------------------------------ τ s ( A 33 + 2A 11 ) τ b ( A 33 + 2A 11 ) 2 --------------------------------------------------------------------------------------------------------------------------------------- 2λ 2 Σ g ( A 33 A 11 ) 2 9λ 2 -------------------------------------- g ( A 33 A 11 ) + ------------------------------- (11) ( + ) 2 ( + ) 2 A 33 2A 11 A 33 2A 11 30 Korea-Australa Rheology Journal
Polyethylene flow predcton wth a dfferental mult-mode Pom-Pom model Fg. 6. Recovery factor L(t)/L 0 versus tme for recovery experments performed at several Hencky strans and stran rates: (a) stran = 3; stran rate = 0.01, (b) stran = 1; stran rate = 0.1 and (c) stran = 2; stran rate = 1. The symbols represent the expermental data, the contnuous curve the pom-pom predcton and the dotted curve the PTT predcton. The dash-dotted lne represents the pom-pom predcton wth the modfed orentaton relaxaton tme (Lee et al., 2001). ble mechansm of molecular dsentanglement due to hgh deformaton undergone by the melt sample durng extenson may possbly be the orgn of ths devaton. Also, t s possble that the presently selected model s not endowed wth the best rreversble mechansm that would be needed n some stuatons. At ths pont t s nterestng to take the recent modfcaton of the pom-pom model (Lee et al., 2001) nto account n the smulaton. Unfortunately, wth ths model, t s no longer possble to fnd an explct formulaton for the stran rate as n formula (11). Therefore, some teratons have to be performed for each tme step n order to obtan the stran rate. Although ths addton to the model has very lttle mpact for moderate stran rate (not represented here for sake of clarty), the predcton for the hgher stran rate (curve c) s clearly mproved (Fgure 6). Another possble explanaton s that the actual stran rate acheved expermentally could be lower than 1 s 1, due to slp or other expermental dffcultes at such hgh stran rates. The pom-pom consttutve model used here yelds comparable accuracy to the PPT model used by Langouche and Debbaut (1999). Ths work s an early ndcaton that the pom-pom model may be sutable to mmc the complex rheologcal behavour of HDPE despte ts predomnantly lnear topology. 6. Conclusons The dfferental pom-pom model was successfully used to predct LDPE behavour under severe reverse smple shear deformatons. The predctve capabltes of the pompom model were shown to be comparable to, and at tmes more accurate than, the Phan-Then-Tanner model. Ths provdes a promsng bass to nvestgate ts behavour n complex deformatons such as planar contracton flow, where establshed models may not be easly applcable to LDPE. The pom-pom model was furthermore shown to predct relatvely well the behavour of a partcular grade of HDPE that exhbts stran hardenng. It appears that despte the expected predomnantly lnear archtecture of ths polymer, ts responses pont at topologcal features that behave lke branch ponts. Despte the encouragng results, there s room for possble mprovement for applyng ths rheologcal model to weakly branched materals. Also there are stll unanswered questons n connecton wth the mathematcal formulaton of the pom-pom model. Fnally, the present work provdes useful results for a possble valdaton of further mprovements made to the model. Acknowledgments The authors thank Fluent Inc., Mcheln, and SK Chemcals for fundng ths research. Ncolas Clemeur wshes to thank Cheng Heng Hye from Ngee-Ann Unversty, Sngapore, for hs precous help n conductng smple shear rheometry at The Unversty of Queensland, and to Andrew Chryss of RMIT Unversty, Melbourne, for RME extensonal rheometry. References Bshko, G.B., O.G. Harlen, T.C.B. McLesh and T.M. Ncholson, 1999, Numercal smulaton of the transent flow of branched polymer melts through a planar contracton usng the 'pompom' model., J. Non-Newton. Flud Mech. 82, 255-273. Blackwell, R.J., T.C.B. McLesh and O.G. Harlen, 2000, Molecular drag-stran couplng n branched polymer melts, J. Rheol. 44, 121-136. Do, M. and S.F. Edwards, 1986, The Theory of Polymer Dynamcs, Oxford, Clarendon Press. Georgou, G.C. and D. Vlassopoulos, 1998, On the stablty of the smple shear flow of a Johnson-Segalman flud, J. Non- Newton. Flud Mech. 75, 77-97. Inkson, N.J., T.C.B. McLesh, O.G. Harlen and D.J. Groves, 1999, Predctng low densty polyethylene melt rheology n elongatonal and shear flows wth "pom-pom" consttutve equatons, J. Rheol. 43, 873-896. Lee, K., M.R. Mackley, T.C.B. McLesh, T. M. Ncholson and O.G. Harlen, 2001, Expermental observaton and numercal smulaton of transent stress fangs wthn flowng molten polyethylene, J. Rheol. 45, 1261-1277. Langouche, F. and B. Debbaut, 1999, Rheologcal charactersaton of a hgh-densty polyethylene wth a mult-mode dfferental vscoelastc model and numercal smulaton of Korea-Australa Rheology Journal March 2002 Vol. 14, No. 1 31
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