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Rheology and processing of high-density polyethylenes (HDPEs) : effects of molecular characteristics Ansari, Mahmoud 2012

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RHEOLOGY AND PROCESSING OF HIGH-DENSITY POLYETHYLENES (HDPEs): EFFECTS OF MOLECULAR CHARACTERISTICS  by MAHMOUD ANSARI B.A.Sc. Amirkabir University of Technology, Iran, 2003 M.A.Sc. Tarbiat Modares University, Iran, 2005  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Chemical and Biological Engineering) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2012 © Mahmoud Ansari, 2012  ABSTRACT In this study, the linear viscoelastic properties of two series of Ziegler-Natta and metallocene HDPEs (ZN-HDPEs and m-HDPEs respectively) of broad molecular weight distribution (MWD) have been studied. Relationships between the zero-shear viscosity and molecular weight (Mw) and molecular weight distribution show that the breadth of the molecular weight distribution (MWD) for m-HDPEs plays a significant role. Other interesting correlations between the crossover modulus and steady state compliance with MWD of both these classes of polymers have also been derived. Finally, the steady-shear viscosities from capillary rheometry are compared with LVE data to check the applicability of the empirical Cox-Merz rule. It is shown that the original Cox-Merz rule is approximately applicable for HDPEs for narrow to moderate MWD and fails for those HDPEs having a wide MWD due to the occurrence of wall slip. The processing behavior of both series of HDPEs was investigated. Their melt fracture behaviour was studied primarily as a function of Mw and MWD, and operating conditions i.e. temperature and geometrical details and type of die (capillary, slit and annular). It is found that sharkskin and other melt fracture phenomena are very different for these two classes of polymers, although their rheological behaviors are nearly the same for many of these. It is also found that critical conditions for the onset of various melt fracture phenomena depend significantly on the type of die used for their study. The slip behaviour of these resins was also studied as a function of Mw and MWD. It is found that the slip velocity increases with decrease of Mw, which expected to decay to zero as the Mw approaches a value with characteristic molecular dimension similar to surface asperities. For HDPEs that exhibit stick-slip transition (narrow to moderate MWD), the slip velocity has been found to increase with increase of polydispersity. The opposite dependence is shown for HDPEs of wider molecular weight distribution that do not exhibit stick-slip transition. A criterion is also developed as to the occurrence or not of the stick-slip transition which is found to depend strongly on Mw and its distribution.  ii  PREFACE The work of this thesis consists of four different manuscripts. Chapter 5 is based on a manuscript that has been published. Ansari, M., S.G. Hatzikiriakos, A.M. Sukhadia and D.C. Rohlfing (2011). “Rheology of ZieglerNatta and metallocene high-density polyethylenes: broad molecular weight distribution effects.” Rheologica Acta 50(1): 17-27. The materials covered in chapter 6 consist of two manuscripts, the first one is already published and the other one is accepted for publication. Ansari, M., S.G. Hatzikiriakos, A.M. Sukhadia and D.C. Rohlfing (2012). "Melt Fracture of Two Broad Molecular Weight Distribution High-Density Polyethylenes." Polymer Engineering and Science 52(4): 795-804. Ansari, M., Y.W. Inn, A.M. Sukhadia, P.J. DesLauriers and S.G. Hatzikiriakos (2012). "Melt Fracture of HDPEs: Metallocene versus Ziegler-Natta and Broad MWD Effects." Accepted for publication in Polymer. Chapter 7 has been prepared based on a submitted manuscript. Ansari, M., Y.W. Inn, A.M. Sukhadia, P.J. DesLauriers and S.G. Hatzikiriakos (2012). “Wall slip of HDPEs: Molecular weight and molecular weight distribution effects.” All of the experiments and data analysis have been conducted by myself. The molecular weights and their distributions have been provided by Chevron-Phillips Chemical Company LP. The manuscripts were a collaborative effort between myself, my supervisor Prof. Savvas G. Hatzikiriakos and scientists of Chevron-Phillips Chemical Company LP who appear as coauthors in the manuscripts. The initial and final drafts of this thesis were prepared by Mahmoud Ansari, with revisions edited and approved by Prof. Savvas G. Hatzikiriakos.  iii  TABLE OF CONTENTS ABSTRACT ................................................................................................................................... ii  PREFACE ..................................................................................................................................... iii  TABLE OF CONTENTS ............................................................................................................ iv  LIST OF TABLES ....................................................................................................................... vi  LIST OF FIGURES .................................................................................................................... vii  NOMENCLATURE ................................................................................................................... xiv  ACKNOWLEDGMENTS ....................................................................................................... xviii  DEDICATION............................................................................................................................ xix  1 INTRODUCTION...................................................................................................................... 1  2 LITERATURE REVIEW ......................................................................................................... 4  2.1 Types of Polyethylenes................................................................................................................. 4  2.2 Molecular Weight and Molecular Weight Distributions .......................................................... 6  2.3 Rheology ....................................................................................................................................... 7  2.3.1 Rheological Experiments ................................................................................................... 7  2.3.1.1 Frequency Sweep .......................................................................................................8  2.3.1.2 Creep and Creep Recovery ........................................................................................ 9  2.3.1.3 Steady Shear ............................................................................................................ 10  2.3.1.4 Uniaxial Extension................................................................................................... 11  2.3.2 Rheometers ...................................................................................................................... 12  2.3.2.1 Parallel Disks ........................................................................................................... 12  2.3.2.2 Extensional Rheometer (SER) ................................................................................. 13  2.3.2.3 Capillary Rheometer ................................................................................................ 14  2.3.3 Zero Shear Viscosity and Molecular Characteristics ...................................................... 16  2.4 Melt Fracture ............................................................................................................................. 16  2.4.1 Sharkskin Melt Fracture .................................................................................................. 17  2.4.1.1 Overview ................................................................................................................. 17  2.4.1.2 Mechanisms ............................................................................................................. 18  2.4.1.3 Effect of Die Geometry ........................................................................................... 18  2.4.1.4 Effect of Temperature .............................................................................................. 19  2.4.1.5 Effect of Molecular Structure .................................................................................. 19  2.4.1.6 Effect of Processing Aid .......................................................................................... 20  2.4.2 Stick-Slip Melt Fracture .................................................................................................. 21  2.4.2.1 Overview ................................................................................................................. 21  2.4.2.2 Mechanisms ............................................................................................................. 22  2.4.2.3 Effect of Die Geometry ........................................................................................... 22  2.4.2.4 Effect of Temperature .............................................................................................. 23  2.4.2.5 Effect of Molecular Structure .................................................................................. 24  2.4.3 Gross Melt Fracture ........................................................................................................ 24  2.4.3.1 Overview ................................................................................................................. 24  2.4.3.2 Mechanisms ............................................................................................................. 25  2.4.3.3 Effect of Die Geometry ........................................................................................... 25  2.4.3.4 Effect of Molecular Structure .................................................................................. 25  2.4.4 Wall Slip of Polymers ...................................................................................................... 26  2.4.4.1 Overview ................................................................................................................. 26  2.4.4.2 Mechanisms ............................................................................................................. 26  2.4.4.3 Effect of Molecular Characteristics ......................................................................... 27  3 THESIS OBJECTIVES AND ORGANIZATION ................................................................ 29  3.1 Thesis Objectives........................................................................................................................ 29  3.2 Thesis Organization ................................................................................................................... 30   iv  4 MATERIALS AND METHODOLOGY................................................................................ 31  4.1 Materials ..................................................................................................................................... 31  4.2 Methodology ............................................................................................................................... 31  4.2.1 Linear Viscoelasticity ...................................................................................................... 31  4.2.2 Extensional Rheology ...................................................................................................... 33  4.2.3 Processing Study.............................................................................................................. 33  5 RHEOLOGY OF HDPEs ........................................................................................................ 34  5.1 Linear Viscoelasticity ................................................................................................................ 34  5.2 Zero Shear Viscosity .................................................................................................................. 36  5.3 Steady State Creep Compliance ............................................................................................... 42  5.4 Summary .................................................................................................................................... 45  6 MELT FRACTURE OF HDPEs ............................................................................................ 47  6.1 Uniaxial Extensional Rheology ................................................................................................. 47  6.2 Capillary Rheometry ................................................................................................................. 48  6.2.1 Melt Fracture and Typical Flow Curves for HDPEs....................................................... 51  6.2.2 Effect of Molecular Weight .............................................................................................. 53  6.2.3 Effect of Molecular Weight Distribution ......................................................................... 54  6.2.4 Stick-Slip: Molecular Criterion for the Onset ................................................................. 56  6.3 Effects of Processing Conditions on Melt Fracture ................................................................ 58  6.3.1 Effect of Temperature ...................................................................................................... 58  6.3.2 Effect of Die Type ............................................................................................................ 61  6.3.3 Effect of Die Entrance Angle ........................................................................................... 65  6.3.4 Effect of Processing Aid................................................................................................... 66  6.4 Correlations for Critical Stresses ............................................................................................. 68  6.5 Summary .................................................................................................................................... 72  7 WALL SLIP OF HDPEs ......................................................................................................... 74  7.1 Cox-Merz Rule ........................................................................................................................... 74  7.2 Wall Slip Measurements ........................................................................................................... 76  7.3 Slip Velocities ............................................................................................................................. 78  7.3.1 Effect of Temperature ...................................................................................................... 78  7.3.2 Slip Velocity of Polymer Exhibiting Stick-slip ( ⁄ ) ............................. 79  ) .............. 82  7.3.3 Slip Velocity of Polymers with Continuous Flow Curve ( ⁄ 7.4 Construction the Flow curves of HDPEs ................................................................................. 84  7.5 Summary .................................................................................................................................... 85  8 CONCLUSIONS AND CONTRIBUTIONS TO KNOWLEDGE....................................... 87  8.1 Conclusions ................................................................................................................................. 87  8.2 Contributions to Knowledge ..................................................................................................... 89  8.3 Recommendations for Future Work ........................................................................................ 91  BIBLIOGRAPHY ....................................................................................................................... 92  APPENDIX A – LINEAR VISCOELASTICITY .................................................................. 107  APPENDIX B – EXTENSIONAL RHEOLOGY .................................................................. 118  APPENDIX C – FLOW CURVES .......................................................................................... 121  APPENDIX D – PROCESSING MAPS.................................................................................. 132  APPENDIX E – MOONEY ANALYSIS ................................................................................ 141  APPENDIX F – FLOW CURVE CONSTRUCTION ........................................................... 143   v  LIST OF TABLES Table 4.1. List of HDPEs used in this study and their different moments of molecular weights. 32 Table 4.2. Characteristic dimensions of capillary, slit and annular dies used in this study. ........ 33 Table 5.1. Rheological parameters of the materials. .................................................................... 40 Table 6.1. Critical apparent shear rates and shear stresses for the onset of sharkskin melt fracture in capillary die for both resins at different temperatures. Note that only gross melt fracture is obtained for ZN-HDPE-0 at high temperatures. ................................................................... 59 Table 6.2. Critical shear stresses for the onset of different types of melt fractures. .................... 61 Table 6.3a. Critical apparent shear rates and shear stress values or the onset of sharkskin for mHDPE-1 resin at different temperatures and different types of die. ..................................... 64 Table 6.3b. Critical apparent shear rates and shear stress values or the onset of sharkskin for ZNHDPE-0 resin at different temperatures and different types of die. ..................................... 64 Table 6.4. The critical apparent shear rate and shear stress values for the onset of sharkskin in capillary extrusion with dies having different entrance angles for resin m-HDPE-1 at two different temperatures. .......................................................................................................... 66 Table A.1. Horizontal shift factors and activation energies for constructing master curve. ...... 115 Table A.2. Parsimonious Relaxation Spectra for all the resins. ................................................. 116 Table D.1. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin ZN-HDPE-5 at different temperatures. ............................. 132 Table D.2. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin ZN-HDPE-6 at different temperatures. ............................. 133 Table D.3. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-8 at different temperatures. ............................... 134 Table D.4. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-9 at different temperatures. ............................... 135 Table D.5. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-10 at different temperatures. ............................. 136 Table D.6. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-11 at different temperatures. ............................. 137 Table D.7. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-12 at different temperatures. ............................. 138 Table D.8. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-13 at different temperatures. ............................. 139 Table D.9. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-19 at different temperatures. ............................. 140   vi  LIST OF FIGURES Figure 2.1. Schematic representation of the microstructure of HDPE, LDPE and LLDPE. ......... 5 Figure 2.2. A typical molecular weight distribution of a polymeric system. ................................. 7 Figure 2.3. Viscoelastic moduli and complex viscosity for a typical polyethylene obtained from frequency sweep experiments (SAOS). .................................................................................. 8 Figure 2.4. A typical result of the creep and creep recovery experiment. ..................................... 9 Figure 2.5. A flow curve of a typical molten polymer obtained from a steady shear experiment. ............................................................................................................................................... 10 Figure 2.6. A typical steady uniaxial shear experiment that shows the tensile stress growth coefficient as a function of time for several Hencky strain rates. ......................................... 11 Figure 2.7. Schematic illustration of parallel plates fixture. ........................................................ 12 Figure 2.8. Schematic of the Sentmanat Extensional Rheometer (SER). .................................... 13 Figure 2.9. Schematic diagram of a capillary viscometer. ........................................................... 15 Figure 2.10. Typical Bagley correction for capillary data. Each line corresponds to a shear rate. ............................................................................................................................................... 15 Figure 2.11. Typical extrudate surface defects. ........................................................................... 17 Figure 2.12. The flow curve of a HDPE extruded in a pressure-driven capillary rheometer as virgin and in the presence of a fluoroelastomer (Adapted from reference [96]). ................. 21 Figure 5.1. Master curves of storage and loss moduli for a metallocene (m-HDPE-8) and Ziegler-Natta (ZN-HDPE-6) polyethylene resins at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE), stress relaxation and creep tests. ............................................................................................................................................... 34 Figure 5.2. The crossover modulus as a function of polydispersity index, Mw/Mn, at T=190°C for several sets of data for ZN and m-HDPEs. The solid line is Eq. 5.1 proposed by Utracki and Schlund [159]. ....................................................................................................................... 35 Figure 5.3. The crossover modulus as a function of Mz/Mw for the HDPEs studied in the present work at T=190°C, showing the existence of a correlation represented by Eq. 5.2. .............. 36 Figure 5.4. The complex viscosity material function of all resins obtained from frequency sweep (LVE), stress relaxation and creep tests at the reference temperature of Tref=190°C. .......... 37 Figure 5.5. The zero shear viscosity dependency to molecular weight at T=150°C. .................. 38 Figure 5.6. The radius of gyration of metallocene HDPEs (filled squares) as a function of molecular weight together with a linear reference, verifying the absence of long chain branching in their structure. .................................................................................................. 40 Figure 5.7. The zero shear viscosity dependency on molecular weight at T=190°C................... 41 Figure 5.8. Correlation of zero shear viscosity with Mw taking MWD into account at T=150°C & T=190°C................................................................................................................................ 42 vii  Figure 5.9. Creep test results for two samples m-HDPE-9 and m-HDPE-11 using a constant shear stress of 10 Pa at T=170°C . ........................................................................................ 43 Figure 5.10a. Correlation between steady state creep compliance and the measure of MWD, ⁄ for several HDPE resins.......................................................................................... 44 Figure 5.10b. Correlation between steady state creep compliance and a measure of ⁄ MWD, for m-HDPE resins. The slope is 2.2 as it is expected according to Eq. 5.7c proposed by den Doelder [164]. .................................................................................... 45 Figure 6.1. The tensile stress growth coefficient of resins m-HDPE-1 and ZN-HDPE-0 at several Hencky strain rates, at T=150°C. The line labeled as LVE 	3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1. ....................................................................................................................... 48 Figure 6.2a. The flow curves of resin m-HDPE-1 in capillary extrusion at different temperatures. ............................................................................................................................................... 49 Figure 6.2b. The flow curves of resin ZN-HDPE-0 in capillary extrusion at different temperatures. ......................................................................................................................... 50 Figure 6.3a. The master flow curve of resin m-HDPE-1 at Tref = 150°C, obtained by superposing the data plotted in Fig. 6.2a by using shift factors determined from LVE. .......................... 50 Figure 6.3b. The master flow curve of resin ZN-HDPE-0 at Tref = 150°C, obtained by superposing the data plotted in Fig. 6.2b by using shift factors determined from LVE. ...... 51 Figure 6.4a. The flow curve of ZN-HDPE-10 at T=190°C which shows the stick-slip transition as a hysteresis loop that connects two distinct branches. Several experiments were performed to completely define the hysteresis loop. The relevant critical stresses are those for the onset of sharkskin, and for the onset of gross melt , for the onset of stick-slip, fracture, . .......................................................................................................................... 52 Figure 6.4b. The flow curve of m-HDPE-8 at T=190°C appears to be a continuous one. The and critical stresses for the onset of sharkskin and gross melt fracture are also shown as respectively. .................................................................................................................... 53 Figure 6.5. The flow curves of some of the ZN-HDPE resins with similar PDI, and different molecular weight at T=190°C. The size of stick-slip discontinuity in the flow curve decreases with decrease of the molecular weight. ................................................................ 54 Figure 6.6. The flow curves of ZN-HDPE-10, ZN-HDPE-5 and m-HDPE-8 at T=190°C that show the effect of polydispersity. The size of the stick-slip discontinuity (size of the hysteresis loop) in the flow curve decreases with increase of polydispersity....................... 