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THE PRESSURE-VOLUME-TEMPERATURE BEHAVIOR AND THE EFFECT OF PREsSURE ON CRYSTALLIZATION KlNETiCS OF POLYETHYLENE REsINS
by • Ludovic CAPT
A Thesis Submittecl to the FICUIty orGraduate 5tudies and Research in Partial Fulfillment orthe Requirements for the Degree ofMuterorEnpneering
Department ofCbemical EnlÏDeering McGiIl University Montreal Ianuary, 1999 National Ubrary BibliothèQue nationale 1+1 of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 385 wellington Street 395. rue WelUngtan OtIawaON K1A0N4 Ottawa ON K1A 0N4 C8nIda c.n.dII
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0-612-50594-4 • ABSTRACT
In plastics manufàeturing. the morphology and crystallinity are important in the determination ofthe properties ofthe final product. They are influencecl by seven1 parameters, in particular the pressure and the temperature history during solidification. Therefore, the study of the influence of pressure and temperature on crystaUization kinetics is important. In tbis study~ the crystaIlization Idnetics and the PVT behavior of various grades ofpolyethylene resins were evaluated and compared using a bigh-pressure dilatometer under bath isothermal and isobaric conditions. A1so, the PVT behavior wu compared to the predictions ofthe Tait equation ofate. It wu round that polyethylene chain structure, u determined by branching and branching uniformity, int1uences nucleation, crysta1lization kinetics, and the effeet of pressure on these processes. In addition, the mults confirm the decrease ofthe surface ftee energy ofthe crystal nucleus • with increasing pressure.
• ü • RtsUMt Dans la fabrication d'articles en plastique, les propriétés du produit fini SOlit principalement déterminées par la morpholoaie et la cristaUinité. Ces demims sont controlées par plusieurs paramètres. en particulier par la pression et la température. pendant la solidification. En collléquence, il est intéreuant d'étudier l'éffet de la pression et de la température sur la cinétique de cristallisation. Dans le cadre de cette étude, la cinétique de cristallisation et les diqrammes de Pression-Volume-Température d'une variété de résines de polyéthylene ont été investisué. et comparés en utilisant un dilatomètre à haute pression. . De plus, le comportement de l'état fondu a été comparé avec les prédictions de l'équation d'état de Tait. D a été découvert que les propriétés structurelles de la chaine moléculaire du polyéthylene, en particulier les branches et leur distribution, influencent 1& nucléation, la cristallisation et l'efret de la pression sur ces deux phénomènes. Les résultats ont aussi confirmé la diminution de l'énergie libre de • surface du nucleus avec l'augmentation de la pression.
• üi •
A la mémoire de mon grand-père PierroL
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• iv • ACKNOWLEDGEMENTS 1 wish ta express my deepest Appreciation and utmost sratitude for the guidance and encouragement received trom my research supervilOr, Prof M. R. ICamaI, who stimulatecl my initial interest in the field ofpolymer science and engineering_ 1 alsa would like to express my sincere appreciation to Mf. M. Samara for his invaluable comments anclsugestïons duriDg this investiption, IJICI to my colleapes for their belp in various upects. 1am indebted to Nova Cbemicals for supplying the materials used in this study. 1 alsa extend my gratetùlness 10 the financial support receiveel Û'om the National Science and Engineering, Research Council ofCanada. the Ministère de l'Education du Québec, and the department of Chemical Engineerins in the fonn of teaching and research assistantships. 1 am especially sratetùl to Miss Gloria Vendrell for her encouragement and ber SUl8estions. Finally, 1 would like to express my love to my parents for their patience and • constant encourqemenl
• v • TABLE OF CONTENTS ABStRAcr ------ii AcxNOWLEDGEMENT ------v TABLB OF CONlENTS vi LIST OF FIGURBS De LIST OF TABLBS xiii
1 IN'I'aOOUcnON 2 TllEOUTICALBACKGROOND 2.1) U••r Law Deult)' Potyetll,.e., 2 2.1.1) Catalysts Cor LLDPE Production 3 2.1.1.1) Ziegler Catalysts 3 2.1.1.2) Metalloœne Cattllysts 3 2.1.1.3) ClroIIIhmI Dnde-BasedCatalysts 4 2.1.2) Polymerization Processes S 2.1.2.1) Gas-phtlse Polymerization 5 2.1.2.2) Solution Polymerization 6 2.1.3) Properties ofLLDPE 6 • 2.1.3.1) Type 01COIIfOIIOIIIer 6 2.1.3.2) Branching lJniformity 1 2.1.3.3) Cryata/linity anddensity 1 2.1.3.4) Mo/eCliIar Weight 7 2.1.3.5) PhysictdProperttes 8 2.2) PoIy.er Cry.....iado. 10 2.2.1) Polymer morphololY 10 2.2.1.1) Po/ytIIer Crystllllograplry Il 2.2.1.2) CItain Folding and iœIIellar erystals 12 2.2.1.3) SpIJenIlites 15 1.2.1) Nucleation and growth 17 2.2.2.1) Nucl«ltion 17 2.2.2.2) Crystal Growth 19 2.2.2.3) Secondtry Crystlll/iZlltion and Crystal Perfection 23 2.2.3) CrystalliJation lCinetics 24 2.2.3..1) TIte Kinetic Equations 25 2.2..3.2) FtlCtors ilflluencingcrystG1/iZlltiOll ii_tics 29 2.2.. 3.3) Otlters .,ecuolCrpltlllimtionKinetics 31 2.3) ...... Vol••e-Te.pen..re Pnperdel.Ip.a,.en 34 • 2.3.1) Pressure-Volume-Temperature Apparatus 34 vi 2.3.2) Equations ofState 34 2.3.2.1) The TlJsoretical Si",1ta-Somcyruky .quation 35 2.3.2.2) The Modified Tait equation 36 • 2.3.2.3) TIte Inverse-Vo"'" Eqwztion 37 2.3.3) Compreuibility and Thermal Expansion Coefficient 38 2.3.4) Literature Review for VariOUI Polyethylenes 39 3 SCOPE AND OBlECl1.VD 4 ExPEIuMENTAL
4.1) Potyedlyleae Relia Propertia .... 4.2) PftlIUn-Volume-Te.perature Apparatul 41 4.3) PVT M..ure••t Technlqu. 43 4.3.1) Sample Preparation 43 4.3.2) Calibration 44 4.3.3) Experiment Description 44 4.3.3.1) lsothel7lltll 1IIelISIUeme"ts 45 4.3.3.2) lsoborlc IlleQSlll'e,.nts 46 4. 3.J.3) lsothermaVIsobaric M.QSllr,,,,,,,ts 46 5 DATAANALYSIS
5.1) lIothermal Masure.eau 50 S.1.1) Determination orthe Additional Volume SO S.1.2) Determination ortheMelt and Solid State Regiona 51 • S.1.3) Determination orthe Parameters ofthe Tait Equation SI 5.1.4) Determination ofthe Parameters ofthe Inverse-Volume Equation 53 S.1.S) Isotbermal Measurement Limitations 53 5.2) IIobarie Meuun.eatl 54 5.3) IIotilenullllobaric Masuremeau 5f S.].I) Crysta1lization durinl Pressurization 56 S.3.2) Induction Time and BeginnÎDI ofCrystallization S8 S.3.3) End ofCrystl1IiRtion S9 S.3.4) Crystallization Temperature ofthe Experiment 60 6 REsULTS AND DISCUSSION
6.1) PVT Propertiel '2 6.1.1) General PVT Behavior 63 6.1.2) PVT Melt Bebavior 66 6.1.2.1) Tait EquDlion 67 6.1.2.2) Inverse-VobIMe EqwztiOll 70 6.1.3) Isothermal Compreuibility and Thermal Expansion Coeflicient 73 6.1.3.1) IsotItmrtal COIIIpI'esri61/ity 73 6.1.3.2) 1'ItmIItIlErponsiOll CœJficient 7$ • 6.1.4) Summary 78 vii 6.2) Preslure Depe.d.ee 01 Crystalizatioa ud Meltia. '9 6.2.1) General Bebavior in Isobaric Experimenta 79 6.2.2) Pressure Dependenc:e ofT. and Tc: 82 • 6.2.3) Me1tina and Crystal1izatiOD curvel und..Pressure 85 6.2.4) Summary 81 6.3) Cry"I'lado. Xûleda .9 6.3.1) SecondaryCrystallization Eftèct 90 6.3.2) Crystallization Kinetic data 92 6.3.3) Effect ofPressure 011 Crysta1lization Kinetics 96 6.3.4) FurtherInterpretation 101 6.3.S) Summary lOS
7 SUMMARY, CONCLUSIONS, AND FURnIER WORIC
7.1) Sam.ary aad C.ad.iou 106 7.2) 1leeo••eadatioa 'orPurther Work 10.
REFERENCES 110
APPENDICES APPENDIXI: PVT SAMPLER.UNS A-t LI) sa.pIe Rua Pnpantioa A-I • U)Callbntioa or"pieRua A'" U) Reca•••datiou A·5 APPENDIX U: ERROR ANALYSIS A-6
8.1) Ernria die VoI••e CIIu.eM...,..eDt A-6 lU) Repeatability .1iIoda..aI ••••n.... A-I APPENDIX m: IsoBARIC ExPERIMENT DATA A-9
DLI) IIobarie Esperi••t Oata A-' ID.2) Cry...... tio. Cu",es A·II
• viii • LIST OF FIGURES FilM" 2.1; Evolution ofthe polyethylene carbon chain. 4
Fipre2.2: PhueDiagram ofPolyethylene. 12
Fipe 2.3: Chanle oflattice puameten ofPolyethylene with Temperature at Skb..Pressure. 13 Fiawe 2.4: A schematic sketch ofthe fiinged micelle model. 13
Fiaure 2.S: Schematic view ofa linsle crystal with possible defects. 14
Fipre 2.6: Polymer Spherulite witb chain·rolded lamellae (schematic). 16 Fipre 2.7: Sphenalite size-preuure relltionship for polypropylene for melt conditions: ., Il min at 2000C; ., 30min at 2000C; ., 120 min at 2000C; ., 120 min at 2200C. Crystallization temperature 13SoC (Reïnshasen and Dunlap, 1973). 17
Fipre 2.8: Temperature dependence of the growth rate of spherulite in isotactic polystyrene (a) and polyamide 6 (h). 21
Fipre 2.9: Dependence of the growth rate on crystallization temperature • in polyethylene...... 21 Pipre 2.10: Dependence of Spherulitic Orowth Rate on Temperature for Different Molecul..Weiahts. 22
Fipre 2.11: Lopritbm of lamell.. powth rate venu. pressure for cil- polyisoprene at OOC (Ref4). 23
Fipre 2.12: Lamell.. crystal thickness versus pressure al aGe for ci.. polyilOprene (llet: 4). 23
Fiaure 2. J3; Schematic diaanm ofthe crysta1lization process u a fùnetion oftime, witb • definitioll ofcertain quantities. 31
Fipre4.1: Schematic drawing showing the usembly of the PVT ap~. ~
Filure 4.2: Temperature venus Time during rapid compression; the IOlid line il for an inc:reue ofpressure ftom 10 to 50 MPa, and the • dubecl-Iine hm 10 to 200 MPL sa Dt Eiaure s. 1: Plot of Specifie Volume versus Pressure for dUFerent ilOtbenns. ~S2 • 'âpre S.2: Cross-plotted isothennal data for Sclair 2908 exhibiting a decreue of volume with increuinl temperature due to measurement error. 55
'âpre 5.3: IDustration of the IfIPhical determination of the end of the melting and the ouet of crystallization &am isobaric experiments ofHDPE under 100 MPa et 2.5°C/min. 56
Fiaure 5.4: Volume Change Rate VerlUS Temperature ftom the melting curves at 2.5°Clmin ofLLDPE-B. 56
'iaure 5.5: Schematie plot of IR ilOthermallilObaric crysta1lization experiment. 58
'âpre 5.6: Schematic diagram of a typical crystallization kinetic experiment. 59
Fâpre 6.1: Cross-plotted isothennal experiments ofrelins A and HDPE. 64
FilUm 6.2: Cross-plotted isothennal experiments ofreliDS A, S, C, and D. 64
FilUm 6.3: Cross-plotted ilOthennal experiments ofrains E, F, G, and H. 65 • Pipe 6.4: Cross-plotted isotherma1 experîments ofrelins 1, J, K, and L. 65 Fil'"6.5: Cross-plotted ilOthennal experiments of ail LLDPE resins; Sehavior in the Mek state under 100 MPL 66
EiRe 6.6: Tait equation parameters vs. density; each data point represents a min. 69 'ipre6.7: Inverse-Volume Parameters vs.. density; each data point represent • rain.. 72
Fjpre 6.8: Experimental Isothermal Compreuibility of the melt state versus temperature Cor rains A, 1, and HDPE.. 74
FilM' 6.9: Isothermal Compreaibility venus Temperature for rain A; ComparilOn of experimental data (Exp) with Tait equation fitting (Tait) and with Invene-Volume tïtting (lV). 75
Fjpre 6.10: Experimental Thermal Expansion Coefficient ofthe melt state venus temperature Cor raiDS A. 1, and HDPE. 76 • x Fipre 6.11: Thermal Expansion Coefticient for raiD A; ComparilOn of experimental data ta Tait equation (salid lines) and to IIlvene Volume equation (clashed line) predictions onder atmospberic • pressure and 200 MPL 77 Pâpre 6.12: Crystallization and melting Il 2.5°Clmin under various pressures for the HOPE rain. 80
PilDD 6.13: Crystallization and meltina at 2.5°Clmin under various pressures for resin B. 10
Pipre 6.14: Crystallization and meltina It 2.S0Clmin under 100 MPa for resiDs S, 1, and HDPE. 81
Pjpre 6.1S: Volume Change Rate vs. Temperature representing the melting ofrains L and HOPE under 10 and 100 MPL 81
Fiaure 6.16: Melting points and crystaIlization temperature venus pressure with the linear and the Simon cquation fitting for resin S. 83
Fiaure 6.17: Volume Change Rate vs. Temperature for the melting under 100 MPa at 2.SoCimin of mini with (a> Butene and (b) Hexene comonomers. 86 Fipre 6.17: Volume Change Rate vs. Temperature for the melting under 100 MPa Il 2.SoClmin ofresinl (c) with Octene comonomers • and (d)F, K, and L. 17 Fipre 6.18: Typica1 LLDPE crystaIlization experiments at constant pressure for various ternperatures, indieating the linear increue ofthe degree ofcrystallinity during sec:ondary crystailizatiOD. 91
FilDD 6.19; Crystallization kinetics experiment for min B under 100 MPa at 144°C, presenting two difl"erent data analyses: Fipre6.21: Crystallization ICinetics experiments under 100 MP. at variOUI temperatures for the resins D and G. 93 Fiaure 6.22: Awami equation oC kinetic experîmentl uncIer 100 MPa Il • various temperature. -:-94 xi fipre6.23: Loprithm ofthe Awami crystallization rate, k1", (obtained for n=l.S) venus crystaIlization temperature under various • pressures for B, D, E, and L ruina. 94 Fipre6.24: Ralf·dme oferystaUization vs. IUpercooling under 100 MPa at variOUI temperatures for all the rainl. 97 Fiaure 6.25: Induction time VI. supercooHnl under 100 MPa al various temperatures for ail the rainl. 97 Fiaure 6.26: Halttime of crystallization vs. supercoolina under Vlrious pressures and temperatures for HDPE. 99 fipe6.27: Halttime of crystallizatioD vs. supercoolïng under various pressures and temperatures for minL. 99 Fipre 6.28: Induction time VI. supercooling under various pressures and temperatures for reliD HDPE. 100 Fipre 6.29: Induction lime VI. supercooling under vuious pressures and temperatures for rosin L. 100 Fipre 6.30: Plots of t1li1 versus undercoalins T~ (AT)1 for the HOPE raiD data and Brown and Jona" data. 102 1 • Fiaure 6.31: Plots of tlli versus undercooling T~ ( AT)1 for the HOPE reliD at various pressures. 102 Fipm6.32: Plots oftl/il venus undercooling T~ (AT)1 for the LLDPE·L resin. 103 Fiaure 6.33: Plots oCtllil versus undercoolinl T.( AT) for the HOPE resin. 103 • xii • LIST OF TABLES Table 2.1; Expressions ofCompressibUity and Thermal Expansion Coefficient for the Tait lIId the Invene-Volume Equation. 37 Table 2.2: Value ranle for the fitting paranteters ofthe Tait Equation in the melt for polyethylene. 37 Table 4.1: Proprieties ofpolyethylene resins. 41 Table 6:1; Onset temperatures orthe melt state Ilvarious pressures. 67 Table 6.2; FiUing results ofthe Tait equation parameten. 68 Table 6.3: Fitting reaalta orthe Inv...Volume equation parameten. 71 Table 6.4: Experimental Isotbermal Comprellibility ofthe melt state for resinl A, 1, and HDPE; Average stands for the average value of the temperature interval. 74 Table 6.5; Experimental Thermal Expansion Coefticient ofthe melt state for resiDl A, 1, and HDPE; Average stands for the Iv...e value ofthe temperature • interval. 76 Table 6.6: Fittinl Resulta ofthe Pressure Dcpenclence ofT. and Tc. 84 Table 6.7: Average values orthe fitting results ofTable 6.5. 84 Table 7.1; Main results achieved in the present study for ail the rains. 107 IableW: Meking and crystallization temperature obtained &am isobaric experiments at 2.SoClmin uncIer variOUI pressures. A-9 • xiii • IN'rRODUcnON Polymen are 10DI molecules composecl oflarge numben ofstnletura1 units that are chemically bonded. Amonl the large vuiety of commercial synthetic polymers, polyethylene is based on the simplest struetural ethylene unit, (-eH2-CH~).. Since 1939, wben it wu mst discovered, polyethylene bas evolved into different molecular structures, which give a wide range of properties and end uses. Consequently, polyethylene resins are now classified in three main groups, dependina on resin density: Rigil Density Polyethylene (HDPE) >0.94 r/cm3 Medium Density Polyethylene (MOPE) 0.926 ta 0.94 Law Density Polyethylene (LOPE) 0.915 0.925 r/cm3 • Elutomers or Very Low Density Polyethylene (VLDPE) < 0.915 rlcm3 The catesory ofLDPE allO includea the linear low density polyethylene (LLDPE) resins, which CID be distinsuished &am LDPE resins by the type of chain branchinl. Special interest is given to LLDPE in this study. LLDPE raiDS are copolymen based on etbylene and a comonomer (for example, butene, hexene, or octene), tbat yield short chain branches. These rains ditTer in the type and content of comonomers in the copolymer, compositional and branching uniformity, crystallinity and density, and molecular weight and molecular weisht distn1Xltion (MWD). This larp variely of commodity and specia1ty rainl cao he obtained by varyïn, the polymerizatioD proceIS conditions, IUch u polymerization medium, eatalysts, and comonomer. In incIustry, end user plastics products Ire manutàctured by diverse polymer • proccssins teebniques, such u injection moldina. blow molcling, blow film extnuion, 1 compression moldin& etc.... Ourlnl these ProcessÎDI operations, polymer rams are exposed ta elevatecl temperatures and pressures, and ta stress, strain, etc.... The • Processinl conditions affect the nature ofthe crystalline structure ofpolymen, referrecl to u morphololY, wbicb is controllecl by the crystallization mecbanisms. For a lDIterial such u polyethylene. morphology and crystallinity are intluenced by the polymer chain structure. Polymer chain structure il defined in tenul of branching size, bJ'IDCbing distribution, molecular weisht, and MWO. Moreover, morphololY and crystallinity determine the ultimate mecbanical and physical properties of the final product, such u tougbness, brittleness, ductility, elasticity, penneability, transparency, etc. In view of the Aboye, it il important ta investipte the etfect of processing conditions and polymer chain structure on crystallization to have a better undentanding ofthe facton affeetinl the u1timate properties ofthe end produets. The present work deall with the study ofthe pressure-volume-tempcrature (pVT) propertiel and the eft"ect ofpressure on crystallization kinetics for l1linear low density polyethylene and one hip density polyethylene ruins. The LLDPE resins coDSist of various grades of resins prepared by either solution or gas phase polymerization with metallocene or Ziegler-Natta eatalysts. They also dift'er by the content and type of • comonomer. The text is divided ioto six chapten. Chapten l and 2 present the theoretical background. They review the cbaracteristics of the LLDPE mina, polymer crystallization, and polymer PVT properties. Chapter 3 stltes the objectives of this project. Cbapters 4 and S describe the experimental procedures and the methods for data analysis. Chapter 6 tint presents the results conceminl the equations of ltate, and subsequently, the pressure dePeDdence of crystaUization and meltiRa temperatures. Furthermore, the effect of pressure on crystallization kinetic is discussed in light of difFerent tbeories. Chapter 7 COntaiDS the conclusioDs and suaestions for fiarther wode. Part of the abundant numerical data pnented by thil work il tabulated in the appendices. An appendix alsa Bives an analysis of error usociated with the measurements. • 1 • CIIAPTE1l2 TllEORETICAL BACKGROUND In this chapter, relevant information reported in the literature is presented in tbree sections. The tirst section descnbes brietly the ddferent polymerization processes and their influences on the physical properties oflinear low density polyethylene resms. The second section reviews the concepts of polymer crystallization and crystallization kinctics. The lut part presents the preuure-volume-temperature propertiel ofpolymen. 2.1) LINEAR Low DENIITYPOLYETHYLENE The large vanety of semicrysta1line ethylene copolymers, which contain small amount of camonomen (a-oletïns), are produced in eatalytic polymerization reactionl. Tbese copolymen have a lower density than those ofethylene homopolymen. They are • usually referred u tinear low-density polyethylene (LLDPE) ta distinguish them !rom conventional low density polyethylene (LDPE). LDPE COntaiDS Ions branches, venus short branches for LLDPE, and it is producecl in radical polymerization reactiODS at high pressure. The Jqe group ofcommodity and specialty nainl collectiveJy Imown u LLDPE iDcludes a wide nn. ofditterent raiDS. Each rain bu düTerent properties that depencl on the tyPe and content ofcomonomen in the copolymer, compositional and branching reguJarity and UDiformity, crystallinity, density, molecu1ar weisbt and molecular weigbt distributioD (MWD). AIl tbue raiD properties are a consequence ofthe type ofeatalyst and polymerizatioD process employ. to produce the l'elin. Figure 2.1 i1lustntes the varioua polyethylene types and tbeir simplifiecl ChaiD structurel• • 3 llWltD.nsaly tfith DInsity liMorlow MI':'a =tfte rui", • o.,i~ 1/ J ICI Phi~. Unioft Ca,bide fJuaDft Dow HaichsI UNJIOL &ad AlFinity 1939 1955 1970'1 1991 1993" FiD'" 2. J: Evolution ofthe polyetlrylene CQ1'1Jon chain. 2.1.1) Catalysu ror LLDPE Production LLDPE relins are procluced in industry with three cluses ofeatalysts: titanium based eatalysts (Ziegler-Natta), metallocene-bued catalysts (Kaminsky and Dow), and chromium oxide-bued catalysts (Pbillips). • 2.1.1.1) Ziegler-Natta Catalysts These catalyst systems represent by far the ar_test sbare of LLDPE resins manufaetured. They conaist oftwo components: the tint contlÎftS u its active ingredient • derivative of. transition metal (usually titanium); the second is an orpnoaluminum compound. Typical beteroleneouJ Ziester-Natta (ZN) eatalysts operate Iltempenture of 70-1000C and pressure of0.1-2 MPL The polymerization reactiona are carried out in 1ft iDert liquid medium (e.a., hexane, isobutane) or in the sas phase. Molecular weisht il controlled by usina bydropn u chain-transfer apnt. Ail solid ZiesJ.-Natta CItIlyltl contain a mixture of severa! typel of active centers wbich lad to IlOn-uniform etbylene-cx-olefin copolymers, and procluce • relatively broad copolymer MWD. • 2.1.1.2) MetaIloccne Catalysts 4 Tbree types of mctaIIocene catalysts are presently usecl in industry: Kaminsky, • ionic combinatioD, and Dow eatalysts. Kominslcy ctItlI1ysts, the oriSÏJlll meta1locene cata1yst systems, coutaiD lM) components: methyllluminoxane and a metI1Iocene complex ofzirconium, titanium, or haIfilium with two cyclopentadienyl rings. Most ofICaminslty eatalysts COntaÎD only one type ofactive center. ThUI, they procluce LLDPE with uniform campo.itiana! distribution and quite narraw MWDs, which, Il their limita, can be characterized by polydispersity indexes (MJM.) ofabout 2.0 and mek tlow ratio (MFll) ofabout IS. These fatures of the cata1ysts determine tbeir use in the synthesis ofeither uniformly branched very low denisty polyethylene (VLDPE) resins or completely amorphous polyethylene plastomen. They have atso been gradually replacinB ZN catalysts for the manufacture of certain commodity LLDPE productl. Ionie eatalysts are a combination of rneta1locene complexes of zirconium or titanium and pert1uorinated aromatic boron compounds. These catalysts operate over a wide ranle of temperatures, -70 to 100°C; however, monomen of high purity are • required ta prevent eatalyst poisaninB. The LLDPE relins procluced by ionic eatalysts have a high compositional uniformity. Dow catolysts, alsa known u constrained-leometry CItaIysts, contain only one type of active center and produce uniform LLDPE. They operate al temperature up ta 16QOC. Dow cataJysts bave a high capacity ta copolymerize Iinear a-olefins witb ethylene. As a result, wbeIl these catalysts are used in solution-type polymerizatioD reactiODl, they 1110 copolymerize ethylene with polymer molecules containinl vinyl double bonds at their ends. This auto-copolymerization reaction cm produce molecules with very rew IODI-ebaïn branches tbat yield IODle beneficial processing properties (et: fil. 2.1, Dow Aftinity 1993). • 2.1.1.3) Cbromium Oxicle-Bued Catalysts 5 Chromium oxide-bued eatalysts were originally developecl by Pbillipi Petroleum Co. for the manuCacture of HDPE resinl. LLDPE resins procluced with these eatI1yltl bave a very broad MWD cbaracterized by Mw/Ma in the 12-35 l'IIIIe and MFR in the 80 • t 200 ranle. 2.1.2) Poly.erizatioD Proaaes The common technologies for LLDPE manufacture include: ps-phase tluiclizecl bed polymerization, polymerization in solution, high pressure polymerization. and sluay polymerization. Most eatalysts are fine-tuned for each partieular process. In tbis present study, special interest will he given to the ps-phue and solution polymerization processes, which are discussed brietly below. 