55 Figure 6.7. The flow curves of the m-HDPEs with PDI>19 at T=190°C. All flow curves are continuous because of the wide molecular weight distribution. ........................................... 55 Figure 6.8. Criterion for the occurrence of stick-slip transition for HDPE resins. Open symbols represent resins with no stick-slip transition while filled symbols those that exhibit this transition. The continuous line is Mw=12×PDI×Me. In general Eq. 6.2 represents the data adequately well. .................................................................................................................... 57  viii  Figure 6.9a. Processability map of resin m-HDPE-1 in capillary extrusion as a function of apparent shear rate and temperature. Various symbols indicate smooth, sharkskin or gross melt fracture appearance of samples extruded at given temperature and apparent shear rate. ............................................................................................................................................... 58 Figure 6.9b. Processability map of resin ZN-HDPE-0 in capillary extrusion as a function of apparent shear rate and temperature. Various symbols indicate smooth, sharkskin or gross melt fracture appearance of samples extruded at given temperature and apparent shear rate. ............................................................................................................................................... 59 Figure 6.10a. Flow curves of the resin m-HDPE-1 for different die types at T=190°C .............. 63 Figure 6.10b. Flow curves of the resin ZN-HDPE-0 for different die types at T=190°C. .......... 63 Figure 6.11. The effect of die entrance angle on the flow curve and processing of resin mHDPE-1 at T=160°C. ............................................................................................................ 65 Figure 6.12a. The effect of 0.1% PPA on the flow curve of m-HDPE-1 at T= 170oC. The open and closed symbols represent smooth and rough extrudates, respectively. .......................... 67 Figure 6.12b. The effect of 0.1% PPA on the flow curve of ZN-HDPE-0 at T= 170oC. The open and closed symbols represent smooth and rough extrudates, respectively. .......................... 68 Figure 6.13. Correlation between critical shear rate for the onset of sharkskin and weight average molecular weights of m-HDPEs according to Eq. 6.6............................................. 69 Figure 6.14. Linear correlation between critical shear stress for the onset of sharkskin and molecular weight distribution parameters for m-HDPEs according to the Eq. 6.7. ............. 70 Figure 6.15. The relation between critical shear stress for the onset of stick-slip with weight average molecular weight for different HDPEs at T=190oC. ............................................... 71 Figure 6.16. Scaling of the ratio of the critical shear stresses for the onset of stick-slip transition and the critical shear stress for the onset of gross melt fracture with zero shear viscosity for high-density polyethylenes. .................................................................................................. 72 Figure 7.1a. Testing the applicability of the Cox-Merz rule for a series of ZN-HDPEs at 190°C. ............................................................................................................................................... 75 Figure 7.1b. Testing the applicability of the Cox-Merz rule for a series of m-HDPEs at 190°C.75 Figure 7.2. The Bagley corrected flow curves of resin m-HDPE-1 at T=190°C for different diameters. The continuous line is LVE data. ........................................................................ 77 Figure 7.3. Slip velocities of resin m-HDPE-1 at T=190°C. The open symbols are obtained from flow curves deviation of LVE data for different diameters. ................................................. 78 Figure 7.4a. The slip velocity of m-HDPE-1 as a function of wall shear stress at different temperatures from 190°C to 230°C. ..................................................................................... 79 Figure 7.5. The slip velocity of selected HDPEs which exhibit stick-slip as a function of wall shear stress at T=190oC. The arrows show transition from weak (closed symbols) to strong slip (open symbols) and vice versa for ZN-HDPE-5. ........................................................... 80 Figure 7.6. Master curve for the slip velocity of HDPEs of the present study and those (resins AF) studied by Hatzikiriakos and Dealy [15] at 190oC. .......................................................... 81 ix  Figure 7.7. The slip velocities versus wall shear stress for the all the HDPE resins at T=190°C. The open symbols correspond to the data after stick-slip phenomenon. .............................. 82 Figure 7.8. Master curves for the slip velocities of m-HDPEs that do not exhibit stick-slip transition at T=190oC. ........................................................................................................... 84 Figure 7.9. Constructing the flow curve of ZN-HDPE-6 starting from LVE data. ..................... 85 Figure A.1. Master curves of storage and loss moduli for resin ZN-HDPE-0 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 107 Figure A.2. Master curves of storage and loss moduli for resin ZN-HDPE-5 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 107 Figure A.3. Master curves of storage and loss moduli for resin ZN-HDPE-10 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 108 Figure A.4. Master curves of storage and loss moduli for resin ZN-HDPE-11 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 108 Figure A.5. Master curves of storage and loss moduli for resin ZN-HDPE-12 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 109 Figure A.6. Master curves of storage and loss moduli for resin ZN-HDPE-13 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 109 Figure A.7. Master curves of storage and loss moduli for resin ZN-HDPE-14 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 110 Figure A.8. Master curves of storage and loss moduli for resin ZN-HDPE-15 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 110 Figure A.9. Master curves of storage and loss moduli for resin ZN-HDPE-16 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 111 Figure A.10. Master curves of storage and loss moduli for resin m-HDPE-1 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 111 Figure A.11. Master curves of storage and loss moduli for resin m-HDPE-9 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 112  x  Figure A.12. Master curves of storage and loss moduli for resin m-HDPE-10 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 112 Figure A.13. Master curves of storage and loss moduli for resin m-HDPE-11 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 113 Figure A.14. Master curves of storage and loss moduli for resin m-HDPE-12 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 113 Figure A.15. Master curves of storage and loss moduli for resin m-HDPE-13 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 114 Figure A.16. Master curves of storage and loss moduli for resin m-HDPE-19 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. .................................................................................................................. 114 Figure B.1. The tensile stress growth coefficient of resin ZN-HDPE-5 at several Hencky strain rates, at T=150°C. The line labeled as LVE	3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1. . 118 Figure B.2. The tensile stress growth coefficient of resin ZN-HDPE-6 at several Hencky strain rates, at T=150°C. The line labeled as LVE	3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1. . 118 Figure B.3. The tensile stress growth coefficient of resin m-HDPE-8 at several Hencky strain rates, at T=150°C. The line labeled as LVE	3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1. . 119 Figure B.4. The tensile stress growth coefficient of resin m-HDPE-9 at several Hencky strain rates, at T=150°C. The line labeled as LVE	3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1. . 119 Figure B.5. The tensile stress growth coefficient of resin m-HDPE-13 at several Hencky strain rates, at T=150°C. The line labeled as LVE	3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1. . 120 Figure B.6. The tensile stress growth coefficient of resin m-HDPE-19 at several Hencky strain rates, at T=150°C. The line labeled as LVE	3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1. . 120 Figure C.1a. The flow curves of resin ZN-HDPE-5 in capillary extrusion at different temperatures. ....................................................................................................................... 121 Figure C.1b. The flow curves of resin ZN-HDPE-5 in slit extrusion at different temperatures. ............................................................................................................................................. 121 Figure C.2a. The flow curves of resin ZN-HDPE-6 in capillary extrusion at different temperatures. ....................................................................................................................... 122  xi  Figure C.2b. The flow curves of resin ZN-HDPE-6 in slit extrusion at different temperatures. ............................................................................................................................................. 122 Figure C.3a. The flow curves of resin m-HDPE-9 in capillary extrusion at different temperatures. ....................................................................................................................... 123 Figure C.3b. The flow curves of resin m-HDPE-9 in slit extrusion at different temperatures.. 123 Figure C.4a. The flow curves of resin m-HDPE-10 in capillary extrusion at different temperatures. ....................................................................................................................... 124 Figure C.4b. The flow curves of resin m-HDPE-10 in slit extrusion at different temperatures.124 Figure C.5a. The flow curves of resin m-HDPE-11 in capillary extrusion at different temperatures. ....................................................................................................................... 125 Figure C.5b. The flow curves of resin m-HDPE-11 in slit extrusion at different temperatures.125 Figure C.6a. The flow curves of resin m-HDPE-12 in capillary extrusion at different temperatures. ....................................................................................................................... 126 Figure C.6b. The flow curves of resin m-HDPE-12 in slit extrusion at different temperatures.126 Figure C.7a. The flow curves of resin m-HDPE-13 in capillary extrusion at different temperatures. ....................................................................................................................... 127 Figure C.7b. The flow curves of resin m-HDPE-13 in slit extrusion at different temperatures.127 Figure C.8a. The flow curves of resin m-HDPE-19 in capillary extrusion at different temperatures. ....................................................................................................................... 128 Figure C.8b. The flow curves of resin m-HDPE-19 in slit extrusion at different temperatures.128 Figure C.9. The effect of die entrance angle on the flow curve and processing of resin m-HDPE1 at T=190°C ....................................................................................................................... 129 Figure C.10a. The apparent flow curves of resin m-HDPE-12 at 190oC for various L/D ratios. ............................................................................................................................................. 130 Figure C.10b. The pressure drop for the capillary extrusion of the resin m-HDPE-12 at 190oC as a function of L/D for different values of the apparent shear rate (Bagley plot). ................ 131 Figure C.10c. The flow curves of resin m-HDPE-12 at 190oC as a function of the apparent shear rate for various L/D ratios corrected for the entrance effects. The data superpose well showing that the pressure effect of viscosity is negligible as expected for HDPE melts. .. 131 Figure D.1. Processability map of resin ZN-HDPE-5 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ......................................................... 132 Figure D.2. Processability map of resin ZN-HDPE-6 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ......................................................... 133 Figure D.3. Processability map of resin m-HDPE-8 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ............................................................. 134 Figure D.4. Processability map of resin m-HDPE-9 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ............................................................. 135 xii  Figure D.5. Processability map of resin m-HDPE-10 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ......................................................... 136 Figure D.6. Processability map of resin m-HDPE-11 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ......................................................... 137 Figure D.7. Processability map of resin m-HDPE-12 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ......................................................... 138 Figure D.8. Processability map of resin m-HDPE-13 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ......................................................... 139 Figure D.9. Processability map of resin m-HDPE-19 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature. ......................................................... 140 Figure E.1. Bagley corrected flow curves of resin m-HDPE-12 for different diameters at T=190°C. The diameter dependence and the significant deviation from the LVE data (failure of the Cox-Merz rule) is consistent with the assumption of slip............................ 141 Figure E.2. Mooney plot using the data plotted in Fig. E.1. The slopes of the lines are equal to 8us (Eq. 7.1) for the corresponding value of stress. The slopes increase with increase of shear stress. ......................................................................................................................... 141 Figure E.3. The slip velocity as a function of shear stress for resin m-HDPE-12 at T=190°C. The solid line represents the slip law given by Eq. 7.3. ............................................................. 142 Figure E.4. The slip corrected flow curve of resin m-HDPE-12 at T=190°C compared with the LVE data. Excellent agreement is shown, demonstrating the validity of the Cox-Merz rule. ............................................................................................................................................. 142 Figure F.1. Constructing the flow curve of ZN-HDPE-0 starting from LVE data. ................... 143  xiii  NOMENCLATURE aT  horizontal shift factor  bT  vertical shift factor  Gc  crossover modulus, [Pa]  J0e  steady state creep compliance, [1/Pa]  ΔPend  end pressure drop, [MPa]  a  Carreau-Yasuda model parameter  a  zero shear viscosity-Mw correlation: PIw exponent  a  slip velocity-Mw pre-factor, [m/s/(g/mole)3]  b  zero shear viscosity-Mw correlation: PIz exponent  Di  die inside diameter (Annulus), [mm]  Do  die outside diameter (Annulus), [mm]  Ea  activation energy, [kcal/mole]  G*  complex modulus, [Pa]  G’  elastic modulus (storage modulus), [Pa]  G”  viscous modulus (loss modulus), [Pa]  gi  parsimonious relaxation modulus, [Pa]  H  die height (slit), [mm]  J  creep compliance, [1/Pa]  K  zero shear viscosity-Mw correlation pre-factor, [Pa.s/(g/mole)a]  L  die length; length of material element, [mm]  L/D  die length to diameter ratio  L0  initial length of material element, [mm]  M0  monomer molecular weight, [g/mole]  Mc  critical molecular weight for the onset of entanglement, [gr/mole]  Me  molecular weight between entanglements, [gr/mole]  Mi  mass of polymer chain with i monomer, [g/mole]  Mn  number average molecular weight, [g/mole]  Mw  weight average molecular weight, [g/mole]  Mz  z-moment of molecular weight, [g/mole] xiv  Mz+2  (z+2)-moment of molecular weight, [g/mole]  n  shear thinning index  ni  number of polymer chains with mass of Mi  PIw  w-moment of polydispersity index  PIz  z-moment of polydispersity index  Q  volumetric flow rate, [mm3/s]  R  die radius, [mm]  r  r-direction distance in annulus, [mm]  Rg  radius of gyration, [m]  t  time, [s]  T  temperature, [°C]  Tref  reference temperature, [°C]  us  slip velocity, [mm/s]  W  die width (slit), [mm]  wi  weight of all polymer chains with i monomer, [g/mole]  ΔP  pressure drop, [MPa]  Greek Letters γA,s  apparent shear rate corrected for slip, [1/s]  γA  apparent shear rate, [1/s]  γc  critical shear rate for the onset of sharkskin, [1/s]  αA,L  slip velocity-Mw pre-factor: lower branch, [m.MPa-3 .s-1  αA,U  slip velocity-Mw pre-factor: upper branch, [m.MPa-3 .s-1  αB  slip velocity-Mw pre-factor: no stick-slip resins, [m.MPa-4 .s-1  γ  shear rate, [1/s]  ε  Hencky strain rate, [1/s] xv  η+E  uniaxial stress growth coefficient, [Pa.s]  ηR  reference viscosity, [Pa.s]  λi  i-th mode of parsimonious relaxation modulus, [s]  σc1  critical shear stress for the onset of sharkskin, [MPa]  σc2  upper critical shear stress for the onset of stick-slip, [MPa]  σc3  lower critical shear stress for the stick-slip effect, [MPa]  σcr  critical reference stress, [MPa]  σW,A  apparent wall shear stress, [MPa]  σW  wall shear stress, [MPa]  τrz  shear stress at location r, [MPa]  ωc  crossover frequency, [rad/s]  2  die entrance angle, [°]    zero shear viscosity-Mw correlation exponent    annulus geometrical parameter    Hencky strain  ∞  ultimate recoil    shear viscosity, [Pa.s]  *  complex viscosity, [Pa.s]  0  zero shear viscosity, [Pa.s]    Carreau-Yasuda model parameter: relaxation time, [s]    shear stress, [MPa]  0  elementary characteristic time [s]    angular frequency, [rad/s]  xvi  Abbreviations BN  Boron Nitride  EPDM  Ethylene propylene diene monomer  ESCR  Environmental Stress Crack Resistance  gmf  Gross Melt Fracture  HDPE  High-Density Polyethylene  LAOS  Large Angle Oscillatory Shear  LCB  Long Chain Branching  LDPE  Low-Density Polyethylene  LLDPE  Linear Low-Density Polyethylene  LVE  Linear Viscoelasticity Envelope  m-HDPE  Metallocene High-Density Polyethylene  MWD  Molecular Weight Distribution  PB  Polybutadiene  PCL  Polycaprolactone  PDI  Polydispersity Index  PDMS  Polydimethylsiloxane  PE  Polyethylene  PI  Polyisoprene  PLA  Polylactide  PP  Polypropylene  PPA  Polymer Processing Aid  PS  Polystyrene  SAOS  Small Angle Oscillatory Shear  SER  Sentmanat Extensional Rheometer  TTS  Time-Temperature Superposition  UHMWPE  Ultra-High Molecular Weight Polyethylene  ZN-HDPE  Ziegler-Natta High-Density Polyethylene  xvii  ACKNOWLEDGMENTS First and foremost I would like to express my gratitude to my supervisor, Prof. Savvas G. Hatzikiriakos who I was honored to work with, for his constant support, skillful guidance, patience and great concern which made this work successful. I would like to thank my committee members for their time, interest, insightful questions and helpful comments. Sincere thanks to past and present group members from Rheology lab that I have had the pleasure to work with, for their inspirational discussions, constructive comments and also their effort to create an enjoyable environment to work. My acknowledgment also goes to Chevron Phillips Chemical Company LP for their research grant that provided the necessary financial support and also the materials for this research. Last but not least I wish to express my thanks to my parents, sisters, brothers, in-laws, friends and co-workers who encouraged, inspired and supported me spiritually. And most of all, I thank the only person who endured this long process with me, my wife for her endless love, great patience, support, encouragement and understanding especially during difficult times.  xviii  DEDICATION  To Sareh  xix  1 INTRODUCTION High-density Polythylene (HDPE) is the most applicable plastic in recent century [1] which can be synthesized by different type of processes and catalysts. Two major industrial catalysts to produce HDPEs are namely Zigler-Natta and metallocene catalysts. The advantage of using metallocene catalysts is to produce a product with controlled molecular weight (Mw) and molecular weight distribution (MWD) [2]. The HDPEs produced by these two methods are far different in mechanical and processing properties. Extrusion is one of the most important routes of production plastic articles such as pipes, films and sheets. From the economical point of view, it is desirable to increase the rate of production without sacrificing product quality. However, this is limited by the occurrence of flow instabilities at flow rates greater than a critical value. These instabilities manifest themselves as surface defects on the surface of extrudates and are collectively known as melt fracture phenomena [3-6]. These include sharkskin (small amplitude periodic distortions) or surface melt fracture, slip-stick or oscillating melt fracture (alternating relatively smooth and distorted portion on the surface) and gross melt fracture (large amplitude periodic and/or non-periodic, chaotic distortions). Melt fracture has been observed in a number of polymers (mainly linear), more frequently on all types of linear polyethylenes including high-density polyethylenes which is the subject of the present work [7]. It has been reported that melt fracture phenomena depend strongly on the molecular characteristics of polyethylenes such as molecular weight and its distribution and levels of long chain branching [3-8]. To assess their processability and correlate it with their molecular structure, rheological methods are frequently employed [9]. In particular shear, extensional and capillary rheometry have been proven to be indispensable methods in assessing the processing behaviour of polyolefins and relate it to rheology [10]. The main objective of this work is to study the melt fracture behaviour of two series of HDPES, namely a series of Ziegler-Natta types (ZN-HDPE) and a second series of metallocene types (mHDPE) all possessing a broad molecular weight distribution (MWD) not studied previously in 1  the literature. As these polymers possess excellent mechanical properties and poor processability it is desirable to understand the causes of this. In order to accomplish this, we first performed a thorough study of the rheology of these broad MWD resins particularly on the effects of molecular weight and molecular weight distribution. For example, relationships between rheological properties such as zero shear viscosity, steady state creep compliance, crossover frequency with molecular weight and its distributions are studied. Consequently, their melt fracture performance in capillary, slit and annular rheometry was investigated. The effect of temperature on melt fracture is also examined which has been reported to be significant in some cases. Their processability is also studied as a function of molecular weight characteristics in capillary extrusion. Several interesting correlations between the critical conditions for the onset of instabilities (slip and melt fracture) and molecular characteristics are derived and discussed which points to the importance of molecular weight characteristics. Polymer processing aids (PPA) are used in order to improve processability of polymer materials. These chemicals enhance the slippage of polymer melt on the die solid surface by forming a lubrication layer in between which results to higher production rates and better processability [11-13]. In this work, a fluoropolymer processing aid is used to examine its ability to eliminate or postpone melt fracture phenomena to higher shear rates. The study of processability has led us to find that slippage plays a significant role in the surprisingly different processing behaviour of these polymers, and their slip behaviour is examined in detail. Slip is neither a recent discovery nor a phenomenon confined to the rheology of complex fluids such as molten polymers. Daniel Bernoulli, Coulomb, Poiseuille, Girard, Maxwell, Navier and Stokes are among those who considered the possible effects of slip in Newtonian fluid dynamics. Simply experimental observations were found consistent with the assumption of no-slip and therefore slip phenomena have received little attention in classical fluid mechanics (see Goldstein [14] for a summary). The molecular dependence of slip velocity for  all available HDPEs was studied. The slip  velocity was determined by performing the Mooney analysis and deviation from their linear 2  viscoelastic behaviour [15]. A criterion for the occurrence of stick-slip transition proposed by Allal and Vergnes [16] is also examined and applied in our case. Based on the findings of this study, the flow curve of high-density polyethylenes (HDPEs) can be predicted solely from linear viscoelastic measurements.  3  2 LITERATURE REVIEW This chapter presents the literature review on the subject of rheology and processing of polyethylenes. More specifically, the various types of polyethylene are presented; Rheological methods and pieces of equipment needed for this study are also discussed. Flow instabilities such as melt surface instabilities are described and the effects of different parameters such as processing conditions and material molecular characteristics are reviewed. Finally, the wall slip of polymer melts and more specifically that of polyethylenes and the effects of molecular characteristics are finally discussed.  2.1 Types of Polyethylenes Plastic materials play a major role in our life through recent decades. One of the most important commodity plastic that is extensively used is polyethylene (PE). The chemical structure of this polymer has the repeating unit of ethylene group (-CH2-CH2-) and it was firstly synthesized by two British researchers, namely Reginald Gibson and Eric Fawcett in 1933 [17]. Thanks to different routes of production, it is rather of low cost to produce it compared to other plastics, and rather easy to process in several applications. Depending on the synthesis processing conditions and catalysts, different types of PEs namely high-density polyethylene (HDPE), low density polyethylene (LDPE), linear-low density polyethylene (LLDPE) and ultrahigh molecular weight polyethylene (UHMWPE) can be produced. These types of polyethylenes have different molecular structures, rheological properties, processing properties as well as mechanical properties [18-20]. Figure 2.1 is a schematic of the molecular structure of various types of polyethylene. It is noted that HDPE, the subject of the present work, has a linear structure.  