2.1.2.1) Gu-phase Polymerization The first ps-pbase tluidizecl-bed process for the production of LLDPE wu • developed by Union Carbide in 1977 (et: fil. 2.1, Unipol 1977). The Unipol process il suitable for the production ofbath HOPE and LLDPE. It usually operates at pressure of l.S-2.S MPa and temperatures of7G-9SoC. The reactor is tilled with a bed ofdry polymer particles vilomusly aaitatecl by a high velocity pl stream, and a mixture of ethylene, comonomen, nitroget1, and hydrolen tbat is used for molecular weight control. Catalyst particles are continuously Ceci into the reactort in which tbey remain, on average, for 2.5-4 houn. The ps-pbue processes are economical, flexible, and wide-ranging in the use of saUd and supported eatalysts. Most of the eataly. used for LLDPE production are heteropneoul ZN eataly. which produce resina contaÎllÏDl • pronounced non-unifonn branchial distribution. The pa p.... cao 1110 ICCOmmodate ..pported metallocene catalysts tbat produce compositionally uniform LLDPE resÎDI• • 6 2.1.2.2) Solution Polymerization • Two types ofsolution polymerization teclmologiel are UJed for LLDPE synthelis: one uses heavy solventl u a polymerization medium, and the other is carried out in mixtures ofsupercritical ethylene and malt.PEe In the tint process, copolymerization reactions are carried out in the temperature range of 130-2S0°C, at reaetor pressure of3-20 MPa, and with an ethylene content of8 lOOA. It usually opentes at residenœ times ofaround 5-10 minutes. Three properties of LLDPE are controlled independently: the molecular weight, the copolymer composition, and, in the case of metallocene eatalysts, the desree of long-chain branching (Dow Aftinity, ct: fig 2.1). The control variables are: temperature, pressure, monomer and polymer concentrations, residence lime, and eatalyst concentration. The second type ofpolymerization concept operates at temperature of 170-3S0°C and under pressure of30-200 MPL The residence time in these reacton is shon, ftom 1 ta S minutes. Exxon uses metallocene eatalysts in a similar solution process ta produce • uniformly branched ethylene copolymers called Exact ruiDl (ct: fig. 2.1, Exxon Exact). Solution proc:esses acquired new importance because of their shorter residence times and ability to accommoclate metallocene eata1ysts. Many heteraleneaus multi center Ziegler eatalysts produce superior LLDPE resins (with heUer branching uniformity) when the eata1yst residence time in the reactor is short. Solution processes, bath in heavy solvents and in the polymer melt, are inherently suitable ta accommodate soluble metallocene eatalysts. 2.1.3) Propertles ofLLDPE 2.1.3.1) Type ofcomonomer • 7 Pour olefinic comonomen are usecl in industry to manufacture LLDPE: I-butene. l-hexene, 4-methyl-l-pentene. and l-octene. The content of comonomer in copolymer • varies greatly from 1 to 2 mol % (characteristic ofmedium density polyethylene raina) ta uound 20 mol % in PE elastomen. The type of comonomer exerts a significant iDfluence on the copolymer properties. For instance, etbylene-l-butene copolymen exbibit inferior mechanical properties to those of ethylene coPOlymers with other comonomers. 2.1.3.2) Branching UniCormity Two classes ofLLDPE resins are on the market. The metallocene catalyst-bued resins have a predominantly uniform branchin. distribution; that il, ail the copolymer molecules have approximately the same composition. In contrait, the Ziea1er-Natta catalyst-based rains bave pronounced non-uniform branchinl distributions. Uniformly and non-uniformly branched LLDPE relins clifFer significantly in phYlica1 and mechanical properties. • As a rule, LLDPE raiDI do not contaiD lons-ehain branches. However, SODle copolymen produced with metallocene catalysts in solution processes CID contain about 0.002 long-ehain branches per 1000 elbylene units. These branches are Cormed in auto. copolymerization, u explained above. 2.1.3.3) Crystallinity and deDSity These two parameten, which are closely rel. depencl moltly on the content of comonomcr in the copolymer. Both density and crystallinity ofLLDPE ft intluenced by the bnnchins unifbrmity. For ÎDStIIlCe, III increue in the comonomer content of a copolymer resuIts in a decreue of bath crystallinity and density, accompanied by a significant reduction ofthe stiflbeu ofthe polymer. • 1 2.1.3.4) Molecular Weigbt • The range ofmolecular weigbts ofcommercial LLDPE relins ilrelatively narrow, usually ftom 50,000 to 200,000. A lypical melt index ranse for LLDPE raina is Û'om 0.1 ta 5.0, but CID reacb over 30 for SOlDe applications. Most commodity-pade LLDPB rains have a narrow MWD, with the MwIMa ratiOI of2.5-4.S and MFll values in the 2. 35 range. However, LLDPE rains producecl with chromium oxide-bued catalysts have a broad MWD, with MwIMa of10-35 and MFll of8Q.2oo. 2.1.3.5) Physical Properties al Meltin6BehtMor Melting points of LLDPE with uniform compositional distributions (prepared with meta1locene eatalysts) decreues almost Iinearly with copolymer composition: for instance, ftom - 120°C - for copolymers containiDg 1.5-2. mol % ofcomonomer - ta IIOoe - for copolymers containing 3.5 mol % ofcomonomer. In contrait, a copolymer with • non-uniform compositional distribution (e.g., • prepared by ZN catalysts) is a mixture that contaiDS copolymer molecules witb a broad ranle ofcompositions, Û'om almalt linear molecules (usually higher molecular weight) ta short macromolecules with quite high comonomer content. Melting of IUch mixtures is dominated by their low branchecl tiactions which ue highly crystalline. As a result, the melting points of LLDPB resinl with DOn-uniform branching distnüution are not too sensitive to copolymer composition and ulUally fall in the temperature range of 125 128°C. il 1'hmrIqItmtl 0IidqtiwDel'Pfhtion LLDPE il relatively stable to heU. Thermal dep'ldation startI Il tempcraturea above 2S0ac and results in • puai decreue ofmolecular weiaht and the formation of double bondi in polymer chain•. Oxidation ofLLDPB cu occur beclUle ofbeatina or • sunlight exposure. Thmno-oxiclatiw dearedation ltaItS IltemperatureI above lSO°C• 9 çJ Casta/Ii'" strMctrre • The presence of short-chain branches in LLDPE molecules inhibits the uniform folCÜlll ofpolymer chains during crysta11izatjon. Such inhibition CID both decreue the lamellar thickness and increue the number ofintermolecular tie molecu1es, thus resulting in a stronger material. The aize ofthe lamelle Cor a c:opolymer ofa given composition dependa on the degree of branching uniformity. If a LLDPE resin bu a unîCorm brancbing, ail its molecules crystallize poorly due to bl"lllChing, Conning very thin lameUae. Such materials bave low rigidity and bigb tlexibility. On the otber band, for a LLDPE resin containing non-uniCorm brancbinJ, its leut-branched components crystalIize in thicker lameUae; consequently, more branched fractions ofthe rain remun amorphoUi and till the voida between lameUIe. A consequence of the difFerent LLDPE crystalline structures ia observecl in optical properties. Resina with a uniCorm branching distribution (metallocene) make highly transparent film with bue u low U 3-4 %. In contrast, filma manufaetured ti'om compositionally non-uniform copolymers (Ziegler-Natta) are much more opaque, with baze of over 10-1S %; the opacily is due ta the presence of larse crystalline lamellu • consisting ofnarly non-bnnched PE chains. dJ Mechanicall'nprties Mechanical properties ofLLDPE are attributed to the complu interaction oftheir structural c~stics: type and content of comonomer, branching uniformity, molecular weight and MWD, and orientation. • 10 • 2.2) POLYMERCRYSTALLIZATION A polymer melt CID be IOliditieci by lowenng the temperature or by increuinS the pressure. The resulting crystallization occurs u the macromolecular cbains lose their mobility and become &ozen in position, wbere chains are more closely packed than in the melt state. Crystallization behavior depencls on the morphological stnIetures that are produced. Morphology is usoclated with features such as crystallite size and shape, agrepte ofcrystallites, crystallinity, ete... The tint section ofthe followinf discussion reviews the current models that are used to describe polymer morpholosy including chain folding, lameUar crystals, and spherulites. The next section discusses brietly the phenomena of nucleation and growth. The lut part introduces the common kinetic models used to describe polymer crystallization. ~ Many reviews on crystaIlization ofpolymers bave been ofconsiderable help in the l preparation ofthe foUowing section. The early book ofMandelkem , wu an invaluable 2 source of details. The reviews by Khoury and PassqIia , HotTman et al.], and Masill't • bave been very helpfW. The three volumes by Wunderlieh'A1 were particularly valuable Cor !heir exhaustive presentation ofpolymer physics. Finally, amans more recent reviews, those by Chu', Mandelkem', and Strobl lOwere very contributive. 1.2.1) 'olymer .0rpliol0l)' The macroscopie fom and structure of the crystal is ils morphology. Under a variety of circumstanees, commonly encountered in pnctice, Iinear macromoleœles crystaIIize into the Conn ofthin pillelets, which ueknown u crysta11ites.. The IlIIe upPer and lower surfaces canaist ofID may ofmacromolecular folds .. The morphology ofIUch polymer crystaIs wu tint investiptecl Cor crystal. grown ftom solution. Nevertheless, crystal. crystaIlizecl fiom the melt and tbeir supermolecular structure are DOW weil understood, thanb to the prevÎOUI stuelies on solution-grown crystals. Even thoup tbis section will mainty outIiDe the morpbololY ofthe structurel obtained &am crystallized • me~ IODle results ftom SOIutiOn-IfOWD crystal studios will "10 be included. Il • 2.2.1.1) Polymer Crystall0Il'lPhy The description of the crystal structure of a polymer il, in molt respects, no difFerent &am tbat ofalow molar mus compound. A crystal may be definecl u • portion ofmatter witbin which the atoms are manSed in a reau1ar three-dimensionally repeating pattern. A crystal may be classified into one ofthe seven large subgroups, referred ta as crystal systems, such u triclinic, monoclinic, orthorhombic, etc... For polyethylene, the orthorbombic crystal form is the malt stable at atmospherïc pressure. Furthermore, melt crystaIlization al pressure above 300 MPa (3kbar) leads to ll the formation ofa hip.pressure phue, which wu discovered by Basaett and Tumer . Figure 2.2 illustrates the phase diqram ofpolyethylene. ~ C".tal :- ,-• • .e!-I ..zr MIIt ,. ;." FoIdId~••,~ ChIIIt.· CMIn ~...... ••• _.....-"----.._... 400 425 450 475 500 525 550 T.mperature (K) Fipre2.2: Phase Diagram 01Polyetllyl'ne~ The hiab-preaure phase is Imown u the disordered hexqonai or pseudo hexqonal phase. The bip preuure bu been Unked ta the particular eue of chain extension on cryataUization under hip pressure. The chan.. of lattice parameten of polyethylene It S kbar pressure lOilll throusb the tranaition ta the mp-pressure phue are shawn in Fipre 2.3. The crystaIloanPhic a axis expancIs continuously to the pseudo • hexqonal phue, wbile the c axis, the chain ai, contnal. 12 -ail---+-+-+--+--+--+..... '- • • «la ~Ul---l~!!!!s&!!!!!IIi! I-I~ t,~~---Io--l-~--a."-I T...... HU Fi.e 2.3: Change ollattice partlllleters 01Porthylene with Temperature at , kbar Pressure 2.2.1.2) Chain Folding and lamellar crystals The chain-roldiDa model wu tint proposed in the yean 1957-1960. Bdore tbat lime, what wu known conceming crystallinity in polymers wu interpreted using the • "fiinSecl micelle" mode~ which is sketched in Fipre 2.4. Fipre 2.4:  schematic .tch01the.fringed"'icelle IIIOdeL The mnsecl micelle model wu supportecl by the tict that one molecule had ta participate in more tban one crystallite, bCClUse the molecular lenph wu much Iqer thIIl the known crystallite lia. No one seems to bave thouabt offoldina the chain back iDto the crystaIIite ta accommoclate the same tàcts.. After iporing for two declda the 12 • early observation ofchain roidi... in tbin filma ofpUa percha by Storb , the aimil., 13 l l yet independent, warka ofKeller', Fisher ", and Till ' on the identification of solution grown polyethylene single crystals led ta the concept of chain foldina. A schematic • cbain-Colded crystallite is illustrated in Pigure 2.5. ~...... VAGMOY \"IUIMTLY ..OUIM 'OLD IUI,acl 'LOO'· CAUSID IV NOII-AD_elT "1-IN11IV Fipre 2.5: Schematic view 0/a single crystal with possible defects. (Ret 2) Figure 2.5 also shows the possible departures trom strietly replar ColdinS and defects in a sinste crystal. The non-adjacent re-entry and loose loop, the slighdy rougit surtàce, cUi&, and row vacancy can be observed tram this sketch. The existence of intercrystalline and interlamellar links shows tbat the non-adjacent re-entry and 10Dg cilla • must certainly accur ta sorne extent in melt-crystallized polymers. It should be Doted tbat this sketch does Dot represent the possible confisuutions of single crystal&, i.e. the hollow pyramid shape that is observed in solution-grown sinsle crystals ofpolyethylene. The etrect ofhydrostatic pressure on crystaIlization oflamellar crystaIs bas been studied for many yean. At sufticiently high pressure (above 300 MPa), polymer crystaIs show considerably increued thiC1me1l alonl macromolecular chains. These much thicker crystaIs are commonly DOwn u chain-extended (CE) crystals. Wunclerlich et al.·6 sullestm that the œ crystaIlization process coosistl oftwo processes. First, chain-folded lamellae are formed with tbiclmess determined by uadercoolin& and tben the chain. are reo...... ized to CE conformation at high temperature mcI pressure. Busett and Turnerl sbowed that tbere are two concurrent competitive processes ofcrystalliation of cbain-Colded and CE lamellae. Under hiper pressure CE crystallization becomes dominant. ft wu...ested that orthorhombic chain ralded lamellae ofpolyethylene are formed directIy ftom the melt, while CE crystal. of ll • polyethylene resuIt via the intermediate hexasonal pbue • MIeda et al. 17 mgested the 14 existence ofditTerent extended-ehain (EC) erystaI fonns: ordinary Ee crystals havinl a lower melting point than that ofhigh melting Be erystaIs. They proposed three ranges of • pressure in which the crystaIlization il lipificantly clitTerent: 1. Up to 200 MPa. Polyethylene crystallizes directly from melt ioto chain rolded lamenae. 2. From 200 to 350 MPL Polyethylene erystaIlizes ioto two forms: chain folded lamellae and extended-ehain lameU.e. 3. Above 350 MP, polyethylene erystalHzes ioto low meltinl ordinary Ee hexagonal crystals. Annealing at elevated pressure causes also thickeninl of lamellae. Nonetheless, at constant temperature, an increase in pressure decreases the amount of thickening obtained. It wu found that the lamellar thickness depends on crystallization temPel'lture and time, molecular weight, and pressure. Busett and Tumerl1 formulated the following conclusions conceming the Iamellar thiclmess: 1- Increases with the increase oftemperature at constant pressure • 2- Increases with the increase ofpressure at constant supercooling 3- Increases with increasing molecu1ar weight in the range 20,000 to 50,000 J'mol al the sarne temperature and pressure ofcrystallization Recently, Bassett et al.ll used high-pressure (about 500 MPa) crystallization to compare three linear low-density polyethylenes, because their potential for fonning lamellae whith tbicknessel equal to the inter-branch separation. Lamellar thickness is l much reduced for branched polyethylene containiDI ethyl or luser branches '. In thil study, major ditferences were revealed between metaIlocene-c:atalyzcd polymers and ZiesJer-Natta-eatalyzcd mins. Butyl-branches were excluded hm the hexqonal and orthorbombic lattices, and made the crystal thidmesa oC the metallocene-eatalyzcd reain invariant. • 15 2.2.1.3) Spberulites • The morpholosy ofmelt-erystallized polymera can be described by the followinl struetural hierarchy: the lamellar crystal u the &andamental unit of the lamellar agreptes, whicb in tum build up vuiouslUpermolecular structures, whieh are senenlly in the fonn of spheru1ite1. Spherulitea coDlist principally of chain-foldecl lamoUae radiatiDs from acentral point, u shown in Figure 2.6. .....--el '... fat , HM. • Ir.cft...... Fime 2.6: Polylller Sphe",/ite with chtlin-folded lamel/ae (schematic). SpberuHtes are optically anisotropie abjects due ta molecular orientations within the microscopie regions. The spherulite size may range ftom submicroscopie to severa! millimeters. The radial expansion ofa spheru1ite is the resuJt ofthe formation oflamellar stICks tbat grow tom a common Dueleatioll site, which could he a sinale crystal or sorne kind of beteropneity. The powinS spherulhes in the crystallizing melt will eventually impinge upon one another. , Pressure, melt tempenture, and lime are the main puameten that CID be used to control spheru1ite dimensions. Spberulite sia increuea at a fixecl pressure. u the melt temperature is rai... It bu been reported that spherulites become sma1Ier witb 4 • increuinS preuure for fixed crystalliatioD conditions • Figure 2.7 reveala how the 16 spherulite diameter decreues with increasinl pressure, which is equivalent to increasing the undercooling. The spherulite dimensions, thus, depend markedly on the degree of • undercooling. Fi., 2.7: Sphe",/ite size-pressure relationshipfor polypropylenefor mel' conditions: ." 11 min al 2000C: • J01rIin al 2000C: ~ 120 ",in al2000C; ~ 120 ",in al 2200C. Crystallizlltion œmperatlll'e 13SoC (Reinshagen • andDrmltlp, 1973)" . Hatakeyama et aI. 19 studicd the crystallization ofpolyethylene up ta SOOO k&'cm2 (about SOO MPa) and reported that polyethylene crystallized into the sphena6tic morphology in the pressure l'IIIIe 1000-2000 kglcmz (100-200 MPa). More recent1y, Kowalewski and Galeski20 observecl spherulitic crystallization up to about 600 MPa. The isotherma1 srowth rate of sphaulites i. a fùnction of crystallization temperature and pressure u weil u the molecular weiaht orthe polymer. These ditferent efrects on the rate ofsphemlite growth will be discussed in Section 2.2.2.2). 2.2.2) Mueleatlo••ad &r0wth • 17 Crystallization is generally divided into two fimelamental processes: Dueleation • and arowth. 2.2.2.1) Nueleation When a polymer melt is supercooled, there exists an interval oftime, dependiDg on crystallization temperature, referred as induction time. durina which nuclei start CorminS randomly and srowina ta stable crystals. The nature ofthe nuelei determines the morphology of the subsequent crystallographie fonn of the growing crystallites'. The emergence of nuclei ftom the melt is called primary nueleation; the Iat« surface nucleation on a crystal face, u a kind of &r0winS mechanism, is referred u secondary nucleation. Nucleation is dividecl ioto two distinct categories: homogeneous and heteroseneaus. HomoSeDeOUS DUeleation occun in the pure molten phase; small portions ofthe polymer chains are oriented ioto resutar lattices wbich sive rise to the nuelei u • result of random fluctuations of arder. Normally, it il unlikely that homoseneoui • nucleation would take place upon coalina ofpolymer melts. For linear polyethylene, only at supercoolinS sreater than SSoC hODlOaeneous nucleation is considered 10 he significant. Heteroseneous nucleation, bowever, usually prevails at low levell of supercooling, and the crystallization process il usually completed beCore reachina a sufticiently low temPel'ltUre for homoSeneoul Dueleation. Ind~ heterogeneoul DUcleation is apparently initiated by heterogeneities present in the melt, lOch u impurities and foreilll surfaces. A foreip surface reduees the nucleus lize needecl for crystal srowth, since the creatiOD of the interface between polymer crystal anclsubstrate may be less hindered than the creation ofthe correspondina hepolymer crystal surfaces. Tumbull and Fistter21 expressed the steacly state DUcleatiOD rate in the rorm: · . ..,.[ ED).J AG_) N:No-rl- kT ~l- kT (2.1) • where Ji il the nueleation rate (m nueleilscc.cm'), Il No is • pre-exponential factor, dependinl on molecular structure and • crystaIlCOmetry ED il the he eneI'IY banier opposiDIJ transport ofseptents &erOIS the melt-crysta1 interface (in erp), AG", is the he energy required to fonn. critical nucleus in the melt (in erp), le is the Boltzmann's constant (in ergs/K), T ilthe crystaIlization temperature. The tenn ~- ~;) expresses the probability tbat a segment of chain of criticallenBth would reach a surface ofa growinl crystal in the local diftùsive transport process at constant temperature lIIld pressure. The nucleation term, erp(- ~;). corresponds ta the probability that • surface nucleus would reach a critical size. Since the 22 influence ofthe term ED becomel sipificant around the slus transition temperature , • and, on the other band, AG. is stronsly dePendent on temperature durinl 23 crystaIlization , the Dueleation rate il dominated by the nueleation term, and subsequently is a fimction ofcrystallization temperature. The critical ftee energy, AG"" of formation of • rectansu1ar heteroseneaus nucleus of thiclmess bo ta • Pre-existilll ftat surface il dependent on supercooling and l surfàce tee energy u shawn below : (2.2) where tT is the specifie ftee eneraY ofthe surfice parallel to the chain axis a. il the specifie heeneI'IY ofthe crystal raid surface !:la is the specifie intedicial heeneraY ditTerence parameter • r: is the equilibrium meltina point 19 AT is the degree ofsupercooling, T: - T, at wbich the crystaDization takes place • Mu is the enthaIpy offusion per unit volume ofa lqe perfect crystal at T: Furthermore, the surface fi'ee enersies, a and oe ofthe crystal nucleus bave been reported ta depend on pressure24.2'.26. 2.2.2.2) Crystal Growth A sphenalite, u described earlier, may be considered u a system consistiDg of lamellae that radiate fi'om a central point. It may he, therefore, usumed tbat spheruJitie growth proceeds through repeated depositions ofmolecular layen ofconstant thickness. This means that Iinear crystal growth rate is Iinked to the secondary or molecular DUeleation rate. 27 The growth rate, G, is described by an equation ofthe fonn : (2.3) • where Go is a pre-exponential constant, which is not stronlly dependent on temperature 21 and on molecular weight • The AG. term describes the he energy of the nucleus growinS in an m-dimensional space. A two-dimensional lamellar crystal model seems ta represent most polymer system adequatel~. Based on the secondary Dueleation growth rate ofthe Lauritzen-Hoftinan model', the linear growth rate cm also be written u: G=Goa{ R(~T..))a{ T(~:)f) (2.4) wbere U* ilthe activation energy which lovems the mobility ofthe polymer with respect to the temperature. T ia the crystallization temperature, T. is the WLF temperature Il which the mobility ofthe molecules converges to zero~ R is the pa constant, K, is the DUeleation factor, AT ia the desree of supercooling, and f is a fictor ta correct the • temperature depenclence ortheheat offùsioD. 20 a) I1Jfh1enq o(Castal/ization Te1lpl1llflre 011 Growth Rate • The Lauritzen-Hotrman model bu contributed to the study of the etfect of temperature or the depee ofsupercooling on the spherulite IfOwth rate. Hoftinan et a1.3 reported tbat the growth rate ofspheruJite u a function ofATpasses thraugh a maximum, u illustrated in Figure 2.8. _."-'Aooo...... ~...... 6 T:tc 0 .10 6 T.-.c 0 Fi"". 2.8: Temperatllre dspendence 01thegrowth rate ofspheru!ite • in isotacticpolystyrene (a) andpolyamide 6 (6). (Rej. J) Fi,.,. 2.9: Dependence ofthe growth rate on crystal/ization temperature in poIyetIryfene. (ReJ. J). ln addition. Hoftinan et al.3 experimentally reported two growth reaimes dependinl on crystaUizatiOIl tempenture. Reaime lleads to axialite morphololY. whereu • reaime n corresponds to spberulitic powth, u shown in Fipre 2.9. 21 • 6) l'!fluenc' o'MolecvR Wei.dt on Growth rat" The isothermal spherulite growth rate is a fimction ofcrystallization temperature. undercoalin&, pressure, and moleculU' weiaht ofthe polymer. The influence ofmolecular 4 weipt wu studied by MqiU • Some ofhis results are presented in Fiaure 2.10. • Fi.e 2.10: Dependence ofSpherv/itic Growth Rat, on Temperature Jor DifferentMo/ccular Wetghts, (Ret 4). From this figure, it can he seen that the growth rate is increases u molecular weipt decreases, at constant temperature. cl 'rrfluenœ o'hem" 011 Growth 'at'· Edwards and Phillips have reporteel tbat the rite of srowth of row·nucleated lameU. in cis-polyisoprene passes though a maximum, u a fimction of pressure al constant temperature. u sbown in Fipre 2.11. • 22 • u..-.---~----..... o Fi., 2.11: Logaritln ofIomellar growth rate Wlrsus pressure lor cis-po/yisoprene al f1JC fRe/4). The growth rate wu about six times 11I'Ie!" tban that experienced at the temperature corresponding ta the maximum me Il atmospheric pressure. This increase cao be explained by an increued Go with pressure and/or by the dependence of meltina point 4 • on pressure • Edwards and Phillips allO round that the lamellar growtb rate is constant for a given supercooling, but that it increases u the pressure is lowered at constant temperature, as presented in Figure 2.12. They alsa observed a similar trend for sphenalite 30 growth • 11.0 ....'--.....~t.a--""!"u~"""-"~u~... .-.... 'arc 2.12: u.el/Qrcrystal tlricbtua wr&llS prusIIN • al fllefor cis-polytsoprene (Re/. 4)• 23 It is Imown that the erystallization ofpolyethylene under pressure is IOvemed by primary 20 • nueleation since the dominant supermolecular structures are spherulites • 2.2.2.3) Secondary Crystallization and Crystal Perfection The procas tbat comprises the initial nucleus formation and the following spheru1itie arowth is senerally called primary crystallization. At this staBe, more tban halfofthe final crystallinity orthe polymer bas been reached. Any further crystallization, wbich is larse1y asociatecl with slow thickeninl and lateral powth ofthe lameUae in the crystal staeks, is labeled secondary crystallization. The tenn sccondary crystallization is ti'equently applied ta ail etTects that increue the crystallinity Ifter the primary crystallization is ov«. The ICCOndary crystallization is charaderized by a linear increase ofthe degree ofcrystallinity with the loprithmic time'l. Secondary crystallization involves two processes: tùrther perfection ofthe poorly crystaIlized polymer in the spherulites and tùrther crystallization of the remaining amorphous phase. • lO Strobl proposed a sliptly ditTerent definition of secondBy crystallization, nunely tùrther crysta1lization that occ:urs when the sample is cooled tiom the crystaIlization tempcrature to room temperature. Investipting the etrect ofthis additional crystaIlization on polyethylene, StrobllO replrted ditTerent behavior for Iinear and branchecl polyethylene. The 'insertion mode' of seconclary crystallization of branched polyethylene provided a larger part (35%) than that of the isothermal primary crystaIlization (15%). On the other band. secondary crystallization oflinear polyethylene wu characterized by a surface crystaIlization-process that only causecl a la-" ine:reue in crystaIlinity. Secondary crystallization will thus cIepend on the bnnching content ofthe macromolecular cbaiDs. • 24 2.2.3) CryltallizatiOD Kinetia • stucly ofcrystallization kinetics at hilh pressure bas been carriecl by • vuiety of techniques. Matsuoka32 studied polyethylene crystaIJization dilatometrically up to pressure of 2 kbar. Martin and Mandelkem33 investipted crystaUization of nabber at pressure up to 800 bar by observinS a dilatometer throup a window in • pressure vessel. The dilatometric technique wu extended ta a maximum pressure of 5.3 kbar with the improvecl high-pressure dilatometen usecl by Hatakeyama et al.1', Kyotani and 24 ICanetsuna , and Hoehn et a1.:M in studies of polyethylene and polyethylene fi'actions. Calvert and UhlmannJ5 concluctecl ditferential scannïnl calorimetry on pressure-quenched 1 25 samples, and Wunderlich and Davidson ' and Sawada and Nose construeted a bigh pressure differential thermal analysis œil Cor their study of polyethylene. Althougb the above methods are the most commonly used techniques, several others have been explored. Edwards and PhiHpS30 obtained Idnetic information on cis-polyisoprene usina transmission electron miCfOSCOPY by chemica1ly staining the samples at high pressure. X ray, infived studies, and Raman spectrosc:opy were also perfonned usins the diamond anvil œil. Eckel, Buback, and Strobl36 performecl a cambioed high pressure Raman and • cIilatometric study on polyethylene. Bridges, Charlesby, and FoUand37 used pulsecl NMR. to meuure crystallization kinetics of polyethylene at atmospheric pressure. Brown and Jonas26 extended the NMR. technique for their study of polyethylene crystallization Idnetics under bigh pressure. To represent erystallization kinetic data al atmospheric or high p~ there are 3 several anetic equations. The theory ofLauritzen, Hoftinan and co-workers CID alsa be appliecl to high pressure crystallization for furthcr interpretation, u reportecl by Kyotani 24 2S and 1CanetJuDa , Sawada and Nose , and Brown and Jonas-, for polyethylene. The inOuences of temperature and pressure on crystaUizatiOIl kinetiCI are reviewed in the second part of this chapter. The induction time dependence on • temperature and the melti. point dependence 011 preaure are also dilCU.... 25 2.2.3.1) The Kinetic Equations • The analysis ofcrysta11ization kinetics ofpolymers cao bued clepending on phase transformation conditions, on isotbermal or non-isothennal experiments. Isotbermal experimental data bave been senerally representecl widl the seneralized Awami equation. The Avrami equation bu been round usetùl to descnDe accurately the primary crystalJization, which starts al the end ofthe induction time, and ends at the beginning of the secondary crystallization. Modifications orthe Avrami equation bave been proposecl ta describe secondary or non-isothermal crystallization. Tobin:J', Di.', Malkin et 11.40 and Kim and Kim41 IUgested models ta describe crystallization kinetics when secondary crystallization wu observed. For non-isotherma1 Idnetics, Nakamura42 and Kamat and Chu4:J developed techniques, atso bued on the Awami equation. a) The GeneralizedAvrami &ration The Avrami....··' equation is applicable to any type ofcrystaIlization. It describes • the time evolution ofthe overaU crystallinity. The Awami equation bas been commonly applied to describe the overall isothennal transformation kinetics, and it is expressed u foUows: r(t) = l-exp(-it") (2.5) where r(1) is the relative crystallinity achievecl Ifter some time t. The constant n represents the mode ofnucleation and k the IfOwth rate, Imown u the rate constant of crystaIlization. It should be DOtecl tbat the Avrami equation wu derived under several assumptions such u tbat the crystaUization aGeS to c:ompletion, and that the kinetics remains the same durinl crystallization. Mandelkem46 introduced, Cor isothermal conditions, the pneralized Avrami equation expreuecl u: X(t). %(1) = l-ap~(T)t"] (2.6) xe(t) where %(1) is the aetual &action of the crystallized polymer at time t, xe(t) il the • ultimate crysta1linity traction IChieved Iltempenture 1, and X(t) is known u the relative 26 crystallinity tbat varies ftom 0 at the beginnina of crysta11ization ta 1 al the end. The • crystaUization balftime tIn : 11I2)1/" t1/2 =(T (2.7) is the time at whieh balfofthe conversion bas taken place, whieh is a convenient meuure for the speed ofcrystallization. The Avrami exponent n wu tirst limited to integer values because it is thought that the values ofn could give an indication ofparticu1ar types ofnueleation and crystal srowtb. In most of recent researeh, the Avrami equation wu used as a convenient equation 10 represent and compare kinetic data, therefore, the restriction on n wu released. hl Tobin's Model Tobin]' proposed a new model based on the Seneralizecl Avrami equation, by pointing out tbat the phase transition problem must he cast ioto the Conn ofthe Volterra integra1 equation. This relation wu derived for two separate cases: homogeneous and • heterogeneous Dueleation and growtb. For conversion below S()I and for ls n S't, Tobin's equation is expressed in the fonn: X(t) _ b" (2.8) I-X(t) The aclvantage of Tobin's model is that it represents the secondary crystallization phenomenon, with a slight1y better accuracy than tbat ofAwami. c) Pietz's IIIOdel DiU' modified the Avrami equation by introducinl an additional parameter to describe the secondary crystallization: dX{t) 1I1t"-I[I-X(t))J-d(tJ] • dt ~l-X(t) (2.9) 17 Here a is the new introduced parameter• • rD Kim and Kj",~ model 41 Kim and Kim derivecl a MW crystallization Idnetics equation by considering the decrease orthe growth rate with crystaIlinity. Tbeir modified Awami equation is: X(I) = l-ap~~(t)"] (2.10) , 1(1) = 1[1- X(I)rdt (2.11) o where X(O and k have the sante definition u those in eq. (14S) and f(t) is a srowth fimetiOD. For m=O, the relation reduces to the Avrami rorm. For 111>0, the growth rate decreases with time. el Hillier's Modd Bued on the assumption tbat the secondary crystallization il a slow tint-order • process, Hillier47 proposed a modified two-Itale model: where the subscripts p and s mer to primary and secondary crystallization, respectively. The secondary crystallization starts when the primary crystallization is usumed to be completed. This il indieated by • change in the order ofcrystallinity-time dependence. 2.2.3.2) Facton inftuencinl crystallization kinetics Crysta1lization tîneties of polymers CID be sipificandy aft"ected by lOverai facton ..ch u crystaIlization temperature, pressure, stress, tlow, and induction tîme. The temperature and preuureetrects will he discuued in the foUowing section. • 21 qJ l'J/luence qfTewpratvre • Crystallization rates in polymers are strong(y dependent on temperature, pressure, aDd stress (molecular orientation). At atmospheric pressure, the crystaIlization rate constant, le, is equal to zero above the equilibrium meltiDg point and below the glas. transition temperature. In the intermcdiate region, it exhibits a maximum. which wu observed29 for a variety of polymers al T... =O.82T~. It bas been shawn tbat the 4l logarithm of the crysta11ization rate versus temPerature appean in • panbolic form • Furthermore, Ziabicld49 proposed an empirical equation for the temperature depend.ce ofthe crystallization ha1f-times u follows: ...L)=(...L) exp[_41n(2/T-T-;-l] (2.13) ('1/2 1112 .. D where (1/1/12)"", TMa and D can be determined ftom experimental data. bl l'!fluence o'Pressure • The Avrami equation bas been commonly used to represent ilOtbermal crystalIization kinetie data al atmospherie pressure and high pressure. The range of Avnmi exponent Il observcd for polyethylene under atmospheric pressure is exceedingly broad, covering the range ftom 1 ta 4'. Hatakeyama et al. 19 observed an Avrami exponent close ta 1 for crystallization ofpolyethylene at SOO MPL Kyotani and ICanetsuna24 found values ofIl decreuing ftom approximately 2 at 840 kbar ta 1.2 Il S and 5.2 kbar. Hoehn et a1.34 observed values centered in the region 2.0-2.2. Sawada and Nose25 found values ofn between 1.8 and 2.3 for extencled-chain crystals in the ranse 2.49-3.0 kbar. Their reaults for folded-ehain crystal. in tbis presaare range Cell between 2.9 and 3.4. Brown and IoJ observee( values of" betweell1.3 and 1.7 in the pressure ranle ftom 1.0 ta 4.5 kbar. They found that the Avrami exponent wu IlOt dePeftdent on pressure, but rather on uncIercooling; it decreased with increuinl ..percoolinl. This wu also observee( by 24 Kyotani and ICanetlllna • Exact apeement aman. the ditferent studies is not expected, sinco crystaUization bebavior il weil known to be dependent on molecuJar weipt and • polydispenjty. 29 The etrect of pressure on the crystaIlization rate is generally explained by its etTect on melt tem~ i.e. an increue ofthe dearee of supercooling with presaare. • However, the crystaIIization rate bu been found ta be depenclent on pressure at constant supercooling. WunclerHch and Davidson·' found that the degree of supercooliDg necesBary for crystallization increued by 500" in goina tram atmospberic pressure ta 5 kbar, indicating that und« identical supercooling, crystallization wu retarded by pressure. On the other band, Hatakeyama et al.·' observed that the crystallization rate, estimatecl nom the ~time ofcrystallization, wu increased when pressure wu raised trom 1.0 to 2.0 kbar and nom 4.0 ta S.O Imar al constant supercooling. The results of Kyotani and Kaneuuna24 were in qreement with this observation. The theory of Lauritzen and Hotrman3 bas atso been used for tùrther interpretation.. Brown and lonas26 ploued the Avnmi rate constant k versus T': / T(AT)' and versus Till /T(AT) to distinguish between rate control by the process ofdiftùsion or by DUeleation and growth.. They observecl that the loprithm ofthe crystaIlization rate wu linearly dependent on both temperature variables, and tbat the slopes ofthac lines decreased by increasing pressure. These resuJts are in qreement with the ones found by 24 25 • Kyotani and KanetsuDa 1Dd Sawada and Nose • Brown and Jonas concludecl that the crystal nucleus surface energies were decreasing with increasing pressure. This 35 observation is in contradiction with the observation of Calvert and UhImann , whicb indicates an increue of the surface energy of the lamellar crystal with increasing pressure. 2.2.3.3) Othen aspects ofCrystailization Kinetics • 30 • al Induction nme X(t) ~. 1 0t---....-=:;;.-I---L----JL-__... Fim' 2.13: Schematic diagram ofthe crystollizatfon process as afimction oflime. with a definition ofcertain quanlities• Figure 2.13 itlustratel the paphica1 inftection metbocl that cm be used to define • tbree characteristic quantities. The tangent to the curve at the int1ection point defines the maximum crystal1ization rate; the half time ofcrystallization, 1'o.s at which balf of the crystallization bu occurred; and the incubation peri~ ri. In this graph, there is another , incubation period, ri. indicating the point al which the curve deviates fi'om zero value. , The induction time, ri. is definecl as the time interval required al a supercoolecl state before the critica1 equilibrium nueleus dimension is established. and ri is defined as the time to reach the steady state of nueleation. There exista 1110 a lime interval before spherulites CID be resolved witb the optical microscope and this is refared as the apparent incubation period. There is no vaUd reIIOn to prefer one detinition ta anotber, except that the induction times defined by the construction in Fipre 1.13 are lesa dependent on experimental technique. The dePelldence ofthe induction time, t, OD the depee ofmpercooling, AT, bu • been rarmulatecl by Godovsky and Slominstty2' u: 31 (2.14) • where P and E are constants. The values of 8 l'IDIed between 8 and 1S for a variety of synthetic polymen. 1J) Pressure Dgmdcnce 'llTe"l"l'4tIIre The main etrect ofp....re on crystallization kinetici il attributed to the etfect of pressure on melting po~ thus the degree ofsupercooling. The increase ofmelting point, T"" and crystaUization temperature, Tc. with pressure has been investipted for many yean. The values ofT.. and T~ exhibit an almolt linear dependence on pressure up to 200 MPI, which is in apeement with the Clausius-Clapeyron equaûon, defined u follows: tH. r.0 --!!!.. =AV; ---=constan t (2.15) dP 4H1 where AV; is the volume change on fusion, MlJ is the hat of fusion, and r: is the equilibrium melting point. For polyethylene. the values of ~ were found 10 he in the • ranle tram O.22°CIMPa for folded-ehain crystal&, to O.35°CIMPa for extended-ebain 16 crystaIs16. Wunclerlich and Davidson ealculated 1 tbeoretic:al value of ~ It atmospheric pressure for a hypotbetical perCect crystal, and reported • value of O.32°CIMpa. In addition, Saeki et aI. 5O demonstnted that the Clausius-ClapeyroD equation wu usefù1 ta calcu1ate values ofthe entropy offusion under bip pressure. However, it bu been shown that for IlI'8e pressure increue, the melting curve of 51 polyethylene decreases in slope with increuing presaare1'-S1. Takamizawa et aI. showed tbat the dependence ofthe melting point for IlI'Ie pressure increues wu best represented by the Simon equation'2: (2.16) • 32 where Til is the meltins temperature at pressureP, and T•.o tbat Il atmospheric pressure; a and c are constants tbat depend on each substance. For polyethylene, the authon found • the that the constants a and c depencled on molecular weisht.. These parameten were round to nDse fi'om 31S to 362 MPa for a and ftom 3.85 to 422 for the parameter c. It must be noted that the Simon equation does not, however, predict the maximum point in the T",..P curve. 2.3) PREssURE-VOLUME-TEMPERATURE PROPERTIES orPOLYMERS A Pressure-Volume-TemPerlture (PVT) study consista of analyzing the signiticant etTect of pressure and temperature on the specifie volume of polymers. The understandins ofthe PVT behavior ofa polymer system is important for the an.lysis and understanding ofprocessing operations, in which shrinkase and compressibility etrects are significant. Compressibility, thermal expansion coefticient, and the temperature • region in which crystallization, glass formation or melting take place are easily assessecl ftom PVT measurements. A PVT investigation also contributes to the understanding of the thennodynamics ofpolymen tbrough the equations ofstate. 2.3.1) Pressure-VoluDle-Te.peratureApparatu. Several suitable techniques aist to study the PVT bebavior of polymers. The latest reported technique uses ID injection moldina machine ta pnerate pvr data Cor thermoplastics'3. Neverthel~ molt ofthe published PVT data were obtainecl using the clusical bellows technique. Thil methodololY il clivided mo two avail.ble methods tbat ditrer by the lIlIIIDef in which pressure is applied ta samples. In the tint methocI, which bu been used mainly in the put, pressure is axially appliecl ta the sample by the use ofa cylindrical piston. However, except for true liquid samples, recent researchS4 reveall _ • generally, thil cIirect meuuriDa technique doea DOt provide ID accurate methocI for 33 determininl the response ofsemicrystalline polymers to pressure, especially for the IOlid state. There is an alternative technique that IUbjects the material to a true hydrolltltic • pressure tbrough the use of. confinÎDg ftuid, usual1y mercury. Thil lut technique, which is utillzecl in the apparatus adoptecl for this wode is descnDect in detail in Chapter 4. The equilibrium pressure-volume-temperature (PVT) properties ofa melt CID be describecl by an equation ofstate: V=V(p,T). Severa! empirica1 and theoretical equations ofstate are avail.ble ta descnDe polymer melts, glasses and crystal&, reviewed by zoner' and Bhateja and Pae". The most usefùl equations are the empirical Tait equation' for melts and Siuses, the theory of Simha and SomCYftsky" for meltl and Slasses, and of Simha and Iain" for crystals. In addition, tbere is the Inverse-Volume equation, developed by Kamal and Levan", which yields consistent values for the prediction ofthe thermal expansion coefficient and compreuibility ofpolymen. For its simplicity and for the sake ofcomparina the accuracy ofour experiments with literawre. the empirical Tait • equation wu chosen ta represent the melt behavior ofthe rains under investiption. In addition, the inverse-volume equation wu utilized in orcier ta describe the compressibility and the thermal expansion coefticient ofthe materials. The empirical Modified Tait equation and Inverse-Volume equation are brietly introcluced in this sectiOD. 2.3.2.1) The Modified Tait equation The Tait equation60 wu oriainally proposecl to explain the compressibility bebavior of seawater. ft wu liter found to describe the PVT response, except al bip pressure, ofsorne liquid hydrocarbona and atlO ofIOlDe cryltalline hyclrocarbons. Simh& et al.57 atso round the Tait equaâOtl to apply weil ta amorphoul polymen IUch u PS, p~ • and PVC. The Tait equation is usuaIly expreued iD the fonn: 34 Y(P,T) = Y(O,T){l-CIn[1+L]} (2.17) B(T) • Y(p, T) is the specifie volume Il pressure P and temperature T, C is a constant, whieh is dependent on the structure and molecular weight in the case ofhydrocarbons, and B(F) il a mataial constanttbat ilwritt.u: B(T) =Boexp(-Bt n (2.18) where Bo, BI and C are the three adjustable Tait parameten. The value ofconstant C wu estimated ta be similar for many hydroc:arbons. Furthermore, the Tait relation wu found ta be valid for high pressure in the glusy state and also for semicrystalline polyethylene. Nowadays, thil equation is commonly used to describe the pvr behavior of a large variety of POlymers, with a lCuniversal" value of C, C=O.0894. However, Beret and Prausntiz'l bave reported that the three-parameter Tait equation represents the PVT data with appreciably better accuracy than the two-parameter Tait equation (where C is equal ta the universal value of0.0894) especially for polyethylene. The volume at atmospherie pressure is calcu1ated byextrapolation. Y(O,T) can be fittecl by a polynomial ofthe fonn: 2 J • Y(O,T) =Ao + Al T+ A2 T + AJ T + ... for T with il, Vo constant thermal expansion coefticient. 2.3.2.2) The Inverse-Volume Equation This equation is based on the observations _ in molt polymers, the inverse volume vs. temperature curve Il constant pressure, (ap1aP)r vs. pressure at constant temperature, and the thermal exp....ion coefticieDt CI VI. temperature can all be approximated by straight Iines. Kamal and Levan" proposed the followinl inverse- volume equtÎOIlofstatewith six Idjustablepuameters P•• (:)1'00'" b, Co lIld d: • 35 p =p.., + (:)p.o xT+(a+6T)P+~(C+dT)P' (2.21) • where p(T,P) is the density al pressure P lftCI tempenture 7: and Pm is the density at zero pressure and O°K. The authon found tbis equation to fit available experimental PVT data ofpolymers very weil. In addition, they made a comparilOn among various equations of state. ft wu round that only the inverse-volume eqUltîon wu observed to predict correctly ID inerease in thermal expansion with increasing temperature. 2.3.3) Co.preuibility and Ther.al E~pan.ionCoeflieient It is common to express certain properties ofthe equatiOD ofstate in the form of derivatives to represent the IOlid-state and the melt behavior ofpolymers. The ilOthennal compressibility K(P,T) is defined u: K(P,T)= -1 (ÔV(P,DJ (2.22) Y(P,T) ôP T and the coefficient ofvolume thermal expansion a(P,T) u • a(P,T) = 1 (ÔY(P, T)J (2.23) y(p,n ôT " The two empirical equatioDs discussed in this paper yield the forma for compressibility and thermal expansion coefficient shawn Table 2.1. For polyethylene, Galeski et al.2O measund the bulk thermal expansion coefficient 1 ofthe melt al atmospberic pressure, ex = 8.SSxIO--.c , and in the IOlid state, a. wu equal to 3.0xIO" ~-l. Compl'ellibility ofpolymen bas senerally the following values: for the 1 1 solid state, ftom lx10'" to 3 x10'" MPa , and ftom lx10'" ta 2Sx10'" MPa- for the melta. • 36 • Equado. Tait Inverse-Volume 1 V[a +cP +T(b +dP)] _v[(aP) +bP+ dP ] aT p.o 2 2.3.4) Literatare Review for Varioui Polyethylenes Many studies investipted the PVT properties ofpolyethylene, and experimental data were predominantly fitteet to the empirieal Tait equation. Since polyethylene rains cao easily vary ftom one type to another, the diff'erent literature values of the Tait equation parameten should not be compared. However, Table 2.2 pments a range in • which those parameters were round. The literature values reported below were calculated using the universal value of the constant C of 0.0894. For the melt, the ditTerent Tait parameters exbibit in the values reported Table2.2: Table 2.2: Valu, rtIIIgefor tIte jittingJXI'tII'Ieters of lhe Tait Equation in the ",eltfor poIyetJrylene Polymer Vo Do J Type (Cllt Ir) (MPa) Law deJ!Sity 0.9190 1.148 6.95 192.9 4.70 Brancbecl 0.9320 0.940 7.34 177.1 4.63 0.9794 0.971 7.81 176.7 4.66 Among the Iarp amount of literature concemina PVT properties of polymen. • few studies comparecl PVT bebavior of similar· polymerie materïals. Nevertheless. 37 ZoUera investipted the PVT bebavior of tbree identical low-density polyethylenes, which had equal density and identical number-avenge molecular weight, but ditfered in • weipt-avenge molecular weiaht and in viny. IfOUpl. These tbree rains w.