4  HDPE  LDPE  LLDPE Figure 2.1. Schematic representation of the microstructure of HDPE, LDPE and LLDPE.  Among all types of PEs, HDPE has the longest history and areas of usage. There are two major processes of production: Ziegler-Natta and metallocene catalyst technologies. Use of metallocene catalysts can produce HDPEs with controlled molecular weight (Mw) and molecular weight distribution (MWD) [2]. This enhances its mechanical properties and as a result can be used in more demanding applications. For example HDPE is widely used in pipe industry, especially in water transportation and sewer applications, due to many advantages such as its chemical and corrosion resistance, ease of installation, and low cost of production and maintenance. This plastic passes the standard acceptable hoop stress (ring stiffness measure) by controlling the thickness; however it suffers from low environmental stress crack resistance (ESCR) [21, 22]. This defect could be solved to some extent by changing the molecular weight distribution from unimodal to bimodal [23]. Using metallocene catalysts it is possible to produce pipe grade HDPEs known as PE100 which exhibit bimodal MWD [24]. Although superior 5  mechanical properties are obtained for this class of polyethylenes, their processability becomes extremely difficult due to certain defects on products’ surface that significantly limits the rate of production in extrusion applications.  2.2 Molecular Weight and Molecular Weight Distributions Polymers are materials with large molecules consisted of many repetitive small simple molecules joint together by simple chemical bonds [25]. In most cases, because of their long length, these molecules are commonly known as polymer chains. In the case of synthesized polymers, because of random feature of the synthesizing process, it is impossible to assign a certain molar mass to the polymer [26]. In these systems, there are chains with different lengths, so there is a distribution of molar mass. To characterize these systems, statistical parameters such as the number average molecular weight, Mn are used: ∑ ni .Mi ∑ wi = ∑ ni ∑ wi ⁄Mi  Mn =  (2.1.a)  where wi =ni .Mi and ni is the number of chains with molecular weight of Mi. This quantity, Mn, individually is not capable to describe a polymeric system. A number of other molecular averages are used, essentially higher moments of average molecular weights, which are defined by: ∑  .  ∑ . ∑ .  (2.1.b)  . .  ∑ ∑  (2.1.c)  ∑ ∑ ∑  . .  ∑  .  ∑  .  ∑  .  ∑  .  (2.1.d)  Where Mw is the weight average molecular weight and Mz and Mz+1 are the z and z+1 average molecular weights. Even for cases where all these averages are known, it is impossible to have a complete picture of the distribution of molecular sizes of a particular polymeric system. Instead 6  the complete molecular weight distribution (MWD) is needed. Figure 2.2 shows a typical MWD of a unimodal HDPE.  0.6  dlog[w(M)]/dlog(M)  0.5  0.4  0.3  0.2  0.1  0.0  3  4  5  6  7  log M  Figure 2.2. A typical molecular weight distribution of a polymeric system.  2.3 Rheology Rheology is the science of flow and deformation of matter. Equations that relate deformation (strain) and rate of deformation (shear strain) to forces (stress) are known as constitutive equations or rheological equations of state. Such constitutive equations need to be compared in terms of their ability to model/predict the rheological response of real fluids/solids. From the scientific perspective, this goal can be attained by first setting up appropriate experiments in which simple flow histories are imposed to the material (such as simple shear or extension) and its response is recorded and compared with that of the model.  2.3.1 Rheological Experiments Polymer melts and solutions are viscoelastic fluids which exhibit both viscous and elastic responses depending on the type of deformation. Due to this complex nature, more than one parameter is needed to describe their rheological properties. These parameters which are known 7  as material functions, including shear viscosity, first and second normal stress coefficients, steady state creep compliance, dynamic moduli and several others. In the following sections, some of the typical experiments in which these material functions are measured, are summarized. 2.3.1.1 Frequency Sweep Frequency sweep is one of the most widely used experiment to characterize the rheological properties of polymers. According to this test the upper plate in parallel plate geometry is set into an oscillatory motion (sinusoidal rotational displacement or sinusoidal stress) and the stress or displacement is recorded [9]. In this type of experiment, the strain (stress) could be either within the linear or the nonlinear viscoelastic regime, which is called SAOS (small angle oscillatory shear) or LAOS (large angle oscillatory shear) respectively. SAOS is used in order to determine the dynamic moduli, G’ (storage modulus), G’’ (loss modulus) and the complex viscosity ( η* ≡ √G'2 G"2 ⁄ω). Fig 2.3 shows typical data of G’, G” and η* for a polyethylene.  Dynamic Moduli and Complex Viscosity  106  105  104  103  102  G' (Pa) G'' (Pa) |*| (Pa.s)  101  100 10-4  10-3  10-2  10-1  100  Frequency (rad/s)  101  102  103     Figure 2.3. Viscoelastic moduli and complex viscosity for a typical polyethylene obtained from frequency sweep experiments (SAOS).  8  2.3.1.2 Creep and Creep Recovery Creep is a transient test in which a constant stress is imposed to the sample and the strain is monitored with time [9]. Depending on the level of applied stress and type of material, a steady state response is reached. Figure 2.4 shows the typical response of a material and the material function of interest which is the steady state creep compliance (J0e ). This can be measured based on the following equation or directly from the graph by extrapolation: ⁄  (2.2.a)    100  Creep Compliance (1/Pa)  10-1  inf 10  -2  10-3  10-4  10-5  Je  10-6 10-3  0  10-2  10-1  100  101  Time (sec)  102  103  104  105     Figure 2.4. A typical result of the creep and creep recovery experiment.  If the stress is removed at a certain time (strain level), then due to the elastic nature of polymeric melts, it starts to recover the imposed strain (recoil); however due to its viscous character, this recoil would be completed after complete relaxation. The amount of strain recovered is called the ultimate recoil,  γ∞ (see Fig 2.4). Based on Boltzmann superposition (valid only in linear viscoelasticity) it can be shown that γ∞ and J0e follow the following equation [9]:  (2.2.b)  9  The steady state creep compliance is a weak function of Mw, however it has a strong dependency on the MWD [9]. There are some different measures of MWD to describe this correlation. Also many experiments have been performed to validate such relationships with different materials such as PS, branched PE & PDMS [27]. It is found based on available data so far that the following scaling is applicable:  ⁄  ∝  (2.3)  which is mentioned by Kurata [28] and has been successfully used by Mills & Nevin [29] for a blend of two narrow molecular weight distributions PS. 2.3.1.3 Steady Shear In this experiment, a constant shear rate is applied to the material and the steady state shear stress (response) is recorded. Repeating this experiment for several shear rates and plotting shear stress as a function of shear rate, defines the flow curve of the polymer. Viscosity is the main material function determined from this type of experiment (shear stress over shear rate). In some cases, the first and second normal stress difference coefficients are also measured. A typical graph for the flow curve for a molten polymer is shown in Fig 2.5.  Zero Shear Viscosity, 0  105  .  Shear Stress, () [Pa]  106  [Pa.s]  105  Shear Viscosity, ()  .  104  103 Viscosity Shear Stress 104 10-2  10-1  100  . Shear Rate,  [1/s]  102  101     Figure 2.5. A flow curve of a typical molten polymer obtained from a steady shear experiment.  10  2.3.1.4 Uniaxial Extension In an industrial polymer processing, the polymer melt or solution experiences a combination of different types of deformations i.e. shear and elongation (or extension). All of the above described experiments include only shear components and they are insufficient to completely characterize the rheological behaviour of a material. In fact elongational experiments have been shown to be more sensitive to subtle changes in the molecular structure and frequently are used to relate rheology and processability with molecular characteristics [9]. In a steady uniaxial elongation experiment, a sample of initial length Lo is stretched at a constant Hencky strain rare   d / dt and its Hencky strain is related to L, the instantaneous sample length, according to: (2.4) Moreover, taking into account the Boltzmann’s superposition principle, it can be shown that in the case of transient elongation viscosity, at short time the following relationship is valid which is known as Trouton’s rule (see Fig 2.6): 3  (2.5)  strain rate = 0.10 s-1 strain rate = 1.00 s-1 strain rate = 10.0 s-1  LVE 3  +    +  E  (Pa.s)  104  10-2  10-1  100 Time (sec)  101  102     Figure 2.6. A typical steady uniaxial shear experiment that shows the tensile stress growth coefficient as a function of time for several Hencky strain rates.  11  2.3.2 Rheometers Rheometer is the apparatus for measuring rheological properties. In the following sections the main rheometers and fixtures which have been used in this study are introduced. 2.3.2.1 Parallel Disks Parallel disks are the most extensively used rheological fixture to produce simple shear (Couette flow). It has two parallel concentric disks with specific diameter and distance (gap) between them. One disk can rotate with respect to the other (see Fig 2.7). In this fixture many types of experiments can be performed including the oscillation test (section 2.3.1.1) to generate the viscoelastic moduli of molten polymers plotted in Fig 2.3. Another advantage of this fixture is that only a small amount of material is needed. On the other hand it has some disadvantages such as the inability to reach high shear rates and strains (due to edge fracture of the sample and presence of secondary flows) and non-uniform strain (strain has a radial profile from zero at centre toward maximum at edges) [30]. In this fixture, the angular rotation is imposed and the torque is measured.  Figure 2.7. Schematic illustration of parallel plates fixture.  12  2.3.2.2 Extensional Rheometer (SER) The Sentmanat Extensional Rheometer (SER) [10] is a suitable and easy-to-use fixture that can be used together with a rotational rheometer to generate uniaxial extensional data [31-33]. This rheometer consists of two drums (see Fig 2.8) where the main drum rotates by the host rheometer shaft and the slave drum counter-rotates due to the existence of intermeshing gears that connects the two drums. The sample could be either in a strip shape [34] or in a cylindrical shape [35]. The sample is held firmly in position by means of two clips. As a result the sample undergoes a uniform uniaxial stretching (extension). The possibility of using small amount of sample which results in better and faster controlling the temperature is one the most important advantages of this fixture. Moreover, contrary to other previous designed fixtures for uniaxial extension measurements, the imposed strain is not finite here. However, sagging is always a source of error which can be avoided by doing experiment as fast as possible. Another method is using an oil bath to immerse the sample in order to avoid sagging due to buoyancy forces.  Figure 2.8. Schematic of the Sentmanat Extensional Rheometer (SER).  13  2.3.2.3 Capillary Rheometer Capillary is one of the most industrially used instruments for measuring the viscosity of polymer melts. As shown in Fig 2.9, it consists of a reservoir (barrel) that contains initially the polymer melt and a piston which forces the material through the die which is located at the end of the barrel. The raw data from this device are the force needed to move the piston (pressure) and piston velocity (melt flow rate). Using the pressure and the velocity of the piston one may calculate the apparent shear stress and the apparent shear rate from the following equations:  4 ⁄  (2.6.a)  Δ ⁄2  (2.6.b)  Where Q, p, R and L are volumetric flow rate, applied pressure, die radius and die length respectively. These data are subject to two corrections. One is the Rabinowitsch correction [9] that corrects the apparent shear rate to the true shear rate. The following equation is used for this purpose.  3  1 4  (2.7)  Where “n” is viscosity shear thinning exponent defined as n= d	logτw ⁄d	logγA . The pressure should also be corrected for the extra pressure needed by the melt to flow from the reservoir into the capillary die (change of cross section). This is known as the Bagley correction. It can be calculated by plotting the pressure versus die length-to-diameter ratios (L/D) at fixed apparent shear rates. Then extrapolating the straight lines to zero L/D the excess pressure ∆ can be calculated (see Fig 2.10). The true shear stress at the capillary wall can be calculated from (see Dealy and Wissbrun [9] for more details): ∆  ∆ ⁄  (2.8)  14  Finally slip at the wall can complicate the experimental data further and there are several studies and methods that can be used to analyze capillary data under slip [3, 8, 15, 36-38].   Figure 2.9. Schematic diagram of a capillary viscometer.  Figure 2.10. Typical Bagley correction for capillary data. Each line corresponds to a shear rate.  15  2.3.3 Zero Shear Viscosity and Molecular Characteristics The melt rheology of entangled polymers is strongly influenced by Mw, its distribution and the level of long-chain branching [9, 39-42]. In particular, increase of polydispersity increases the level of shear thinning [42], while it has been reported that it has no influence on the zero shear viscosity, although some reports are contradictory. For example, Wasserman and Graessley [43], Garcia-Franco and Mead [44] and Kazatchkov et al [45] have reported that the zero shear viscosity of linear polymers does depend on the breadth of molecular weight for Ziegler-Natta HDPEs and LLDPEs with polydispersity index (PDI) in the range of 3-12. A weak dependence of the zero shear viscosity of linear polymers has also been predicted by the molecular theory of Pattamaprom and Larson [46]. It has also been reported that at a given molecular weight, the zero shear viscosity of m-HDPE are higher compared to that of ZN-HDPE and this is possibly due to the presence of a small undetectable level of long-chain branching [47-49]. The zero shear viscosity of m-HDPEs has been found to depend on Mw with an exponent of greater than 4, compared to the accepted value of 3.6 reported for ZN-HDPEs [50-52]. It is believe that for the case of broad molecular weight distribution HDPEs, the role on high Mw chains in the relaxation process is significant and therefore, the effect of z-moment of polydispersity should be considered [53].  2.4 Melt Fracture Industrial polymer processing includes many different types of shaping processes including extrusion, injection molding, film blowing, fiber spinning and many others. From the economical point of view, it is desired to increase the rate of production without sacrificing product quality; however the rate of production is frequently limited by flow instabilities that lead to commercially unacceptable final products due to surface defects. These instabilities are collectively known as melt fracture [3, 54]. They are more specifically categorized into sharkskin or surface melt fracture, stick-slip or oscillating melt fracture and gross melt fracture. Typical images for each are presented in Fig 2.11.  16        Smooth   Sharkskin      Stick-Slip      Gross melt fracture   Figure 2.11. Typical extrudate surface defects.  2.4.1 Sharkskin Melt Fracture 2.4.1.1 Overview Sharkskin is commonly the first observed instability that occurs during polymer processing (particularly linear polymers) at relatively low shear rates. This phenomenon manifests itself by small amplitude periodic irregularities on the surface of extrudates i.e. size are typically of about less than one-tenth of extrudate diameter [55]. It is believed that the first report on sharkskin observation is the experiments of Garvey and his co-workers in 1942 [56] on the extrusion of synthetic rubber compound. There are several thorough reviews available in the literature on the sharkskin phenomenon [57-59]. In the following section, a review of the sharkskin phenomenon is presented to outline the critical parameters that play a role on its origin and manifestations. 17  2.4.1.2 Mechanisms Two different mechanisms have been proposed to explain the mechanism of occurrence of sharkskin. The first mechanism which is known as exit stick-slip was firstly proposed by Vinogradov [60]. According to this, the stress at die exit (die lips) is greater compared to that of the die walls. Therefore, by increasing progressively the throughput, at a certain point the stress at the exit region reaches a critical value that promotes slip at the wall [3]. In this case, the melt starts to slip at the exit region of the die to relieve the extra stress. Taking into account that slip leads to less deformation and this decreases die swell [61], the die swell under slip is less than that under no-slip; thus the valleys in sharkskin defect are formed. Therefore according to this mechanism, sharkskin is due to a periodic slip boundary condition at the exit to the die. The experimental evidence that supports this mechanism is not convincing and the communication between Cogswell and Wang [62] discusses this. The second mechanism for sharkskin which is the most accepted one has been proposed by Cogswell [63] and further explained by Migler et al [64]. This mechanism focuses on the fact that the velocity profile of polymer melt undergoes a sudden change from a parabolic shape inside the die region to an almost plug profile as it emerges from the die. This sudden change causes a stretching on extrudate that leads to surface rupture and thus surface melt fracture appears. 2.4.1.3 Effect of Die Geometry Many groups have investigated the effects of capillary die dimensions on sharkskin [55, 65-68]. Experimental results show that at a fixed capillary length, the critical shear rate for the onset of sharkskin times the die diameter is a constant [69] while the severity of this instability (amplitude of distortions) increases linearly with the die diameter [65]. It has also been reported that at a fixed die diameter, the severity of sharkskin decreases with increase of the die length for sufficiently long dies (L/D>10 to achieve fully developed flow) [66]. The entrance angle has no significant effect on sharkskin [66]. 18  The type of die (capillary versus slit versus annular) has been reported to have a significant effect for the onset of sharkskin. The results show that the critical stress for onset of sharkskin is higher in slit and annular dies compared to that for capillary dies [6, 70-72]. Such studies are extremely important from the industrial point of view as most of industrial processes such as pipe extrusion, blow molding and film blowing are performed with annular geometries rather than capillary ones. Therefore these studies can correlate results obtained in the laboratory with the industrial performance of the polymers. 2.4.1.4 Effect of Temperature Considering the mechanism of sharkskin discussed above, it is clear that by increasing temperature, melt viscosity and other viscoelastic properties decrease and thus sharkskin would be postponed to higher shear rates. Therefore, one may expect that one way to eliminate sharkskin is to increase temperature. Although this is helpful to a certain extent, it has been shown that the critical stress is a weak function of temperature [60, 69, 73, 74]. Some studies also have shown that the die temperature (particularly at the exit) has a greater effect on sharkskin rather than the melt bulk temperature (barrel temperature) [75, 76]. Finally some experimental results have shown that there is a narrow window at the low temperature melt region in which smooth extrudate can be obtained [63, 77-80]. Crystallinity here is playing a significant role. 2.4.1.5 Effect of Molecular Structure Regarding the molecular effects on sharkskin, it seems that by increasing the Mw, the critical stress for the onset of sharkskin decreases [74, 81-83]; while some other studies have shown a weak dependence on Mw [45, 84, 85]. Deeprasertkul et al [86] have reported that increase of Mw leads to more sever sharkskin; while it seems from their paper that there is no strong dependency on Mw. Moreover, it is clear that considering the power-law relation between zero shear viscosity and Mw (with a power index of 3.4), in all cases the critical shear rate for the onset of sharkskin is a decreasing function of the molecular weight.  19  The effect of MWD on critical shear stress for the onset of sharkskin is more complicated. It has been shown that with broadening the breadth of the MWD, the critical stress increases [87], or decreases [74] or is independent [69, 73, 88]. A possible reason for the existence of this complexity is that it becomes difficult to change the breadth of the MWD while keeping the Mw constant. Kazatchkov et al [45] found that for a series of LLDPEs with constant Mw, the critical shear rate is an increasing function of molecular weight distribution. These authors have indicated that at a critical value of polydispersity index and above, sharkskin disappears completely. Fujiyama et al [84] found that the critical shear rate increases proportional to the square of  ⁄  . Yamaguchi  et al [89] have also shown that by increasing the molecular weight between entanglements (decreasing entanglement density), the critical stress decreases. Allal et al [90] using an elastic theory provided a criterion for the onset of sharkskin based on a critical elongational stresses which depends on the plateau modulus, the Mw, the molecular weight between entanglements, Me, and the polydispersity (will be discussed in chapter 6). Extending this study Allal and Vergnes [91] have proposed a molecular criterion for the occurrence and existence of sharkskin, including the parameters of the Mw and MWD and the entanglement characteristics. The theory was validated for data available for PS, PP and PE. 2.4.1.6 Effect of Processing Aid In order to overcome sharkskin and to render the processes economically feasible, polymer processing aids (PPAs) are frequently used. PPAs can eliminate the flow instabilities known as sharkskin melt fracture and stick-slip or postpone them to higher flow rates. The end result is an increase of the productivity as well as an energy cost reduction, while high product quality is maintained. The additives mostly used for polyolefin extrusion processes are fluoropolymers [92-94]. Additional examples of other conventional processing aids such as stearates, silicon based polymers, hyperbranched polyethylenes and various polymer blend combinations have also been used. Finally Boron Nitride (BN) based PPAs have recently been proposed [95]. However, the most commercially relevant example is the use of fluoropolymer based PPAs to eliminate sharkskin melt fracture in linear polyethylenes. 20  Figure 2.12 shows the flow curve of a HDPE with and without fluoropolymer. It can be seen how typically a fluoropolymer acts in the extrusion of polyethylenes. As seen the addition of 1000ppm of fluoropolymers, significantly decreases the wall shear stress (due to significant slip) and postpones the onset of instabilities at significantly higher shear rates typically up to one order of magnitude (depending on the temperature and the type of polymer being extruded).  Wall Shear Stress, W (Pa)  100  Onset of Sharkskin (c1)  c2  10-1  Onset of Gross MF (c3)  HDPE Virgin Oscillation Loop HDPE (1000 ppm Fluoroelastome) 10-2 100  101  102  103  104  105  . Apparent Shear Rate,  (1/s)  Figure 2.12. The flow curve of a HDPE extruded in a pressure-driven capillary rheometer as virgin and in the presence of a fluoroelastomer (Adapted from reference [96]).  2.4.2 Stick-Slip Melt Fracture 2.4.2.1 Overview Extruding a polymer with linear molecular architecture such as LLDPE and HDPE, becomes unstable under certain flow conditions i.e. oscillations in pressure in spite of the face that the piston velocity is kept constant. This instability which is called stick-slip or oscillating melt fracture manifests itself by a discontinuity in the flow curve (Fig 2.12). The surface of the obtained extrudate exhibits a periodic appearance, namely alternating rough and smooth portions (Fig 2.11).  21  The works of Tordella [54] and Bagley et al [97] on the HDPEs are the first reports on this instability. However, more systematic studies on this subject have been done by Lupton and Regester [98] and Myerholtz [99]. Other terms rather than stick-slip and oscillating melt fracture have been used for this phenomenon such as spurt flow by Vinogradov et al [100, 101], main flow instability by Weill [102] and cork flow by El Kissi and Piau [103]. Stick-slip usually happens after sharkskin and before the gross melt fracture as flow rate increases. This discontinuity subdivides the flow curve into a low and a high flow rate branch. The important features of this instability are the critical shear stresses for these two branches ( and  in Fig. 2.12), the width of the discontinuity and the size and frequency of the pressure  oscillations. 2.4.2.2 Mechanisms The accepted mechanism for oscillating melt fracture consists of two major elements i.e. melt compressibility and melt slippage over the die solid wall. The melt compressibility compresses the material in the reservoir during the stick portion of the oscillation, which flows as extra when slip occurs. The main cause of slip is a sudden disentanglement of the bulk of the polymer from a monolayer of polymer molecules strongly adsorbed at the die wall. This leads to massive slip [62, 65, 104]. 2.4.2.3 Effect of Die Geometry Most of the previous studies have been done with capillary dies, although there are few works which reported that the critical shear stress for the onset of oscillation increases by switching from capillary to slit dies and the phenomenon totally vanishes in the case of annual dies [6, 70, 71]. The effect of L/D on the critical shear stress for the onset of stick-slip instability is not conclusive and depends on the experimental conditions and material under investigation. For the case of HDPE, it has been reported by different researchers [8, 99, 105, 106] that the upper critical stress, c2 (Fig 2.12) and also the discontinuity gap in the flow curve and the critical 22  stress difference (c2-c3) increases by increasing L/D, while the critical shear rate decreases. Vergnes et al [107] also reported a similar trend for EPDM. Wang and Drda [79, 108] for entangled linear polyethylenes and Ramamurthy [3] for LLDPE showed that c2 is almost independent of L/D. However, Kalika and Denn [109] observed that c2 reduces as L/D increased from 33.2 to 66.2 and 100.1. El Kissi and Piau [110] reported similar observation for PDMS. For materials like HDPE [8, 15, 98] and LLDPE [3, 109] which exhibits stick-slip defect, the flow curve is subdivided into two branches as discussed above (Fig 2.12). In the low flow rate branch (before the discontinuity of stick-slip) the slip is weak due to adhesive failure of the interface. However at the high flow rate branch, it is massive and the velocity profile is almost plug flow. Lupton and Regester [98] reported that the slip velocities are 10 times higher in the high branch compared to that in the low flow rate branch. For data reported by Hatzikiriakos and Dealy [8] on HDPE, this ratio is even higher for some cases. The plug or nearly plug flow in the high flow rate branch have also been quantified for HDPEs [105, 111], LLDPEs [109] and EPDM [107]. 2.4.2.4 Effect of Temperature Previous studies with different polymers showed that temperature has a negligible effect on the critical stress for the onset of stick-slip, although the critical shear rate increases considerably [3, 63, 99, 100, 112]. For HDPE, it was observed that the high flow rate branch of the flow curve is almost insensitive to temperature, although the lower branch shifts to higher shear rates [8, 113]. Effect of temperature on the c2 is different for different polymers. For example it has been shown that for LLDPEs [3] and PDMS [110], that c2 is independent of temperature. Moreover, for the cases of PBs [100, 114], PIs [100] and fluoropolymer resins [12, 54] it is a weak increasing function, while for the HDPEs [8, 79] and ethylene–propylene copolymer [101] the dependency can be much stronger.  23  2.4.2.5 Effect of Molecular Structure Reducing the Mw and broadening MWD leads to suppression of the stick-slip instability and the flow curve becomes a continuous one (with no discontinuity) for polymers with sufficiently low Mw and broad MWD. This has been verified for different polymers such as HDPEs [8, 99, 115, 116], LLDPEs [3], PBs [100, 112, 114, 117], PIs [100, 112, 118, 119], PDMS [110] and PSs [118, 119]. For the case of LLDPE different effects have also been reported i.e. c2 decreases with increasing Mw [120]. Broadening the MWD has the same effect as lowering the Mw [3, 99, 112, 114]. Experiments by Myerholtz [99] on HDPEs with different MWD showed that by broadening the MWD, the hysteresis loop in the flow curve becomes smaller until completely vanishes for very broad MWD, producing a continuous flow curve. Similar observations has been reported by Vinogradov et al [112] for PBs. Wang et al [108, 114] reported that for monodisperse PB melts the c2 is independent of Mw, while for HDPE they found the following scaling  ∝  .  .  This scaling is also valid for the data of Hatzikiriakos and Dealy [8] on HDPEs.  2.4.3 Gross Melt Fracture 2.4.3.1 Overview Gross melt fracture (gmf) occurs at sufficiently high shear rates regardless of the type of polymer. In this phenomenon, the extrudate exits the die in an irregular manner and exhibits severe distortions. These instabilities happen in the order of extrudate volume, and therefore sometimes gmf is referred to as volume melt fracture in order to be distinguished from sharkskin or surface melt fracture [3, 69, 121]. The cross section of extrudate is also not constant within its length. As it is mentioned before, the main difference between the previous discussed melt fracture instabilities (sharkskin and stick-slip) and gmf is that it can happen for all types of polymers. A thorough review in this subject has been done by Kim and Dealy [122].  24  2.4.3.2 Mechanisms While the origin of sharkskin is located at the die exit, the gross melt fracture initiates at the die entry region [54, 59, 123-127]. In the entry area into the die, the polymer melt experiences a considerable amount of elongational stress in the direction of flow, near the flow centerline [128]. At sufficiently high flow rates, melt strands can no longer bear the elongational stress and break. At this point the streamlines become irregular and flow breaks into several layers where each layer moves with its own velocity causing the chaotic extrudate appearance of the melt as it exits the die. 2.4.3.3 Effect of Die Geometry Since the origin of gmf is at the die entrance, the die diameter does not affect the phenomenon. However, capillary die can alter gmf in two ways. Firstly, the severity of extrudate distortion decreases with increasing of the die length. Due to larger length of the die the melt partially relaxes and perhaps partially restores its continuity inside the die. Secondly, by increasing the die length, the pressure in the die entrance region increases and therefore, at a given shear rate, flow is shifted toward the stick-slip flow region [129]. The die entrance angle has a more accountable effect on gmf. It has been shown [130] that reducing the entrance angle, causes gmf to be postponed to higher flow rates and it also reduces the severity of the extrusion distortion. This is due to lower tensile stress experienced by the melt when the entry angle decreases. Kim and Dealy [122] reported that this effect is only observable when the entrance angle is less than 90°. 2.4.3.4 Effect of Molecular Structure Piau et al [131] found that the gmf is almost independent of die material. This supports the idea that gmf is mainly a function of polymer nature and therefore, the molecular structure of the polymer plays a significant role.  25  Available data on the effect of Mw on the critical shear stress for the onset of gmf showed that it has a decreasing effect, although the effect of MWD is confusing. Vlachopoulos et al [73, 82] for HDPE, PP and PS reported that c3 scales inversely with Mw, while MWD had practically no effect. Baik and Tzoganakis [74] reported the same trend for Mw on polypropylenes, although they also found that MWD cause a decrease on c3. Data for HDPEs reported by Kim and Dealy [132] showed a completely different trend, i.e. c3 was independent of Mw, and varied linearly with MWD.  2.4.4 Wall Slip of Polymers 2.4.4.1 Overview Unlike Newtonian fluids, polymer melts slip over solid surfaces violating the classical no-slip boundary condition of fluid mechanics [3, 15, 133-136]. According to the slip theory developed by Brochard-Wyart and de Gennes [137], polymer melts slip no matter how small are the applied forces (shear stresses). This has been verified by direct slip measurements using optical techniques [138, 139]. From macroscopic slip measurements i.e. capillary extrusion, a critical shear stress for the onset of slip, c1 is frequently defined for practical reasons depending on the work of adhesion of the interfaces involved [4, 136, 140-143]. 2.4.4.2 Mechanisms Two are the main mechanisms of slip, namely direct desorption of molecules from the interface and disentanglement of molecules from a monolayer adsorbed on the interface. The first slip mechanism accounts for relatively small deviations from the no-slip boundary condition, and it has been observed at relatively small values of the wall shear stress. In this region, the mechanism of slip is direct detachment/desorption of a few chains from the wall that leads to measurable slip velocities even from macroscopic rheological measurements [140, 141, 144]. At a second critical wall shear stress value, c2 a transition from weak to strong slip has been observed and this causes the discontinuity in the flow curve discussed above in Fig 2.12 [3, 8, 79, 108, 145, 146]. The velocity profile in this slip regime approaches that of plug flow. The 26  mechanism of slip in this regime is sudden disentanglement of the polymer chains in the bulk from those in a monolayer of polymer chains adsorbed at the wall [91, 124, 147]. The transition from weak to strong slip has been observed in the capillary flow of linear molten polyethylenes [3, 109]. This transition is mainly a characteristic of linear polymers typically with relatively narrow molecular weight distribution, including linear polyethylenes such as high density (HDPEs) and linear low density polyethylenes (LLDPEs) [3, 147], polyisoprenes (PIs) [112, 148], polybutadienes (PBs) [149], fluoropolymers [12], polydimethylsiloxanes (PDMSs) [110, 150] and lately polycaprolactones (PCLs) [151]. As a result, the flow curve of such polymers is a discontinuous one, consisted of two branches, namely a low flow-rate branch (weak slip) and a high flow-rate branch (strong slip) as discussed above in the context of Fig 2.12. On the other hand, linear polymers (high-density polyethylenes) of significantly wide molecular weight distribution do not exhibit this transition [99]. 2.4.4.3 Effect of Molecular Characteristics Several studies have attempted to quantify the slip velocity as a function of wall shear stress, wall normal stress, and temperature [3, 15, 98, 133, 135, 152]. However, very few studies have attempted to address the dependence of slip velocity on molecular weight characteristics, such as molecular weight (Mw) and polydispersity index (PI≡ Mw ⁄Mn ). Mhetar and Archer [153, 154] and Park et al [149], for the case of nearly monodisperse polybutadiens, Awati et al [155] and Mackay and Henson [156] for nearly monodisperse polystyrenes and Othman et al [157] for narrow molecular weight distribution polylactides (PLAs) (polydispersity less than 2) have shown that the slip velocity increases with decrease of molecular weight. Considering the convective constraint release mechanism for bulk and tethered chains, Joshi et al [158] proposed some scaling relationships which are in agreement with these experimental observations. Hatzikiriakos and Dealy [15] studied the slip velocity of a series of polydisperse HDPEs (polydispersity index in the range from 3.2 to 9.4) and scaled their slip velocity data with a critical shear stress for the onset of slip, c1 (determined from the macroscopic dependence of the flow curve on capillary diameter) and polydispersity to produce a master slip curve which was a decreasing function of polydispersity. 27  As stressed by Brochard-Wyart and de Gennes [137] polymers slip no matter how small are the applied forces and therefore the critical shear stress, c1, cannot be easily defined. However, c1 is simply a convenient parameter which can be defined from macroscopic rheological experiments, where the flow curves start exhibiting a diameter dependency. Alternative interpretation of the data reported by Hatzikiriakos and Dealy [15] is discussed below and it shown that the slip velocity of the HDPEs studied by Hatzikiriakos and Dealy [15] increases with decrease of molecular weight and with increase of polydispersity.  28  3 THESIS OBJECTIVES AND ORGANIZATION 3.1 Thesis Objectives The main of objective of this study is to understand the rheology and processing (melt fracture phenomena) of two series of broad molecular weight distribution Ziegler-Natta (ZN) and metallocene HDPEs as functions of molecular parameters such as MW and MWD. In other words to understand the critical parameters that play a role in the melt fracture phenomena of wide molecular weight distribution HDPEs and correlate them with molecular parameters and rheological properties. The particular and detailed objectives of this project are listed below: 1- Study the shear and extensional rheology of two series of metallocene and ZN HDPEs with a relatively broad molecular weight distribution over a wide range of temperatures using a parallel plate and an extensional rheometer. This includes the effects of molecular weight and molecular weight distribution on the rheological properties such as zero shear viscosity, steady-state compliance and crossover frequency. 2- Study the processing behaviour of the HDPE resins over a wide range of temperatures by using a capillary rheometer and a variety of capillary, slit and annular dies. In other words the objective is to assess the effects of die geometry (including die diameter, length-to-diameter ration and entrance angle) on the critical parameters for the onset of sharkskin melt fracture. 3- Identify theoretical and phenomenological relationships which relate the rheological parameters such as zero shear viscosity and steady-state creep compliance to Mw and MWD. These relationships can be extended to relate critical conditions for the onset of the flow instabilities (mainly surface melt fracture or sharkskin) with measures of Mw and its distribution.  29  4- Identify the critical parameters (essentially rheological) that play a role in the processing of these polymers as this can be assessed by their melt fracture performance. The question here is how to explain the large differences in the processability of the ZNHDPEs and m-HDPEs, although their rheological properties are not much different. 5- Study the effect of molecular weight and its distribution on the slip of the available resins in order to shed light on the role of slip on the melt fracture behaviour. Moreover, it is an objective to develop a criterion based on molecular characteristics for the occurrence of stick-slip melt fracture.  3.2 Thesis Organization The organization of this dissertation is as follows. A brief introduction to the problem of interest is presented in chapter 1. This is followed by introducing the different types of polyethylenes and their molecular characterization in chapter 2. This chapter also presents a literature review on the subject including different rheological experiments and rheological instruments used to carry out the experimental work. Moreover, previous reports on the rheology of HDPEs their slip and melt fracture behavior are also reviewed chapter 3 presents the objectives and the organization of the thesis. In chapter 4, the materials and methods used in this research are discussed. In chapter 5, the linear viscoelastic properties as well as extensional rheology of all the resins are presented. The effects of molecular characteristics on the rheological parameters are discussed (Objective 1). Chapter 6 provides all the results on the melt fracture properties of these resins considering the effect of molecular characteristics and processing conditions (Objective 2). The effects of Mw and MWD on the critical conditions for the onset of melt fracture (Objective 3) along with their proposed correlations with rheological parameters (Objective 4) are discussed. Chapter 7 reports the slip velocities and its dependency on Mw and MWD (Objective 5). This chapter also includes a discussion on the slip mechanisms in order to develop a molecular based criterion for the onset of stick-slip melt fracture. Finally, chapter 8 summarizes the important conclusions from the results of this research and provides recommendations for future work.  30  4 MATERIALS AND METHODOLOGY 4.1 Materials Two groups of high density polyethylenes (HDPEs) are studied, namely a group of Ziegler-Natta referred to as ZN-HDPEs and a second group of metallocene referred to as m-HDPEs, all provided by Chevron Phillips Chemical Company LP. A characteristic difference of these polymers with those used in other studies [47, 48] is that they all possess a much wider molecular weight distribution. Table 4.1 lists all resins used in the present study along with various measures of their molecular weights. These values are carefully measured using high temperature GPC by Chevron Phillips Chemical Company LP. As can be seen, the polydispersity index, PDI, varies from 11 to 17 and 12-42 for the ZN-HDPEs and m-HDPEs, respectively. Among the m-HDPEs, the last two ones (m-HDPE-13 and m-HDPE-19) have PDI values in the range of the ZN-HDPEs (10-15), while the polydispersity of the rest of m-HDPEs are well above these, typically greater than 19. In addition, for many m-HDPEs the number average molecular weight (Mn) changes while weight average molecular weight (Mw) is approximately the same. This enables us to study the effects of molecular characteristics of these samples in a more systematic way.  4.2 Methodology 4.2.1 Linear Viscoelasticity The linear viscoelastic properties of all resins have been studied using the Anton Paar MCR501, a stress/strain-controlled rheometer. Two types of tests have been performed; small amplitude oscillatory shear (SAOS) and creep tests. Both types of experiments have been performed with the 25mm parallel disk geometry and a gap of about 1 mm. For this type of experiment, the reproducibility is within ±3%. To avoid degradation particularly for the case of creep tests, all experiments have been run under nitrogen environment. The frequency sweep tests were carried out at temperatures ranging from 150°C to 230°C, over 20oC increments. The various curves were shifted by means of applying the time-temperature superposition (TTS) in order to generate the master curves at the reference temperature of Tref=190°C. All samples were found to be 31  thermorheologically simple as they obeyed the time-temperature superposition (TTS) principle. The resulted master curves did not contain the terminal zone at low frequencies for many polymers and thus the zero shear viscosity could not be inferred with certainty solely from these data. One way to extend the data to even lower frequencies is using a stress relaxation (step strain) experiment run for long time. Then the stress relaxation data were used to calculate the relaxation moduli using the Boltzmann superposition principle [9]. This procedure has been applied for all polymers and found that the generated data agreed well with G’, G” from SAOS, while resulted the zero shear viscosities in many cases. Furthermore, creep experiments have been performed with stress level of 10 Pa as another means of determining the zero-shear viscosities. The zero-shear viscosity values were found to be in agreement with those generated by stress relaxation. Table 4.1. List of HDPEs used in this study and their different moments of molecular weights. Resin Mn (Kg/mole) Mw (Kg/mole) PDI=Mw/Mn Mz (Kg/mole) Mz/Mw ZN-HDPE-0  23.0  328.4  14.3  1541.0  4.7  ZN-HDPE-5  18.7  328.7  17.6  1500.7  4.6  ZN-HDPE-6  26.8  296.2  11.1  1244.3  4.2  ZN-HDPE-10  29.6  320.3  10.8  1443.4  4.5  ZN-HDPE-11  26.6  290.4  10.9  1372.3  4.7  ZN-HDPE-12  22.8  291.0  12.8  1588.8  5.5  ZN-HDPE-13  21.8  270.4  12.4  1449.9  5.4  ZN-HDPE-14  21.7  259.6  12.0  1477.2  5.7  ZN-HDPE-15  18.0  228.3  12.7  1319.8  5.8  ZN-HDPE-16  16.0  193.3  12.1  1277.5  6.6  m-HDPE-1  5.3  222.5  42.0  876.7  3.9  m-HDPE-8  10.8  272.4  25.2  972.5  3.6  m-HDPE-9  12.8  269.9  21.1  1081.0  4.0  m-HDPE-10  12.5  266.2  21.3  1049.0  3.9  m-HDPE-11  13.0  248.5  19.1  1041.0  4.3  m-HDPE-12  11.6  229.8  19.8  1048.0  4.6  m-HDPE-13  14.5  210.5  14.5  678.9  3.2  m-HDPE-19  18.1  212.7  11.8  579.9  2.7  32  4.2.2 Extensional Rheology Their extensional behaviour in simple uniaxial extension has been studied using the second generation Sentmanat Extensional Rheometer (SER2) housed in the Anton Paar MCR501 rotational rheometer. More information regarding this fixture can be found elsewhere [10, 31, 32]. The tests were conducted at T=150°C, over a wide range of Hencky strain rates, namely 0.01, 0.1, 1.0, 10.0 and 20.0 s-1. Rectangular samples with 6-8 mm width and thickness of 0.5– 0.7 mm were used. It should be noted that due to their high viscosities, the samples have shown no sign of sagging. The reproducibility of this experiment is within ±10%.  4.2.3 Processing Study The processing behaviour was characterized by using an Instron constant speed capillary rheometer fitted with several capillary dies of various lengths, diameters and entrance angles. For example, to determine the slip behaviour of the polymers, capillary experiments were performed using dies having the same length-to-diameter (L/D) ratio and different diameters [8]. Furthermore, determination of Bagley correction required the use of capillary dies having the same diameter and different length-to-diameter (L/D) ratios [9]. Slit and annular dies were also used in this study in order to examine the effects of die geometry type on the processability of polymers. Table 4.2 summarizes the dies used in this study along with their characteristic dimensions. It is worthwhile to mention that the barrel of the capillary rheometer used has four individual heating zones from barrel to the die land in order to keep temperature uniform along the barrel. The reproducibility of this experiment is within ±10%. Table 4.2. Characteristic dimensions of capillary, slit and annular dies used in this study. D (mm) L/D 2α Capillary die  Slit die Annular die  0.51 0.79 0.79, 1.22, 2.11  15o, 30o, 45o, 60o & 90o 180o 180o  20 5, 16, 32 16  H (mm)  L (mm)  W (mm)  L/H  0.47  20.68  2.54  44  Do (mm)  Di (mm)  Di/Do  L/(Do-Di)  2.54  1.54  0.61  10 33  5 RHEOLOGY OF HDPEs In this chapter, the rheological properties of ZN-HDPEs and m-HDPEs are presented. As all these resins have relatively large molecular weights and broad molecular weight distributions, it is interesting to study the effects of molecular characteristics on their rheological parameters such as zero shear viscosity and steady state compliance.  5.1 Linear Viscoelasticity Fig. 5.1 depicts the master curves of the dynamic moduli G’ and G” and the complex viscosity for two representative samples of m-HDPE-8 and ZN-HDPE-6 at the reference temperature of 190°C (the viscoelastic moduli and complex viscosity of all the HDPEs studied in this work are plotted in Appendix A.1, Figs. A.1-A.16). As discussed in section 4.2.1, to reach the Newtonian regime in order to infer the zero shear viscosity, stress relaxation results were used along with creep tests. As seen the experimental data obtained from these three independent tests agree well.  106  (Pa.s)  G' G"  105  105 104 103  104 m-HDPE-8: LVE ZN-HDPE-6: LVE m-HDPE-8: Stress Relaxation ZN-HDPE-6: Stress Relaxation m-HDPE-8: Creep ZN-HDPE-6: Creep  102 101  T = 190°C  100 10-5  10  -4  10  -3  10  -2  10  -1  10  0  10  1  10  2  |*|  10  3  103  10  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  4  Frequency, aT. (rad/s) Figure 5.1. Master curves of storage and loss moduli for a metallocene (m-HDPE-8) and Ziegler-Natta (ZN-HDPE-6) polyethylene resins at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE), stress relaxation and creep tests.  34  From the linear viscoelastic moduli curves the crossover frequency, frequency at which  , where  , can be defined as the  is referred to as the crossover  modulus. This parameter is important because it separates the viscous and elastic behaviour and it has been used in the past to relate it with various measures of the molecular weight distribution [48, 159]. It has been shown that this parameter is a decreasing function of polydispersity index [48]. This is due to increase of elastic modulus with polydispersity. Utracki and Schlund [159] found that the following equation holds for  ≡  and the polydispersity index  ⁄  for a  series of linear low-density polyethylenes (LLDPEs): 8.4  10  ⁄  .  (5.1)  Vega et al [48] have compared this equation to their data for ZN and m-HDPEs. Fig. 5.2 depicts the experimental results obtained in the present work along with those of Vega et al [48] and Raju et al [50]. Eq. 5.1 proposed by Utracki and Schlund [159] is also plotted on Fig. 5.2 as a straight solid line. It is interesting to note that the data obtained in the present work for both mHDPEs and ZN-HDPEs are in general agreement with those of Vega et al [48], pointing to the correct trend that  is a decreasing function of the polydispersity index, as Eq. 5.1 implies. 106  Gc (Pa)  105  104  m-HDPE (Present work) ZN-HDPE (Present work) Vega et al [48], Unimodal-ZN Vega et al [48], Bimodal-ZN Vega et al [48], Metallocene Raju et al [50] Utracki & Schlund [159] Gc = 8.4×105×(Mw/Mn)-1.385  103  102  1  10  100  Mw/Mn  Figure 5.2. The crossover modulus as a function of polydispersity index, Mw/Mn, at T=190°C for several sets of data for ZN and m-HDPEs. The solid line is Eq. 5.1 proposed by Utracki and Schlund [159].  35  For broad molecular weight distribution resins, higher moments of the molecular weight distributions can correlate better with of  ⁄  . Fig. 5.3 plots the crossover modulus,  , as a function  , where it is shown that a power law relationship (Eq. 5.2) fits well the data for the  HDPEs studied in this work. 3.1  10  .  ⁄  (5.2)  Gc (Pa)  106  105 m-HDPE ZN-HDPE Fitting Gc = 3.1×106×(Mz/Mw)-3.1  104  1  2  3  4  5  6  7 8 9 10  Mz/Mw Figure 5.3. The crossover modulus as a function of Mz/Mw for the HDPEs studied in the present work at T=190°C, showing the existence of a correlation represented by Eq. 5.2.  5.2 Zero Shear Viscosity Fig. 5.4 shows the complex viscosity of all resins at 190°C obtained from frequency sweep (LVE), stress relaxation and creep experiments as described above. For the sake of clarity each curve has been multiplied by a factor listed in Fig. 5.4. The Newtonian flow regime has been reached in most cases with a few exceptions particularly for the high Mw resins. For these cases creep tests were run to determine these values which are also plotted in Fig. 5.4. In addition, the data were fitted with the Carreau-Yasuda model Eq. 5.3 to obtain the zero-shear viscosity as a model parameter.  