-e round to exbibit similar PVT properties in the melt, i.e. comparable Tait puameters. However, variations in the zero-pressure compressibility were reportecl. • • 38 • lleferences 1L. Mandelkem, Crystallization 01Polymers, McGraw-Hil~ New York, 1964. 2 P. IChoury and E. P.S"8til, l'realise on SolidSlale Che",lstry, vol. 3, N. B. Banny, Ed., Plenum Press, New York, 1976. 31. D. Hoftinan, G. T. Davis, lIld 1. L. Lauritzen, Treatise on SoIid State CIIe",istry, vol. 3, N. B. Haony, Ed., Plenum Press, New York, 1976. 41. H. Malill, Propmies 01solidpolymerie lIItIteria/s, Part.4, Treatise on Materials Science and Technology, 10, (JM. Schultz Ed.), Academie Press, New York. 1977. 5 B. Wunderlicb, Crystal Structure, MOI'phology, Defects, Macromolecular Physics, vol. l, Academic Press, New York, 1973. 6 B. Wunderlich, Crystal Nuc/eatton, Growth, Annealing, Macromolecular Physies, vol. 2, Academic Press, New York, 1976. 7 B. Wunderlich, Crystal Melting, Macromolecular .Physics, vol. 3, Academie Press, New Yo~ 1980. 8 E. Ch~ Master ofEngineering. McGill University 9 L. Mandelkem, Physical properties ofPolymers, 2nd eci., ACS Professional R.efBook, ACS, Washinston, 1993. 10 G. R. Strob~ Conceptslor understtlllding their stnIctrlres and behavior, The Physics ofPolymen, Springer Ed., Berlin, 1996. Il O. C. Hasselt, B. Turner, On Chain-extended andchain-foldedcrystallization 01 Polyethylene, Phil. Mag., 19, 285 (1974). 12 K. H. Stark&, A.n Electron-DiJfraction ÛDIIIinatlon ofSome Lillear Riglt Po/ynlers, L Am. Chem. SOC., A, 1753 (1938) • 13 A., Keller, Phil. Mag., 2 1171 (1957) 14 E. W. Fischer, Z. Nlturforsh, 12A, 753 (1957) 15 P. R t Ir. Till, 1. Polym. Sei., 24, 301 (1957) 16 T. Davidson, B. Wunderlich, DifferentiaiAnalysis ofPolyethylene rmder High Pressure, 1. Polym. Sci.. Part A2, 7, 377 (1969) 17 Y. Maeda, KICanetsu_1C, Tapshira, and T. Takemura, J. Polym. Scï. Polym. Phys. Ed., 19,1313 (1981) 11 J. A. Parker, D.C. Bassett. R. H. Olley, and P. Jaaikelainen, On higlt preSSllre crysta1lization andthe chanJcterization ollm.arlow-densitypolyetlrylenes, Palymer. 35, 4140, (1994) 19 T.1IItakeyama, H. Kanetsuna, H. Kaneda, lIICl T. Huhirnoto, FleetofPru.wue 011 the CryslllllizatiOli Kinetics ofPolyethylene, J. Maqpmol. Sei. PhYl., 810(2), 359 (1974) 20 T. Kowalewski and A. Galeski, Crystalizalion 01Une.Polyethylenefrom Meil in lsotlterMal Compression, ,. AaPI. Pop. Sei.• 44, 95 (1992). 21 D, Tumbull and I.e. Fisher, Rate ojNflclealion in COIIdensed Systems, 1. Cbem. 2mL 17, 71 (1949) 22 F. P.. Priee. Nllc/«ltion in Polymer Crystal1iza1ion, in "Nucleation" (A. C. Zettlemoyer, ed.), pp. 405-488 Marcel Dekker, New York(1969) 23 F. P. Priee. KineticsofSp/lmllite FonIIIItion, 'e Pop. Set, C.3, 117 (1963). 24 M. Kyotani and H.1C.net.IDI, C",stalliZtltion Ki_tics01Polyethylerre JlIIder • PreSSllre, 1. PolYOl, Sei, Polyml. PJays.. Edt! U, 2331 (1974). 39 2S S. Sawada and T. T. NOie, C"naIlization Kinetlcs ofFractionated Polyethylenes lit hippressures by a DrA metlttJ( Polym. ,., Il, 477, (1979). • 26 D. Il Brown and J. Jonas. NMR Study ofPolyethylene Cryattlllization Kinetics ""., Hig#J Pressure, J. Polym. Sei. Polym. PhYL Ed., 22, "655 (1984). 27 H. D. Keith. and F.I., Jr., Padden, Spherulitic Crysltlllizationjrolll the ".It IL l"jluence ofFrtlCliontltiOll andllIIpIIrity Segregation on the Kinetics olCrystllllization, la ABI. PbYI., 3S, 1270 (1964). 28 D. W. van Krevelen, Crystallinity 01polyme,s tmd the "'etI1IS to i"puence the crystalltZlltionprocess, Chimia, 32, 279 (1978). 29 Y. K. Godovsky and G. L. Sloni"'My. Kinetics ojCrystal/izotionfrom the Melt (Calorl",etry Approach), L PoIym. Sei PoJym. Physu Ed., 12, 1053 (1974). 30 B. C. Edwards and P. J. Phillipl, High-Pressu,e Phases in Polyme,s Il... 1. Polym. Sei.; Palym. PhYl. Ed., 13,2117 (1975). 31 J. M. Schultz. J. S. Lin, and R.. W. Kendrick&,  dynamic Study ofthe CrystIlllization ofPolyethylenejrolll the ",elt, J. ARPI. Ctyt., Il,551 (1978). 32 . Matsuoka, Pressure-indMced C,.,stal/ization in Polyethylene, 1. PoIYm. Sei., 42, 5II (1960). 33 G. M Martin and L. Mandelkem, 1. Appl. Phyl, 34, 2312 (1963). 34 K K Racho, ll. C. Ferpnon, and ll. ll. Herbert, Effect ofMo/ecultu wighton htgh pNSSIII'e crystallization ofli",.polyethylene. 1ki",tics andgross morphology, polym. Ena. Sei., II, 457 (1978). 35 P. D. Calvert and D. ll. UhImann, Crystal/ization ofpolyetlrylene lit high premues:  ktneticview, 1 Polym. Sei A-2, 10, 1111 (1972). • 3611. Ecke~ M. Buback, and G. R. Strobl, Coligid PolYOl. Sei., 2S9, 326 (1981). 37 B. J. Bridges, A. Cbarlesby, anclIl. Folland, Pme. R. Soc. Lgdon Sers A 367, 343 (1979). 38 M.C. Tobin, The Theory ofPhDse Transition Ki",tics with Growth Site Impmgement. n. Helerogeneous NIIC/eation, 1. 'olym. SR.: Palym. PhYl. Ed., 14,2253 (1976). 39 W. Dietz, SphtJrolitllwaclutlmt in Polymeren, Coll Polym. Sei., 2!9, 413 (1911). 40 Y. A MaIki~ vo. P. Beghishev, 1. A. Keapin, and BollOV S. A, General tl'eatment of polymer crystallizlltlon iinettcs-Part}. A new 1IIQC1'oIdnetic equation and ilS experillle"tal verlfictltion, Polym. Ena. Sei.. 24, 1]96 (1984). 41 S. Po. Kim and S. C. ICim, C"ystallizatiOll ofPolyethylene Terephthalate. Part 1: KinettcEquation witll Varlahle GrowtltRate, Polym~ Eni. Set, 31, 110 (1991). 42 K. Nakamura, T. Watanabee, and K. Katay""l. AlDI. 'olym. Sej., 16, 1077 (1972). 43 M. R. ICamal and E. Chu, lsotllmrtal tIIIfINan-isol1leP1lltll Crystallization of PoIytelrylene, Polym. Bq. Set, 23, 27 (1983). 44 M. Awami, 1. Chem. Phys., 7, 1103 (1939). 45 M. Avram~ 1. Chem. PhYl., l, 112 (1940). 46 L. Mandelkem. J. _Ir Phyl. 26, 443 (1955). 47 L R Hiller, 1. Plym. Sei"  3, 3069 (1965). 48 B. D. Bumett, and W. F. McDevits 1. ARDt. Phyl., 21, 1101 (1957). • 49 A. Ziabieki, Colloid Palymo. Sei.. 252,207(1974) 40 SO S. s~ S. T~ Y. Ookubo, M. Tsubokawa, T. Yamapchi, and T. Kikegawa, Pressrne dependenœ ofmeltingtemperaIIIrU in branclledpolyethylene up to 2 GPa, • PoIymer. 39,4267 (1991). SI IC. Takamizawa, A Ohno, and Y. Unbe, FJfect 01Pre."., on Melting TemperatfD'es ofPolyethylene FrtlCtions, PoIym. J., 7, 342 (1975) S2 VOD F. Simon and G. GIatze~ lJMwbmgen ZIIT Scheme1zdnlckbnw, Z. Ancq. Alli. Cbem" 171,309 (1929). 53 C. P. Chiu, K. A. Liu, and 1. H. Wei,.4 Methodfor MetlSllring PITJœlatlonships of 1'IIerIIIoplastics Usingan Injection MoldingMachine, PoJym. Eni, Sei., 35, IS0S (1995). 54 Y. A Fakhreddine and P. Zoller, Presswe-Vohlllle-TempertIhITe Shldies: A Versatile .4nalytical Tooi/or Polymers, ANTEC. ConfercnceProc:eedinll. 1991, Soc. o(Plasties Engineen, Brooldielcl. CT, USA., 1642 (1991). 5S P. ZoUer, "Polymerie Equations ofState," Polymer Handbook. 3rd Edition, 1. Brandup and E. H. Immerpt, Ecliton: Wiley" Sons, (1989). 56 S. K. Bhateja and K. D. Pse, J. MQC1'OIIIOL Set. /leva. Mqqomol. Che",., 13, 77 (197S). 57 R. Simha and T. Somcynslty, MqcromoIeÇfl/es, 2, 342 (1969). 58 R. Simha and ll. K.1ain, J. PO". Sei. Polym. '#lys. &1., 16, p1471-89 (1978). 59 M. R. Kamal and N. T. Levan, Polym. Eœ Sci., 13, p131-138 (1973). 60 P. G. Tait, "Physics andche",iSlry 0/the VO)'Qge 0/H. M. S. chal/enger," 2, Pt.4, University Press, Cambridae, pl900 (1888). 61 S. Beret and 1. M. Prausnitz, Densities 0/LiquidPolymers atHigh Pressure. PreSSllre-VoIUllle-Te",peratllre Meœwementsfor Polyethylene, Polyisobutylene, Poly(vinyl acetate) andPoly(dilllelhy/silomne) to 1iIJar, Macromolecules. l, 536 • (1975). 62 DilDtometry, Enc;yc;Icmedia ofPoIymcr Science and Soaineerinl- 2nd Edition, H. Marks N. Bikales, Ch. Overberger, Ed.; Wiley & Sons, New York, 1986. 63 P. Zotler, The pressure-lIOlale-telllperalllre properties 01tlne well- characterized low-density polyethylenes, l Appt Polym. Sei., 23, lOS 1(1979). • 41 • CBAPrER3 OBJECTIVES The present workis part ofa more Seneral research protp'Blll, currently in prosress in this laboratory, to study the etrect of processinS conditions, such u pressure, temperature, stress, and defonnation history, on morphololY and crystallinity, and, subsequently, on the optical and mechanical properties of the final product. Usina a pressure-volume-temperature apparatus, the present research project attempts to study and compare the PVT behavior and the etrect of pressure on crystallization kinetici of various grades ofpolyethylene resins. The main specifie objectives ofthe present study are outlined u follows: (1) To carry out PVT meuurements up to 200 MPa for the 13 resins in the isothennal and isobaric modes ofinvestigation. • (2) To perform isotherma1 crystallization studies under various pressures for the mins. (3) To obtain the equations ofstate of the PVT melt behavior ofthe résins usina the isothermal experiment data. (4) To usess the rneltins and crysta1lization temperatures up ta 200 MP. and their dependence on pressure. (5) To fit kinetic data with the Awami equation and compare the crystaIlization kinetics ofthe variOUI paofpolyethylene rains. (6) To evaluate the influences of the polymerization process parametcn and the stnIc:tural parameters of the polymer on the crystaIlization behavior under pressure ofpolyethylene resinl. Some polymen are Imown to IOlidify in a two-step crystaUization process, referrecl to u primary and secondary crystallization. Rowever, the Avrami equation • CIIUIOt be usecl ta represent the two-step proCess. Nevertbeless, primlry crystallization is 39 the malt kinetica1ly importlllt !tep durinB solidification. Therefore, the present work bu also involved the development ofID anaIytical technique to usess consistently the end of • primary crystallization for fittinl purpose. • • 40 • CBAPrER4 ExpERIMENTAL This chapter is devoted to describing the physical properties ofthe materials used in the present study. Detail description ofthe pressure-volume-temperature apparatul, its principle and the measurement procedure employed in this worlc are also presented. 4.1) Polyethyleae Resia Properties The 12 LLDPE ruins, used in the present study, were provided by Nova Chemica1s, u weil u the commercial HOPE resin, Sclair 2908. The LLDPE mins are experimental resins that were prepared under dift"erent polymerization conditions. The rains were copolymerizecl in SU phue or solution medium, usins Ziegler-Natta or Metallocene catalysts with ditTerent comonomer types (butene, hexene or octene). The • properties resulting hm the vuious polymerization conditio~ u supplied by NOVA, are listed in Table 4.1. Table 4.1: Proprieties ofpolyetltylene ruina• c.___...... CatIII.JIt"""" __... Ma Mw MwlHl CNe c.-;__ Tyfe wcJ) W1"') .... w-t) BDPE 1 1 1 0,961 7,4 22300 74500 3J 1 A Hcxcae Oaa ZN 0,9208 0,90 30000 111000 3,7 3,94 B ButeDD OU ZN 0,9194 0,94 24200 98700 4,1 4,03 C Hexeœ Solution Met 0,9234 0,85 36000 111300 3,1 3,77 D Hcxcae Ou MIt 0,9191 1,00 44000 91000 2,1 3,01 E Hexeœ Gu Met 0,9194 1,03 43000 94000 U 2,56 r LOPE Ou 1 0,9190 2,30 12000 l1OOO 7,3 1 G Octcae Sa.... ZN 0,!200 1,00 17000 106000 6,2 3,20 R Buteœ SollliDa ZN 0,9190 0.75 24900 120000 4,1 3,10 1 Oc.. Sa..-. Met 0,9010 6,50 22000 53000 2,4 5,00 1 Octeae 50..-. Mat 0,9110 1,10 31000 70000 1,1 3JO Je LDPB Ou 1 0,91. 1 1 1 1 1 • L UDPE Sa.... 1 0,9221 1 1 1 1 1 41 • A fully automatic GNOMIX high-preuure elilatometer, bucd on principles described by Zol1er" wu used and is presented in this parqraph. A schematic drawing ofthe PVT apparatus is given in Fipre 4.1. In this apparatul, the sample is contained in a piezometer coll tOlether with a confining ftuid (in this study, mercury), which must not interact with the sample. Volume changes ofthe sample are calculated by subtracting the known volume cb&nle of ftuid ftom the combined volume chanSei ofthe sample and the confining tluid. The .PPHllos uses the bellows technique. The SIIIlple and the comning liquid are enclosed in a rigid sample cell, the piezometer cell, one end ofwhich is closed by a flexible Metal bellows. A hydrostatic pressure il applied outside the coll. The hydrostatic pressure ofsilicone ail (Dow Cami..21011) is produced with a high-pressure motorized (or manual) pump. The pump CID produœ pressures up to 200 MPL The pressure is sealecl in bath endl by the vertical force of a 2S-ton hydraulic ram. The pressure is tranlmitted ta the contents by the Oexible bellows, which expands undl the pressure in • the œil equals the applied pressure, except Cor a sinall pressure difTerenœ resultinl hm the spriug constant ofthe bellowi (amounting to les. than 0.1 MPa"). The length choie ofthe bellows is a measure for the volume change ofthe contents, provided the effective cross-sectional area ofthe bellows is Imown. The length change is measured by a linear variable differentiai transformer (LVDT), located beneath the pressure vessel. The hiabest part ofthe sample œil contains the polymer sample and is separated ftom the bellows usina a nickel sample cup, which prevenu molten polymer ftom getting into the corruptions ofthe bellows and maintains a hydroltltic pressure" . • 42 • Dl TO" ....AT. i IlA. aiL CATCH / 14. Til .... • IUbll • COII"'Ol. TC Iltl ....LOW. ".aO.ITlIl • 1 Di ilOT•• ",.J " 1 ~ 'UIIIT'Il'• 'IIAlla...... IH'.LD ClAU,•• C- ~ ... '.AL" '" .11 AI. YALY&...... - 1 .mL P9d ., .:-: __.. .Iii1..,.ç:.....:,.z-: 7.: 5 ,-., •• ·1J... _ Fi"", 4./: Sche1UicdI'tlwing sIIowing the ...",blyoflite PYTtIplJtI1YltIIS • 43 A bigh-pressure thermocouple (type K), which bas its sheathed end located close to the sample in the œil, measures the sample temperature. A separate thermocouple • (type K) is used to control the temperature. It is locatecl in a hole drillecl about 1/2 in. into the lide ofthe pressure vesael. A temperature caIibntion is needecl ta relate the sample temperature and the cantrolled temperature. The vessel il equipped with a 1700-1800 Witt e1ectrical heater. To prevent excessive heating ofthe bouom plate, the base ofthe usembly is water-cooled. A fan provides cooling to the upper part orthe assembly. The measurement data including sample temperature, pressure, and voltage fi'om the LVDT (representing bellows detlection), are read by a data acquisition board, located in the MS-DOS computer. The computer allO guides the user through the pmper operation ofthe equipment. 4.3) PVT Masure.eat Techniques There are three main modes ofoperatinS the PVT apparatus: isothennal, isobaric, and isothermalJisobaric. In the isotbermal mode, the sample is heated and then • maintained at a constant temperature. Subsequently, the pressure is increased, with specifie volume recorded Il pressure intervals set by the operator. The second mode of operation is the isobaric mode, in which the pressure is maintained Il a certain value, white the temperature il changinl Il a certain beatinl or cao1inl rate, with volumes recorded in selectable ûme intervals. The lut mode of operation il the ilOthermallisobaric crystallization mode. In this mode, specifie volume is recorded u a tùnction oftime at constant pressure and temperature. 4.3.1) Sample Preparation PVT experiments permit the study ofvuiOUl Ih.pes ofpolymer samples IUch u peUets, pieces ofmoldinPt lIld liquida. The u.offine powclen and fiben are, however. not stronaly recommended sinee IODle problema miPt he encountereel durinl the fiUinl • of the ceU with a confininl fluid. ConsideriDg the volume of the œil and the 6mitcd 44 bellows detlectioDS, the suitable amount ofpolymer is about 1.5 cm' for semicrystalline samples and 2 cm' for amorphous materials. Consequently, the sample masses employed • in tbis study were in the raDge tram 1.15 ta l.l0 1fIIIIS. Samples were used in form of pellets, u provided by the company. The complete sample preparation procedure CID be found in Appendix L 4.3.2) Calibration A calibration ND is requirèd for each measurement, if the bellows, one piece of the cel~ or the confining ftuid is replaced. The volume changes measured with the PVT apparatus depend on the relative position of the core and the coil of the LVDT. This position is not sole1y inftuenced by the volume chanle of the polymer. Therefore, & calibration nm consista ofmeasuring the volume changes due ta the confining tluid in the cen, the bellows, the vessel and ail the other parts ofthe apparams that cao be intluencecl by temperature and pressure increases. Ali these panmeters are accounted for by the correction fùnction dl(P,T) tbat is obtained in a separate experiment carried out with the • cell filled with the confining Ouid only. Calibration points are talcen trom room temperature upta 400°C along ilOtherms every 1SoC and for pressure between 0 and 200 MPa every 10 MPL These data are fitted to the correction funetioD with the followins relation, dl(P,T) = 10 + alP + (&2 + &3T)P (l.l) which is subtraeted Itom the radina during a sample NIL Acalibration run is canied out by following the same methodology u with & sample run, except tbat the œil is tilled with the confining tluid only. A detailed procedure of a SImple ND is described in AppendixI. Due ta the specifications orthe PVT appantul, the hi....temperature is limited • ta 405°C and the pressure maximum is 200 MPL However, the temperature nnp tbat 45 wu chosen for PVT experiments wu mainly determined by the depadability of materials studied. Thermal-oxidation ofpolyethylene wu repolted to occur over 2S0°C, • at atmospberic presmre". Intbis study, polyethylene samples were immerged in mercury and under pressure o( al leut, 10 MPL A maximum tempenture of240oe wu therefore a reuonable choice to avoid any degradation. 4.3.3.1) Isothermal measurements An isothermal PVT measurement consisted ofrecording volumes along isotherms with increasins pressure fi'om 10 MP. ta 200 MP. by increments of 10 MPL Ail the parameters of isothermal experiments, such u pressure increment, isotherm spacing. holding lime and pressure and temperature range, were set before each experiment. In this wode, isotherms were canied out fi'om 30 ta 220°C in order ta observe the polyethylene sample bebavior, up to 200 MP.. in the semicrystalline, the transition and the melt regions. Isotherms were spaced by 100 e trom 30 ta 90°C, by SoC trom 95 ta 17SoC, and by 1QOC fiom 180 ta 220°C. IIOtherms were spaced by SoC in the transition • region due ta the large volume changes induced by Metting or crystallization. A typical i5Othenna1 experiment started at room temperature under 10 MP.. the sample wu tint beated ta the desired isotherm under 10 MPa, and, Ifter temperature stabilization, pressure wu next increased up ta 200 MPa, by increments of 10 MPa. The holding âme, which ilthe time that the program will pause al each ofthe pressure points requested, wu set Il a compromised value of20 seconds, talcing into account accuraçy and residence time (degradation). This boldinl âme wu al50 recommended in the PVT apparatus manual. Aler having reached 200 MPa, pressure wu tben releasecl and the sample wu hated to the next ilOtherm. The Prolfllll could not control the heating rate. One cycle, trom one i50therm ta another, lutecl approximately 20 to 30 minutes. Aiter • fWl isothermal ND, the sample hacI to be replaced since it could be degndecl. Bach isotherm wu .veel u a separate AScn file. It wu ellCntial to record severa! ilOtherma around room tempenture. Indeed, tbese data were utilizecl ta calculate • 46 the additional volume ofthe sample, which allowed the determination ofthe exact sample • specifie volum~ at the initialization temperature and Zero pressure (et: Section S.I». 4.3.3.2) Isobaric meuurements Isobaric experiments consisteel of recording volume in selected time intervals while heating or caoling the sample al a constant, preset rate and pressure. The bell compromise, between speed and accuracy, wu obtaineel for a rate of 2.SoCimin. The melting and erystallization curves were obtained for each rain at 2.SoC/min und« the pressures ofla, 50, 100, and 200 MPL Specifie volume ehange data were recorded every 10 seconds, and saveel in a separate file for each melting and crysta1lization curve. Prior to each melting experiment, the sample wu 6rst heated al 4°C/min under a pressure of10 MPa ta 200°C, at which it wu maintained for 20 minutes. The sample wu Dat cooled ta room temperature at 2.SoC/min under 10 MPL This methocl a1lowed the reuse ofthe same sample for different isobaric Nns u long u each coolina experiment at 10 MPa, between the NU at higber pressure, revealed the same erystaIlization behavior • u that ofa ftesh sample. This method alsa producecl an identica1 thermal history ofthe sample prior ta each new pressure experiment. 4.3.3.3) IsothennallIsobaric Meuurements In the isathennallisobaric experiments, volume chaftles were recorded u a fimction of âme al constant pressure and temperature. In this mode, pressure and temperature were controlled by the operatorthrough the software. al ae_rqlProcedMre. This procedure wu bued on the one developecl by Samara" for bis study on cryataIIization kinetics uncIer pressure of the sune HOPE resin. In this procedure, the sample wu first heated Il 4°C/min ta 200°C under 10 MPL Ailer 20 minutes. which il • the time nec:essary to stabUize the volume ofthe melt (indieating complete melting), the 47 sample wu then coolecl under 10 MPa ta the clesired crysta1lization temperature. The coaling rate wu approximately S-6°Clmin, which correspondecl ta the maximum rate • provided by the apparatus. The sample wu maintainecl al tbis specific temperature for another 20 minutes ta allow stabilization ofthe temperature. Prior ta pressurization, data acquisition wu set to record volume, temPenture, and pressure at minimum intervals of lOs, which is the shortest interval oflime a1lowed by the software. This limitation il probably due ta the time needed ta obtain a representative signal ftom the IW'dware. Next, the pressure wu increased manually ftom 10 MPa to the desirecl crystallization pressure. Manual compression wu preferred ta the computer-controlled pressurization because pressure could he raisecl considerably faster. For instance, pressurizatiOD tiom IOta 200 MPa luted about six minutes when it wu computer controlled, and about 30 seconds for manual compression. This becomes of major importance for low crystallization temperatures, i.e. high degree of supercoolins, for which crystallization is likely to occur during slow pressurization. It wu convenient to employ the online piotter ta follow the progress of the crystallization. At the end ofthe crystallization kinetic experiment, data acquisition wu stopped and pressure wu brought down manually to 10 MPa. Ifthe same sample wu • reused, the temperature wu then raised back to 200°C, and another isothermal could he performed. Ifthe sample wu degraded, the ternperature wu then decreasecl ta 20°C, the pressure to atmospberic pressure, and the œil wu cleaned and a new sample wu prepared. At leut three different crystallization temperatures were employed at each ofthe following pressures SO, 100, ISO, ancl200 MPa. Due ta the ntber slow maximum coalinl rate, experiments at 10 MPa couId IlOt he carried out, since crystallization would probably occur during coaling. bA Choo.rin6"" Castallizqtion Conditions: T"'PrtItIn'e andPreSSIIN At atmospberic pressure. the tlUe supercooHng is theorelically defined u:AT =T: -T,. t where T: il the equilibrium meltins point and Tapis the crystallization temperature ofthe experiment. However, values ofthe equih"rium mehina • point unef« preuure have DOt beeIl reported in the literature; tberefore, in this work, the 48 supercoo6ns WIS defined u follows: AT =r.! - T.., wbere Ti is the meltins point • obtained &am the isobaric experiments al the experimental pressure, and T.. is the experimental crystallization temperature. This methocl bu been succeufially used by l lJ other workers 6. • At each dift'erent pressure, crystaIlization kinetics experiments were performecl in the same range of depee of supercooling, in order to compare experimentl al distinct pressures and for various raina. In addition, because of the fast crystallization of polyethylene, the set ofexperiments wu conducted in a limited supercooling range. The supercooling wu typically in the range ftom 2 to 10°C. Under these conditions, crystaIlization wu rather slow, it took about 20 minutes ta 24 hours ta he completed. The supercoo1inS range wu alsa chosen to senerate an induction lime 1111er than 80 seconds. This interval time was necessary for better accuracy in detectins the heginning of crystaIlization (ct: section 5.3.2). Under very low supercooling, AT<2°C, the volume change due to crystallization wu ofthe sune order ofmagnitude u the sensitivity ofthe appuatus, resulting in large error in the results. Under hish supercooling, AT>1aoc, crystaIlization wu 50 fut tbat • sorne phase transformation occurrecl during pressurization, and the analysis of these experiments wu therefore complex and errar sensitive. Crystallization temperatures were thus selected ta give a certain depee of supercooling. The compression heating etTect bad to he considered in the estimation of crystaI1ization temperatures. For instance, for an exjJeriment under 100 MPa, the crystaIlization temperature set in the software wu: TÏJGIIImII =r!-/OO - AT - ATCtIIIIpN8f_ (4.1) wbere AT~" is the approxirnate expected increase of temperature due compression and AT il the desired supercoolinB. W'lth this mcthod, the supercooling wu predicted within 1°C. • 49 1'71 140 1'77 f, 139 \ • .1 ~, . 131 ql'M 1 . • • q ~ \ , 131 ~ i 115 •, \ i 1 114 '. 1~ • 1 " • J 173 135 J 134 j 172 âT.-2.5OC fil ... 111 133 110 .....-.. 132 18 131 -'GO .. ·200 0 200 400 100 lOGO T_(I) Fia" 4.2: Temperatrll'e versus Time during rapid compression. The solidli,. isfor an increase o/pressurefrom JO to 50MPa, and the dœhed-line.from JO to 200MPa. The rapid manual pressurization caused • temperature increase (and thus, • instability oftemperature), which wu predietably dependent on the pressure inaease, u illustrated in Figure 4.2. The temperature increued by 2 to 7°C durinS the first minute and then decreased and stabilized in S to 10 minutes depending on pressure. It wu noticed that the temperature stabilized al a value hiSher than the temperature of the isotherm before pressurization. An augmentation of -1°C wu observed for an experiment al sa MPI, and of-2.SoC for one al 200 MPa. • so • 64 P. ZoUer, P. BoUi, V.. Pabud, and K Ackerman, ÂPfJ'I'QIIISfor metISII1'ingpressure volllllfe-te",peratrue relationsltips 01polymers 10 Jso(e and 2200 iglC1II2t Jleyt Sei. InstnJm" 47,948 (1976).. 6S 1. He, Y.. A. Fakhreddine, and P. ZoUer, le Appl. Palym. Sei., 4St 4S (1992). 66 Linear Law DensityPolyethylene, Ençyclgpedia ofPoIymcr Science and Enaincerina. 2nd Edition, H.. Mark, N. Bibles, Ch. Overberger, Ed.; Wiley et Sons, New York, vol. 17, 756, 1986. 67 M. Samara, PhD's Thesis, McGill University, in prim. • • 51 • CllAPl'ER5 DATA ANALYSIS This chapter is devoted ta describing the analysis ofthe experimental data files. The procedure to obtain the parameters of the equations of state trom isothermal experiments is first presented. The evaluation ofthe melting points and the crystallization temperatures trom isobaric experiments is next explained. Finallyt a detailed description of the method used to extract the crystalliz.ation kinaics data trom isothermallisobuic experiments is discussed. 5.1) ISOTHERMAL MEAsUREMENTS This parqraph is dedicated ta explainina how to relate the recorded volume • chaDges ta the tnle specifie volumes, and how to fit tbese adjustecl volumes to the Tait equation and the Inverse-Volume equation. 5.1.1) Determination ofthe Addition•• Volume The additional volume is added to the volume change data to yield the true specific volume values. The additional volume wu calculated &am isothennal ND data and the value ofthe specific volume (inverse ofdensity) at 23°C and zero pressure. The zero pressure volume changes were ploUed venus tcmperature (hm lOoe to 100°C). The additional volume ofthe sample wu obtained by extnpolating the zero pressure data ta the temperature al which the exact spceific volume ofthe sample wu known and by uling the Collowing equation: (5.1) • 51 5.1.2) Determination ofthe Melt and Solld State Relionl • The firlt step ofthe aaalysis wu to identify the temperature range in wbich the transition takes place. This transition reaion cm he eaily observecl by plotting specifie volume data venus pressure for each isotherm, as shown in fipre S.l. Melt Tnmsitiœ Salid o 50 1. 150 r...re - Fi.,e S. J: Plot ofSpecifie Volume verSllS PresSInfor different isothemu. • This figure IUgeats that the isotherms between 140°C and 16SoC should he omitted nom PVT relations for the liquid and melt phases for the pressure range up ta 200 MPa. Indeed, these isotherms exhibit a volume drop alons the curve that corresponds ta the occurrence ofcrystal1ization during preuurizatioD. The PVT meh data, which were obtained for a cha..pressure range. were tben tilted 10 an equation ofstate. Data Cor the solid state region couId alsa he used to evaluate thermal expansion coet1icient and isothennal compressibility ofthe semicrystalline phase. 5.1.3) Determination orthe Pan.eten oftheTait Equatlon Two techniques were testecl to fit experimental data 10 the empirica1 Tait equatîon, eq. (2.17). In bath CIIeSt it wu found tbat the univenal value of0.894 for the • constant C yielded resuhs witb a comparable accuracy to those obtained with C u S2 variable. The tirst fitting technique WIS composed ofthree steps, whereas the second one had only one step. • For a better undentanding ofthe Tait equation, a three step non·linar regression wu tint used ta determine the four paranteters ofthe Tait equadon: Yo, a, Bo and BJ. The experimental specifie volumes were tirst tittecl to the folloWÏDI equation: (5.2) with C equal to the standard value of 0.0894. For each isotherm, a value of the Tait parameter B wu then obtained. B(lj wu next titted by a non-Iinear regression to an exponential functioD, as follows: B(lj = Boexp(-BJ1} (5.3) In addition, the specifie volume Ilatmospherie pressure wu fitted by the same method ta the Collowing equation: Y(O.T)=YQnp(aT) (5.4) A1though this three-step methocl yields a precise representation ofthe experimental data, and a Sood overview oCthe Tait equatiOD, it is âme consuming. • On the other band, the second method used to analyze the PVT data wu less time demanding. Indeed, the specifie volume data were titted directly to the Tait equation, as Collows: The Cour parameten, Vot a, Bo and Bl~ were thus obtainecl in one non·lioear regression. This methocl yielded 1 slightly bètter fittinS than the three-step rqression describecl earlier. The followinl standard deviations were obtained for the tittinls ofthe PVT melt bebavior ofa LLDPE relin: a= 0.0003 cm]/g (orthe three-step method a=0.0002 cm]/1 for the direct non-Hnar repesaion. Because ofits convenience and its accuncy~ the second and direct method wu used ta • analyze the PVT melt bebavior ofall isotbermal experiments. S3 • 5.1.4) Determination ofthe Panmeten ofthe laver••Volume EquadoD The melt resion wu tirst obtainecl usins the same methocl describecl in the previous section. The Inverse-Volume equation wu atso fittecl Ulina two difTerent methods, a two-step and a direct nonlinear resression. The two-step procedure wu applied as detined by Levan". The experimental densities (the inverse of the specifie volumes) were tirst fittecl by a linur resression for each isobuu follows: (S.6) The second observation made by Levan is that the density can he representecl at constant temperature by a second order polynomial. Consequendy, Ql and Q2 wer,a fittect ta the two followiDg equations: (S.7) Q, =(ôp) +bP+dP' (S.8) ôT 1'=0 This Proœdure yieldecl the six parameten of the Inverse-Voiume equation, which are • p Pc, (ô ) , a, h, c, and d. However, itwu possible ta obtain these parameters usins a ôT 1'=0 non-linear regression by directly fittina data ta one equation, u Collows: P=P"+(:)r-o xT+(a+bT)p+(c+dT)p2 (5.9) Similarly u for the Tait equation fitting, the direct estimation gave a sliaht1y better fit than that ofthe non-clirect calculation. ThereCore, this technique wu usecl ta evaluate the panmeters ofthe invene-volume equation. 5.1.5} IIothermal MalurementLlmitatioDl ln the transition regioo. meltilll and crysta1lization occur cIurina ilOtherma1 experiments. At the olllet ofcertain isothenns WlCIer 10 MP, the IIIDple i. in the melt, • but, with increuing pressure, the sample starts crystaIlizing due ta c:Jumae in the S4 thermodynamie equilibrium (see isotherm at 140°C in Fipre S.I). At 200 MPa, the sample miSht he partly or completely solidifieddepencliog on the isotherm temperature. • A partly 106dified sample implies that crystallization wu already in progress when the volume value wu recorded, which is probably a tint source oferror. Crystallization duriDS pressurizatiOD can influence the specifie volume values recorded al pressure increments &fter crystaIlization. For instance, in Figure S.2, the HOPE resin investipted in this work seems to possess a neptive thermal expansion coefficient al certain pressure. This etrect is due ta the measurement technique and will be discussed in more details in Chapt«6. 1.35 ~------:''':o=---' 1.30 1.25 11.2·il i 1.15 .. 1.10 ~ • 1.05 1.. 'MeMu...... t error 0.95 +----.,...... ------,r-----~---__T"---__I o 100 150 200 250 T...r....re(oC) Fi"" 5.2: Cross-plotted isolhmlltlldlltafor Sciai, 2908 u/ti1Jitinga decrease of vo1ullle wiM increasing I,IIIpII'tIIIIN dM. 10 ",•.",.",.",.rroT. 5.2) ISOBARIC MEAsUREMENTS A lypical isobarie experiment is shawn in Fipre 5.3. From this plot, it is possible to estimate the temperatures correspondina ta the 0"- T., and the end, T... of • melting. iD beatîDI ruai• 5S • J 1 J Fi.,., 5.3: lilustmtion oft.graphiClll dete""ination ofthe end ofthe melting andthe onset ofcrystallizationjroltl isolJaric experllllents ofHDPE under 100 MPa at 2..5 CC'nrin. • 1. 1. 1• • TrC) Fi." 5.4: yoœ. Change RIII, ver..., Te"".ratwefrom the ",.lting CII1Wa al 2.StCmtin ofLLDPE-B • 56 Similarly, it is possible to _mate the temperatura corresponcling to the beginning, To.ct and the end. T..Ct oC crystallization, in coaling ruas. Tbese temperatures are grapbically • acquirecl hm the beating and COGlinl NIlS, U illustrated in Figure 5.3. The (A) Hne il the tangent ofthe curve in the melt reaion, the (8) lines are the tanpnts al the inflexion point ofthe curves in the transition reaiOD, and the (C) line is the tangent ofthe curve representing the solid state behavior. The melting poÛltl and crystallization temperatures were defined u the temperature at which the maximum volume change rate wu observed during meltinl and crystallization, respectively. Figure 5.4 shows typic:al melti. curves of the volume change rate versus temperature for a LLDPE resin. These curves were also usefùl for comparing the ditrerences in crystallization or melting behavior ofthe various grades of polyethylene resins. 5.3) IsOTHERMALllsoBARIC MEAsUREMENTS A methodological analysis ofthe isobaric:lisothermal experimental data bad ta he • developed. Indeed, the temperature ofthe experiment, the occurrence of crystallization during pressurizatioD, the induction time, and the bepnniDI and end ofcrystallization are quite sensitive to error durinl anaIysis. In order ta obtain crystallization data tbat could he fit ta kinetic equatioDS, all the above ntentioned quantities had ta be known. The tbree main parts of • kinetics experiment are c:ooling, pressurizatioD, and crystallization. Tbese sesments are easily determined, and they can he observed by plotting volume changes venus âme, u shawn in Fipre S.S. 5.3.1) Crystaillzalion durial Preuurlzation It il ofprimary importance ta determine when crysta1lization startI, especiaIly for low erystallization tempenture. Ind~ crystalIization is likely to occur during pressurization at bigh ..perc:ooliDl. When no induction period wu ob~ IWO • metbocll were used for verifying the occurrene:e ofc:rystallization 57 • -Tilt ..... CaoUa. -Es...... C.,..."ip.... Fipre 5.5: Schematlc plotofQ1I isothermtlVisoboric crystallization aperiment Firstly, usins the PVT melt data trom the isothermal experimenu. the volume values ofthe melt were extrapolated to the temperaturea and pressures considered in the kinetica experiment. Figure S.S presents a ND in which the Tait equation wu employed to estimate the ditTerent volume chanses. Deviation ftom the Cluve representing the Tait • estimation will indieate crystalIization, as shown in the lut putofthe plot in this figure. A second complementary method coftSisted ofcomparins the volume chanse due ta the increase of pressure with other experiments carried out at higher temPel'ltul'eS, wbich exhibit an incubation period. In fact, the presence ofan induction period implied that no crystaIlization had occurred durinl preuurization. The slilht temperature dependence ofthe compression volume chanpwu nealigible. When crystallization wu detected durinl pressurization, experimental data were analyzed, but the errar senented by the anaIysis wu larp. In Rich eues. a trial and errar approach wu used, employinl extrapolation ofIII Avrami type tùnction. ln this work. aystallization temPentures were pnenlly cholen ta enaare that DO crystaIlization wouId occur beCore the end of preuurizatîOIl, and tbat the experimentl would exluDit ID induction time Iarger than 80 seconds. The incubation periocl of80 1 wu determined ta ensure tbat crystilliZltion wouId start der the volume stabilization • due ta npid compression wu over, as explained below. 51 5.3.2) IDducdoD nme••d BealDDiDI ofCryltaUlzatioD • The besiJming ofcrystaUization, wbich corresponds to the end ofthe induction time, wu definecl at the point at which the curve deviates &am the straight line observecl during the inductioll time, u shawn in Fipre 5.6. J CrylltlDizalian 1 ...... ~~~~------•••••• J •••••• ValIDe •••••• SlIbiIizaIion •••• • ..--lDductionlÎlM - ...... Fi.",.. 5.6: Sclte1lltJtic diagrtllll ofa typicalcrystll/lization kinetic erperlment This figure a1so illustrates the changes observecl during the induction time. The volume decreue, referred u volume stabilization, wu observecl during the tirlt 80 seconds (averase value) ofthe induction lime. The decreue in volume is probably due to a combinatioll of severa! facton such u the viscoelastic relaxation of the silicon oil, pressure stabilization, and the bellows responae. This decreue is certainly not attributed to any crystallization sinee ilwu alsa observed when • polymer melt wu pressurizecl ta • pressure under which the sample remains in the melt. This volume decreue wu in the range fiom 1.0 ta l.Smm:J for pressurization ta 50 MPa, and ftom 4.0 ta 4.5mm.:J for preuurization to 200 MPL In addition, sUght variations were observed for 10DI induction periods due ta tanperature fluctuatioDl. • S9 -. 5.3.3) End ofCry.gllization Crystallization kinetics experimental results are usuaIly represented by the relative crystallinity versus time. In dilatometric studies, the relative crystaUinity, X. is defined u: ~-JI'o x= (s. 10) VCl) -J'O The value Vo is the volume at which crystallization begins, corresponding to the end of the induction tim~ u defined earlier, and the value Jl'aD is defined u the end of erystallization. In theory, the end ofcrystallization does DOt seem difticult to evaluate. However, certain polymeric materials such as, for exampl~ polyethylene terepbthalate and linear low-density polyethylene exhibit secondary crystallization. which is a very slow process that cao lut days ta he completed under certain conditions. Sinee the objective ofthis work wu ta compare crystaIlization Idnetics ofVlrious grades of polyethylene resins, the development of an anaIytical method that wu repeatable and convenient, wu of utrnost importance. The method developed by Samara" for his study on crystallization under pressure ofHDPE, which wu bued on • the assumPtion that, al constant pressure, the maximum volume change • due to complete erystallization - wu not dependent on temperature, wu not applicable ta LLDPE resins for two reasons. Firstly, secondary crystallization wu extremely slow, and ns completion bad never been reacbed, even after 24 houn. Secondly, the maximum volume chanle wu dependent on crystallization temperature. These two observations are discussed and illustnted in section 6.3.1). The main idea ofthis method wu ta exclude the data corresponding ta sec:ondary erystallization sinee the completion ofcrystaIlization couId DOt be observed in reasonable time, and sinee the Avrami equation omy fit acc:untely primary crystallization. The value ofV., which thus corresponds ta the end ofprimary crystallization, wu detennincd by a 44 methodololY bued on a principl. tbat wu tint indieated by Avnmi • He sugested tbat it wupossible to sbift isotbermal kinetics data to a single mutercurve with respect ta 1ft ubitrarily cbosen reCerence temperature.. ICinetics data of ail experiments (Il different • pressures and temperatures and Cor ail resins) were sbifted to one sinsle muter curve by 60 varying the value of J'ID. This wu achievecl usinga pro..... that compared the Avrami fitted values of the reCerence experiment to the run under analysis. The reference • 8()1~ experiment wu cho.. to represent of crystaUinity and yielded a good Iinar correlation coefficient (Il?o.99) ofthe Avrami equatiOD fit. Using the utimatecl value of the end ofprimary crystaUizatio, the relative crystallinity wu calcuJated, and data were then fitted ta the Avrami equation by a non-linear repasion. 5.3.4) CryltallizadoD Te.penture ofthe Esperiment The temperature Il the ilOtherm before pressurization, the maximum temperature atthe peak due ta compression heating, the temperature al the beainning ofcrystallization or the average temPerature during crysta1lization were ail the possible values that could have been usecl to represent the crystailizatiOD temperature ofan experiment However, the use ofthe ternperature Ilthe isotberm would bave neglected the temPerature rise due ta compression heating. The value at the peak would have been imlevant for slow experiments in wbich crystallization occurs Ifter temperature stabilization. The • temperature at the onset of crystallization would have not represented accurately fut experiments during wbich tempcrature wu importantly varying. Final1y, the temperature ofthe experiment wu detinecl u the average value ofthe temperatures recorded during crystallization ftom Yountil JI•• • 61 • RESULTS & DISCUSSION The results for the re&iDl investipted in this study ue presented in three sections, corresponding to the three types of PVT experimentl carriecl out In each section, a comparative presentation of the results for the variOUI resinl is repol1ed and then discussed. The PVT properties for the melt and the salid states, as fitted by the Tait and Inverse-Volume equations, are given in the tirst part. The dependence on pressure ofthe crystallization and melting processes, characterized by the meltiog points and crystallization temperatures are then presented. The final section presents a discussion of • the resu1ts ofthe crystallization kinetiCl experiments under pressure. 6.1) PVT PROPERTIES This part il devoted to presenting the differences in PVT bebavior ofall the resins in the temperature and pressure range under investiption in the isothenna1 experiments. The PVT melt behavior and its representation by the Tait and Invene-Volume equations is &1so discussed. The empirical Tait equation wu chosen bec:ause it is the most commonly used equatiOIl of state in polymer procesaina computer simulations. In addition, the inverse volume equation wu employed to compare ita predictions ofthe tbermocIyJWIÙc parameten, ..cil u the isothermal compressibility and thermal expansion coefticient, with the estimates orthe Tait equation. • • 6.1.1) General PVT Behavlor lsothennal experiments are commonly presented in the form of ctOss-plotted isobars, u shawn in Figures 6.1 to 6.4. Figure 6.1 i1lustrates the PVT behavior oftypical HDPE and LLDPE raiDS up ta 200 MPL Bach cross-plotted isobu curve CID be divided into tbree resions. The high temperature region, in which the specifie volume is linearly dependent on temperature, corresponds ta the bebavior of the melt state. The low temperature resion is referred to u the semicrystalline resioo. Between bath sections, there is the transition region in which large volume chaftles are observed due to erystallization alooS the isotherm during pressurization. The large specifie volume ditference in the salid state rqion in Figure 6.1 is due to the density clifference between the A and HOPE rains. In contrut, no obvious distinctions caR he Doticeel in the semicrystalline state rqion in fipres 6.2-6.4 because of the simitu densities of the LLDPE rosins. In addition, it is interesting ta note that these fisures indieate an almast identical PVT behavior of the melt state for ail the rains, includins HDPE. This indieates tbat the polymer chain stNcture, such u branching s~ branching distribution, • molecular weight, and polydispersityt does not affect strongly the PVT behavior of the melt state. The major differences in comportment are observed in the region where erystallization occun. For i~ the HDPE rain bas a narrow transition region, similar ta a first-order transition, whereu the LLDPE rains have tendency to exhibit a broader transition resion. Moreover, some distinctions can be reportecl between LLDPE resins. Resins F, and 1have certainly distinct behaviors, which are related ta their density and molecular structure. The rain F is a low density polyethylenet and the resin 1 is • very low density polyethylene. The uncertainty (ct: Appendix Il) and the reproducibility ofthe results are 0.003 and 0.002. cm'/., respectively. The apparent cIecreue ofthe volume observed for the HOPE resin in Fipre 6.1 il merely due to the mealUrement procedure. At certain ilO~ crystaIlizatioll occun cIuring presauization yielding a denser sample. Thil wu confirmed by the tact that a nm curied out with a sample prepared by crysta1lization under 200 MPa did IlOt exhibit IUch • abehavior. 63 1.35 .,.------A • 1.30 ••• HOPE ~1.2S ~ 11.20 u 1.15 J /. 200MPa ~ • t1.l0 .•.. ,.,• . ..•..•• ••• •• (12 LOS . ... -..-...... 4 •..- 1.00 ·...... •..•..--- 0.95 +---~----I~-__+----+----f.--__+--_+--__t 25 50 75 100 12.5 150 175 200 22.S Temperature (C) Fipre 6.1: Cross-p/oned isothe17Pltl/ aperlmentsolresins  andHDPE. 1.35 .A • • • aB 1.30 • •OMPa AC ••• ~ ••• {1.25 "oD ,.••• OD° ...... o,, • • •• J1.20 ,1 ••••• lOOMPa , 0\•••• ~ • ••• 1.15 - i' •••• 200MPa Co) • 1 o' • • .1 l" 11.10 . . • • l ,1 U) • • • • • • • ....,. LOS • • • • • • • • • • • . . 1.00 T · T 2S 50 7S 100 125 ISO 175 200 22S Temperature (C) • flDre 6.2: Cross-plotted isot1temla/ezperiments a/reains .4, B, C, andD.. 64 1.35 ~------,---. -E 1 t • aF ••t OMP, 1.30 ~ .0 • , ••••••• ~ aH • -!tl.25 '" DD ! a t .., 81.20-· a;'· aa'....•..• • l00MP, .a • a' ~ 1.15 - 1 J a. a a •••t t •200MP.••• 1~ 1 1 Ji" aa,' ~ 1.10 •• ::•• 1 • : , f JJ;J" l.OS -.: •••••• 1.00 -+------oj~---+-:---+---~:--+---+----+--~ 25 50 75 100 125 150 175 200 Temperature (C) Fime 6.3: Cross-plotted isothermal experime"ts 01resins E, F, G, andH. 1.35 ~------.---'111_ -1 • • aJ ._.- OMP, 130 -~ •• . aK ._- iD aL •••• ",'" 1.25 -.. _ ~.. .. S; 0 • a a •• ~ • • 0 •••••• lOOMPa 1.20 . ~ D 0 .-•• III J • 0·0 \.. :>1.15o .. • • '0.1 •• •aD 0 a.. ••••••••• 200MPa•• u •• .' • aD 0 •• 0 0 ~..· .' .. .&.' 0 •• .a 0 1.10 •• .'cr •• ..C& 0 t~ t~· ••• •• 1 .':..•• •••• 0 1. • LOS'";:• •• ' •• 1.00 +--~:~---+-:--O+:---ll'---+l--~I---+:---I 2S 50 7S 100 125 ISO 175 200 225 Temperature le) • 6S 1.21 ...... ------+-.,....., • Error bIr=OJ)02 cult. ~ 1.20 1...... 