36  ⁄  1  (5.3)  The values obtained by this method were in agreement with those obtained independently from the combined frequency sweep, stress relaxation and creep data. The determined parameters of the Carreau-Yasuda model for best fit to the experimental data are listed in Table 5.1 and model fits are plotted in Fig. 5.4 as continuous lines. As confirmed by Stadler et al [160], the CarreauYasuda model works well for representing the viscosity curve of linear polydisperse HDPEs (similar to our materials). Table 5.1 lists the values of zero-shear viscosity at 150°C and 190°C, the steady state creep compliance,  and the energy of activation for all samples obtained from  applying the TTS principle (see Appendix A.2 for the list of shift factors).  Dynamic or Shear Viscosity (Pa.s)  1013 1012  Tref = 190°C  1011 1010 109  6  108  ZN-HDPE-6: ×3×10 ZN-HDPE-5: ×106 5 ZN-HDPE-0: ×3×10  107  m-HDPE-19: ×3×10  6  m-HDPE-13: ×6×10  10  103 10  2  101  3  3  m-HDPE-8: ×10  105 104  4  m-HDPE-9: ×300  LVE Creep Carreau-Yasuda Model  m-HDPE-10: ×80 m-HDPE-11: ×20 m-HDPE-12: ×5 m-HDPE-1: ×1  10-5 10-4 10-3 10-2 10-1 100 101 102 103 104 105 106  Frequecny (rad/s) Figure 5.4. The complex viscosity material function of all resins obtained from frequency sweep (LVE), stress relaxation and creep tests at the reference temperature of Tref=190°C.  The dependence of the zero shear viscosity,  , on Mw has gained a great interest in literature,  because it could be used as an evidence of LCB and broadness of the MWD [42, 160, 161]. It is generally accepted that  depends on the Mw according to Eq. 5.4, with an exponent  equal to  1 for Mw below the critical molecular weight for entanglements, Mc, and with an exponent of  equal to about 3.4 for Mw above Mc.  37  (5.4) Wood-Adams et al [52] reported a value of  6.8  10  and an exponent  3.6 at 150°C  for a series of linear polyethylenes (LLDPs). Stadler et al [161] found a higher coefficient for K for linear HDPEs, that is  9  10  3.6 at 150°C. The same exponent  with an exponent  of 3.6 has also been reported by Wasserman and Graessley [43] and Raju et al [50] and therefore it is the accepted universal exponent in the literature for linear polyethylenes. Fig. 5.5 summarizes these findings in the literature along with the data of the present work. It can , of m-HDPEs in the present work are higher than those  be seen that the zero shear viscosities,  reported by Wood-Adams et al [52] and Stadler et al [161], which is typically found for metallocene HDPEs [48, 49]. The power exponent of our correlated data shows an exponent of 4.5 as in Fig. 5.5. This is an indication of either of the presence of a small amount of Long Chain Branching (LCB) or an effect of the high polydispersity or a combination of both. To investigate this further, the energy of activation Ea determined from the application of the TTS principle is a useful quantity (all listed in Table 5.1).  Zero Shear Viscosity, 0 (Pa.s)  107  m-HDPE ZN-HDPE Wood-Adams et al [52] : h0 =6.8×10  106  -15  ×Mw -15 3.6 Stadler et al [161] : h0 =9.0×10 ×Mw Our Fitting : h0 =3.4×10-19×Mw4.5  3.6  105  T = 150°C 104 100  150  200  300  400  500  Molecular weight, Mw (kg/mole)  Figure 5.5. The zero shear viscosity dependency to molecular weight at T=150°C.  38  First, the application of (TTS) to obtain the master curves did not require the application of vertical shift, which is frequently required for highly branched polymers [41, 162]. High values of the activation energy Ea imply the presence of LCB or polydispersity [41]. The values determined in the presence work were higher than 7 kcal/mol for ZN-HDPEs, and in the range of 5.5-7 kcal/mol for the m-HDPEs. Values for Ea for narrow molecular weight HDPEs in the absence of LCB are expected to be in the range of 5-5.5 kcal/mol [51, 125, 163]. Vega et al [48] reported values between 5-5.5 kcal/mol for conventional bimodal PEs, which were explained to be due to low molecular weight tails and not to LCB. Furthermore, their reported values of Ea for m-HDPEs were in the range of 7-9 kcal/mol which were possibly attributed to the presence of low levels of LCB [49]. In our case the Ea values for the m-HDPEs are lower and possibly not due to LCB. Hatzikiriakos [41] found a correlation of polydispersity with Ea for a series of LLDPEs and reported values between 6-8.5 Kcal/mol. Therefore, in the present case the relatively high values of Ea seem to be due more to the broadness of the MWD and less to the presence of LCB. A similar comparison and discussion for the zero shear viscosity of various sets of data reported in the literature at 190°C follows. The presence of long chain branching can be also verified by comparing the radius of gyration (Rg) of m-HDPEs detected by light scattering with a reference for linear polymers. Fig. 5.6 shows such a comparison and gives a good agreement with the linear reference. It can be concluded that the m-HDPEs studied in this work do not possess any LCB. Raju et al [50] studied a series of fractionated polydisperse linear HDPEs and reported an exponent of 3.6 and a proportionality constant of et al [48] reported a power exponent of 4.2 with  3.4 2.3  10 10  for Eq. 5.4. However, Vega for nearly monodisperse  metallocene HDPE of low levels of LCB. Fig. 5.7 depicts our data at T=190°C (Table 5.1) and the equations of Raju et al [50] for linear monodisperse HDPEs and that of Vega et al [48] for nearly monodisperse branched m-HDPEs. Our data lie between these two sets indicating that levels of LCB (if present) are much less than those in the m-HDPEs studied by Vega et al [48] who reported that they were undetectable with analytical methods.  39  Radius of Gyration, Rg (nm)  103  102  101  Linear Reference m-HDPE 100 101  102  103  104  Molecular weight, Mw (kg/mole) Figure 5.6. The radius of gyration of metallocene HDPEs (filled squares) as a function of molecular weight together with a linear reference, verifying the absence of long chain branching in their structure.  Table 5.1. Rheological parameters of the materials. Resin  Ea (kcal/mol)  Carreau-Yasuda Model Parameters (T=190°C)  (T=170°C)  (T=150°C)  (1/Pa)  (Pa.s)  (Pa.s)  (s)  n  a  ZN-HDPE-0  7.6  653639  314290  1.60  0.14  0.29  ZN-HDPE-5  7.5  480690  226620  0.82  0.11  0.30  ZN-HDPE-6  7.3  894972  348190  0.07  -0.12  0.20  m-HDPE-1  6.9  339908  168990  0.91  0.11  0.49  -5  729620  350700  1.51  0.10  0.52  m-HDPE-8  6.9  1.11×10  m-HDPE-9  7.0  1.62×10-5  582082  278270  1.90  0.12  0.53  5.5  1.90×10  -5  598497  307760  2.37  0.12  0.51  -5  421544  218770  2.14  0.14  0.55  368463  191660  2.40  0.15  0.55  m-HDPE-10 m-HDPE-11  6.0  2.33×10  m-HDPE-12  5.7  2.67×10-5  m-HDPE-13  6.3  202839  102760  0.18  0.02  0.42  m-HDPE-19  6.8  184135  91364  0.06  -0.05  0.39  40  Zero Shear Viscosity, 0 (Pa.s)  107  m-HDPE ZN-HDPE Raju et al [50]: 0 =3.4×10-16×Mw3.6 Vega et al [48]: 0 =2.3×10-17×Mw4.2  10  Our Fitting : 0 =6.0×10-19×Mw4.4  6  105  T = 190°C 104 100  150  200  300  400  500  Molecular weight, Mw (kg/mole)  Figure 5.7. The zero shear viscosity dependency on molecular weight at T=190°C.  Wasserman and Graessley [43] have presented a more general correlation for the zero shear viscosity of linear polymers that allows the zero shear viscosity to be not only a function of Mw, but also a function of different measures of polydispersity. Their proposed correlation is: ⁄  ⁄  (5.5)  Previous studies on polyethylenes have revealed that for polyethylenes the coefficients “a” and “b” are 0 and 1 respectively [43, 44, 161]. Stadler et al [161] has shown that at T=150°C, Eq. 5.5 is valid with  4.1  10  ,  3.6, a=0 and b=1. den Doelder [164] also has proposed  exponent b as 0.9 using the double reptation single exponential model for the case of polydisperse polymers. These results are also consistent with modelling efforts by Nobile and Cocchini [165, 166] and Cocchini and Nobile [167]. Fig. 5.8 shows that by considering this modification (effect of polydispersity), our data fall on a straight line with a slope of 3.6 at both temperatures of 150°C and 190°C (accepted universal exponent for linear polymers) and agrees remarkably well with the correlation derived by Stadler et al [161] using their own experimental data. The fact that the new slope of the corrected data for polydispersity effects is 3.6, is perhaps an indirect indication of the absence of LCB from our m-HDPEs. It is noted that the range of 41  ⁄  covered by Stadler et al [161] was from 1.5 to 3.8, whereas in this case this is extended  from 2.7 to 4.7 indicating that our resins have a much broader MWD and thus the effect of polydispersity might be more significant.  106 -15  Stadler et al [161]: 0 =4.1×10 ×Mw -16  0/(Mz/Mw) (Pa.s)  Our Fitting : 0 =9.1×10 ×Mw  3.6  3.6  ×(Mz/Mw)  ×(Mz/Mw)  m-HDPE (150°C) m-HDPE (190°C) ZN-HDPE (150°C) ZN-HDPE (190°C)  105  104 100  150  200  300  400  500  Molecular weight, Mw (kg/mole)  Figure 5.8. Correlation of zero shear viscosity with Mw taking MWD into account at T=150°C & T=190°C.  5.3 Steady State Creep Compliance Fig. 5.9 shows typical creep test results for two samples (m-HDPE 9 and 11) using a 10 Pa shear stress value. From creep tests, the steady state creep compliance ( ) can be determined by extrapolating the steady state linear portion of the compliance curve to time zero, according to the following equation [9]: lim →  ⁄  (5.6)  To verify that steady state conditions have been reached, Eq. 5.6 has been used to check the linearity of the final portion of the curves. 42  Creep Compliance, Je (1/kPa)  12 10 8 6 4  m-HDPE-9 m-HDPE-11  2 0  0  1000  2000  3000  4000  Time (s)  Figure 5.9. Creep test results for two samples m-HDPE-9 and m-HDPE-11 using a constant shear stress of 10 Pa at T=170°C .  Rather than viscosity which is a function of molecular weight, it is widely accepted that the steady state creep compliance does not relate to Mw, but it is very sensitive to measures of the molecular weight distribution for molecular weights greater than the critical molecular weight for entanglements, Mc [42]. There are many relations (empirical and theoretical) proposed to relate with molecular weight distribution. Two of them are the most accepted in the literature since they have been verified experimentally. The first one was proposed by Mieras and Rijn [168] (Eq.  5.7a)  and  has  been  tested  successfully  for  polystyrene  (PS),  HDPEs  and  polydimethylosiloxanes (PDMSs) by Mills [12, 169]. The second relation is that proposed by Agarwal [27] (Eq. 5.7b) and explained theoretically by Kurata [28]. This was found to work well for PSs [29, 170-173]. ∝ ∝  .  ⁄ ⁄  (5.7a) (5.7b)  Fig. 5.10a depicts the experimental values for m-HDPEs of the present work together with Eqs. 5.7a and 5.7b and the experimental data reported by Mills [174]. As shown in the figure, our data 43  follow Eq. 5.7a well. However, it should be noted that our data are lower than reported by Mills [174] by about 1 order of magnitude at the same molecular weight distribution. Gabriel and Münstedt [175] and Resch et al [176] have shown that molecular structure such as LCB plays a significant role in melt elasticity i.e. steady state creep compliance. However detailed information on the molecular structure of HDPEs used by Mills [174] is lacking and the origin of this difference cannot be concluded. Another correlation has been also proposed by den Doelder [164] based on the double reptation single exponential model (Eq. 5.7c): ∝  .  ⁄  .  ⁄  (5.7c)  As it is clear from Fig. 5.10b, this correlation also describes well the experimental data of the present work.  Steady State Creep Compliance, Je0 (1/Pa)  10-3 m-HDPE (Present Work, T=170°C) Mills [174] (T=190°C) 10-4  Slope=3.7  10  -5  Slope=3.7  10-6 1.0  1.5  2.0  3.0  4.0  5.0  Mz/Mw  Figure 5.10a. Correlation between steady state creep compliance and the measure of MWD, several HDPE resins.  ⁄  for  44  m-HDPE  0  Steady State Creep Compliance, Je (1/Pa)  10-4  10-5  T = 170°C  Slope=2.2  10-6  4  5  6  7  8  9 10  15  20  (Mz+1/Mw)×(Mz+1/Mz)  ⁄ Figure 5.10b. Correlation between steady state creep compliance and a measure of MWD, for m-HDPE resins. The slope is 2.2 as it is expected according to Eq. 5.7c proposed by den Doelder [164].  5.4 Summary In this chapter, the rheological properties of broad molecular weight distribution ZN and mHDPEs have been examined. The zero shear viscosity showed a power-law correlation with molecular weight, but the exponent was greater than the universal exponent of 3.6. As it was shown that there is no branching for m-HDPEs resins, it was assumed that for these high molecular weight polymers, the effect of very long molecules needs to be considered. This was done by introducing the z-moment of molecular weight into the correlation. Once the effect of molecular weight distribution has been taken into account, the exponent of 3.6 was recovered. It should be noted that the effect of polydispersity was in accordance with previous studies. The crossover modulus also shows a power-law dependence on the z-moment polydispersity, ⁄  .  45  Finally, the correlation of steady state compliance with different measures of polydispersity was checked. Our experimental results for m-HDPE resins were in good agreement with different proposed correlations in the literature.  46  6 MELT FRACTURE OF HDPEs In this chapter, the melt fracture instabilities of ZN-HDPEs and m-HDPEs are studied. The effects of temperature, addition of a processing aid, type of die (cylindrical, slit and annular) and geometrical details, as well as molecular characteristics such as molecular weight and its distribution on the different modes of melt fracture are investigated. Some interesting correlations for the onset of instabilities and molecular and rheological parameters are also derived.  6.1 Uniaxial Extensional Rheology According to the accepted mechanisms for the different types of melt fractures [31, 63, 64], the elongational part of the flow in the die inlet (gross melt fracture) and outlet (sharkskin) plays a key role in the phenomena. Therefore, studying the uniaxial extensional properties would be a matter of interest in order to have a better understanding of the processing properties of polymer melts [32]. Fig. 6.1 depicts the uniaxial extensional results of resins m-HDPE-1 and ZN-HDPE-0 at T=150°C and different Hencky strain rates. Similar graphs for several other ZN and m-HDPEs can be found in Appendix B. The continuous lines represent 3  determined from linear  viscoelastic measurements by calculating the parsimonious relaxation spectrum  ,  of both  resins [177, 178] from the following expression (Parsimonious relaxation spectra for all the resins are listed in Appendix A.3):  3  1  ⁄  (6.1)  It is noted that both resins follow Trouton’s rule at short times. As it is shown in Fig. 6.1, the LVE 3  for the metallocene resin is higher than ZN-HDPE-0 in  the shown time span. The resin m-HDPE-1 ruptures at relatively longer times, which shows its higher extensibility and capability to store energy. Another interesting feature of this figure is 47  that the m-HDPE-1 shows a necking behaviour at Hencky strains higher than 1.0 s-1. This necking is accompanied by a two steps rupture. The first rupture is incomplete and happens at the edges of the sample (necking), although the sample still withstands the tensile stress. The necked sample is further extended, becomes opaque, which is sign of crystallinity until it eventually breaks up. Based on these results, the ZN-HDPE is expected to exhibit better processability.  Tensile stress growth coefficient (Pa.s)  107  T = 150°C   0.1 s 1  106    1 s 1   10 s  LVE 3     0.01 s 1  1  5  10    20 s 1 104  m-HDPE-1 ZN-HDPE-0 103 10-3  10-2  10-1  100  101  102  103  104  Time (s)  Figure 6.1. The tensile stress growth coefficient of resins m-HDPE-1 and ZN-HDPE-0 at several Hencky strain rates, at T=150°C. The line labeled as LVE 3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1.  6.2 Capillary Rheometry Figs. 6.2a and 6.2b depict the capillary flow curves for the resins m-HDPE-1 and ZN-HDPE-0, respectively at different temperatures, namely from 140 to 230°C. Similar graphs that depict the flow curves of several other ZN and m-HDPEs can be found in Appendix C.1. The temperature dependency of metallocene resin is slightly milder than its ZN counterpart. This can be deduced by the activation energy values obtained by the application of TTS principle to obtain the LVE master curves presented in chapter 5 (Table 5.1). These are 6.90 kcal/mole for m-HDPE-1 and 7.29 kcal/mole for ZN-HDPE-0, typical values for linear polymers with a very broad MWD [41]. 48  The same shift factors were used to superpose the capillary data plotted in Figs 6.2a and 6.2b. The results are shown in Figs 6.3a and 6.3b. The superposition is excellent, except for the flow curves obtained at low temperatures and high apparent shear rates. This is clearly due to the effect of flow induced crystallization (FIC) which seems to be more dominant for m-HDPE-1 compared to ZN-HDPE-0. For example Fig 6.3a shows that the flow curves for m-HDPE-1 even at the temperature as high as T=160°C shows lack of superposition due to FIC. It is noted that the melting points for the two resins are T=126.7°C and T=127.5°C for the m-HDPE-1 and ZNHDPE-0 respectively. As it was discussed in the previous section, FIC crystallization was found to be more dominant for m-HDPE-1 from the uniaxial experiments that leads to necking. The higher heat of fusion for the metallocene resin which is 174.5 J/gr compared to the one of ZNHDPE-0 which is 143.6 J/gr would be another sign of its higher tendency to crystallize.  Apparent Shear Stress, W,A (MPa)  0.60  m-HDPE-1 L/D = 16 0.40 D = 0.79 mm die 0.30  2 = 180°  0.20 T = 140°C T = 150°C T = 160°C T = 170°C T = 190°C T = 210°C T = 230°C  0.15 0.10 0.08 0.06  100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure 6.2a. The flow curves of resin m-HDPE-1 in capillary extrusion at different temperatures.  49  Apparent Shear Stress, W,A (MPa)  1.00 0.80  ZN-HDPE-0 L/D = 16 0.60 Ddie = 0.79 mm 0.40  2 = 180°  0.30 0.20  T = 140°C T = 150°C T = 160°C T = 170°C T = 190°C T = 210°C T = 230°C  0.15 0.10 0.08 0.06 0.04 100  101  102  103  .  104  Apparent Shear Rate, A (s-1)  Figure 6.2b. The flow curves of resin ZN-HDPE-0 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.60  m-HDPE-1 L/D = 16 0.40 D = 0.79 mm die 0.30  2 = 180°  0.20 T = 140°C Tref = 150°C  0.15  T = 160°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06  100  101  102  .  103 -1  Apparent Shear Rate, A (s )  Figure 6.3a. The master flow curve of resin m-HDPE-1 at Tref = 150°C, obtained by superposing the data plotted in Fig. 6.2a by using shift factors determined from LVE.  50  Apparent Shear Stress, W,A (MPa)  1.00 0.80  ZN-HDPE-0 L/D = 16 0.60 Ddie = 0.79 mm 0.40  2 = 180°  0.30 0.20  T = 140°C Tref = 150°C  0.15  T = 160°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 0.04 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure 6.3b. The master flow curve of resin ZN-HDPE-0 at Tref = 150°C, obtained by superposing the data plotted in Fig. 6.2b by using shift factors determined from LVE.  6.2.1 Melt Fracture and Typical Flow Curves for HDPEs Linear polymers like HDPE and LLDPE of narrow molecular weight distribution generally show a discontinuity in their flow curve that separates their flow curve into two distinct parts. The flow in this regime is unstable, in other words the flow rate and pressure drop oscillate between two extreme values, a phenomenon known as stick-slip or oscillating melt fracture. Fig. 6.4a shows such a behavior for resin ZN-HDPE-10 at T=190°C. As seen the flow curve is divided into two branches, namely a low and a high flow-rate branches [34, 179]. In the low flow-rate branch, the velocity profile of the melt in the die region is almost parabolic and the magnitude of slippage is typically low. On the other hand, in the high flow-rate branch the velocity profile is almost plug like. The two branches are connected to each other through a hysteresis loop as shown in Fig 6.4a. Three are three relevant critical shear stresses that are of importance in the melt fracture phenomena under discussion, namely the critical shear stress for the onset of sharkskin (  ), the 51  critical shear stress for the onset of stick-slip (  ) and the critical shear stress for the onset of  gross melt fracture (  ). The first two critical shear stresses,  rate branch, while  to the high flow-rate branch. During the pressure oscillations the shear  stress in the wall of the capillary oscillates between  and  and  , refer to the low flow-  . The difference  defines  the magnitude of the oscillations.  0.80  ZN-HDPE-10 T=190°C  Shear Stress, W (MPa)  0.60 0.40 0.30  c2 c1  0.20 0.15  c3  Smooth Sharkskin Gross  0.10 0.08 100  101  102  103  104  . Apparent Shear Rate,  (s-1) Figure 6.4a. The flow curve of ZN-HDPE-10 at T=190°C which shows the stick-slip transition as a hysteresis loop that connects two distinct branches. Several experiments were performed to completely define the hysteresis loop. The relevant critical stresses are those for the onset of sharkskin, , for the and for the onset of gross melt fracture, . onset of stick-slip,  Contrary to the ZN resins, most of the metallocene resins, particularly those with PDI>19, do not exhibit any stick-slip instability and their flow curve is rather a continuous one. Fig. 6.4b shows such a typical flow curve for resin m-HDPE-8 at 190°C. In this case, there are two relevant critical shear stress values, namely the critical shear stress for the onset of sharkskin ( the critical shear stress for the onset of gross melt fracture ( in this case  is always less than  ), and  ). It is worthwhile to mention that  . However, this is not true for the case of HDPEs that  exhibit stick-slip transition (see Fig. 6.4a).  52  0.30  Shear Stress, W (MPa)  m-HDPE-8 T=190°C 0.20 0.15  0.10 0.08  c3  Smooth Sharkskin Gross  c1  0.06 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure 6.4b. The flow curve of m-HDPE-8 at T=190°C appears to be a continuous one. The critical stresses for the onset of sharkskin and gross melt fracture are also shown as and respectively.  6.2.2 Effect of Molecular Weight Fig. 6.5 shows the effect of molecular weight on the shape of flow curve for resins ZN-HDPE-10 to ZN-HDPE-16. These resins have almost the same polydispersity and different molecular weight, Mw. It is clear that the size of the discontinuity (size of the hysteresis loop) decreases by decreasing the molecular weight. In other words, as the molecular weight decreases, the difference  (magnitude of the oscillations) decreases and the two branches come closer.  Essentially, the hysteresis loop shrinks in both the horizontal and vertical directions. Also the critical shear stress for the onset of stick-slip transition appears to slightly decrease with increase of molecular weight as will be discussed later. Below a certain molecular weight, Mw, the flow curve becomes continuous (no stick-slip transition) as was also reported by Myerholtz [99] and Blyler and Hart [115]. Another interesting feature of this figure is that the high flow rate branch of the flow curve is independent of Mw.  53  1.00  Shear Stress, W (MPa)  0.60  T=190°C  0.40 0.20 0.10 ZN-HDPE-10 ZN-HDPE-11 ZN-HDPE-13 ZN-HDPE-15 ZN-HDPE-16  0.06 0.04 0.02  101  102  .  103  104 -1  Apparent Shear Rate, A (s )  Figure 6.5. The flow curves of some of the ZN-HDPE resins with similar PDI, and different molecular weight at T=190°C. The size of stick-slip discontinuity in the flow curve decreases with decrease of the molecular weight.  6.2.3 Effect of Molecular Weight Distribution Fig. 6.6 depicts the flow curves of three different HDPEs having almost the same Mw and different polydispersities, namely 11, 18 and 25. As it is clear, the effect of increasing the polydispersity is similar to the effect of decreasing the molecular weight. In other words as PDI increases, the size of the hysteresis loop decreases in both horizontal (the two distinct branches come closer) and vertical directions and the magnitude of the oscillations  also  decrease. For polymers with PDI>19 the stick-slip transition disappears. Note the flow curve of HDPE-8 in Fig. 6.6 and several other flow curves for HDPEs with PDI>19 in Fig. 6.7. Finally, the high flow rate branch of the flow curve appears to be independent of polydispersity as well. For example see how they all come together at high shear stress values in Fig. 6.7. In this flow regime, the HDPEs slip is considerably high (nearly plug flow) and the friction of monomer with the wall becomes the relevant parameter that controls the level of stress developed [8]. 54  0.60  Shear Stress, w (MPa)  0.40 0.30  ZN-HDPE-10, PI=11 ZN-HDPE-5, PI=18 m-HDPE-8, PI=25  0.20 0.15  0.10  T=190°C  0.08 0.06 100  101  102  .  103 -1  Apparent Shear Rate, A (s )  Figure 6.6. The flow curves of ZN-HDPE-10, ZN-HDPE-5 and m-HDPE-8 at T=190°C that show the effect of polydispersity. The size of the stick-slip discontinuity (size of the hysteresis loop) in the flow curve decreases with increase of polydispersity.  0.40  Shear Stress, W (MPa)  0.30 0.20 0.15  m-HDPE-1 m-HDPE-8 m-HDPE-9 m-HDPE-11 m-HDPE-12  0.10 0.08 0.06  T=190°C  0.04 0.03 100  101  102  .  103 -1  Apparent Shear Rate, A (s )  Figure 6.7. The flow curves of the m-HDPEs with PDI>19 at T=190°C. All flow curves are continuous because of the wide molecular weight distribution.  