1 j 1.19 :> 1.11 ~~_-~~'--_+~-_-_--_-_----4 130 135 140 145 150 1~5 160 165 170 T_peratuœ (oC) Fipre 6.S: Cross-ploned isothennal experiments 01ailUDPEresins: Behavior in the • Meil state "nder 100 MPa. 6.1.1) PVf Melt Behavlor The PVf melt bebavior of the isothcrma1 experiments wu defined for the pressure range up ta 200 MPI, and subsequently for a certain raDie oftemperature. At 200 MPa, the onset ofthe melt state region for the resins wu found to lie between 142°C and 177°Ct as shawn Table 6.1. Fipre6.5 illustrates tbese differences in the onset ofthe melt resion ofail LLDPE resinl under 100 MPL It can he observed that raiD 1 exhibit. the lowest ouet temPerature, Collowed, at slightly hi'" temperatures, by the IJ'OUP of resÎDs Ft 1, and K, and tinally, ail the othen LLDPE raina show 1 similar high olllet temperature. This is mainly explained by the dilferences iD density. Indeed, it is weil DOwn tbat the meltinl of polyethylene rainl occun at hi'" temperature for bialter density. The melt resion definecl up to 200 MPa wu tben fitted by two equations ofstate: • the Tait ancIlnverse-Volume equations. Table 6. /: Onset te1llpBratures ofthe melt staI, alvarious pressures• .!.!!!J!!) A B C D E r G R 1 1 K L BD. • 0 138 132 132 132 131 121 137 132 112 127 123 133 142 100 148 142 147 142 142 137 142 142 126 127 133 148 162 200 173 167 167 168 162 152 167 162 142 152 153 168 177 6.1.2.1) Tait Equation Using the fiUing method described in Chapter St the PVT melt data for ail the rains were fitted to the empirical Tait equation: Y(P.T}=Voexp(aoT}x(l-Cln{l+ p 1) (6.1) Boexp(-BJT} In this study, the "universal" value of C (C=O.0894) wu employed 10 fit the experimental datL The fitting results for the four parameten ofthe Tait equation for the various grades under study are given in Table 6.2. The results for the exponential parameter Vo and the thermal expansion • coefficient (Jo are in good agreement with the values found in the Iiterature, as shown in section 2.3.4). However, the values of Bo and BIt whieh represent the pressure dependence ofthe specifie volume, are sliahtly higher than the Iiterature data. This small differenee produces higher values for the specifie volume ofthe seme order ofmagnitude of the uncertaiDty in the results. Moreover, the results are slightly dependent on the measurement technique, the tiaing method, and the min grade. Therefore, these results are considerecl consistent with the values in the literature. In Table 6.2, the symbol Cf represents the standard deviation ofthe ditTerence between experimental and estimated values, and it indieates a remarkably good reproducibility ofthe experimental data by the Tait equation. The parameters of the Tait equation contirmed that tbese polyethylene rains bave ID almolt identical PVT behavior in the melt state. Actua11y, all the parameten only ditfer one hm the otber by a maximum of6%. Table 6.2 lists densities ofthe rains by dacendinl value. It CID be seen that, althoup they bebave similarly• • there exista a tendency in the dependence ofthe parameten on density. The coefficient Cl 67 • Tahle 6.2: Fitting results o/the Tait equation parameters. DeaIity V. CIo .. lit fi Code (a/œi) (~/g) (-10' (MPa) (-Uf) (aJ/l! HOPE 0.9620 1.130 7.68 197.1 5.11 0.0003 C 0.9234 1.134 7.54 191.7 S.Œ 0.0002 L 0.9222 1.140 7.56 203.7 5.21 0.0003 A 0.9208 1.136 7.52 194.5 4.99 0.0004 G 0.9200 1.145 7.55 198.3 5.04 0.0003 K 0.9198 1.143 7.53 198.0 ,.os 0.0002 • E 0.9194 1.138 7.44 196.3 4.99 0.0002 B 0.9194 1.137 7.46 194.4 4.93 0.0002 D 0.9192 1.138 7.42 199.5 s.œ 0.0002 B 0.9190 1.141 7.39 199.8 5.01 0.0002 F 0.9190 1.143 7.40 197.0 5.03 0.0002 1 0.9180 1.144 7.35 195.2 4.91 0.0002 1 0.9070 1.148 7.~ 194.1 4.91 0.0002 • 68 • (a) (b) 1.160 .,...------, 7.1 ~------, .rr.r-'.s~ 7.7 1.150 7.6 1.140 7.5 7.4 1.130 .rr"-'.l~ 7.3 1.110 +---r---_---r----i 7.2 ~--_--__- __---.f 0.9000 0.9200 0.9400 0.9600 0.9800 0.9000 0.9100 0.9400 0.9600 0.9800 {cl {dl 110 ....------. S.3 ....------. .rr.,-ltM. 105 S.1 5.1 100 S 195 .".,-l~ • 4.9 190 +---.,.....--r----~-~ 4.1 +------...-----1 0.9000 0.9100 0.9400 0.9600 0.9800 0.9000 0.9100 0.9400 0.9600 0.9800 Fi"", 6.6: Tait equation ptI1'tIIIIetera vs. density: eœh dtlttI point represents a nsin. • seems to increase with increuing density, u illustratccl in Figure 6.6 (b). This is in • agreement with the observation ofZoner--. Figure 6.6 (c) and (d) show tbat the Tait parameten, Bo and BI, are not clearly dependent on density. Furthermore. tram these figures, it can be noticed tbat relin L exhibits a singular behavior, (the highest value ofBQ and BI), wbich indicates tbat rainL probably possesses a slightly difTerent responae ta pressure. No fürther conclusions on chemical structure CID be drawn because of the variety in molecular weight and molecular weight distribution ofthe mins. 6.1.2.2) Inverse-Volume Equation The experimental data were alsa fitted to the Inverse-volume equation, which is detined as: P=Pe+(Ôp) xT+(a+bT)p+l..(c+dT)P1 (6.2) • ôT p=o 2 where Pœ., (:t.o' a, h, c, and d are the six parameters tbat are obtained by fitting, u explainccl in section 5.1.4. The fitting results of the Inverse-Volume equation are summarized in Table 6.3. As for the Tait equation nUina, the parameters are almost identical with some variations depending on density, U shown in Figure 6.7. For p instance, the thermal density expansion, (ô ) , plotted in Figure 6.7 (iv) exhibits a ôT P-o similar bebavior u the tbermaI expansion coefficient. However, a liD8'llar behavior il at50 observed in Figure 6.7 (üi), (v), and (VI) for the resin A The parameten tend ta be limilar ta the ones for the HOPE rain. No tbeoretical explanation cao be IÏven Cor this observation onthe bUÎl ofthe available data. • 70 • • • Table 6.3: Fitting Tesults ofthe Inverse-VohmJe eqll(ltion parameters .. DeMity pw • c dplR b d cr 1 ~arts., ~aD,> (J.an·'.MPa· ) (a).OC·' (a...·'.OC·') (a.clI{'.MP..; (bJ.OC·' (Wall,> HDPI 0.9620 0.175 2.65E-04 5.17E-OI -5.67E-04 2.43E-06 -6.6E-09 0.0009 C 0.9234 0.172 2.94E-04 -1.13E-07 -5.S6E-04 2.268-06 -S.IE-09 0.0009 L 0.9222 0.167 2.11E-04 -9.5IE-OI -S.S3E-04 2.31E-06 -5.1E-09 0.0007 A 0.9201 0.171 2.72E-04 4. 17E-OI -S.S7E-04 2.31E-06 ~.6E-09 0.0007 G 0.9200 0.164 2.91E-04 -1.09E-07 -S.S3E-04 2.22E-06 -S.6E-09 0.0007 le 0.9191 0.166 2.95E-04 -1.15&07 -S.S6E-04 2.23E-06 -S.7E-09 0.0001 B 0.9194 0.170 2.91E-04 -9.97E-01 -S.51E-04 2.218-06 -S.7E-09 0.0001 E 0.9194 0.170 2.91E-04 -7.61B-OI -S.51E-04 2.26E-06 -S.9E-09 0.0007 D 0.9192 0.169 2.82E-04 -S.92E-OI -5.47E-04 2.31E-06 -6.0E-09 0.0001 H 0.9190 0.161 2.80E-04 -4.IIE-OI -S.41E-04 2.218-06 -6.0E-09 0.0009 r 0.9190 0.167 2.98E-04 -1.32E-07 -5.41E-04 2.23E-06 -S.6E-09 0.0009 J 0.9180 0.166 3.02B-04 -1.34E-07 -S.4SE-04 2.17E-06 -S.SB-09 0.0009 1 0.9070 0.163 3.11E-04 -1.76E-07 -S.39E-04 2.14E-06 -S.4E-09 0.0010 ..... (1) p.., (II) • • 0.171 3.20E-cM 0.17C 3.10E-cM 0.172 3.00E-cM 0.170 2.IOE-cM 0._ 2.IOE-cM O.... O.au 2.70E-cM 0.112 2.IOE-cM 0.10 0.12 O.M O•• O•• 0.10 0.12 O.N O.• O.• (ili) ct (Iv) dp/dT 1.0E-G7 -5.35E-cM S.OE-G8 -5••0E-eM o.oe+oo -5.•5E-04 -S.50E-04 -5.0E-oa -S.SSE-GC -1.0E-07 -S.IOE-04 .1.5E-07 -5.I5E-GC ·2.0E-07 -S.70E-GC • 0.10 0.12 O.M O... O.• 0.10 0.12 O.,. O... O•• (v) Il (vI) d 2••5E-GI -5.0E-oI 2•.aE-GI -5.2E-oI -5.•E-oI 2.35E-GI -5.IE-ol 2.30E-GI -5.IE-ol 2.25E-GI ".OE-oI ".2E-oI 2.20E-oe ".CE" 2.15E-GI ".IE" 2.10E-oe ....E.. 0.10 0.12 O... O.• O... 0.10 0.12 O." O•• O•• Fi,.,., 6. 7: Inverse-YohoIIe Partllllet,rs 1'1 deruity: etICII datapoint"",."ta ruin. • 72 • 6.1.3) IIotber..al CODlpr..lblllty and Ther.a. Espan.lon Coefllclent The isotbermal compressibility, K, ancl thermal expansion coefficient, a, are the derivatives of the equation of state Y(p, 1}. They are important in the computer simulation ofvarious polymer processinl operations. Tbe PVT data were differentiated numerica11y to obtain the values ofŒ and K. The numerical differentiatïon consisted of fiUinS polynomials to isotherms, and exponentials to isoban fol1owed by ditrerentiation of the polynomials and exponentials. The results are presented for three representative resins: A, 1, and HDPE. The resin 1 bas the lowest density, whereas HDPE bu the highest, and A represents the typical behavior of the other LLDPE rains. Indeed, a similar behavior wu observed for ail the resins. In addition, experimental data are compared ta the Tait and Inverse-Volume predictions. • 6.1.3.1) lsothermal Compressibility The formulu of the derivative for bath equations of state, which are Biven in Table 2.1, were used ta calculate the predictions of experimental isothermal compressibility, K. Table 6.4 presents some values ofK at ditrerent temPerltures and pressures for the three resins considered in this section. At fint, these values are consistent with the ones founcI in the Iiterature (ct: section 2.3.4). The compressibility wu logically round to increase with temperature and to decreue with risinS pressure. However, the values calcuIated fi'om experimental data showed a decrease ofK witb increasing temperature Il200 MPa, which can a1so he observed in Figure 6.8. This efFect is merely due to the use of a seconcl-depee polynomial equation to fit the isothermal data, which induces Iarpr error st biah pressure. fiDally, il cao be seen tbat the ilOtberma1 compreuibility oCthese three ditl'erent polyethylene resias il alma. identical. This ilOtherma1 compreslibility il thereCore IlOt stroaaty influencecl by the chain stnIeture • ofpolyethylene resinl. 73 Tabl, ft 4: &pert",."tallsothe11lltllCOIIIJINssibillty 01the INltstaI, for resimÂ, /, andHDPE: Avemge sttIIIdsfor the average WÜfle of • the telllperature intervaL o 100 8.47 5.70 2.39 A{A:' 9.25 6.13 2.33 224 10.20 6.59 2.21 8.56 5.72 2.35 BD,..fA:" 9.30 6.15 2.17 LW 10.10 6.50 2.17 177 8.57 5.80 2.49 BD' Av...... 9.26 6.18 2.41 222 10.10 6.59 2.16 1.2 -""HDPE ~ • O· A OMPa .., •.....- 1.0 -6-1 ..... •• a ~ ~ 0.8 •••• • lOOMPa 'W 5 a ca·· • 10.6 0.4 200MPa 1O~ ••• • . • -- 0.0 165 175 185 195 105 215 Tem...... rC) Fipre 6.8: Erperimentallsotltermal COIIIJWessibility ofthe meltsttIt, versus t"",..."for reslnsÂ, /, Q1II/HDPE. • 74 1.6 .....------, • "0 - 1.4 lU•".... i 1.0 ... := _-----_.------~ ~ OMP••..... '1 0.8 ! 0.6 ------~~~-- ~ ------10.4 ~------i 0.2 •• • 0.0 +----+---0+---...-.+----+---...-.+------1 165 175 185 195 205 215 225 Temperature rC) Fi., 6.9: Isothermtll COIIIpressibi/ity versus Temperature/or resin Â; Comparison 01 uperimental data (Exp) with Tait equationftttingrrait) andwith lmrerse-Vo/umefitting • (lY). Figure 6.9 illustrates the predictions ofisothermal compressibility by the Tait and Invene-Volume equations for lleliD A, which are typical of data for all rains. In this figure, it is apparent that bath equations of state yield poor predictions of isothermal comprelsibility. The averaae ablOlute percentage deviation, .4.., defined in Appendix II, wu found ta be equal ta 21% and 35% for the Tait and Inverse-Volume equatio~ respec:tively. 6.1.3.2) Thermal Expansion Coefficient • 75 The thermal expansion coefficient, a, is plotted venus temperature at different pressures for rains A, 1, and HOPE, in Figure 6.10. SolDe of these values are also • presented in Table 6.S. Table ~s: &peri1lle"tal1'lJe1'1rltl1 E:qxmsion Coefficient ofthe melt sltlte.for nainsÂ, 1, andHDPE,· Averap stand.rfor the average vaille oflite lemperatll1'e i"terval. re·t)-IO" , 1.. Z.. 1,. 9.17 5.13 4.34 A Ave..... 10.0 5.79 4.37 224 10.2 5.16 4.41 117 9.50 5.61 4.32 1 Av... 9.6S S.61 4.3S 223 9.12 S.73 4.39 177 9.95 S.71 4.45 BD' Ave..... 10.1 5.76 4.48 222 10.3 S.13 4.'3 1.1 ,...... ------. -s 0 MPa • ...... -=a. '.0 • 1.0 o U"'" IIF=-'--==-...... e= _.-.-._.-.8 _.& • .:' _.-.. _. e.. 0.9 1 0.8 , 0.7 f.J 100 MPa 0.6 ...... • • • ..• •• V · _. 1:D • _. TC· -• ..., t 0.5 200MPa .:t ..... ~&...... = ... Si le 10.4 a0.3 0.2 -f-----t----+----t------t----+----+'----t 160 170 180 210 220 230 Fi"". ~JO: üperlllle1lttJITltenlttJlE:qxmsion Coefficient ofthe ,.11state ver.sus • t'Illperalrnforresim.A, 1, andHDPE. 76 The thermal expansion coefficient a. wu round to be similar for ail the raiDS under study. However, IODle differences were observecl, u shawn in Figure 6.10 and • Table 6.S, in contrait ta ilOtberma1 compressibility. Rain 1, which bu the lowest density, exhibited the Iowest value ofCI, whereas the HOPE raia, which bu the hipest density, showed the hilhest value ofa. These resu1ts are consistent with the tendency reported earlier Cor the estimated Tait equation parameter, CIo. Figure 6.11 shows the predictions ofbath equatiODS ofstate Cor a in the melt resion. It il apparent tram this fipre tbat both equations of state yield poor predictions of a.. The average absolute percentase deviations are 190" and ]]% Cor the Tait and Inverse-Volume equations, respectively. However, it is clear that only the Inverse-Volume equation prediets a positive sign for da/dT, which is in agreement with experimental data. 1.2 ~------. tQ.... OMPa ~• 1.0·-. ••• • • • • 1J • ~ 1 0.8-~ 13 _-_-_-_-_-_------~-~-~-!!!"""l!II-P"'!I-!P ~ tJ 0.6 -+-Exp -Tilt --IV 1 8 0 4 • • • • 1. •••• l°.2-';;:------ 0.0 +----+----....-.---..ol.~--~--+__-___I 165 175 185 195 205 115 115 Temperature rC) Fi"", 6.11: 17IemItIlûptInsion Coe.fIicientp rain.4; COIIIptlI1son ofI%pBillle1lttll data to Tait equtJtiOll (solid lInu) andto Inwr.-YohDIIe equation (dashedline) • predictions ",..,tIt1IIoSphericJ1I'ISSIft tIIIIl200 MPa. 77 • 6.1.4) Su••ary Severa! conclusions can be drawn trom the study of the PVT properties ofthe polyethylene resins investipted in this wode. At tint, the PVT experimental data are in good agreement with those reponed in the Iiterature, and are reproclucible witbin 0.2%. The general PVT behavior is similar for ail grades ofLLDPE resiDs investipted in this project. The HOPE resÎn cliffers tram the LLDPE resins by its behavior in tho solid state region and in the meltinwcrystallization processes. Ali the rains bacI an almost identica1 PVT melt behavior, includina the HOPE rain. It CID be tberefore concluded tbat the PVT melt bebavior is not strongly affected by the molecular weight, the branching size and their distnbutions. These observations were confirmed by the titting results ofthe Tait and Inverse Volume equations. They both prediet the experimental data with a bener accuracy than the uncertainty in the measurements. From these predictions, the thermal expansion coefticient at atmospberic pressuret (Jo, wu found to be dependent on density; il increases with density. The sante tendency was observed for the thermal expansion • coefticient. For the latter and the isotbermal compressibility, the predictions by both equations ofstate were rather poor. However, oRly, the Inverse-Volume equation could prediet the observed tendency for increasing thermal expansion coefficient with temperature. • 78 • 6.2) PREssURE DEPENDENCE 01' CRYSTALLIZATIONAND MELTING The isobaric experiments, in whicb the samples were heatecl and cooled al 2.SoCimin at constant pressure, were employecl to study the eff'ect of pressure on the melting and crystallization processes. Characteristic features, sucb as meltiDg points and crystallization temperatures evaluated.. The pressure dependence ofthese temperatures, together with their tendencies, are discussed. Theo, the curves ofthe volume change rate versus temperature are presentecl and compared for ail the LLDPE resins. 6.1.1) General Behavior in lIob.rie Experimenu Fipres 6.12 and 6.13 display a set of isobaric experiments for the HOPE resin and a typica1 LLDPE resin. Only reliD B is presented because ail the LLDPE resins exhibit the same curve shape, as shown in Figure 6.14. In contrat, the difYerences in • crystallization and meitiDI behavior between LLDPE and HDPE resina are quite clear ftom these figures. For instance, the end ofcrystallizatioD, whieh is observed when the crystallization curve becomes parallel to the melting curve in the IOlid state reaion, is markedly ditrerent. HDPE crystallization occurs in a narrow temperature range, whereas LLDPE relins crystallize in a mueh broader temperature intaval, over sooe. For LLDPE resiDs, the oMet ofcrystallization is followecl by a tint rapid diminution ofthe volume, and then by a slow decreue process. This is the typical bebavior ofa resin crystaIIizing in two kinetic steps, refemd ta u primuy and seconclary crystallization. Another cbaracteristic of the imPE resin is that crystallization under high pressure seems to yield rnaterial with bigher density. Indeed, the crystallization curve al 200 MPa (m Fipe 6.12) is observed to he below the meltin. curve, &fter crystallization wu complete. On the otber band, the opposite behavior wu observecl for LLDPE resins, particularlyat low pressure, as illusttated in Fipre6.13. • 79 1.35 10MPa • 1.30 ~ 1.25 ~ looMPa ! 1.20 200MPa 11.15 -----...... - 11.10 ,. 1.05 1.00 0.95 0 50 100 ISO 200 250 T(C) Fipre 6.12: Crystallization and me/lingal 2.5'C'min under vorioupressuresfor the HDPEresin. 1.35 ..,.....------, • 10MPa 1.30 ~1.25 looMPa i...., 61.20 200MPa j 1.15 ~ 1.10 1.05 1.00 -+-----+-----+----~~--~~----t o 50 100 150 200 250 T(C) • Fi"", 6.13: Crystallization tIIIIl.,Itingal2.Scc:m.in under WJl'ÎOIISprusIII'esfor reain B• 80 • 1.25 ,...------, 1.20 ~ 1.15 ~ ! LLD'E ... 1.10 1.05 1.00 +------t~--__+_---_+_---~--__+-- 30 65 100 135 170 205 T(C) Fi., 6.14: Crystallization andnre/ting al2.5'C/min 1Int/er 100 MPafor ,enns B, 1, andHDPE. 7.&04 ,....-.------, • BOPE 1••• LLD'I-L 1.&04 OB+OO ~--.....------.-----...-----~---- 50 70 90 110 130 150 170 190 T(aC) Fipre 4/5: YohnrIe Change Rate VI. TIIIIpmlture ,.",.."tingthe .'tingofresinsL • andHDPE"".. 10 and100MPa. Il • 6.2.2) Pressure Dependenee ofT. and Tc The melting and crystallization temperatures under pressure were defined as the temperature al wbich the volume change rate wu maximum during melting and crystaI1ization, respectively, under constant pressure, u descnDed in section S.2. An example of a plot of volume chanle rate versus temperature showing the pressure dependenœ of the melting points for resins L and HDPE is presented in Figure 6.15. Melting points and crystallization temperatures under 10, SO, 100, and 200 MPa were determined for each min. Tbese values are reported in tables in Appenclix m. Ta evaluate the pressure dependence of Till and Tc. temperature data were ploueet versus pressure, u illustrated in Figure 6.16. From this graph, which is illustrative ofthe results for ail resins, it can be seen that experimental melting and crystallization temperatures are almost linearly dependent on pressure. Data were thus titted to a linear equation that yielded good fitting results, within 1%. The slope ofthis line represents a thermodynamic quantity expressed by the Clausius-Clapeyron equation as fol1ows: O. =1.0 AV; (6.3) • dl' • MID 1 where AV; is the volume change on fusion, ÂH~ is the hat offusion, and r: is the equilibrium melting point. However, it bu been shown that for large pressure increase, l6J2 the meltins curve of polyethylene decreases in slope with increasing pressure • Takamizawa et a1.52 showed that the dependence ofthe melting point for large pressure increases wu best represented by the Simon equation": (6.4) where T. is the meltina temperature Ilpressure P, and T•.II is the meltina temperature al atmospheric pressure; a and c ueconstants that clepend oneach substance. Subsequently, • data were fittecl ta the Simon equation. The reproducibility ofthe tempenture data by the 82 Simon equation wu extremely aceurate (0<0.1OC), which is better than the error in the temperature measurements (O.2°C). The results ofboth correlations are given in Table • 6.6. From this table, it CID be ..n that the fittinl puameten are comparable for ail the polyethylene resins investiptecl in tbis study. The average values ofthe fitting results are given in Table 6.7. For the pressure range studied, the linear resreuion yieldecl a quai identical slope for ail rains, which wu equal to O.2soClMPL This value il in aareement l with the ones reported by Wunderlich and Davidson ', and it il closer ta the value they found Cor Colded-ebain crystal. tban that for extendecl-chain crystal•. This wu logically expected sinee no extendecl-chain crystal formation bas been reported for polyethylene below 200 MPL From Table 6.7, the followinl parameters ofthe Simon equation, a=320 and c=4.0, can be used to represent the pressure dependence of T", and Tc: for all 10sins witb an accuracy of0.8%. These parameters are in agreement with the values reponed in 25 the Iiterature • 180 170 • 160 .-. 150 t) \., l- 140 130 - -Simon 120 -LiIlelr() 110 0 so 100 150 200 ~o P(MPa) Fipre 6.16: Me/tingpoints tIIIdcryslDlliZGtion t~ versuspresswe. wilh ,he li,..and the SÎIrIOII equtllionfittingjor ruinB• • 83 Table 6.6: Fitting Results ofthe Pressure Dependence DIT", and Te:- __ - o. CIY...... T tare P.t • RelIa • c T. dT/" T. • c T. dT/" T. (NPa) (C) (ota) (C) ~) (C) (cVa) (C) A 318 3.9 115.0 0.26 116.5 407 3.3 126.9 0.26 128.0 B 277 4.3 111.0 0.25 112.9 316 3.4 124.6 0.26 125.9 C 296 4.1 112.3 0.25 114.0 319 4.0 125.3 0.26 126.9 D 339 3.7 111.4 0.26 112.2 311 4.1 122.7 0.25 123.9 E 260 4.5 109.3 0.2.5 111.3 202 5.7 120.7 0.24 123.2 F 411 3.4 99.4 0.23 100.05 2105 4.5 111.1 0.23 112.8 G 254 4.5 109.9 0.26 111.9 308 4.1 126.1 0.26 127.8 B 222 05.1 106.05 0.2.5 108.8 451 3.0 124.3 0.25 125.4 1 281 4.5 89.4 0.23 91.1 244 5.2 103.3 0.22 105.2 J 326 3.9 101.6 0.24 102.7 499 2.8 114.8 0.24 115.5 K 331 3.9 99.1 0.24 101.2 583 2.6 113.1 0.23 114.5 • L 331 3.7 112.4 0.26 113.9 541 2.5 125.0 0.26 125.8 BOPE 249 4.7 111.5 0.26 120.6 352 4.0 135.4 0.24 136.8 Table 6.7: ~verage valuuoft1teftttingreSllltsofTable 6.6. c -- .. M...... Bedl Av. Avdev. Av. Avdev. Tot. Av. c 4.2 0.4 3.8 0.8 4.0 • 300 40 377 94 338 J'Itted _da e-4.o • 310 8 1 327 12 1 319 ...... Iq••d_ ft'" 0.25 0.01 1 0.25 0.01 [ 0.25 • 84 • 6.2.3) Meltinl and Cry_tallizadon curv_ uader Pr••ure By plotting the volume change rate versus temperature to assess the transition temperatures, we observed that the melting and crystallization curves variecl sipificantly trom one resiD to another. Figures 6.17 Three typical curve shapes can be observed trom these fisures. Firstly, the characteristic melting behavior of a relin that possesses a non-uniform branching distribution (Ziesler-Natta catalyzed rains) can be seen for mins B and G. Tbese curves exhibit a peak, between 125 and 128°C, attnDutecl to the melting of the almost Iinear molecules, and a broad and Oat part, corresponding to the meltin! ofthe highly branched molecules ofvlrious sÎZes. The metallocene catalyzecl rains display the two other types of melting curves. The resins 1 and l exhibit one simple peak, corresponding to the melting of the unifonnly branched molecules. Finally, some other metallocene based • rains, such u resins D and E, feature one peak with a shoulder al a lower temperature. This indicates the presence of two dominant molecular structures with a ditferent brancbing distribution. Figure 6.17 (a> shows the melting behavior oftwo butene- ethylene copolymers, which difTer mainly by their polymerization procesa. Resin B, which wu manufaetured by a ps-phase process, display a more distinctive peak and a higber melting point tban that for resin H, which is a solution-polymerized min. This is consistent with the tict that solution polymerization yields a more uniform branching distribution than that for ps-phase rains". • 15 1.4E-04 ------• 1.28..Q4 (a) Butene ~ ~ "'! l.OB-04 ~ ! 8.08..QS ! 6.0E~S 1 4.08-oS L....,..-~!II' ~ 2.08-0S 0.08-+00 +----+---+--.--..---+----1----+---+-----1 90 100 110 120 130 140 ISO 160 170 T(C) 1.8E-04 ------, 1.6E..Q4 (b) Resene ~ • ~ 1.4E-04 ~ ! 1.28-04 a .1 1.0E..Q4 ! 8.OE~S 16.0E~S > 4.0E-oS t..~...... 1.0E..QS O.OE-tOO +----+---+----i---+-----t---+---+-----4 90 100 110 120 140 ISO 160 170 Fi"". 417: Yohmie CIttInp Rate ys. TelllperaIIIrefor the IIIe/ting fIIIder 100 MPa al • 2.SU.in ofruillswitll (a) Bill"and (6) Hezene COIIItNIOIIIe1' 86 2.0B-04 • 1.8E-04 (e) Octene i 1.6B-04 ~ ....! 1.4E.04 ! 1.2E-04 • 1.0E-04 ~ 8.0E-oS 6.0E-OS J4.0E-oS 2.0E-OS O.OB-+OO 90 100 110 120 130 140 150 160 170 TrC) 2.0E-04 1.8E.Q4 (d) i 1.6B-04 • ~ ...,! 1.4E-04 ! 