55  6.2.4 Stick-Slip: Molecular Criterion for the Onset As discussed in previous sections, the magnitude of the oscillations in the stick-slip transition is a function of Mw and its distribution. The most acceptable mechanistic molecular explanation for this phenomenon is the entanglement-disentanglement transition between the tethered molecules attached to the solid wall and the molecules in the bulk [137]. According to these authors, the tethered chains can be stretched in the direction of flow since they are entangled to the bulk molecules. After a critical shear stress, these entanglements could not sustain the load, since the tethered chains cannot be stretched further due to their contour length limitation. As a result of the sudden disentanglement, a transition from weak to strong slip is obtained at  (see Fig.  6.6). In this state the tethered chains slowly recover their random coil shape until they reentangled with surrounding chains at  (see Fig. 6.6) and be stretched again. The disentangled  bulk chains also can slip over them during this period. This cyclic process leads to an oscillation in shear stress exerting by molecules, essentially a hysteresis loop defined by the two arrows in Fig. 6.6 for ZN-HDPE-10. Based on this analogy, it is expected to suppress stick-slip effect by introducing short chains (e.g. increasing the molecular weight distribution), as they prefer to migrate to the wall and act as tethered chains [153, 180]. These short tethered chains do not have the ability to be oriented much by flow and therefore the bulk chains can slip over them gradually (instead of catastrophic disentanglement and abrupt slippage in the case of long tethered chains) by increasing the shear stress [158]. This has been also verified experimentally by Myerholtz [99] and Blyler and Hart [115]. As it is depicted in Fig. 6.7, it appears that once PDI increases above 19 and closer to 20, the stick slip flow regime disappears and the flow curves are monotonically increasing functions of the shear rate. However, PDI is not the only criterion for the occurrence of stick-slip. Allal and Vergnes [16] have argued that in order to eliminate this defect, the entanglement at the solidpolymer interface should be removed. These authors argued that the size of these interfacial molecules must be lower than a few times of the molecular weight between entanglements, Me, with the exact number depending on the chemical structure of the polymer and nature of the solid material. They have developed the following criteria for the absence of stick-slip flow from a class of linear polymers: 56  ⁄  (6.2)  (no stick-slip)  where PDI is the polydispersity index and  is a constant depending on the polymer chemical structure and the solid wall material. Assuming a value for Me=1.2 kg/mole for polyethylenes and Mc=2Me, a value of 12 is obtained for where Mc is the critical molecular weight for the onset of entanglements [16]. Fig. 6.8 demonstrates that the criterion of Eq. 6.2 based on the weight average molecular weight and polydispersity index for stick-slip existence can describe the flow behaviour of HDPEs examined in this work as well as those studied by Hatzikiriakos and Dealy [8]. Eq. 6.2 as an equality is represented by the continuous straight line in Fig. 6.8 and separates the domain (PDI, Mw) in two areas, namely the stick-slip and the no stick-slip domains.  800  Mw = l×Me×PDI=14.4×PDI  Molecular Weight, Mw (kg/mole)  700 600  Present Study Hatzikiriakos & Dealy [15]  500  STICK-SLIP  400  NO STICK-SLIP  300 200 100 0  0  10  20  30  40  50  Polydispersity Index, Mw/Mn  Figure 6.8. Criterion for the occurrence of stick-slip transition for HDPE resins. Open symbols represent resins with no stick-slip transition while filled symbols those that exhibit this transition. The continuous line is Mw=12×PDI×Me. In general Eq. 6.2 represents the data adequately well.  57  6.3 Effects of Processing Conditions on Melt Fracture 6.3.1 Effect of Temperature Fig. 6.9a and 6.9b illustrate the processability maps of resins m-HDPE-1 and ZN-HDPE-0 in capillary extrusion experiments as a function of apparent shear rate and temperature. Similar processability maps for several other ZN and m-HDPEs can be found in Appendix D. The various symbols on the map indicate smooth, sharkskin and gross melt fracture appearance of samples extruded at a given temperature and apparent shear rate. It is clear that the processability of ZN-HDPE-0 is superior compared to that of m-HDPE-1. This can be also seen from the critical shear rates and stresses for the onset of melt fracture at various temperatures listed in Table 6.1 for both polymers (The critical apparent shear rates and apparent shear stresses for most of the resins are listed in Appendix D). Therefore, the processability of the two resins with similar rheology is distinctly different mainly due to the broadness of their molecular weight distribution. m-HDPE-1 230  Temperature (°C)  210 190 170 smooth sharkskin gross  150 130  2  5  10  20  50  100  .  200  500 1000 2000  Apparent Shear Rate, A (s-1)  Figure 6.9a. Processability map of resin m-HDPE-1 in capillary extrusion as a function of apparent shear rate and temperature. Various symbols indicate smooth, sharkskin or gross melt fracture appearance of samples extruded at given temperature and apparent shear rate.  58  ZN-HDPE-0 230  Temperature (°C)  210 190 170 Smooth Sharkskin Stick-Slip Gross  150 130  2  5  10  20  50 100 200  .  500 1000 2000 5000  Apparent Shear Rate, A (s-1)  Figure 6.9b. Processability map of resin ZN-HDPE-0 in capillary extrusion as a function of apparent shear rate and temperature. Various symbols indicate smooth, sharkskin or gross melt fracture appearance of samples extruded at given temperature and apparent shear rate.  Table 6.1. Critical apparent shear rates and shear stresses for the onset of sharkskin melt fracture in capillary die for both resins at different temperatures. Note that only gross melt fracture is obtained for ZN-HDPE-0 at high temperatures. m-HDPE-1 ZN-HDPE-0 Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 3.9 0.11 3.8 0.09 140 6.1 0.11 21 0.15 150 13.7 0.12 50 0.19 160 13.7 0.12 50 0.18 170 13.7 0.11 312 0.29* 190 21.0 0.11 312 0.26* 210 32.2 0.12 312 0.24* 230 * Gross melt fracture  59  At low temperatures, the first extrudate defect that is observed is sharkskin; while by increasing the temperature, sharkskin is obtained at higher and higher apparent shear rates, it eventually disappears at temperatures greater than about 190oC. At these high temperatures gross melt fracture occurs. Moreover, inspecting the critical shear rates and stresses listed in Table 6.1, it can be concluded that the critical shear stress for the onset of sharkskin is a weak function of temperature, also in agreement with other reports in the literature [15, 181]. Although the critical stress for the onset of sharkskin is a weak function of temperature, it is sensitive to molecular weight distribution [3, 45, 69]. Kazatchkov et al [45] showed that for linear low-density polyethylenes having PDI>9, sharkskin instability was not a serious problem. While this is the case in the present study for 9<PDI<19, for resins with PDI>19, sharkskin becomes a serious problem as it occurs at critical shear stress values of significantly less than 0.1 MPa [3, 4, 45, 64]. Finally, the critical shear stress values of 0.24-0.30 MPa for the onset of gross melt fracture also agree well with previous reports [3, 58, 63, 100, 112]. For example Hatzikiriakos and Dealy [8] have reported critical shear stress in the range of 0.22-0.50 MPa for various HDPEs. While the results for ZN-HDPE-0 overall agree with those reported in the literature, the results for the m-HDPE-1, are quite surprising. Given its high PDI of about 40, the critical shear stresses for the onset of sharkskin at around 0.1 MPa is typical for very narrow molecular weight distribution HDPEs [3, 7, 45, 134]. It is also noted that no stick-slip instability was obtained which is mainly due to the high PDI. Moreover, as it will be shown in the next chapter, the mHDPE-1 resin exhibits significant slippage on the die wall surface and this has also an impact on its processability in the presence of processing aids. Table 6.2 summarizes the critical shear stresses for the onset of various instabilities for all HDPEs at various temperatures. Once again these values are independent of temperature. One striking difference between the HDPEs with PDI>19 (first group) and the second group is the distinctly different critical values. The first group exhibits sharkskin at relatively low critical shear stress values, typically less than about 0.1 MPa. They exhibit no stick-slip and their flow curve is monotonically increasing, exhibiting significant slip and failure of the Cox-Merz rule as 60  reported by Ansari et al [182]. The second group exhibits critical shear stress values above 0.15 MPa and in most cases above 0.20 MPa. They all exhibit stick-slip transition with low slip at the low flow curve branch. Table 6.2. Critical shear stresses for the onset of different types of melt fractures. Critical shear stress (MPa) Material T (°C) Gross Sharkskin, Stick-slip, ° 0.29 0.29-0.34 150-230 0.15-0.18 (150-170 C) ZN-HDPE-0 170-230 N/A 0.30 0.28-0.31 ZN-HDPE-5 170-230 0.16-0.20 (150-170°C) 0.33 0.29-0.32 ZN-HDPE-6 190 0.27 0.30 0.15 ZN-HDPE-10 190 0.28 0.33 0.20 ZN-HDPE-11 190 0.27 0.34 0.21 ZN-HDPE-12 190 0.27 0.35 0.22 ZN-HDPE-13 190 0.26 0.36 0.25 ZN-HDPE-14 190 0.24 0.36 0.33 ZN-HDPE-15 190 0.26 N/A 0.42 ZN-HDPE-16 150-230 0.10 N/A 0.16 (150-170°C) m-HDPE-1 150-210 0.08 N/A 0.10 m-HDPE-8 150-210 0.07 N/A 0.09 m-HDPE-9 150-210 0.07 (150-190°C) N/A 0.09 (150-190°C) m-HDPE-10 150-210 0.06 (150-190°C) N/A 0.09 (150-190°C) m-HDPE-11 ° 150-210 0.05 (150-190 C) N/A 0.1 (150-190°C) m-HDPE-12 170-210 0.10 0.36 0.20 m-HDPE-13 170-210 0.14-0.15 0.37 0.25 m-HDPE-19  6.3.2 Effect of Die Type The type of die (capillary versus slit versus annular) has been reported to have a significant effect on the critical conditions for the onset of sharkskin. The results show that the critical stress for onset of sharkskin is higher in slit and significantly higher in annular dies compared to that in capillary dies [6, 70-72, 183]. For example Delgadillo et al [6] have performed extrusion experiments for a linear low-density polyethylene with capillary, slit and annular dies. They have reported that the critical stresses for the onset of sharkskin, oscillating and gross melt fracture differ significantly depending on the die geometry used. Oscillating melt fracture was obtained with capillary and slit dies but it was absent in annular die extrusion. Such studies are extremely important from the industrial point of view as most of industrial processes such as pipe extrusion, 61  blow molding and film blowing include annular flow geometries rather than capillary/slit ones. Therefore, such studies can correlate results obtained in the laboratory with the industrial (production scale) performance of the polymers. Three types of die have been used to check the effect, i.e. capillary, slit and annular. The die dimensions are listed in Table 4.2. Figs. 6.10a and 6.10b depict the flow curves of m-HDPE-1 and ZN-HDPE-0 respectively, through these dies. The apparent shear rates are defined as 8 ⁄     6 ⁄  (6.3)     48 ⁄     where Q is the volumetric flow rate. The wall shear stress for the annular die is not uniform across the thickness and therefore we have calculated two flow curves for this case, which correspond to the shear stress values at inner and outer surfaces. The stresses have been calculated using the following formula, which is based on the assumption of a power-law fluid [184]: ∆ 4 where and  2 2  (6.4)  is the shear stress at radius r, ∆ is the pressure drop, L is the length of the die land, is the parameter depending on the geometry and the power law index [184].  62  Apparent Shear Stress, W,A (MPa)  0.40 0.30  T=190°C m-HDPE-1  0.20 0.15  0.10  Capillary (D=0.79mm, L/D=16) Slit Annular - Inner surface Annular - Outer surface  0.08 0.06 100  101  102  103  .  Apparent Shear Rate, A (s-1)  Figure 6.10a. Flow curves of the resin m-HDPE-1 for different die types at T=190°C  Apparent Shear Stress, W,A (MPa)  0.50 0.40  T=190°C ZN-HDPE-0  0.30 0.20 0.15 0.10 0.08 0.06 0.05 100  Capillary (D=0.79mm, L/D=16) Slit  101  102  103  .  -1  Apparent Shear Rate, A (s )  Figure 6.10b. Flow curves of the resin ZN-HDPE-0 for different die types at T=190°C.  63  First the flow curves of the slit die lie above those obtain from the other dies. Once all corrections (Bagley, Rabinowitsch and slip) are applied these flow curves are expected to superpose at least over the low shear rate range. Similar observations have been reported by Delgadillo et al [6]. Tables 6.3a and 6.3b list the critical shear rates and stresses for all types of die geometries used for resins m-HDPE-1 and ZN-HDPE-0, respectively. For the case of capillary dies both the Bagley corrected and uncorrected critical stresses are reported (typical Bagley correction results are presented in Appendix C.2). They differ by roughly 10%. Table 6.3a. Critical apparent shear rates and shear stress values or the onset of sharkskin for m-HDPE-1 resin at different temperatures and different types of die. Capillary Slit Annular Temperature -1 -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) , 	(s ) , (MPa) 3.9 0.11 3.9 0.08 140 6.1 0.11 6.1 0.10 150 13.7 0.12 13.7 0.10 160 13.7 0.12–0.11* 13.7 0.10 170 13.7 0.11–0.10 * 16 0.09 50.6 0.13 190 21.0 0.11–0.10* 21 0.09 80 0.13 210 32.2 0.12 32.2 0.09 230 * Bagley Corrected  Table 6.3b. Critical apparent shear rates and shear stress values or the onset of sharkskin for ZN-HDPE-0 resin at different temperatures and different types of die. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 3.8 0.09 2.4 0.06 140 21 0.15 – 0.14* 32.2 0.16 150 50 0.19 160 0.24 160 50 0.18 250 0.28 170 312 0.29 – 0.23* 312 0.26 190 312 0.28 501 0.27 210 501 0.28 – 0.23* 501 0.26 230 * Bagley Corrected  64  For the case of metallocene resin, there is no clear difference in its processing in capillary and slit dies, but a considerable improvement in its processability has been achieved in the annular die. Slit over capillary die also causes a small improvement in the processing of ZN-HDPE-0 compared to capillary die. The higher critical shear rates and stresses obtained in annular dies is believed to be due to the fact the extrudates at the exit possess a higher surface area to volume aspect ratio. This provides the ability of flow to have a 3D spiraling movement that provides additional degrees of freedom for the stress concentration at the die exit to be relieved as it is opposed in the case of capillary where the stress can be relieved only in one direction [6].  6.3.3 Effect of Die Entrance Angle Fig. 6.11 depicts the flow curves for resin m-HDPE-1 at relatively low temperature T=160°C in capillary dies with different entrance angles namely 15°, 45° and 90° (data for 30° and 60° were obtained but not plotted for the sake of clarity. Other data are plotted in Appendix C.1.2). As it was discussed before with reference to Fig. 6.2a, this resin exhibits FIC at high rates with the effect to be more dominant for extensional flows. The same effect can be inferred from Fig. 6.11. The effect is mitigated by decreasing the entrance angle simply due to a decrease in the average Hencky strain rate in the entry region of the die.  Apparent Shear Stress, W,A (MPa)  0.60 0.50 0.40 0.30  m-HDPE-1 Ddie = 0.51mm L/D = 20 T = 160°C  0.20 0.15  2 = 15° 2 = 45° 2 = 90°  0.10 0.08 100  101  102  103  .  -1  Apparent Shear Rate, A (s )  Figure 6.11. The effect of die entrance angle on the flow curve and processing of resin m-HDPE1 at T=160°C.  65  Table 6.4 lists the critical apparent shear rate and shear stress values for the onset of sharkskin for resin m-HDPE-1 extruded through dies having different entrance angles. It appears that use of a low contraction angle die improves processing significantly (see Table 6.4). There are a few studies on the effect of die entrance angle on the sharkskin and most of them not systematic although they appear to support the findings of the present work [3, 69]. According to Moynihan et al [55] for the case of fully developed flows (L/D > 10) [7], sharkskin is influenced by stresses encountered by the melt in the upstream area. Since a low contraction angle keeps extensional stresses to lower levels compared with those at larger contraction angles, a relative improvement is expected. Table 6.4. The critical apparent shear rate and shear stress values for the onset of sharkskin in capillary extrusion with dies having different entrance angles for resin mHDPE-1 at two different temperatures. T=190°C T=160°C Die Entrance -1 -1 Angle (°) , 	(s ) , (MPa) , (s ) , (MPa) 15 30 45 60 90  50.6 32.2 32.2 21.0 21.0  0.13 0.12 0.10 0.10 0.10  20.0 13.7 13.7 13.7 9.1  0.14 0.14 0.13 0.11 0.11  6.3.4 Effect of Processing Aid In order to overcome sharkskin and to render the processes economically feasible, polymer processing aids (PPAs) are frequently used. PPAs can eliminate the sharkskin and the stick-slip or postpone them to higher flow rates. The end result is an increase of the productivity as well as an energy cost reduction, while high product quality is maintained. The additives mostly used for polyolefin extrusion processes are fluoropolymers [92-94]. Amos et al [185] and Achilleos et al [96] have reviewed the formulation, applications and performance of various processing aids. The fluoropolymer particles in the melt flowing next to the solid wall gradually coat the die wall thus providing a layer for polyethylene to slip [4, 94, 186].  66  Figs. 6.12a and 6.12b illustrate the effect of adding 0.1 wt% of a fluoropolymer (referred as PPA) on the flow curves of m-HDPE-1 and ZN-HDPE-0, respectively. The PPA has no significant effect on the flow curve of m-HDPE-1 in terms of both decreasing the pressure drop needed for flow and the processability. On the contrary it appears to be more effective for the case of ZN-HDPE-0. Mavridis and Fronek [187] have reported that PPA efficiency increases for the broader MWD resins. While this is true for ZN-HDPE-0, it is not for the m-HDPE-1. It will be shown in the next chapter that the m-HDPE-1 slips significantly on the die wall. Hence, addition of PPA does not cause further slippage. However, in the case of ZN-HDPE-0, the slip in the presence of PPA is significant as the pure melt does not slip at all. Therefore a large reduction in the pressure drop is expected. This pressure (or shear stress) reduction causes the significant improvement in the processability of ZN-HDPE-0 in the presence of PPA.  Apparent Shear Stress, W,A (MPa)  0.50 0.40 0.30  T = 170°C L/D = 16 Ddie = 0.79mm 2 = 180°  0.20 0.15 0.10 0.08 0.06 0.05 100  m-HDPE-1 + 0.1% PPA m-HDPE-1 101  102  103  .  -1  Apparent Shear Rate, A (s )  Figure 6.12a. The effect of 0.1% PPA on the flow curve of m-HDPE-1 at T= 170oC. The open and closed symbols represent smooth and rough extrudates, respectively.  67  Apparent Shear Stress, W,A (MPa)  0.80  T = 170°C L/D = 16 Ddie = 0.79mm  0.60 0.40  2 = 180°  0.30 0.20 0.15 0.10  ZN-HDPE-0 + 0.1% PPA ZN-HDPE-0  0.08 0.06 100  101  102  103  .  104  -1  Apparent Shear Rate, A (s )  Figure 6.12b. The effect of 0.1% PPA on the flow curve of ZN-HDPE-0 at T= 170oC. The open and closed symbols represent smooth and rough extrudates, respectively.  6.4 Correlations for Critical Stresses Using a development for the critical shear stress for the onset of sharkskin proposed by Graessley [39] and data reported in the literature for m-HDPEs [109, 188-190], Allal et al [90] derived Eq. 6.5 to correlate the critical shear rate for the onset of sharkskin with molecular parameters: 1⁄ Where  0.16  .  ⁄  ⁄  is the critical shear rate for the onset of sharkskin,  .  (6.5) is an elementary characteristic  time independent of molecular weight, which includes the effects of temperature and chain microstructure [191] and  ,  and  are monomer molecular weight, average molecular  weight between entanglements and weight average molecular weight, respectively. In Eq. 6.5, except Mw, all the other parameters in the right hand side are independent of molecular weight. Therefore, by rearranging Eq. 6.5, the following correlation can be easily derived:  68  .  ∝  (6.6)  Critical Shear Rate, cr (1/s)  104 slope = -4.3 103  .  102 slope = -4.3 101  m-HDPEs ZN-HDPEs 100 100  150  200  300  400  500  Molecular Weight, Mw (kg/mole)  Figure 6.13. Correlation between critical shear rate for the onset of sharkskin and weight average molecular weights of m-HDPEs according to Eq. 6.6.  Fig. 6.13 depicts such a scaling for the cases of the m-HDPEs and ZN-HDPEs resins. Although both groups follow the same scaling, the pre-factors for the Eq. 6.5 are different. The critical shear rates for the onset of sharkskin for the ZN-HDPEs appear to be one order of magnitude higher than those for the m-HDPEs. The critical shear stress for the onset of sharkskin,  , can be correlated with the molecular  weight distribution parameters as is illustrated in Fig. 6.14. This correlation can be written as below: .  4 Where  ≡  ⁄  (6.7)  ⁄877,000  and the number 877,000 is used as a reference  value.  69  Sharkskin Critical Shear Stress, c1 (MPa)  0.20  m-HDPEs 0.15  slope = 0.25  0.10  0.05  0.00 0.0  0.2  0.4 0.5  0.6  -1  PIw ×PIz ×(Mz/877,000)  0.8  -1  Figure 6.14. Linear correlation between critical shear stress for the onset of sharkskin and molecular weight distribution parameters for m-HDPEs according to the Eq. 6.7.  The values of the critical stress for the onset of gross melt fracture for the HDPEs with PDI<19 are in the range of 0.20-0.32 MPa which is in agreement with the reported values for HDPEs [3, 8, 58, 63, 100, 112]. However, these values for the case of m-HDPEs with PDI>19 are well below the values reported in the literature. Regarding the critical shear stress,  , for the onset of stick-slip transition, based on the  cohesive failure molecular mechanism (disentanglement), it has been shown that  ∝  .  [65, 108]. This correlation appears to hold as demonstrated in Fig. 6.15 where data from the present study as well as from other studies on HDPEs are plotted. The continuous line in Fig. 6.15 is: ⁄56,000 where  .  (6.8)  is a critical reference stress equal to 0.71 MPa and 56,000 is a normalization  molecular weight in g/mol.  70  Stick-Slip Critical Stress, c2 (MPa)  1.00  0.80 0.60 0.50 0.40 0.30 slope = -0.5  0.20 0.15 0.10 100  Hatzikiriakos & Dealy [15] Drda & Wang [147] Current study 150  200  300  400  Molecular Weight, Mw (kg/mole) Figure 6.15. The relation between critical shear stress for the onset of stick-slip with weight average molecular weight for different HDPEs at T=190oC.  Moreover, as it has been reported in previous studies [3, 100, 112, 148], the critical shear rate for the onset of stick-slip instability is an increasing function of temperature, while the  is  practically a weak function of temperature and about equal to 0.3 MPa. This value is in the range of 0.22-0.50 MPa reported by Hatzikiriakos and Dealy [8] for HDPEs of various molecular characteristics. These observations are also in agreement with present experimental results. For a polymer of known molecular weight,  ,  can be calculated from Eq. 6.8. Then to  complete the prediction of the transition from weak to strong slip an equation for  is needed.  Such an expression was proposed by Hatzikiriakos [143]. It was assumed that after the sudden disentanglement at the critical shear stress of  , the stress relaxes exponentially from  to  for the reestablishment of entanglements with a characteristic time proportional to the longest relaxation time. As zero shear viscosity is also proportional to the longest relaxation time, the following expression can be readily derived: ⁄  ⁄  (6.9) 71  Where  is some reference viscosity (a constant) equal to 20 Pa.s. Fig. 6.16 plots  ⁄  as a function of zero shear viscosity for high-density polyethylenes studied by Hatzikiriakos and Dealy [15] and data from the present work. The continuous line represents Eq. 6.9.  100  log (c2/c3)  Hatzikiriakos and Dealy [15] Present Study 10-1  10-2 Slope=1 10-3 103  104  105  106  Zero Shear Viscosity, 0 (Pa.s)  Figure 6.16. Scaling of the ratio of the critical shear stresses for the onset of stick-slip transition and the critical shear stress for the onset of gross melt fracture with zero shear viscosity for high-density polyethylenes.  6.5 Summary In this chapter, the effects of processing conditions and molecular parameters on the melt fracture of broad molecular weight ZN-HDPEs and m-HDPEs have been discussed. The results showed that although these resins exhibited similar rheology both in shear and extension, they exhibited markedly different melt fracture behaviour. In contrast with previous studies, the mHDPEs which had a broader molecular weight distribution, showed markedly worse processing properties compared to their ZN counterparts.  72  According to the experimental results, the critical shear stresses for different modes of melt fracture were almost insensitive to the temperature, but the critical shear rates shift to the higher values with temperature. The type of die has also been shown to affect the processing properties of the resins, i.e. better processability was found by switching from capillary dies to slit dies and much better in the case of annular dies. Based on the findings of this work, it can be concluded that the die entrance angle has a significant effect on the processability of HDPEs particularly at low entrance angles of 15 to 30°. Regarding the effect of PPA, it was shown that it is not effective in the case of m-HDPE resins, and it works better for the case of ZN-HDPEs. This is due to the significant slip of m-HDPEs which is the subject of study of chapter 7. It was also found that the HDPEs with PDI<19 exhibit stick-slip instability, however, this is not the case for the m-HDPEs with PDI>19. A molecular mechanism was discussed that describes this behaviour quite well i.e. a criterion was derived for the occurrence of the stick-slip transition. Therefore based on these findings, resins with polydispersity in the range 9<PDI<19 exhibit superior processability compared to those with relatively narrow (PDI<9) and those of very broad MWD (PDI>19). Finally, some interesting correlations were derived between the critical shear stress and shear rate for the onsets of sharkskin, stick-slip transition and gross melt fracture and molecular parameters. These correlations have been verified by experimental findings.  73  7 WALL SLIP OF HDPEs Slip plays an important role in rheology and processing of polymers. In this chapter, the wall slip of Ziegler-Natta and metallocene HDPE resins is studied. The Cox-Merz rule for these resins is tested and possible origins of its failure are discussed. The Mooney analysis for slip is performed by using capillary data for dies having the same L/D and different diameters. The effects of molecular weight and molecular weight distribution on slip are also studied. Finally, slip models are proposed that can be used to completely predict the flow curves of HDPEs.  