1.2E-04 J1.0E4t 8.0E..oS 16.0E..oS ~ 4.0E..oS 2.0E-OS O.OB-+OO 90 100 110 120 130 140 150 160 170 TrC) Fi., 6./7: Yo"'" C1Itlnge RJlte Ys. Telllpel'Qturefor tlte,.ltingrmder /OOMPaat 2.SC(Ym;n ofruiM(c) witIJ Octe. COIIIOIIOIIIerS and(d) F, K, andL • 81 Figure 6.17 (b) presents the melting behavior Cor the hexene-ethylene copolymers und« 100 MPL The two meta1locene catalyzed and ....phue processecl rains, D and E, • bebave ditferendy in meltinl than the metaUocene eata1yzed min, the reliD C, which wu solution polymerizecl. Both resins D and E exhibit a peak with a sboulder indicatinl the existence of two dominant branchinl compositions. This confirms that the sas medium seems to induce leu uniConn branching. Figure 6.17 (c) shows the Metting behavior for the octene-ethylene copolymers. The three polyethylene pes were solution polymerized, but resin G wu Ziegler-Natta eatalyzed and resins 1 and 1were metallocene eatalyzed. From the curves ofresins G and J, it is clear that the catalyst type affects the molecular stnJeture, and therefore the meltinl behavior. The etfect of the comonomer content on the melting behavior oC meta1locene catalyzed resins, i.e. decrease of the melting point with increasing comonomer content, is quite apparent trom this fipre. Figure 6.17 (d) illustrates the melting behavior ofthe LOPE rain F, and mins K and L. The details oC the stnIeture ofrains F and K are not known, except that they are LDPE miDS prepared in the gas phase. llesin L is a solution LLDPE rain. • Unfortunately, the composition ofthis resin is also not known. 6.2.4) Summary The isobaric experiments yielded important information conceming the mehing and crystallization under pressure ofpolyethylene rains. Ali the LLDPE relins displayed a secondary crystallization behavior durinl caoURS under pressure. In contrast, the HOPE resin did DOt exbibit such • bebavior, crystallization occured in a narrow temperature raDse. The meltina points and crysta1lization temperaturel ofail anda of polyethylene resins învestipied in this worlt were afFected limâlarly by pressure. Up ta 200 MPa, T. and Tc: were observed to be linarly depenclent on pressure. The pressure • dependence of tbese transition temperatures. tR/dP, wu tbuncl to be equal ta 88 O.2S0CIMPL Up to 200 MPa, T. and Tc ofail rains cao be preclictecl with an error of1% • by the following Iinear equation: r:-or-c =O.2SxP+T.1J (6.4) where T..o is the melting tempenture Il atmospheric pressure. However, the Simon equation, which predicted experimental data with a slightly better accuracy tban that for a Iinear fit, should he considered for higber pressure. Finally, the investigation of the plots of the volume change rate versus temperature clearly demonstrated the difFerences in meltin. and crystallization bebavior oftbese various grades ofpolyethylene rains. The differences in the melting curve shape are attributed to the differences in molecular stnIeture. The differences result Û'om variations in the crystalline packing ability ofpolyethylene due to the ditTerences in the branching size and distn"ution. Remarkable variations were observed between polyethylene grades catalyzed by Ziegler-Natta or metallocene catalysts. The etrects of the polymerization medium and ofthe comonomer content on crystallization and melting • behavior were also observed trom these isobaric experiments. 6.3} CRYSTALLIZATION KlNETICS To study crystallization kinetics, ilOthermalJilObaric experlments were carriecl out accordinl ta the procedure described in Chapter 4. The sample wu pressurized hm 10 MPa to the crystaIlization pressure at constant temperature. Volume data were thm recordecl u a fimction of tïme. Utilizing the procedure detailed in Chapter S, the induction lime and the relative crystallinity were evaluated ftom experimental data. The relative crystallinity wu clefined as: (6.S) • 89 The value Vo wu the volume at which crystallization begins, corresponding to the end of the induction time, and the value V. wu defined at the end of primary crystallization. • ICinetics data were then fiued by the Avrami equation: x =1-erp(-/ct" ) (6.6) where k and n are the Avrami parameten. The constant, k, represents the overall crystaUization rate, and " is an exponent that cmbe related ta the mode ofBUcleation and growth rate. Even thougb tbis work presents solely the Idnetics results of the primary erystallization, this section mat discusses the secondary crystallization bebavior of the LLDPE resins. Crystallization kinetics curves for some polyethylene resins and their representation by the Avrami equation are presented nm. Theo, the etTects ofpressure and ofpolymer type on the Avrami parameters are discussed. The etrect ofpressure on balf·time of crystallization and on induction lime at constant supercooling is al50 considered. Finally, further interpretation of the pressure etrect on crystallization rate utilizing the Lauritzen-HotTman theory is discussecl. • 6.3.1) Secondary Crystallization In this work, the completion ofcrystallization wu not observed. The secondary crystallization wu still in progress even 8fter more than 20 hours, u shown in Fiaure 6.18. The linear iDCrease ofthe degree ofcrystallinity with the logarithm oflime for the 31 secondary crysta1lizatioD, u reponecl by Schultz et al. , is clearly observed in this tipre. In addition, the secondary crystallization rate seemed ta be not straDaly dependeDt on temperature al constant pressure. Inadclition, the volume chanae il dependent on temperature. Fipre 6.19 demonstrates the importance of the end-of-erystallization etfect on the Avrami parameten. Fipre 6.19 displays the kinetica data for. NIlthat wu analyzed in the roUoWÎDI ways. The upper data curve correaponds to the case for which • erystaIliDity wu calculated witb Y. hein. the end ofprimary crystallization (Y. Il20 90 0.045 • 0.040 0.035 ",.~ ...,B0.030 0 025 1• 0.020 10.015 :iilo 0.010 0.005 0.000 ooססoo 10סס1 1000 100 10 TIJM(I) Fi." 6. lB: TypiCQI UDPE crystallization eqJerlments atconstant pressIINfor various te"'pe,aturesJ ind/ClIIing the linear Increase o/the degree 0/crystallinity durlng secondary crystallization. 1.0 • 0.9 0.8 •• 0.7 k-l.50E-83 0.6 ~ 0.5 0.4 0.3 0.2 0.1 0.0 0 200 400 600 800 1000 1200 1400 "n-(.) Fi., 6.19: Crystllllizlltion ki,.licsaperl"."t/orrai"B lI1Ider /OOMPaat 144tCJ presentiflg IWo diJ/erent dtIIa QllQ/ysu: (a) Y.atendofpritIttIt'y erySlll1liZlltlon. and (6) Y.alend0/secondary crystll1lization. Sylftbols are experi",."ttIl..salidlines are ,Ile • AWtllfli efllllllionfit 91 minutes), and the lower curve for the eue for which Y. is the encl ofthe experiment, (V• at 20 hours). From this figure, it is obvious that the two-step crystallization cannat be • represented by the Avrami equation because it affects the Awami parameters, resuIting in a lower value of ft, tasether with a slower crystallization rate. This is in agreement with the observations ofother worken·'. Because of the slowness of secondary crysta1lization, and the difficu1ty ta represent it by the Avrami equation, the present worlt bu focused on the study of primary crystallization. 6.3.2) Crystallization Kinetic data Figure 6.20 illustrates the typical curves tbat were obtained for isothermallisobaric experiments, and it also shows the end ofprimary crystallization of each experiment, indicatins the curve section that wu considered for kinetie! analysis. Volume chanse data were employed to calculate the relative crystaIlinity, whieh • wu plotted versus time, u shawn in Figure 6.21. This figure atsa shows that the isotherms are superimposable upon shiftinS the lime axis for the same material, with the possible exœption of the lowest supercooling. Errar is greatest for the lowest supercooling owinS ta uncertainty in Y... The slopes ofthe curves were aImost identical fiom 50 ta 200 MPa for a sune material, indicating a consistent meçhanism offormation of the crystaIs. DitTerences in slopes were observed between materia1s, u shown in Fisure 6.21. This suggests tbat the polymer chain structure affect the mecbanism of erystallization. Data were next titted by non·linear rearession ta the Avrami equation. Ali the resuJts ofthe Avrami cquation fittin& the values ofn, k, induction time ',,11.5, and balf· tinte ofcrystallization 1112. may be obtained ftom Profeuor M. Il. ICamal. The Avrami coDltaDt k1.s wu obtaùled fiom the Avrami fittina with " equal ta 1.5. The cboice ofthe • value of1.5 is explained below. Fipre6.22 presents an example ofthe Awami titting oC 0.110 • 0.105 'i 0.100 "'-" , O.09~ 146.rc ...10.090 ~ 0.015 0.010 0 5000 10000 15000 20000 VB: W.(I) Fipre 6.20: Crysttlilization Idnetics aperimenlsfor resin H rmder 100 MPa at varlous crystallizalion te"'pe1'alrlres. Bo/dlines indieate the part 01the curve that wasfined10 tlle Avramiequation. • 1.0 -r------~~~._..--~-~ 0.9 100 MP. 0.1 LLDPE-D 1::: 142.0°C 144.0°C y 0.5 o 145.0 C LLDPE-G ~ l44.0G 0.4 146.5°e e t G l46.0 C i 0.3 141.5GC 0.1 0.1 0.0 -1-.-I-=;;iiÏII.....IIII!I!Ii~~+----+---+-----+-----J 2 3 4 6 7 1 9 10 l.II(t) Fi." 6.21: CrysltllliZlltion Kinetics experlments rmder 100 MPa alWlrious • te"'fJ'D'tlllll'sfor tlte ruinsD andG. 93 1.0 • 153S'C 0.1 154S'C IS5.9'C 10.6 1S1.3·C t- (.J u j 0.4 0.2 0.0 ....- ...•• 10 100 1000 10000 Fime 6.22: ÂWQllli.fttsfor kinetics experl",enls rmder 100 MPa at varlous te"'perature • 1.&02 -r------~------__. SlMP. ISOMP. _MP. 1.&07 +---+---+---+-----+---+---+--1---+----+---( 125 130 135 140 145 ISO 155 160 165 110 175 T(oC) Fi.", 6.23: Logarlt"'" oftheÂvrtI1IIi crystailiZlltiOll rate. il.!. (obtainedlor n=}.j) versus crystallization temperatflre lIIIder WIrlOllSprasIIIYsfor ruimB, D, E, andL • 94 NDS carried out under ISO MPa at ditferent temperatures, for rain E. Thil figure also shows that the Avrami equation predicts the experimental data with IOme error, wbich • wu tarae for sorne resina, including D, E, F, and IC. Larp error wu &Iso observed for these raiDs when computing the end ofcrystallization volume using the "muter curve" analysis. It is interesting ta note tbat resins D and E are the only two resins prepared in a pl phase medium with metallocene catalysts., and resins F and K are two LDPEs. The values ofn were round to be in the ranse ftam 1.1 to 1.9, and the average values ofn for each resin were between 1.2 and 1.8. These values are in good agreement 26 with the ones reportecl by Brown and Jonas • The lowest average value ofn, 1.2 wu observed for resin D, and the highest, 1.8 wu for resin G. The pades D, E, F, and K, had Avrami exponents between 1.2 and 1.4; for re&iRS 1 and 1, n wu around I.S; for the othen resina, n wu in the range tram 1.6 ta 1.8. With the exception ofmin C, ail the grades exhibitins a high value of n, were prepared by Ziegler-Natta catalysts. On the otber band, the LOPE resin and the metallocene polymers displayed a value ofAvrami n below 1.5. Originally, the value of Awami n wu considered to be a characteristic of the • growth mechanism. In this study, the value of n could he idealized to l, 1112, or 2 depending on the rain grade. This soggests that the nature ofthe eatalyst, which controls brancbing uniformity, influences the growth habit ofcrystals. This CID be related ta the observation of Bassett11 that the lamellar thickness wu affected by the size and distribution of branches. In addition, the values of n were round to increase with increasing supercooling. This is in agreement with other reported datal.26. From the previous figures, it cmhe observecl that crystaIlization is accelerated by decreuing temperature Il constant pressure. This can he verifiecl in Fipre 6.23, illustratiDI the Avrami crysta1lization rate, kl,S, venus crystallization temperature under variOUI pressures for sev..t mins. The value orthe Awami cryltallization rate constant is strongly related to the Awami exponem, n, whicb is in tum affectecl by temperature·, 14 resia pacte, and pouibly by pressure • Kinetic data were dus c:ompareel for the sune value orthe Avrami exponeat n, which wu cha..ta be equal to the average value of • 1.S. From Figure 6.23, the crystallization rate orthe various mini seem to be atrected 9S similarly by temperature Il constant pressure. The effect of pressure on crystallization • rate is DOW discussed in the section. 6.3.3) Effed o,p,.ure OD Cry.tallization K1netia In asreement with Brown and Ionas-, the Avrami exportent wu round to be independent ofpressure in the range investipted. In contrait, for example for rain B in Figure 6.23, it is apparent that the crystallization rate Il 150 MPa is significantly fast« than that at 100 MPa at constant temperature. In general, this eifect is simply and sufficiendy explained by the etrect ofpressure on the meltin. temperature, whicb atfects the degree of supercooling. Pressure increues the dearee of supercooling. However, it bas been reported that pressure affects crystalUzation kinetics even at constant supercoolingl~19.2'. Inthis project, crystallization halftime and induction time were used ta represent crystallization kinetics data. This section considera the influence ofpresaare on half-tîme ofcrystallization and induction time at constant supercooling. • The supercooUng wu defineci as the ditrerence between the melting point obtained ftom isobaric experiments at the pressure investi8atecl and the crystaUization temperature. The half-time of crystallization wu calcul.ted using the Awami parameters, as foUows: 1 1112=T"(ln2J- (6.7) Fipre 6.24 displays the results ofall the experiments caniecl out under 100 MPa for eacb resin. This Saure shows explicitly that rains D and G bave the two extreme bebavion. llesin D exhibits the fastest crystaIlization, whereu raiD G shows the slowest transformation procesa. The HDPE arade display. a crystallization behavior as slow as tbat for the G raiD. The sante rate orcier wu observed throupout the pressure ranp învestipted. The same IfIPh wu plotted for the induction âme versus sapercooling, u clisplayecl in Figure 6.25. Intbis Saure. resin D shows the abortelt ÏDduetion time, 10000 ~·"·~A ~8 -.-C ~D ___Il ___0 ~F ··O··K ~ .BDPE • ~ .... ~I ··6··K ~ ··.··r -'-L " ~ 1000 1Y :,'; 1 100 3 4 S 6 7 8 9 10 11 AT Fipre 6.24: Half-time ofcrystallization vs. supercoo/ing under 100 MPa al various temperaturesfor ail the resins• 10000 ~------....., • - ... HOPE ~B --c ~D ~~.··E ~F -"'0 ··a··H -+-1 -&-1 -.-K - -t.-·L S 1000 M 1100 10 -4----~--+__-___lfo_.-~--_+--___+_--_+_--'"""" 2.5 35 4.5 S5 65 7.5 1~5 9.5 105 AT(C) Np" 6.25: lndllction li". va. Sllpercooling 1I1ItIe, /00MPa alWI1'ÎOIIS teltlpmltlns/or • lIIl ,he reslns. 97 confirming ilS rapid crystaUization bebavior. Resins G and H were round ta display the IODgest induction time. Thil IUl8ests that the Dueleation and crystallization might be • intluenced by molecular ehain structure, and especially polydispersity. HaJt:time of crystallization and induction time were then plottecl versus supercoo6ng Cor each raid At 50, 100, 1sa, and 200 MPL Solely two resinl, L and HDPE, showed their crystaIlization half-time significantly atTected by pressure at constant supercooling, u shown in Fipres 6.26 and 6.27. In additio~ these two polyethylene grades exhibited an opposite behavior. Indeed, the value of tm for the L rain wu observed to increase with increasing pressure, whereu tlJ2 ofBOPE raiD wu lowerecl by increasing pressure. This opposite behavior wu contirmed by plotting induction time versus supercooling Cor both resins, u illustrated in Figures 6.28 and 6.29. The effect ofpressure on induction âme appears to he IOmewhat more notable than tbat on crystallization halC-time, sinee the induction times ofrelins A, B, G, and H were also increased by pressure. From these results, it is DOW clear that pressure affects crystallization kinetics even at constant supercooling. Pressure appears ta influence more sipificant1y induction • time, thus, nucleation mecbanisms, than crystallization rate. Nevertheless, at constant supercooling, crystallization wu accelerated by pressure for the HDPE resin, as hu been 1 4 6 reported by several othe.- workers '.2 .24.2 • On the contrary, the conclusion that crystallization wu retarded by pressure for polyethylene bas only been mentionecl (not l6 investipted) by Wunderlich and Davidson • This contradictory bebavior is certainly attributed ta the respective sinsuIar polymer chain structure ofboth resins. Since the L and HOPE miDI probably ditrer munly by tbeir branching content (the L resin structure WIS not Imown at the time ofwriting), this implies that branching sipificandy influences the eff'ect ofpressure on kinetics. In addition, it CID be concludecl tbat the andes prepared by Ziegler-Natta catalysts rapond difl'erently to pressure than the metalloeene eatalyzecl raina.. For Ziester-Natta raiDs, the onset of crystaIlization is delayecl by pressure, wbereu the metallocene grades are not sipificantly affected by pressure up to 200 MPL • 98 5000 • P 4-50 4000 4-100 3000 ---1'0 ":' ,.. ---200 2000 1000 0+-----+---+----+---+----+---1 4.0 '.0 6.0 7.0 8.0 9.0 10.0 AT(C) Fipre 6.26: Half-time 01crystal/ization vs. supercoo/ing unde, various pressures and te"'pe,aturesfor HDPE. 1600 • 1400 .....200 1200 -6-1'0 1000 -0-100 ....• S.. 800 600 400 ---II 200 0 3 4 6 7 1 9 AT(C) Fipre 6.27: HaIf-ti..ofcrystaRimtion ys. SIIptII'COOiing""''' Wll'i0llS pressures and • teltlpmltrlresforrainL 99 600 • -50 500 p .-. ~400 -'100 ! ....150 .1 300 :1 200 '-200 100 0 4.0 5.0 6.0 1.0 1.0 9.0 10.0 AT(C) Fipre 6.28: Induction lime vs. supercooling""" variouspressures andtemperatures for resin HDPE. 1200 • P ~200 1000 • -'150 ,~ 800 "'100 600 04-50 i 400 200 0 4.0 4.5 5.0 5.5 6.0 6.5 7.0 1.S 1.0 1.5 AT(C) Fi.,., 6.29: lndllction m. ys. SIIperCOOiing lI1Ider Wll'i0ll8 fJNSSIIrU and temperalJlres • forresinL 100 This su_esta tbat the etfects of the Don-uniformity of the branehinl and • molecular weight distribution on the Dueleation mecbanisms are magnified by pressure. 6.3.4) 'urther Interpret8tioD For further interpretation, the theory ofLauritzen, HoiTman and coworkers3 and the treatment by Mandelkem1 could he usetùl hue. Mandelkem uses eq. (2.1) for the nucleation rate (for a homoseneaus procas) ta express the overall crystallization rate constant for a rectangular nucleus u k =koexp(-nE12 [l+("-l)ii KT~ ,) (6.8) RT T(T,.T) where ED is DOW the he energy ofactivation for transport across the interface between the nucleus and the polymer melt, li is the ratio of critical tiee energy of secondary nucleation (addition ofchains) ta that ofprimary Ducleation, and where " is the Avrami • exponent. The tenn 1C contains the surface energy and heat ofNsion terms (6.9) where !Jlfl is the hat offusion per repeat unit, tSIf is the eneIJY ofcrystal-melt interface panllel ta the chain axis, and tr. is the surface he energy ofthe crystal fold surface. Mandelkem at50 described the situation where growth is controltecl not by the primary Dueleation but by the rate of crystal srowth. This seconduy Dueleation rate (two dimensional DUcleation) wu aiven by Sawacta and Hose2! U (6.10) wbere b ilthe width ofthe nucleus. • 101 • 1.E-02 -r------__. a c J 1.E-03 a • 1.E-G4 +------,-.---r----~---_..._--' 0.5 0.7 1.3 1.5 Fipre 6.30: Plots 01tll2-1 verSlls rmdercoolingT~( AT)1 for the HDPE resin dIlta and Brown andJontJS'6 data. • LE.Q2 ------., ~ j l.E-63 lSOMP. lOOMP. l.E-04 -I------....------T"'"---.,..----...,-----.--~ 0.5 0.7 1.3 1.5 Fipre 6.Jl: Plots 0/1/12-' wrsIIS Illldercool;ngT~(AT)1 for IlleHDPE resin alVQl'Îous • presSlUeS- 101 • l.E.Q2.,...... ------. s j l.E~3 'OMPa lOOMPa l'OMPa l.E-04 +----~---r__--__.,---___r-----I 0.1 0.3 0.6 0.7 Fi., 6.32: Plots oftll2-1 ""SIlS undercool;ngT~( AT)2 for the UDPE-L"sin. • l.E.Q2 -r------....., 1 j l.E~ 50MPa lOOMPa l.E-G4 +---,...... ---T'---.....--~--,...... -___,----t 6.0 6.5 7.0 7.5 1.0 1.5 9.0 9.5 TJI'{AT) • 103 In arder ta discem if the crystallization rate wu controllecl by the procesa of 1 diflùsion or by the nucleation and arowth process, the plots oflog 1112- versus r,!(AT)' • 26 were tint traeed for the HOPE rain and compared with the results ofBrown and Jonu • u shawn in Figure 6.30. 1 The plots presented here ue known to he strongly dependent on temperature • In l this study, the values ofT. were taken tiom the data ofWunderlich and Davidson '. The melting points reported by these autbors are for extended-chain crystals that were prepared by crystaUization under S Imar at 214°C. These tempentures are an approximation of the equilibrium melting point under pressure. These data were al50 2 chosen for the sake ofcompari5On ofour values with those ofBrown and Jonu '. Figure 6.30 shows that the kinetics results trom this study are in good agreement 26 with those ftom literature • The straight lines, which are an exponential fitting ofBrown and Ionas data. indicate that the rate is controlled by the surface energy and heat of fusion term in eq. (6.8). This fiaure and Figure 6.31 show a decrease ofthe slopes with inereasina pressure, implying a decrease in the energetic tenu in eq. (6.9). OntYthe surface energies ue likely ta change. This is in agreement with the decrease in the 24 25 • surface enerBY observed by Kyotani and Kanetsuna • Sawada and Nose • and Brown and Jonas:z6. However, tbis indicated decrease contradicts the observation ofCalvert and Ubmann35 ofan increase ofsurface energy o. with pressure. 1 The plot of108 1112- versus T~ (AT)' wu tben traeed for ail the LLDPE resins. They all exhibited the same tendency u above. The decreue ofthe slopes with pressure WIS more or less noticeable dependina on the min. As an example, Figure 6.32 presents the results for the L rain. 1 The plots oflog 1112- venus T",IT(tJ1') alsa yielded straigbt lines, as illustratecl for the HDPE resin in Fiaure 6.33. However. the decreue ofthe sJopes with pressure wu less obvious or eYen not present for the LLDPE resinl. From eq. (6.10). this sUght decrease iIldicates that the secondary nucleation is controUecl by the surtàce energy term • and Dot by diftbsioD. 104 • 6.3.5) Summary The investiption ofcrystallization kinetics fumished notable information on the efrect ofpressure and chain stnIeture on crystaIlization. Itwu confirmed that the LLDPE resins exhibit a two-step crystallization process, primary and secondary crystallization. The analysis of the Awami fitting showed that the Avrami parameter n wu influenced by the nature ofthe catalyst employed in the polymerization reaction. The study ofthe efrect of pressure on half·time of crystallization and induction time demonstrated that resins D and G had a noticeable distinct bebavior. Rain D exhibited the fastest crystallization bebavior, whereas resin G displayed the slowest crystallization. The investigation of the effect of pressure at constant supercooling led ta contradictory results, depending on resin grade. Crystallization was accelerated by pressure for the BDPE resi~ while on the contrary it wu delayed for only one LLDPB resin, the L grade. In addition, the onlet of crystallization wu only observed to he • retarded by pressure for the resins prepared by Ziegler-Natta eatalysts. Finally, the interpretation of the results by the Lauritzen-Hoftinan theory indicated tbat some resina, such u BOPE and L, exhibit a decrease in the surface he energy ofthe crystal nucleus with pressure. It also appears that, Cor most LLDPE resiDs, this eifect is not observable. The above results indicate that the polyethylene chain structure influences crystallization kinetics in ditl'erent ways. It CID be concluded tbat the branching non unifonnity and molecular weipt distribution. which are determined by catalyst type and polymerization medium have a sipificant influence on the nucleation and growtb mecbanisms. Furthermore, this effect on DUelcation seems to he magnitied by pressure. Pressure etl'ects on crystaIlization kinetics depend on chain stnJ~ particu1arly branching and ist uniforDlity and molecular weiaht distribution. Finally, pressure causes the lowering of the surface ftee eDergy of the crystal nucleus. This etl'ect of preuure • depends on the chain structure aacI molecular weisht distribution. lOS • CHAPTER7 SUMMARY, CONCLUSIONS, AND FURTBER WORK 7.1) Su••ary aDd CODdusio. The main results obtainecl in the present study for ail the resins are summarizecl in Table 7.1. It iDCludes the thermal expansion coefficient, Table 7. J: Mai" Nsulta achieved in the presen' stIIdyfor aH the resins. .... • eIyaL C....,.. c• Dllldty CIe .. .T,JdP • BlJict cfPN... • Cade Medl_ Type ..