7.1 Cox-Merz Rule The Cox-Merz rule is an empirical relationship stating that the shear viscosity is identical with complex viscosity at corresponding shear rates and angular frequencies [192]. This rule is practically important, because one may use it to predict shear viscosity using dynamic data (easier to measure). The Cox-Merz rule has been verified experimentally for many materials including PCLs [193], PEs [120], PSs [192, 194], metallic glasses [195] and many others. On the other hand, Venkatraman et al [196] and Vega et al [48] have reported that the Cox-Merz rule failed for HDPEs and m-HDPEs respectively and the origin of this failure remained unknown. Figs. 7.1a and 7.1b plot the complex modulus | ,  ∗  | and the apparent wall shear stress,  , obtained from capillary rheometer for both ZN and m-HDPEs. For the sake of clarity  each curve has been multiplied by a factor listed in Figs. 7.1a and 7.1b. It is clear that the CoxMerz rule approximately applies for the case of ZN-HDPEs at all shear rates and frequencies. On the other hand, the Cox-Merz rule fails for all m-HDPEs. Capillary data for stresses greater than 200 s-1 where gross melt fracture effects are significant are not plotted. In addition the data have not been corrected for entrance pressure effects (Bagley correction) which are insignificant at these relatively small shear rates.  74  Complex Modulus, |G*| or Apparent Shear Stress, W,A (MPa)  101  T = 190°C 100 ZN-HDPE-6: × 10  10-1  ZN-HDPE-5: × 3  ZN-HDPE-0: × 1  10-2 10-1  100  LVE Capillary 101  102  103  . Frequecny (rad/s) or Apparent Shear Rate, A (1/s)  Figure 7.1a. Testing the applicability of the Cox-Merz rule for a series of ZN-HDPEs at 190°C.  *  Complex Modulus, |G | or Apparent Shear Stress, W,A (MPa)  105  T = 190°C  104 m-HDPE-19: × 30000  10  3  102  m-HDPE-13: × 6000 m-HDPE-8: × 1000 m-HDPE-9: × 300  101 100  m-HDPE-10: × 80 m-HDPE-11: × 20 m-HDPE-12: × 5  10  -1  10-2 10-1  m-HDPE-0: × 1  100  LVE Capillary 101  102  103  . Frequecny (rad/s) or Apparent Shear Rate, A (1/s)  Figure 7.1b. Testing the applicability of the Cox-Merz rule for a series of m-HDPEs at 190°C.  75  The failure of the Cox-Merz rule might originate from a number of reasons which are examined next. First, when capillary experiments are performed at temperatures close to melting point, stress-induced crystallization originating at the entrance to the capillary where strong extensional effects apply, might cause an increase of  compared to |  ∗  failure of Cox-Merz rule we are familiar with). A similar effect due to a strong dependency of viscosity on pressure [11]. The  | (which is opposite to the |  ∗  |	 might exist  should also be obtained at  rates below those where instabilities such as stick-slip or oscillating melt fracture occurs. to be lower than |  Viscous heating effects might cause  ∗  | [37]. For HDPEs the effects  of pressure and viscous heating on viscosity are not significant due to its low sensitivity to temperature and pressure [163]. Finally, failure of the Cox-Merz rule might be observed if the  is not corrected for possible  slip effects [149]. If this is the case, therefore, it is possible to calculate the slip velocities considering the deviation of flow curves from LVE data. This will be examined in the following sections.  7.2 Wall Slip Measurements Fig. 7.2 depicts the diameter dependency of the flow curves for the m-HDPE-1 at T=190°C. The continuous line represents the LVE results presented in chapter 5 and Appendix A.1 plotted as flow curve (|  ∗  | versus  ). The diameter dependence of the flow curves is clear. This  diameter dependence is consistent with the assumption of slip and the Mooney technique [3, 15, 36] can be used to determine the slip velocity as a function of shear stress by plotting  versus  1⁄ (details for Mooney analysis are presented in Appendix E):  ,  8 ⁄  (7.1)  76  T = 190°C L/D = 16  10-1  *  Complex Modulus, |G | or Shear Stress, W,A (MPa)  100  D=2.10 mm D=1.20 mm D=0.79 mm LVE 10-2 10-1  100  101  102  103  104  . Frequecny (rad/s) or Apparent Shear Rate, A (1/s)  Figure 7.2. The Bagley corrected flow curves of resin m-HDPE-1 at T=190°C for different diameters. The continuous line is LVE data.  Moreover, the deviation of each flow curve of Fig. 7.2 that corresponds to a different diameters from the LVE curve can also be interpreted as wall slip by using ⁄8 at a given wall shear stress value that corresponds to the frequency,  4 ⁄3  1  . If the assumption  that Cox-Merz rule failure is mainly due to slippage, the slip velocities from the two methods should be the same. Fig. 7.3 depicts slip velocity of m-HDPE-1 at T=190°C obtained from Mooney analysis along with the ones calculated from flow curves deviation of LVE data for different diameters. As it is clear from this figure, the data agree well which supports the previous assumption i.e. slippage is responsible for Cox-Merz failure.  77  Slip Velocity, us (mm/s)  102  T = 190°C L/D = 16 101  100  10-1 0.07  D=2.10 mm D=1.20 mm D=0.79 mm Mooney Analysis 0.08 0.09 0.1  0.15  0.2  Shear Stress, W (MPa)  Figure 7.3. Slip velocities of resin m-HDPE-1 at T=190°C. The open symbols are obtained from flow curves deviation of LVE data for different diameters.  7.3 Slip Velocities 7.3.1 Effect of Temperature Fig. 7.4a plots the slip velocities of resin m-HDPE-1 at different temperatures from 190°C to 230°C. The slip velocity increases with temperature consistent with the time-temperature superposition (TTS). It is noted that calculated slip velocities less than 1 mm/s have been neglected as insignificant (within experimental error). Using the values of the shift factors obtained from linear viscoelasticity, the slip velocity curves can be brought together to obtain a master slip velocity curve which implies a power law trend as,  . This is shown in  Fig. 7.4b where a reasonable superposition has been obtained. Based on the slip velocity calculations, the HDPEs can be placed into two major groups: In the first group belong to these HDPEs of typically wide molecular weight distribution (PDI>19) that slip readily and significantly at relatively low values of the wall shear stress. These resins exhibit no stick-slip transition and the Cox-Merz rule typically fails due to significant slip. The second 78  group of resins exhibit stick-slip transition. The Cox-Merz rule is approximately obeyed for low polydispersity polymers, where the stick-slip is dominant and occurs over a wide range of apparent shear rates. For the second group of resins both their flow curve and slip velocity are subdivided into two distinct branches (double valued function over certain range of rates) [179].  Slip Velocity, us (mm/s)  103  102  D = 0.79 mm L/D = 16  101  T=190°C T=210°C T=230°C  100  10-1 0.08 0.09 0.1  0.15  0.2  0.25  Shear Stress, W (MPa)  Figure 7.4a. The slip velocity of m-HDPE-1 as a function of wall shear stress at different temperatures from 190°C to 230°C.  7.3.2 Slip Velocity of Polymer Exhibiting Stick-slip (  ⁄  )  Fig. 7.5 plots the slip velocities for some of the resins which show stick-slip instability. The slip velocities for the low-flow branch show that the magnitude of slip increases with increase of polydispersity. For the high-flow rate branch, the slip velocities are essentially of the order of plug flow and they seem to be independent of the molecular characteristics of the polymer [8]. The jumps from the lower branch to the upper and from the upper to the lower occur discontinuously in the oscillatory flow region, and are shown by the two arrows for ZN-HDPE-5 in Fig. 7.5. These jumps seem to scale with PDI, although molecular weight has an additional effect. Roughly as PDI decreases, the range of shear rates over which stick-slip flow increases and this also makes the magnitudes of the jump to increase [8].  79  103  Slip Velocity, us (mm/s)  Resin,  102  PDI  ZN-HDPE-6, ZN-HDPE-0, ZN-HDPE-5,  11 14 18  101  100 0.07  0.10  0.15  0.2  0.3  0.4  0.5  Shear Stress, W (MPa)  Figure 7.5. The slip velocity of selected HDPEs which exhibit stick-slip as a function of wall shear stress at T=190oC. The arrows show transition from weak (closed symbols) to strong slip (open symbols) and vice versa for ZN-HDPE-5.  It is interesting to examine how the slip velocity scales with molecular characteristics of the HDPEs. For this purpose we will also use the data reported by Hatzikiriakos and Dealy [15] for a series of HDPEs of various molecular weights and polydispersities. Fig. 7.6 plots a master curve for the slip velocity for resins of low to moderate polydispersities ⁄  up to 19 (HDPEs which obey (  , see Fig. 6.8) including those reported by  Hatzikiriakos and Dealy [15] shown as Resins A-F. An overall good superposition is obtained. It can be observed that the slip velocity increases with increase of polydispersity and decreases with increase of molecular weight. The numerical value of 56,000 is chosen as a convenient normalization value for the molecular weight. A slope of roughly 3 can be obtained from Fig. 7.6, which results in the following scaling for the slip velocity model for these polymers (straight line in Fig. 7.6):  ,  ⁄  ⁄56,000  2⁄3 3  (7.2)  80  here  ,  9.3  10 	 .  .  is the proportionality constant of the lower slip velocity  branch for the first group of resins (A group) with PDI<19. The slip velocity for the upper branch is essentially nearly plug flow (as the flow mean velocity is about 90% of the slip velocity) slip velocity and it can be described by  ,  where  ,  5.4  10 	 .  .  is  the proportionality constant of the upper slip velocity branch for the first group of resins (A group) with PDI<19. This Equation can be transformed into a flow curve to describe the upper branch for all HDPE’s by substituting  with  ⁄8. It is noted that the upper branch of the  flow curve is of no importance as the polymers in this flow regime exhibit gross melt fracture and no process operates at such high shear rates. As also pointed out above at very high frequencies, the complex viscosities of the various resins converge, as well as their flow curves and thus their slip velocities become independent of molecular characteristics.  Slip Velocity, us (mm/s)  40 30 20 15 10 7 5 4 3  A B C D E F ZN-HDPE-0 ZN-HDPE-5 ZN-HDPE-6  2 1.5 1 0.4  0.5  0.6  1  1.5 2/3  W×PIw/(Mw/56,000)  2  (MPa)  Figure 7.6. Master curve for the slip velocity of HDPEs of the present study and those (resins A-F) studied by Hatzikiriakos and Dealy [15] at 190oC.  As discussed above the increase of slip velocity with decrease of molecular weight has been observed for several nearly monodisperse systems, namely PSs [155, 156], PBs [149, 153, 154] and recently for PLAs [157]. The observation that the slip velocity increases with polydispersity is a new, although it is in order with the effects of polydispersity on the flow curves of HDPEs. As polydispersity increases from very low values, the size of the stick-slip transition (distance 81  between the two flow curves at the transition point) decreases (see Fig. 7.6). Moreover, the range of apparent shear rates over which stick-slip occurs decreases as can be seen to a certain extend from Fig 6.6 and as explained by Myerholtz [99].  7.3.3 Slip Velocity of Polymers with Continuous Flow Curve (  ⁄  )  Fig. 7.7 plots the slip velocity of m-HDPEs at 190°C, those that do not exhibit stick-slip transition. The slip velocities were calculated based on the Mooney technique and based on the difference between the LVE master and the capillary flow curves. The slip velocities exhibit a power-law dependence on wall shear stress as also shown before [15]. On the same graph we have also plotted the slip velocities of the high flow rate branch of the HDPEs that exhibit stickslip (plug flow slip velocities). It can be seen, the slip velocities of the m-HDPEs tend to attain almost plug flow at high enough levels of shear stress and that at these levels slip velocity becomes independent of molecular parameters.  Slip Velocity, us (mm/s)  103  102  101  Resin, Mw (kg/mol), PDI m-HDPE-8, 272, 25 m-HDPE-9, 270, 21 m-HDPE-10, 266, 21 m-HDPE-11, 249, 19 m-HDPE-12, 230, 20 m-HDPE-1, 223, 42 m-HDPE-19, 213, 12 m-HDPE-13, 211, 15 ZN-HDPE-0, 328, 14 ZN-HDPE-5, 329, 18 ZN-HDPE-6, 296, 11  100  T = 190°C 10-1 0.01  0.02 0.03  0.05  0.10  0.2  0.3  0.5  Shear Stress, W (MPa) Figure 7.7. The slip velocities versus wall shear stress for the all the HDPE resins at T=190°C. The open symbols correspond to the data after stick-slip phenomenon.  82  The slip data for the m-HDPE resins can be brought into a master curve by normalizing the vertical and horizontal axis by appropriate molecular parameters. The superposition is satisfactory. As seen in Fig. 7.8 the slip velocity increases with molecular weight also found for the previous set of resins (stick-slip) although this time the dependence is much stronger. Surprisingly, the polydispersity for these resins have an effect opposite to what have been seen for the stick-slip HDPEs. The slip velocity decreases with increase of PDI. Apparently chains with molecular sizes far apart (high polydispersity) interact in a different manner as far as slip velocity concerns compared to molecular sizes that are much closer (low polydispersity). It appears that for a given molecular weight, there is a distribution that maximizes the slip velocity (around the point where the stick-slip transition occurs). A slope of roughly four can be obtained from Fig. 7.8, which results the following Equation for the slip velocity, ⁄ Where  4.47  10 	 .  .  ⁄56,000  .  4  (7.3)  is the proportionality constant of the slip velocity for  the B group of resins with PDI>19. The power dependence of the slip velocity on wall shear stress is now 4 compared to 3 found before. Powers between 3 to 6 for the slip velocities of polyethylenes have been reported by various authors summarized by Hatzikiriakos [136]. The power of 10 for the molecular weight might seem unreasonable. For these wide molecular weight distribution resins, higher moments of molecular weight should play a role as these distributions are not typically symmetric i.e. normal distributions. However, the objective here was to find a scaling that involves the minimum number of molecular parameters. Another possible reason is that all these resins have molecular weights in the short range of 220,000 to 270,000 and possibly this causes the high exponent. At any rate, this study appears to be the first to consider the combined effects of molecular weight and polydispersity on the slip velocity for several HDPEs of practical importance, namely HDPEs that exhibit stick –slip transition (low to moderate molecular weight distribution) and HDPEs which do not exhibit such transitions (wide molecular weight distribution and wide molecular weight distributions). Certainly more studies are welcome in the area as was recently stressed in a review article by Hatzikiriakos [136].  83  106  us×PIw4 (m/s)  105  T = 190°C  104  103  m-HDPE-8 m-HDPE-9 m-HDPE-10 m-HDPE-11 m-HDPE-12 m-HDPE-1  102  101 0.02  0.04  0.06  0.1  W/(Mw/223,000)  2.5  0.2  0.4  (MPa)  Figure 7.8. Master curves for the slip velocities of m-HDPEs that do not exhibit stick-slip transition at T=190oC.  7.4 Construction the Flow curves of HDPEs Based on developments through this study, the flow curves of HDPEs can be constructed. An example is shown in Fig. 7.9 for ZN-HDPE-6 that exhibits stick-slip transition. Starting from its LVE behaviour, its low flow rate branch for a capillary die of a certain diameter, D, can be 4 ⁄3  predicted by utilizing Eq. 7.2 for the slip velocity coupled with ⁄8. This part of the flow curve is stopped at The value of  , which value can be determined by Eq. 6.8.  can be calculated from Eq. 6.9 using the value of  flow curve of the polymer can be calculated from  1  ,  . The approximate upper  by replacing  with  ⁄8.  As it can be seen from Fig. 7.9, the overall representation of the experimental flow curve is good. Similar results were obtained for all HDPEs studied in this work (more constructed flow curves can be found in Appendix F).  84  Shear Stress, W (MPa)  0.50  0.30  ZN-HDPE-6 T=190°C L/D = 16 Ddie = 0.79 mm  Calculated C2  0.20 0.15  Calculated C3  0.10  Experimental Calculated LVE  0.07 0.05 100  101  102  . -1 Apparent Shear Rate, A (s )  103  104  Figure 7.9. Constructing the flow curve of ZN-HDPE-6 starting from LVE data.  7.5 Summary Wall slip of ZN-HDPEs and m-HDPEs was the subject of this chapter. Firstly, the validity of Cox-Merz rule for available resins was assessed and it failed for m-HDPE resins with PDI>19. To check the idea that slip is responsible for such a failure, slip velocities were calculated by means of Mooney analysis as well as the flow curves deviation from LVE data. The agreement of the calculated slip velocities from both methods supports the mentioned hypothesis. The effects of molecular weight and molecular weight distribution were also studied. It was shown that for the resins which show stick-slip instability, there are two distinct slip regimes i.e. the weak and strong slip regimes attributed to the low and high branches of flow curve, respectively. The slip velocity in the strong mode is a plug/semi-plug flow and almost independent of molecular characteristics. However, the slip velocity in the weak regime was found to increase with decrease of molecular weight and polydispersity. Moreover, the gap size between these two regimes was found to increase with decrease of polydispersity. 85  Contrary to the ZN-HDPEs, the m-HDPEs with continuous flow curves exhibit a continuous slip velocity as a function of wall shear stress. Their slip velocity was also found to increase with decrease of molecular weight, but contrary to the stick-slip resins, increases with the polydispersity index. Finally, starting from LVE results and based on the derived correlations, the flow curves of some stick-slip resins were constructed which were fairly in agreement with experimental results.  86  8 CONCLUSIONS AND CONTRIBUTIONS TO KNOWLEDGE 8.1 Conclusions The rheological properties of ZN and m-HDPEs of broad molecular weight distribution have been investigated. The relation between the zero shear viscosity and the weight average molecular weight showed a power exponent which was initially found to be higher than the experimentally accepted value of 3.6 reported for linear PEs in the literature. However, taking into account the effect of broad molecular weight distribution and applying appropriate corrections the universal exponent was recovered. The steady state compliance showed a powerlaw dependence on M ⁄M M  ⁄M  as expected. It was also a power-law function of M  ⁄M  with power of 2.2, as previously found based on molecular theories. The crossover  modulus has been found to exhibit a similar dependence on M ⁄M . The processing behaviors of these resins (two different series of Ziegler Natta and metallocene) have also been studied. Although these resins exhibited similar rheology both in shear and extension, they exhibited markedly different melt fracture behaviour. The ZN-HDPEs were found to fracture at significantly higher critical shear stresses exhibiting superior processability compared to the m-HDPEs. Based on the extensional results and slip analysis, it was found that the origin of these differences lie in the difference in the shape of broadness of MWD of these two polyethylenes. The type of die has also been shown to affect the processing properties of the resins. For example it was found that the critical stresses for the onset of sharkskin are higher in the case of a slit die and much higher in the case of an annular die, when compared with the values obtained in capillary extrusion. The enhanced processability in the case of an annular die is due to the high surface area to volume aspect ratio which provides degrees of freedom for the stress concentration to be relieved easier at the die exit. It has also been shown that the die entrance angle had a significant effect on the processability of high density polyethylenes particularly at low entrance angles of 15 to 30°. The validity of Cox-Merz rule was also tested for all HDPE resins. It was shown that this rule is approximately valid for ZN-HDPEs and apparently fails for m-HDPEs. However, wall slip measurements have shown that slip effects are present in m-HDPEs and this is the reason for the apparent failure. Once capillary data were 87  corrected for slip effects, the Cox-Merz rule was shown to be valid for the m-HDPEs as well. It was also shown that the PPA is not effective in the case of the m-HDPEs due to significant slippage of these polymers in their absence. Therefore, addition of PPA cannot further increase its slip which can cause a drop in the shear stress which in turn can improve their processability. On the other hand, the ZN-HDPEs did not slip in the absence of PPA or slip very little. Then PPA itself can cause significant slippage and drop in the shear stress that is the main reason for its effectiveness. Regarding the oscillation defect, the results showed that the HDPEs with PDI<19 exhibit a stick-slip transition and as a result their flow curve consists of two distinct branches, namely a low flow rate and a high flow rate. These resins exhibited significant slip in the high flow-rate branch where their slip approaches almost plug flow. The HDPEs with PDI>19 exhibited no stick-slip transition and a molecular mechanism was discussed to this effect that describes this behaviour quite well i.e. a criterion was derived for the occurrence of the stick-slip transition. Although the m-HDPEs had a broader molecular weight distribution compared to their ZN counterparts, their processing properties were much worse. Therefore based on these findings, resins with polydispersity in the range 9<PDI<19 exhibited superior processability compared to those with relatively narrow (PDI<9) and those of very broad MWD (PDI>19). Moreover, some interesting correlations were derived between the critical shear stress and shear rate for the onsets of sharkskin, stick-slip transition and gross melt fracture and molecular parameters. These correlations have been verified by experimental findings and have been proven to hold true. The slip velocities of all available HDPEs were studied in an attempt to elucidate the effects of molecular weight and its distribution on their slip behaviour. Two distinctly different flow behaviors were observed. For HDPEs obeying the simple rule (M ⁄M  λ  PDI) which are  essentially HDPEs of high molecular weight and low polydispersity, stick-slip transition occurs that subdivides both the flow curve and the slip velocity as a function of wall shear stress into two branches (essentially double-valued functions). For such HDPEs the slip velocity was found to increase with decrease of molecular weight and polydispersity. Moreover, the jump in terms of slip velocity was found to increase with decrease of polydispersity. The higher slip velocities obtained in the upper branch at high shear stress values were essentially plug flow and roughly independent of the molecular parameters of HDPEs. On the other hand, HDPEs obeying the 88  simple rule (M ⁄M  λ  PDI) essentially HDPEs of medium to high molecular weight and  high polydispersity exhibited both a continuous flow curve and a continuous slip velocity as a function of the wall shear stress. These resins did not exhibit stick-slip transition. Their slip velocity was also found to increase with the decrease of molecular weight, although the effect of PDI was surprisingly found to be opposite to that PDI was found to have for the HDPEs exhibiting stick-slip transition. It is expected that as molecular weight decreases to sizes comparable to these of surface asperities, the slip velocity should start declining to zero. More studies are needed in this area to fully explore and understand the effects of molecular weight parameters such as molecular weight and its distribution on the slip behaviors, particularly for multimodal distribution polymers.  8.2 Contributions to Knowledge The present work has yielded the following contribution to knowledge: 1- From the linear viscoelastic measurements, for m-HDPEs with very broad molecular weight distribution, the zero shear viscosity is not only a power-law function of molecular weight, but also molecular weight distribution (M ⁄M ). Based on our finding in this study, the universal exponent of 3.6 for the Mw dependence was confirmed, along with proposed linear dependence of viscosity on M ⁄M . The new pre-factors for this correlation at different temperatures also have been found. Moreover, the available theoretical and experimental correlations of steady state compliance with different measures of molecular weight distribution were verified. 2- According to the results of this study, the processing properties of these two types of resins are completely different, while their rheological characteristics are almost similar. It was concluded that the shape of broadness of MWD is responsible for such an observation. 3- This study adds new contribution to the few available data in the literature regarding the effect of die type on the processing characteristics. The findings support previous observations that processability can be enhanced by using slit dies compared to capillary ones and can be enhanced much more for the case of annular dies. It was also shown that the processability would be better for low entrance angles capillary dies.  New 89  correlations between critical shear rates and shear stresses with molecular characteristics and rheological properties for the onset of instabilities were also proposed. 4- Although the Cox-Merz rule is approximately valid for HDPEs of narrow to moderate MWD HDPEs, it fails for m-HDPEs with a wide MWD. It has been shown in this work that slip effects are present in m-HDPEs and this is the reason for the failure. Once capillary data are corrected for slip effects, the Cox-Merz rule is shown to be valid for the m-HDPEs as well. So it is found in this study that the failure of the empirical Cox-Merz rule is an indication of considerable slip. Therefore, it is proposed that slip velocity can be calculated from deviation from LVE results (Cox-Merz rule). 5- Although there is considerable number of reports on the effect of Mw on the stick-slip defect, there are a few on the effect of MWD. It was shown that by increasing the MWD, the stick-slip transition decreases in size until disappearing for HDPEs with a polydispersity (PDI) roughly greater than 19. The results of this project have shown to support the criterion for the occurrence of stick-slip transition developed by Allal and Vergnes [16] based on the hypothesis of disentanglement between tethered chains on the surface and molecules in the bulk. 6- It has been shown convincingly in this work that the slip velocity increases with increase of temperature and decrease in Mw. It has also been shown that the slip velocity decreases with increase of PDI for resins with PDI>19 (no stick-slip transition) and decrease of PDI for resins with PDI<19 (stick-slip). Possible reason for this change might be interaction between the long and short changes in the broad MWD HDPEs (possible phase separation phenomena).  