~ CCl) (MPa) CClMPa) tlIJ ft A 0- ZN Hexene 0,9208 1,52 195 0,26 1,6 ~ B Gas ZN Butene 0,9194 1,46 194 0,26 1,6 ~ C Solution Met Hexene 0,9234 7,54 199 0,26 1,7 D Gas Met Haune 0,9191 7,42 200 0,2.5 1,1 E 0- Met Hexme 0,9194 7,44 196 0,24- 1,4 P Ga LDPE 0,9190 1,40 197 0,23 1,3 G Solution ZN Octene 0,9200 1,55 198 0,26 1,8 ~ H Solution ZN Butene 0,9190 7,39 200 0,25 1,7 ~ 1 Solution Met Octene 0,9070 7,26 194 0,22 1,5 J Solution Met Oc:tene 0,9110 7,35 195 0,24- 1,4 le Gas LOPE 0,9191 7,53 191 0,23 1,3 L Solution LLDPE 0,9222 1,56 204 0,26 1,6 ~ BD'E 0,9620 7,68 197 0,24- 1,7 ~ - • 106 From the i50thermal experiments, it wu observecl that the Seneral PVT behavior wu similar for ail the LLDPE grades, but it ditfered for the HDPE relin in the transition • region. The PVT melt behavior ofaU resins wu found to he almost idcntical. This wu verifiecl with the fitting readts of the Tait and Invene-Volume equations, which represented accurately the bebavior in the melt statc. In addition, the isothennal compressibility and thermal expansion coefticient were al50 not especially afFected by the pe of polyethylene resm. The Tait and Invene-Volume equatioftl were found to poorly prediet these two derivatives. However, only, the Inverse-Volume equation could predict the tendency ofthe thermal expansion coefficient ta increase with temperature. From these results, it can be concludecl that the polyethylene chain structure, as ret1eeted by comonomer content and type, branehina size, and molecular weiaht and their distributions, does not affect significandy the PVT behavior of the melt state, the compressibility and the thermal expansion. The investiption of the isobarie experiments lead to the conclusions that the meltina and crystallization temperatures of LLDPE and HOPE resins are Iinearly and similarly dependent on pressure up to 200 MPL This dependence wu estimated It O.2soCIMPL Moreover, the curve shapea oC the melting and crystallization processes of • the LLDPE resina contributed ta a better understanding of the difFerences in crystaIIization and melting behavior attn"buted ta their branehing size and uniformity, molecular weight and polydispersity. The titting oC the crystallization kineties data to the Avrami equation yielded interesting results. The average value ofthe Avrami exponent n wu Cound to vary &om 1.1 ta 1.8 depending on the polyethylene resin. Furthennore, Ziegler-Natta catalyzed resins elearly exhibited a value oC" superior ta 1.6, whereas the metaUocene catalyzed resins showed values ranging between 1.1 and 1.5, with the exception ofresin C. This sugests tbat the branchiDl size and its uniformity, detcrmined by the type of eatalyst, affect crystallization, and especially, the crystal growth and ÎtI shape. The study of the haIf time of crystaIlization and induction âme at constant supercooliDl indicated and contirmed tbat the metallocene resin D and the Ziegler-Nana • resin G exhibit ditferent and extreme crystallization kiIletics properties, compared ta the 107 other grades. Thil implies tbat the polymer chain properties, and mbsequently the polymerization process parameten influence crystallization kinetics. • The evaluation of the etfect of pressure on crystallization kinetics at constant supercooling yieldecl the foUowing results: crystallization wu accelerated by pressure for the HOPE resiD, and in contrait, crystaJlization and more lignificantly crystal1iDtion onset were delayed by pressure for the Ziegler-Natta eatalyzecl LLDPE raiDs. This indicates that the etfects of the Don·uniformity of bnnching on the nucleation mechanisms are magnified by pressure. The interpretation ofthe results by the Lauritzen-HoftiDan theory pointed out tbat pressure induces a decrease ofthe surface &ee eneru ofthe crystal nucleus. This could explain the increase in crystallization rate with pressure for the HOPE resin. A less noticeable decrease ofthe surface enersY wu observed for most LLDPE rains. For these resins, this suggests that the influence ofpressure on the surface energy is less important than the other etTects ofpressure on the nucleation mechanisms. From the present worlc, the ultimate conclusions concerning the effect ofpressure and pu:.11Der chain structure on crystallization are summarized below. The • polymerization process conditions, such u polymerization medium, catalyst, and comonomer, affect the polymer chain structure, which in tum atTect crystaUization kinetics. With pressure, the retardation etrect ofbranching and branching non-unifonnity on the nucleation and crystallization process seems ta he magnifiecl and ta overcome the other, opposite efFect of pressure on crystallization. Indeed, pressure also influences crystaJlization by decreuing the surface energy ofthe crystal nucleus, which seems to induce an acccleration ofcrystallization. 7.2) Reco••eadatioa ror FartherWork lnorder ta tùrther explain the resuIts obtained iD the present study, and to extend the inveltiption oC the etrect ofproceuing and polymerization conditions on crystallization • mecbanil~ some morphological and thermal ItUdies sbould be carriecl out. 108 As for thermal studies, the numerous data generated in this work: should be employed ta investigate the non-ilOtherma1 kinetic behavior ofthe raiDS under pressure. The larae • range of undercooliDg covered by non-isothermal experiments under pressure would certainly brinl valuable information to confirm and, eventually, to extencl the conclusions ofthis report. Furtbermore, using the same experimental technique, the study ofthe etrect ofpressure on the secondary crystallization process could provide important information for the undemanding of LLDPB crystallization. In addition, an extensive work usina differential scanning calorimetry should be performed to assess isotherma1 and non isothermal crystallization kinetic data at atmospheric pressure, for comparison with hip pressure data. Ilcould alsa he employed ta measure the melting behavior ofhigh pressure crystallized sample. Morphological investiption ofthe samples crysta1lized under pressure using optica1 microscope and X-rays would he very helpful ta observe the correlated etrect ofpressure and chain structure on morphology and crystaUinity. • • 109 • REl'ERENCES 1) L. Mandelkem, Cryattlllization 01Po/ytllers, McGraw-BilI, New York, 1964. 2) F. ICboury and E. PUSlllia, Treatise on SoIid S,ale Chemistl'y, vol. 3, N. B. Banny. Ed., Plenum Press, New York, 1976. 3) J. D. Hoftinan, G. T. Davis, and 1. L. Lauritzen, Treatise on So/id State Che",istry, vol. 3, N. B. RanDy, Ed., Plenum Press, New York, 1976. 4) J. H. Magil~ Properties 01 m/id po/yrIIeric """mals, Part A, Treatise on Materials Science and TecbnololY, 10, (lM. Schultz Ed.). Academie Press. New York, 1977. S) B. Wunclerlieh, Crystlll SlnIcture, Morpho/ogy, Defeets, Macromolecular PhysÎes, vol. l, Academic Press, New York, 1973. 6) B. Wunclerlieh, Crystal Nucletllion, a,owth, Annealing. Macromolecular Physics, vol. 2, Academic Press, New York, 1976. 7) B. Wunclerlieh, Crystal Melting, Macromolecular Physies, vol. 3, Academie Press, New YorIe, 1980. • 8) E. Chu, MuterofEftlÎneering, McGiIl University, 1984. 9) L. 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Simon and G. GIatze~ Bemerbmgen ZII1" Schemelzdruclrbow, z. ADam. Alla. Cbem., 17', 309 (1929). 53) C .. P. Cbiu, K.. A. Liu, and 1.. K We~  Methodlor MetlSllrlng PYTIœlationsllips of The,.",oplastics Using an l''jection Moiding Machine, Polym. Eni. Sei., 35, 1505 (1995).. 54) Y. A Fakhreddine and P. ZoUer, Pressure-VollIIIIe-Temperatllre Studies: If Versatile Analytical Toolfor Po/ymers, ANJEÇ. Conference Proceedinll. 1991, Soc.. ofPlasticl Engineers, Brooldield, CT, USA, 1642 (1991). 5S) P. ZoDer, "Polymerie Equations of State," Po., Hqndbook, 3rd Edition, 1. Brandup ancIE. K Immergut, Editon: Wiley& Sons, (1989).. 56) S .. K.. Bhateja and K.. D. Pu, J. MtlCrotIIOl. Sei. Rlvs. Mqcromol Che",., 13, 77 (1975). S7) R. Simha and T. Somcynsky, Mqcromol,cul,s. 2, 342 (1969).. 58) R. Simha and R. K.. Jain, J. Polym. Set. Po_. Phys. E4., 16, p1471-89 (1978). 59) M. R. Kamal and N. T. Levu, Polym. Enz. Sci., 13, p131-138 (1973). 60) P. G. Tait, "Pltysics and che"'istl'y o/the voyage ofH. M. S. challenge,," 2, Pt..4, • University Press, Cambridge, p1900 (1888). 61) S. Beret and 1. M. Prausnitz, Macromolecules. l, 536 (1975). 62) Di/atometl'y, EnC)'CI01Iedia ofPoIymcr Sejcnçe and Enaineerina, 2nd Edition, H. Mark, N. Bibles, Ch. Overberpr, Ed.; Wiley & Sons, New York, 1986. 63) P.. ZoDer, TIte preSSlUe-voIvme-telllpUtltll1" prope11ies 01 tin. well characterized low-density polyetlrylenes, L ADI. Pon. Sei., 23, 1051 (1979).. 64) P. ZoDer, P. Boll~ V. Pallud, and H. Ackennan, Rey. Sei. Instrum.. 47, 948 (1976). 65) J. He. Y. A Fakbreddine, and P. ZoUer, I, ADJ. PoIym. Sei., 45, 4S (1992). 66) Lille. Low Deruity Polyethylene, EgçycJQpedil OC Pqlymcr Science and Enaineerina. 2Dd Edition, K Mark, N.. Bibles. Ch. Overberpr, Ed.; Wiley ~ Sona, New York; voL 17" 756, 1986. 67) M. Saman, 'hO". Thais. McGiIl Univ..ity, inprint. • 68) Levants MalterTbesil, McGill University, 1973 .. 113 • APPENDIXI PVT SAMPLE KUNS This appendix il devoted ta describing the detailed procedures of each &tep requirecl prior to each sample NIl. ft details the procedures for the nickel cup and œil preparation, the œil tilliD& the apparatul assemblinsand disassembliDg, the œil cleaning, and the LVDT calibntion. Some recommendatioDS conceming calibration are also included. 1.1) Sa.ple RUD PrepantioD Note: - refers to the page ",."be,. ofthe PITmanua/. 1.1.1) - Nickel cup preparation (page 52)* Ta maintain a hydrostatic pressure on the sample in the saUd state and ta protect the bellows, a nickel cup is required. A rectangular pieee (30xSO mm) is tint cut inta a • 0.001" thick nickel foil. The cup shape is tben fonned with the use of an aluminUDl conical mandrel provided with the apparatus. The nickel cup is next weighed to one milligram. Ail the parts ofthe cellt including the two pskets and the steel sealing disk, are also weighecl to 0.01 grams. The nickel cup is then entirely filled with the material, which is weighed to one milligram. The cup and the sample are next placecl in the sample compmment. Ll.2) - BelloWi pnpandoR (pale 64)* The length of the bellowi fixes the ovenU volume of the ceU. It is thus an important step ta control the bellows lenstb. The bellows are tint mounted on the tiller bue.. The bellows misbt need to be compressed or stretehed in arder to obtain the fixed • lengtb cletermined by the set screw of the tiller bue. Indeed, the set screw shouId A-l compress the bellows slightly (about 112 mm) when the screw is eIIsased in the sroove of • the constraining rode Ll.3) - Celi preparadon (1llustnted pale 54-(2)* The different parts of the cell, including the bellows, the central piece, and the sample compartment (containing the sample and the cup), are assembled with the help of the ring DutS. The two pet rings are placed between parts ta ensure a proper sealing of the usembly. The cell pans are tiptened with a wrench applied to the central piece. The assembled œil is then mounted on the tiller base. Ta fix the overall volume oC the ceU, which is extremely important, the constraining rod is inserted and threads iota the end of the bellows. The rod position is then adjusted sa that the set screw engages exaetly the groove in the constraining rode The œil is then attached to the reservoir with a tlexible tube. • 1.1.4) - Fililai procedure (pale 6S)· Ailer having placed the tiller bue in a tray in a fiune hood, ti'esh, clean confining fluid (about 10-12 cm') is poured in the reservoir through the tiller base hale, using a sYringe. The œil, mounted on the tiller bue. is next placed in the vacuum vessel, with the rescrvoir on top and the piezometer cell on the bottom. The vacuum vessel is kept in the 45° position and connected to a vacuum pump in arder ta obtain a vacuum ofbetter tban 4x10.2 mm ofHg. When the required vacuum is attained, the vacuum vessel is cautiously tilted put the horizontal, until it comes ta rest apinst the stop provided. Keeping the vacuum pump on the system, the vacuum vessel is maintained ttlted for 5 minutes ta permit the contining fluid to tlow fiom the reservoir into the sample œil. Without movinl the vacuum vesse!, the vacuum pump is tumed off: Over a period of 1S seconds, the air is admitted tbrough the valve to reduce vacuum ta about 15 inches of HI_ The vacuum vessel il tben moved bacle to its original 450 position, and the valve is next opened • completely ta equilibrate to atmospberic pressure. The tiller is DOW withclrawn caretùlly A-2 ftom the vacuum vessel. The tlexible tube il disconnected ftom the reservoir, and the remainins confininS tluid is poured into the end ofthe œil. Employing a sumon device, • the excess ofcanfininS tluid il Rlcked until the meniscus ofthe tluid il exactly at the top ofthe Iman bore leading iDto the sample compartment. The steel sealing disk tbat wu previoully weigbecl is placecl ioto the end piece and centered, and the œil closure DUt comel to seal the end of the ceU. The set screw ofthe tiller bue il now unscrewed to messe the constnining rad that can be removed nom the bellowi. The assemblecl œil il tben removed trom the tiller base. The fidly tilled ceU il finally weighed ta 0.01 grams, ta evaluate the mus ofthe confining tluid used. The filliog ofthe cell can he cbecked with the help of the "fill procedure" section of the PVT software. The repeat fillings ofthe same œil should have the same volume within 0.03 ta O.OS cm'. (.I.S) - Aue_blini ofthe .pparatui At thil step, the usembled cell, containing the sample, the nickel cup, and the confining fluid, must be mounted on the PVT apparatul. The LVDT core, which il mounted at the end ofa long thin tube, il Iifted up, far enougb, to he capable ofscrewing • ils end ioto the beUOWI. The sample œil il screwed to the adapter in the bue ofthe PVT apparatus. The pressure vessel, which is a thick steel cylinder with heaters. is cautioully lowered over the sample œil. The control thermocouple il then placed ioto the side hole of the pressure vesse" The thermocouple black is next placccl on top ofthe pressure vesse., ensuring that the thermocouple tip enters the conical opening ofthe œil closure nut. The spacer is added ta the top ofthe thermocouple black and finally the fiMed aluminum hat sink il placed on top ofthe Ipacer black. The ram piston is then lowered using the hancI pump untiI the pressure reacbes the value of9OOO Pli (around 62 MPa). A pressure of2oo MPa is applied manually ta the pressure with the higb-pressure pUlDp to till the vessel with the lilicon oil and Cor inspecting for any possible Ieakage. The PVT apparatul is DOW ready Cor a calibration orsample run (et: Panpaph1.2» . • A-3 LI.') Bia__bli. ofthe PVT .ppantui • One must be aware ofthe possible bazards ofa PVT ND, for example, a leak of confining 8uid or generation of gues by decomposition or by chemical reaetion. The safety information chapter of the PVT manual should he rad befote disassembling the apparatus. The disassembling should be done at ambient temperature. Firstly, the electronic boxes are turncd ott: Seconclly, the pressure is releasecl by openina the bottom drain valve. The high-pressure thermocouple and the heater connector are unplugcd. and the control thermocouple is removed. Subsequently, the nm piston is raised by releasing the raID pressure slowly. Thirdly, the tinned heat sink, the spacer black, the thermocouple black, and the pressure vessel are removed. Finally, the œil is caretùlly unscrewecl and sa is the LVDT core. Ail parts of the PVT apparatus, including the œl~ are cleaned with paper towels and acetone. 1.1.7) Cleanina 01 the cell The cell can DOW be disassembled. The ccli must he untied by placing it in a strong • tlexible polyethylene bag in order ta prevent spillina ofconfining tluid, usually Mercury. Working over a tray in a fume hood, the bellows are first removed, and then the confining tluid is drained aimait completely. After having removed the centtal part and the ring nuts and drained the remaining confining tluid, the sample, whicb is usually beavily contaminatecl with confining tluid, is easily removed thanka 10 the use ofa nickel cup. Finally ail parts are abundantly rinsecl with acetone, or other suitable salvent, until DO more confining tluid can be seen on any part. L2) CaUbntioa or S••ple Rua Ifthe computer and the two electronic units are turned on, ail switches are in the correct position, and the stroke traIIIducer indicatCl the correct value of45Y.. the PVT apparatul is ready to start • new sample run. Every new sample rua starts with a LVDT • calibntion. A-4 • U.l) - LVDT Calibration (pale 27)* The position ofthe LVDT, ret1ecting the position ofthe bellows end, is meuured with a resolution of 0.001 mm, repreaenting a volume change of approximately 0.0001 cm3/g. The sensitivity of the PVT is calibrated against a precision digital micrometer, which is attached ta the LVDT coil. Consequently, before &taning a NIl, the LVDT calibration must be IChieved in order ta obtain the Hnur relationship between the output voltage and the micrometer readings. Henee, the output voltage ofthe LVDT is tint set to zero with in 0.001 volt by moVÎng the micrometer. The micrometer is next moved in steps of 1.0 mm (within 0.1 mm) within a range of ±4 mm trom the zero point; the corresponding voltage and micrometer readings are recorded. After the final reading, a linear regression is calculated on these recorded data by the PVT software. The LVDT calibration is acceptcd ifthe deviation between the experimental and tïtted values is lelS tban 0.003 mm; ifnot, anotber calibration must be repeated. • 1.2.2) - Run information required The inputting of the ND information does not need many explanations. It il, however, important that the specifie volume of the sample at ambient temperature il known with reasonable accuracy (2%). A1though only volume changes are recorded during experiments, the prosram still requires the 1% lCCUl'IC)' in order ta make certain calcu1ations such u Tait extrapolation to zero pressure. L3) Recom.eDdatioas The temperature calibntioll must be performed at leut once a montb, especially if the apparatus is nmnïng constantly. In addition, the data acquisition (DASCON) board • must be calibrated every six months., otherwise it couId result illlarae data reading error.. A-S • APPENDIXll ERROR ANALYSIS Error analysis is the study and evaluation ofuncertainty in measurement. Experience bas shawn that no measurement, however carefùlly mad~ can he completely he of uncertainty. This appendix will thereCore report bow the uncertainties in the PVT measurements were evaluated. Moreover, tbis section will include the formulae utilized ta obtain the uncertainties and standard deviations ofthe parameters reported. II.1) Error in the Volume Cllaale MeasureDleat The chanse in position ofthe LVDT core is a measure ofthe volume chanse. In fact the volume change corresponds to the dilference of position of the LVDT core • between a calibration Nn and a sample ND. The ditTerence between the positions ofthe LVDT core is solely causecl by the fact that a certain amount ofsample (plus nickel cup) have replacecl an amount of confining tluid, and that the sample, nickel cup, and the confining Ouid have ditTerent PVT properties. The following equation expresses that mathematically: AV = Y,(p,1) - Y,(pit TJ (fi.l) with V.(Pt T) = [llmJ [~(PJT) d(P, T)J- "'NI VNI(Pt T) + "'CF yafP, T) (U.2) where 1i and Pt He the initialization conditions ofthe sample ND, A(p,T) is the cross sectional area ofthe bellows, d(P. T) is the LVDT core position, Vis the specifie volume and mthe mus ofthe ditl'erent species; subscript s inclieatel the sample, Nt indieates the • nickel cup and CFthe confinina tluid. A-6 Equation (ll.2) stands at all pressure and 10 at the initialization conditiODS. Accordin8 ta equation (D.I), the uncertainty in the volume change CID be written u • foUows: (U.3) By u8UDlÏng tbat erran in the volume values are not clependeDt on temperature and pressure, it cm be written that: 3(AV) =2 aV.(p,T) (11.4) For convenience, the volume cao al50 be written: V,(p, 1) =(1) + (2) + (3) whicb yields the ColloWÎng equation for the uncertainty in the volume: 6(Y.) ~ 6(1) + 8(2) + 6(3) (D.S) Jo': (1) (2) (3) with 6(1) =6nt, + &t(P,T) + 6d(P,T) (ll.6) (1) "'. A(P,T) d(P,T) 6(2) 6nJ 6YNf(P,T) --=--+.--_~NI (n.7) (2) "'Ni VNf(P,T) CI'( • __6(3) = __6mCl' + --=:...0-';"""';"av P, T) (n.I) (3) "'CI' VCI'(P, T) Providing the foUowiDI uncertaÎnties: 6m. = 0.0001 & 6mNI = 0.0001 & 6mCF=O.Ol & and usuming tbat the erron in the specifie volume orthe ditferent species equal 0.1%, the Collowing values are obtained: 6(1)/(1)= O.IS% 6(2)/(2) = 0.3~" 6(3)/(3) = 0.14%. • Ityields A·7 6fJ':)y ~.". • • Therefore, the volume accuraçy in the temperature range 2S-220°C and the pressure range 10-200 MPa is equal ta 0.03 cm'/a. It should he noted that the above value of the uncertainty corresponds ta the systematic error component ofthe total uncertainty ofthe volume change. The random error companent ofthe uncertainty, the standard deviation, will then be discussed. 0.1)Reputability ofisother." malurement! Isothermal experiments coDSist ofrecordiog the volume changes a10ng isotherms with increasing pressure by increments of 10 MPL In arder to estimate the repeatability of those experiments, the data points must he recordecl at the exact sante tempenture. Indeed, a temperature difference ofO.SoC will consequently result in a volume change of 0.0004 cm3/g. The sensitive part of the experiment is in the meltina transition. The bigest discrepancy will be obtained in that reaion. • The followiDg standard deviatioo wu used to evaluate the repeatability: lN .. CI = --I(x; - Y/)· (ll.9) N -};=1 The averase absolute percent deviation, which wu utilized ta assess the prediction ofthe isothermal compressibility and thermal expansion coefficient, is definecl as: l N (r. r )2 ..... = -~ ap-'" xlOO (ll.10) N ;:.1 %..2 where Xap is the experimental value, XJII'fIII is the value calculated for x.. &am the corresponding equatiOD ofstat~ and N is the DUmber ofx..uncIer investiption. • 8.3) A-a • APPENDlXm ISOBARIC EXPEIUMENT DATA m.I) MeltiDlaad CrystaUizatiOD TelDpentures The following table presents the melting and crysta1lization temperatures obtained ftom isobaric experiments carriecl out al 2.SoC/min under 10, SO, 100, and 200 MPa. Table lU: Melting andcrystallization te"'peratul'e obtained.from isolJaric experi",ents at 2.5CClmin under varlous pressures. r(MPa) 10 50 lOG 200 10 50 100 200 A B T. 118.4 130.2 143.2 166.8 109.9 121.7 135.6 157.7 T. 130.0 141.8 154.1 178.3 127.2 138.3 151.4 175.4 1 T. 114.2 126.1• 139.6 162.8 92.2 103.1 114.8 135.7 • T. 127.7 139 152.7 176.7 106.0 117.6 128.4 149.1 C S Tc I1S.5 127.2 140.4 163.4 104.5 llS.S 128.1 150.3 T. 128.2 140.8 153.3 177.4 117.7 127.7 140.9 163.7 D K Tc 114.4 125.9 139.2 162.7 102.6 113.7 125.8 147.9 T. 115.9 136.3 150.3 172.3 116.3 126.1 137.9 159.9 E L Tc 112.5 124.6 137.8 160.8 115.5 127.3 140.9 164.9 T. 124.2 135.1 149.7 170.9 127.9 139.0 152.5 177.4 l' HOPE Tc 102.1 112.2 124.4 146.0 122.2 134.1 148.1 171.4 T. 113.8 125.5 137.0 158.9 131.2 149.5 161.7 184.3 G Tc 113.1 125.5 131.9 162.3 T. 129.2 141.2 154.5 178.2 • A-9 • DL2 CrysallizatioD Curves O.OE+OO .,------~ ~ i ..S.08-0S r...... -...-.... __ j ~ i.. -1.0E.()4 a-1.5E-04 J-%.0B-04 (a)_e -2.5B-04 +----__------...------.----4 90 100 110 120 130 140 ISO T(C) • O.OE.oo .,....------.-, i ·S.OE-OS -w~ ! .1.0&04 !-1.5E-04 J-%.0B-04 (b)Ba.e .2.5&04 ~--~--__r_--___r---~--~---t 90 100 no 120 130 140 ISO T(C) Fi." IU.I: Yolute C1ItInge RIlle ys. T."".rtltWe.for" crystaIlization "",.,.100 MPa • at 2.Seem.in ofresimwit" (a) Butene and(6) Hsene ~ A-IG a.OB+OO ~------, ~ ~~~ • -S.OMS r....=::::::::-..... __ :a!"il; )-1.0&04 ! l-l.sE-04 1-2.0&-04 1 ;! -2.5&04 (c:)OâeIIe J -3.0&-04 +----~-- ...... --_.,---_--_,....--__t 90 100 110 120 130 140 ISO T(C) O.OB+OO ------..--, i -S.OE-OS • t.... ! -1.0&04 X-l.sE-04 1 ~ -2.0&04 (cl) -2.5&04 +----...----_--_--_---~---t 90 100 110 120 130 140 150 T(C) Fi.,.. UI.1: Vohmte Change Rate vs.. TelllperatJuefor the crystalliztltion "".,100 MPa at 2.SU.;n ojrestns (c) witJI Octerre COIIIOIIOIIIerstllld (d) F. X. antIL • A-Il