90  8.3 Recommendations for Future Work There are several aspects of this work that could be investigated in the future. Some suggestions and recommendations on future studies are as follows: 1- In capillary rheometry or in many polymer processes polymer melts flow through converging zones. In these zones the melts experience considerably high tensile stresses which may possibly lead to flow induced crystallization (FIC). 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Master Curves  (Pa.s)  G'  T = 190°C ZN-HDPE-0  105  106  G" 105  104 104 103  101 10-3  |*|  LVE Stress Relaxation  102  10-2  10-1  100  101  102  103  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.1. Master curves of storage and loss moduli for resin ZN-HDPE-0 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  (Pa.s)  T = 190°C ZN-HDPE-5  105  106  G' G"  105  104 104 103  10  101 10-3  |*|  LVE Stress Relaxation  2  10  -2  10  -1  10  0  10  1  10  2  10  3  103  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.2. Master curves of storage and loss moduli for resin ZN-HDPE-5 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  107  T = 190°C ZN-HDPE-10  106  (Pa.s)  106 G' 105  G"  105 104  104  103 *  102 101 10-3  10  -2  10  -1  10  0  10  1  103  | |  LVE Stress Relaxation  10  2  10  3  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  107  102  Frequency, aT. (rad/s) Figure A.3. Master curves of storage and loss moduli for resin ZN-HDPE-10 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  (Pa.s)  G'  T = 190°C ZN-HDPE-11  105  106  G" 105  104 104 103  101 10-3  103  LVE Stress Relaxation  102  10-2  10-1  100  101  *  | |  102  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.4. Master curves of storage and loss moduli for resin ZN-HDPE-11 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  108  (Pa.s)  T = 190°C ZN-HDPE-12  105  106  G' G"  105  104 104 103  10  101 10-4  103  LVE Stress Relaxation  2  10  -3  10  -2  10  -1  10  0  10  *  | |  1  10  2  10  3  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.5. Master curves of storage and loss moduli for resin ZN-HDPE-12 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  (Pa.s)  G'  T = 190°C ZN-HDPE-13  105  106  G" 105  104 104 103  101 10-3  103  LVE Stress Relaxation  102  10-2  10-1  100  101  *  | |  102  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.6. Master curves of storage and loss moduli for resin ZN-HDPE-13 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  109  (Pa.s)  10  T = 190°C ZN-HDPE-14  5  106  G' G"  105 104 103  104  102 101 100 10-4  103  LVE Stress Relaxation  10  -3  10  -2  10  -1  10  0  10  *  | |  1  10  2  10  3  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.7. Master curves of storage and loss moduli for resin ZN-HDPE-14 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. 105  T = 190°C ZN-HDPE-15  (Pa.s)  105  G' G" 104  10  4  103 103 LVE Stress Relaxation  102 101 10-3  10-2  10-1  100  101  *  | |  102  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.8. Master curves of storage and loss moduli for resin ZN-HDPE-15 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  110  G' G"  T = 190°C ZN-HDPE-16  105  (Pa.s)  105  104 10  4  103 103 LVE Stress Relaxation  102 101 10-3  10  -2  10  -1  10  0  10  1  *  | |  10  2  10  3  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.9. Master curves of storage and loss moduli for resin ZN-HDPE-16 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation. 106 G'  105  105  G"  104  104  103  102 10-2  103  LVE Stress Relaxation  10-1  100  101  *  | |  102  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  T = 190°C m-HDPE-1  (Pa.s)  106  102  Frequency, aT. (rad/s) Figure A.10. Master curves of storage and loss moduli for resin m-HDPE-1 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  111  T = 190°C m-HDPE-9  105  (Pa.s)  106 G' G"  105  104 104 103  10  101 10-4  103  LVE Stress Relaxation  2  10  -3  10  -2  10  -1  10  0  10  *  | |  1  10  2  10  3  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.11. Master curves of storage and loss moduli for resin m-HDPE-9 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  T = 190°C m-HDPE-10  105  (Pa.s)  106 G' G"  105  104 104 103  101 10-3  103  LVE Stress Relaxation  102  10-2  10-1  100  101  *  | |  102  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.12. Master curves of storage and loss moduli for resin m-HDPE-10 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  112  T = 190°C m-HDPE-11  105  (Pa.s)  106 G' G" 105  104 103  104  102 101 100 10-4  103  LVE Stress Relaxation  10  -3  10  -2  10  -1  10  0  10  *  | |  1  10  2  10  3  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.13. Master curves of storage and loss moduli for resin m-HDPE-11 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  T = 190°C m-HDPE-12  5  (Pa.s)  10  106 G' G" 105  104 103  104  102 101 100 10-4  103  LVE Stress Relaxation  10-3  10-2  10-1  100  101  *  | |  102  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.14. Master curves of storage and loss moduli for resin m-HDPE-12 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  113  (Pa.s)  T = 190°C m-HDPE-13  105  106  G' G"  105  104 104 103  10  101 10-3  103  LVE Stress Relaxation  2  10  -2  10  -1  10  0  10  1  *  | |  10  2  10  3  10  4  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  106  102  Frequency, aT. (rad/s) Figure A.15. Master curves of storage and loss moduli for resin m-HDPE-13 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  T = 190°C m-HDPE-19  6  (Pa.s)  10  106 G' 105  G"  105 104  104  103 102 101 10-3  10-2  10-1  100  101  103  *  | |  LVE Stress Relaxation  102  103  104  Complex Viscosity, |*|/aT  Dynamic Moduli, G', G"  (Pa)  107  102  Frequency, aT. (rad/s) Figure A.16. Master curves of storage and loss moduli for resin m-HDPE-19 at Tref = 190°C. Data were obtained from frequency sweep linear viscoelastic measurements (LVE) and stress relaxation.  114  A.2. Shift Factors Table A.1. Horizontal shift factors and activation energies for constructing master curve. Horizontal Shift Factor, aT Ea Resin T=150°C T=170°C T=190°C T=210°C T=230°C (kcal/mole) ZN-HDPE-0  2.17  1.40  1.00  0.71  0.51  7.56  ZN-HDPE-5  2.08  1.50  1.00  0.69  0.52  7.46  ZN-HDPE-6  2.27  1.59  1.00  0.75  0.59  7.29  ZN-HDPE-10  2.06  1.40  1.00  0.74  -  6.93  ZN-HDPE-11  2.07  1.49  1.00  0.74  -  7.04  ZN-HDPE-12  2.05  1.40  1.00  0.74  -  6.89  ZN-HDPE-13  2.12  1.45  1.00  0.75  -  7.07  ZN-HDPE-14  2.13  1.43  1.00  0.69  -  7.56  ZN-HDPE-15  2.07  1.36  1.00  0.75  -  6.84  ZN-HDPE-16  1.94  1.34  1.00  0.71  -  6.73  m-HDPE-1  2.06  1.39  1.00  0.73  0.56  6.90  m-HDPE-8  2.13  1.43  1.00  0.77  -  6.90  m-HDPE-9  2.07  1.45  1.00  0.74  -  7.03  m-HDPE-10  2.01  1.38  1.00  0.92  -  5.49  m-HDPE-11  2.03  1.37  1.00  0.85  -  5.98  m-HDPE-12  2.0  1.37  1.00  0.87  -  5.72  m-HDPE-13  1.91  1.35  1.00  0.73  0.58  6.34  m-HDPE-19  2.03  1.37  1.00  0.75  0.57  6.75  115  A.3. Parsimonious Relaxation Spectra Table A.2. Parsimonious Relaxation Spectra for all the resins. ZN-HDPE-0 ZN-HDPE-5 ZN-HDPE-6 Mode Gi (Pa) Gi (Pa) Gi (Pa) i (s) i (s) i (s)  ZN-HDPE-10 Gi (Pa)  i (s)  1  0.000679  236375.7  0.000745  256309.2  0.000657  295177.1  0.000711  342347.9  2  0.004687  119993.9  0.005228  133955.3  0.004628  155683.5  0.005364  217558.1  3  0.025803  83941.61  0.029119  92935.65  0.025754  102084.6  0.033395  159770.3  4  0.136957  51196.86  0.153983  55001.55  0.138294  54071.65  0.200159  88998.29  5  0.717454  24323.07  0.800486  24159.86  0.745673  21322.23  1.193298  35367.43  6  3.751479  8582.533  4.104464  7480.318  4.090299  6308.725  7.56262  8931.907  7  21.33558  2623.237  23.37502  1879.899  23.7603  1740.091  50.93401  1527.885  8  146.5475  424.0777  181.9244  320.7249  198.4493  370.9492  387.3278  138.0883  Mode  ZN-HDPE-11 Gi (Pa) i (s)  ZN-HDPE-12 Gi (Pa) i (s)  ZN-HDPE-13 Gi (Pa) i (s)  ZN-HDPE-14 Gi (Pa) i (s)  1  0.000789  310768.1  0.000847  309566.8  0.000744  310067.1  0.000629  262915.7  2  0.005772  178274.4  0.006807  166115.8  0.00557  158964.5  0.004589  137940.8  3  0.034286  124485.1  0.044399  106083.6  0.033504  103555.9  0.027792  90467.74  4  0.196429  68475.55  0.279046  51716.46  0.194462  54270.72  0.162565  48126.22  5  1.111369  27392.82  1.72403  17669.13  1.111972  20932.34  0.928967  18879.81  6  6.225188  7167.717  11.20091  4020.929  6.446502  5295.38  5.237733  5067.522  7  33.42536  1593.01  77.03012  591.025  34.14986  1067.25  38.25527  1008.886  8  231.4075  174.7466  593.3561  68.62508  257.8695  97.41813  463.1736  56.2897  Mode  ZN-HDPE-15 Gi (Pa)  i (s)  ZN-HDPE-16 Gi (Pa)  i (s)  1  0.000538  289612.6  0.00053  263501.3  2  0.003722  132123  0.003711  102779.5  3  0.020789  82917.65  0.020594  57984.49  4  0.112638  44992.76  0.1131  28470.74  5  0.598268  19071.13  0.605889  11209.88  6  3.111558  5790.563  3.149166  3163.732  7  16.92205  1402.951  16.30383  759.652  8  107.9036  165.1019  98.58297  102.9341  116  Mode  m-HDPE-1 Gi (Pa)  i (s)  m-HDPE-8 Gi (Pa)  i (s)  m-HDPE-9 Gi (Pa)  i (s)  m-HDPE-10 Gi (Pa)  i (s)  1  0.000652  138464.3  0.000781  173981.6  0.000915  157658.3  0.00083  161704.4  2  0.004062  76619.69  0.007566  108566.6  0.009964  82743.72  0.008412  74074.87  3  0.01966  80614.43  0.051275  124598.6  0.075295  88850.09  0.058835  78231.77  4  0.088475  76837.51  0.308882  106458.4  0.492641  69868.74  0.362638  67498.28  5  0.383417  52753.99  1.727492  51695.54  2.949796  29356.31  2.074226  34189.03  6  1.611592  22962.76  10.10858  11670.6  17.84629  5248.18  11.62367  8779.244  7  6.46897  6004.133  81.40472  703.4459  162.0862  289.8034  74.57821  857.4981  8  34.81225  1088.862  828.6629  37.62244  1597.216  26.00886  691.4772  54.14863  Mode  m-HDPE-111 Gi (Pa) i (s)  m-HDPE-12 Gi (Pa) i (s)  m-HDPE-13 Gi (Pa) i (s)  m-HDPE-19 Gi (Pa) i (s)  1  0.000692  151086.4  0.000877  138000.9  0.001216  190695.1  0.000632  321847.3  2  0.006232  64215.47  0.008623  50780.37  0.009128  151029.6  0.004626  283051  3  0.0413  65509.82  0.058747  49289.99  0.050401  137146.6  0.024851  247904.1  4  0.243975  60145.76  0.344953  43762.19  0.260753  73058.5  0.12211  133085.7  5  1.3339  35068.01  1.856976  23966.08  1.352669  20168.35  0.595997  38574.47  6  6.957693  10394.29  9.657726  6827.35  7.853792  3055.936  3.098442  6523.295  7  36.4237  1575.595  58.03224  831.1806  58.42575  354.1737  20.00976  885.5724  8  447.1503  62.85716  636.6833  51.33302  504.6236  71.9296  210.0838  155.0879  117  APPENDIX B – EXTENSIONAL RHEOLOGY  Tensile stress growth coefficient (Pa.s)  107  T = 150°C ZN-HDPE-5 106  105  104 10-2  LVE 3     10 s 1    0.1 s    1 s 1  10-1  100  101    0.01 s 1 1  102  103  104  Time (s) Figure B.1. The tensile stress growth coefficient of resin ZN-HDPE-5 at several Hencky strain rates, at T=150°C. The line labeled as LVE 3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1.  Tensile stress growth coefficient (Pa.s)  107  T = 150°C ZN-HDPE-6 106  LVE 3     0.01 s 1  5  10    1 s 1  104    0.1 s 1   10 s  103 10-2  10-1  1  100  101  102  103  104  Time (s) Figure B.2. The tensile stress growth coefficient of resin ZN-HDPE-6 at several Hencky strain rates, at T=150°C. The line labeled as LVE 3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1.  118  Tensile stress growth coefficient (Pa.s)  107    0.01 s 1 LVE 3  106    10 s 1    1 s 1    0.1 s 1  105  T = 150°C m-HDPE-8 104 10-2  10-1  100  101  102  103  104  Time (s) Figure B.3. The tensile stress growth coefficient of resin m-HDPE-8 at several Hencky strain rates, at T=150°C. The line labeled as LVE 3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1.  Tensile stress growth coefficient (Pa.s)  107  T = 150°C m-HDPE-9    0.01 s 1 LVE 3   106    10 s 1 105  104 10-2    1 s 1  10-1  100    0.1 s 1  101  102  103  104  Time (s) Figure B.4. The tensile stress growth coefficient of resin m-HDPE-9 at several Hencky strain rates, at T=150°C. The line labeled as LVE 3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1.  119  Tensile stress growth coefficient (Pa.s)  107    0.01 s 1 106  LVE 3     0.1 s    10 s 1  1    1 s 1  105  T = 150°C m-HDPE-13 104 10-2  10-1  100  101  102  103  Time (s) Figure B.5. The tensile stress growth coefficient of resin m-HDPE-13 at several Hencky strain rates, at T=150°C. The line labeled as LVE 3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1.  Tensile stress growth coefficient (Pa.s)  107    0.01 s 1    0.1 s 1  106  LVE 3     1 s 1   10 s 1 105  T = 150°C m-HDPE-19 104 10-2  10-1  100  101  102  103  104  Time (s) Figure B.6. The tensile stress growth coefficient of resin m-HDPE-19 at several Hencky strain rates, at T=150°C. The line labeled as LVE 3 line has been calculated from fitting linear viscoelastic measurements with a parsimonious relaxation spectrum and use of Eq. 6.1.  120  APPENDIX C – FLOW CURVES C.1. Flow Curves C.1.1. Effect of Temperature  Apparent Shear Stress, W,A (MPa)  1.00 0.80 0.60 0.40 0.30  ZN-HDPE-5 L/D = 16 Ddie = 0.79 mm 2 = 180°  0.20 0.15 T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 0.04 100  101  102  .  103  104  -1  Apparent Shear Rate, A (s )  Figure C.1a. The flow curves of resin ZN-HDPE-5 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  1.00 0.80 0.60 0.40  ZN-HDPE-5 L/H = 16 H = 0.47 mm W = 2.54 mm  0.30 0.20 0.15 T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 0.04 100  101  102  .  103  104  Apparent Shear Rate, A (s-1)  Figure C.1b. The flow curves of resin ZN-HDPE-5 in slit extrusion at different temperatures.  121  Apparent Shear Stress, W,A (MPa)  1.00 0.80 0.60 0.40 0.30  ZN-HDPE-6 L/D = 16 Ddie = 0.79 mm 2 = 180°  0.20 0.15 T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 0.04 100  101  102  .  103  104  Apparent Shear Rate, A (s-1)  Figure C.2a. The flow curves of resin ZN-HDPE-6 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.60 0.40 0.30 0.20  ZN-HDPE-6 L/H = 16 H = 0.47 mm W = 2.54 mm  0.15 0.10 0.08  T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.06 0.04 0.03 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.2b. The flow curves of resin ZN-HDPE-6 in slit extrusion at different temperatures.  122  Apparent Shear Stress, W,A (MPa)  0.60 0.40 0.30  m-HDPE-9 L/D = 16 Ddie = 0.79 mm 2 = 180°  0.20 0.15 T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 0.04  100  101  102  103  .  Apparent Shear Rate, A (s-1)  Figure C.3a. The flow curves of resin m-HDPE-9 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.40 0.30 0.20  m-HDPE-9 L/H = 16 H = 0.47 mm W = 2.54 mm  0.15 0.10 0.08  T = 170°C T = 190°C T = 210°C T = 230°C  0.06 0.04 0.03  100  101  102  .  103 -1  Apparent Shear Rate, A (s )  Figure C.3b. The flow curves of resin m-HDPE-9 in slit extrusion at different temperatures.  123  Apparent Shear Stress, W,A (MPa)  0.60 0.40  m-HDPE-10 L/D = 16 Ddie = 0.79 mm  0.30  2 = 180°  0.20 0.15  T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.4a. The flow curves of resin m-HDPE-10 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.40 0.30 0.20  m-HDPE-10 L/H = 16 H = 0.47 mm W = 2.54 mm  0.15 0.10 0.08  T = 170°C T = 190°C T = 210°C T = 230°C  0.06 0.04 0.03  100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.4b. The flow curves of resin m-HDPE-10 in slit extrusion at different temperatures.  124  Apparent Shear Stress, W,A (MPa)  0.60 0.40 0.30  m-HDPE-11 L/D = 16 Ddie = 0.79 mm 2 = 180°  0.20 0.15 T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 0.04 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.5a. The flow curves of resin m-HDPE-11 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.40 0.30 0.20  m-HDPE-11 L/H = 16 H = 0.47 mm W = 2.54 mm  0.15 0.10 0.08  T = 170°C T = 190°C T = 210°C T = 230°C  0.06 0.04 0.03  100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.5b. The flow curves of resin m-HDPE-11 in slit extrusion at different temperatures.  125  Apparent Shear Stress, W,A (MPa)  0.60 0.40 0.30  m-HDPE-12 L/D = 16 Ddie = 0.79 mm 2 = 180°  0.20 0.15 T = 150°C T = 170°C T = 190°C T = 210°C T = 230°C  0.10 0.08 0.06 0.04 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.6a. The flow curves of resin m-HDPE-12 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.30 0.20 0.15  m-HDPE-12 L/H = 16 H = 0.47 mm W = 2.54 mm  0.10 0.08 T = 170°C T = 190°C T = 210°C T = 230°C  0.06 0.04 0.03  100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.6b. The flow curves of resin m-HDPE-12 in slit extrusion at different temperatures.  126  Apparent Shear Stress, W,A (MPa)  0.60 0.40  m-HDPE-13 L/D = 16 Ddie = 0.79 mm  0.30  2 = 180°  0.20 0.15  T = 170°C T = 190°C T = 210°C  0.10 0.08 0.06  100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.7a. The flow curves of resin m-HDPE-13 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.60 0.40 0.30  m-HDPE-13 L/H = 16 H = 0.47 mm W = 2.54 mm  0.20 0.15  T = 170°C T = 190°C T = 210°C  0.10 0.08 0.06 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.7b. The flow curves of resin m-HDPE-13 in slit extrusion at different temperatures.  127  Apparent Shear Stress, W,A (MPa)  0.60 0.40  m-HDPE-19 L/D = 16 Ddie = 0.79 mm  0.30  2 = 180°  0.20 0.15  T = 170°C T = 190°C T = 210°C  0.10 0.08 0.06 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.8a. The flow curves of resin m-HDPE-19 in capillary extrusion at different temperatures.  Apparent Shear Stress, W,A (MPa)  0.60  0.40 0.30  m-HDPE-19 L/H = 16 H = 0.47 mm W = 2.54 mm  0.20 T = 170°C T = 190°C T = 210°C  0.15  0.10 0.08 100  101  102  .  103  Apparent Shear Rate, A (s-1)  Figure C.8b. The flow curves of resin m-HDPE-19 in slit extrusion at different temperatures.  128  C.1.2. Effect of Die Entrance Angle  Apparent Shear Stress, W,A (MPa)  0.40 0.30 0.20  m-HDPE-1 Ddie = 0.51mm L/D = 20 T = 190°C  0.15  2 = 15° 2 = 30° 2 = 45° 2a = 60° 2a = 90° 2 = 180°  0.10 0.08 0.07 0.06 0.05 0.04 100  101  102  103  .  -1  Apparent Shear Rate, A (s )  Figure C.9. The effect of die entrance angle on the flow curve and processing of resin m-HDPE-1 at T=190°C  129  C.2. Bagley Correction As it has been noted in section 2.3.2.3, in order to perform Bagley correction, one should collect data for 3 different dies with the same diameter, but different length to diameter ratios. Then by plotting pressure drop values versus L/D at a given shear rate and extrapolate the data to the zero L/D, one can calculate the Bagley end pressure (Eq. 2.8). This procedure is shown below for one of the HDPE resins as an example.  Apparent Wall Shear Stress, W,A (MPa)  0.60 0.40 0.30  m-HDPE-12 T = 190°C D = 0.79 mm 2 = 180°  0.20 0.15 0.10 0.08  L/D=5 L/D=16 L/D=33  0.06 0.04  101  102  103 .  -1  Apparent Shear Rate, A (s )  Figure C.10a. The apparent flow curves of resin m-HDPE-12 at 190oC for various L/D ratios.  130  35  .  A (1/s)  Pressure Drop, p (MPa)  30  5 11 26 64 160 390 1000  25 20  m-HDPE-12 T = 190°C D = 0.79 mm 2 = 180°  15 10 5 0  0  10 20 30 40 Die Length to Diameter Ratioe, L/D Figure C.10b. The pressure drop for the capillary extrusion of the resin m-HDPE-12 at 190oC as a function of L/D for different values of the apparent shear rate (Bagley plot).  Wall Shear Stress, W,A (MPa)  0.20 0.15  0.10  m-HDPE-12 T = 190°C D = 0.79 mm 2 = 180°  0.08 0.06 L/D = 5 L/D = 16 L/D = 33  0.04 0.03  101  102  103 .  -1  Apparent Shear Rate, A (s )  Figure C.10c. The flow curves of resin m-HDPE-12 at 190oC as a function of the apparent shear rate for various L/D ratios corrected for the entrance effects. The data superpose well showing that the pressure effect of viscosity is negligible as expected for HDPE melts.  131  APPENDIX D – PROCESSING MAPS  Capillary  ZN-HDPE-5  Slit  Temperature (°C)  230  210  190  170  (a)  150 10  100  1000  (b) 10000  10  .  100  Smooth Sharkskin Stick-Slip Gross  1000  Apparent Shear Rate, A (s-1)  Figure D.1. Processability map of resin ZN-HDPE-5 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.1. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin ZN-HDPE-5 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 32.2 0.19 32.2 0.17 150 312 0.34 200 0.26 170 312 0.31 312 0.26 190 501 0.34 442 0.28 210 630 0.33 610 0.31 230  132  Capillary  Slit  ZN-HDPE-6  Temperature (°C)  230  210  190  170  (a)  150 10  100  1000  (b) 10000  10  .  100  Smooth Sharkskin Stick-Slip Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.2. Processability map of resin ZN-HDPE-6 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.2. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin ZN-HDPE-6 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 21 0.21 55 0.21 150 126 0.27 140 0.27 170 550 0.41 305 0.31 190 530 0.36 490 0.33 210 860 0.4 670 0.33 230  133  Capillary  Slit  m-HDPE-8  Temperature (°C)  230  210  190  (a)  170  1  10  100  1000  (b) 10000  10  .  100  Smooth Sharkskin Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.3. Processability map of resin m-HDPE-8 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.3. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-8 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) No smooth extrudate got 1.73 0.08 170 No smooth extrudate got 6.12 0.10 190 No smooth extrudate got 13.67 0.12 210 No smooth extrudate got 21.02 0.13 230  134  Capillary  Slit  m-HDPE-9  Temperature (°C)  230  210  190  170  (a)  150 10  100  1000  (b) 1  .  10  100  Smooth Sharkskin Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.4. Processability map of resin m-HDPE-9 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.4. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-9 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 2.25 0.07 Not Done 150 4.47 0.08 1.73 0.05 170 6.12 0.08 9.08 0.08 190 6.12 0.08 13.7 0.08 210 13.7 0.08 21 0.09 230  135  Capillary  Slit  m-HDPE-10  Temperature (°C)  230  210  190  170  (a)  150 1  10  100  1000  (b) 1  .  10  100  Smooth Sharkskin Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.5. Processability map of resin m-HDPE-10 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.5. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-10 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 2.45 0.07 Not Done 150 3.46 0.07 1.73 0.05 170 6.12 0.08 13.7 0.09 190 803 0.28 502 0.21 210 502 0.23 312 0.18 230  136  Capillary  Slit  m-HDPE-11  Temperature (°C)  230  210  190  170  (a)  150 1  10  100  1000  (b) 1  .  10  100  Smooth Sharkskin Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.6. Processability map of resin m-HDPE-11 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.6. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-11 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 3.46 0.07 Not Done 150 4.47 0.07 6.12 0.07 170 9.08 0.07 9.08 0.07 190 312 0.19 312 0.16 210 312 0.17 312 0.16 230  137  Capillary  Slit  m-HDPE-12  Temperature (°C)  230  210  190  170  (a)  150 1  10  100  1000  (b) 1  .  10  100  Smooth Sharkskin Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.7. Processability map of resin m-HDPE-12 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.7. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-12 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 3.87 0.06 Not Done 150 6.12 0.06 6.12 0.05 170 9.08 0.06 9.08 0.06 190 502 0.20 312 0.14 210 502 0.17 312 0.13 230  138  Capillary  Slit  m-HDPE-13  Temperature (°C)  210  190  170  (a) 1  10  100  1000  (b) 10  .  100  Smooth Sharkskin Stick-Slip Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.8. Processability map of resin m-HDPE-13 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.8. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-13 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 3.87 0.11 6.12 0.12 170 3.87 0.11 6.12 0.11 190 6.12 0.11 9.08 0.11 210  139  Capillary  Slit  m-HDPE-19  Temperature (°C)  210  190  170  (a) 10  100  1000  (b) 10  .  100  Smooth Sharkskin Stick-Slip Gross  1000  Apparent Shear Rate, A (s ) -1  Figure D.9. Processability map of resin m-HDPE-19 in extrusion through capillary and slit dies as a function of apparent shear rate and temperature.  Table D.9. Critical apparent shear rates and shear stresses for the onset of melt fracture in capillary and slit dies for resin m-HDPE-19 at different temperatures. Capillary Slit Temperature -1 -1 (°C) , 	(s ) , (MPa) , (s ) , (MPa) 6.12 0.17 13.67 0.20 170 9.08 0.16 13.67 0.18 190 13.67 0.16 21.02 0.20 210  140  100  m-HDPE-12 T = 190°C L/D = 16 10-1  *  Complex Modulus, |G | or Shear Stress, W,A (MPa)  APPENDIX E – MOONEY ANALYSIS  10-2 10-1  100  101  102  103  104  . Frequecny (rad/s) or Apparent Shear Rate, A (1/s)  Figure E.1. Bagley corrected flow curves of resin m-HDPE-12 for different diameters at T=190°C. The diameter dependence and the significant deviation from the LVE data (failure of the Cox-Merz rule) is consistent with the assumption of slip. 700  -1  Apparent Shear Rate, A (s )  600 .  500  m-HDPE-12 T = 190°C L/D = 16 W (MPa)  400 300 200 100 0 0.0  0.2  0.4  0.6  0.8  1.0  1.2  1.4  -1  1/D (mm )  Figure E.2. Mooney plot using the data plotted in Fig. E.1. The slopes of the lines are equal to 8us (Eq. 7.1) for the corresponding value of stress. The slopes increase with increase of shear stress.  141  Slip Velocity, us (mm/s)  102  m-HDPE-12 T = 190°C L/D = 16 101  100  5  us(mm/s)=1.88×10 [W(MPa)]  4  Mooney D=0.79 mm, Deviation from LVE D=1.22 mm, Deviation from LVE D=2.11 mm, Deviation from LVE  10-1 0.03  0.04  0.06  0.08  0.10  0.15  Wall Shear Stress, W (MPa)  Figure E.3. The slip velocity as a function of shear stress for resin m-HDPE-12 at T=190°C. The solid line represents the slip law given by Eq. 7.3.  0.30  *  W or |G |, (MPa)  0.20  m-HDPE-12 T = 190°C L/D = 16  0.10 0.07 0.05 Shifted LVE Mooney D=0.79 mm D=1.22 mm D=2.11 mm  0.03 0.02  0.01 10-1  100 .  101  102  103  -1  A (s ) or (4n/3n+1).LVE , (rad/s)  Figure E.4. The slip corrected flow curve of resin m-HDPE-12 at T=190°C compared with the LVE data. Excellent agreement is shown, demonstrating the validity of the Cox-Merz rule.  142  APPENDIX F – FLOW CURVE CONSTRUCTION  Shear Stress, W (MPa)  0.50  0.30  ZN-HDPE-0 T=190°C L/D = 16 Ddie = 0.79 mm  Calculated C2  0.20 0.15  Calculated C3  0.10  Experimental Calculated LVE  0.07 0.05 100  101  102  103  . -1 Apparent Shear Rate, A (s )  104  Figure F.1. Constructing the flow curve of ZN-HDPE-0 starting from LVE data.  143  

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