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Rheology and capillary flow of sodium and zinc ionomers Zuliki, Muaad 2020

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RHEOLOGY AND CAPILLARY FLOW OF SODIUM AND ZINC IONOMERS by  Muaad Zuliki  B.A., King Saud University, 2017  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF   MASTER OF APPLIED SCIENCE  in  THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Chemical and Biological Engineering)   THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)   April  2020   © Muaad Zuliki, 2020     ii  The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, a thesis entitled:  Rheology and capillary flow of sodium and zinc ionomers  submitted by Muaad Salah Zuliki in partial fulfillment of the requirements for the degree of Master of Applied Science  in Chemical and Biological Engineering   Examining Committee: Dr. Savvas G. Hatzikiriakos, Chemical and Biological Engineering Department, UBC Supervisor  Dr. Ian Frigaard, Mechanical Engineering Department, UBC Supervisory Committee Member  Dr. John Frostad, Chemical and Biological Engineering Department, UBC Supervisory Committee Member   iii   Abstract Ionomers are an important class of polymers that contain a small number of ionic functionalities capable of forming reversible ionic associations. They are used in many applications such as self-healing materials, semi-permeable membranes, and in coating and food packaging. It is important to understand the interplay between the dynamics of these ionic functionalities and their role on the rheological and mechanical properties in order to explore their potential for new commercial applications. Using a parallel-plate rheometer equipped with a partitioned plate, and the Sentmanat extensional rheometer (SER) fixture, a full rheological characterization of several sodium and zinc poly(ethylene-co-methacrylic acid) and their corresponding parent copolymers has been carried out. Particular emphasis has been placed on the distribution of the relaxation times to identify the characteristic times, such as the reptation, Rouse, and lifetime of associations that are associated with entanglements, segmental dynamics and ionic and hydrogen bonding associations respectively. It was found that ionic interactions increase the linear viscoelastic moduli and the viscosity by up to one order of magnitude and cause significant strain hardening effects in the uniaxial extension of ionomers.  The time-strain separability that can be used to formulate a K-BKZ constitutive equation (Wagner damping function), was found to hold for all ionomers and their corresponding copolymers. Their damping function was found to have different values indicating that their relaxation depends on the number of ionic associations. The relaxation of copolymers is faster mainly due to the absence of ionic interactions. iv  Finally, the capillary flow properties of these ionomers were studied to assess their processability in terms of instabilities such as wall slip and melt fracture. It was found that the no-slip boundary condition is a valid assumption for these polymers due to the strong ionic associations and strong interactions with the capillary wall. All ionomers were found to exhibit gross melt fracture and no sharkskin, a characteristic of strain-hardening polymers. The critical shear stress for the onset of gross melt fracture was found to depend on the lifetime of associations, independent of temperature, molecular weight and type of ion.            v  Lay Summary   Ionomers are a class of  polymers that contain a small amount of metal ions (sodium, zinc) that interconnect the polymer chains, creating strong networks. The relatively simple process of adding these metal ions generate materials with evident optical clarity, improved mechanical, and other excellent physical properties compared to their original copolymers. This work answers the following questions: How the amount and type of metal ions (sodium or zinc) affect their physical, rheological and mechanical properties? How the amount and type of metals (sodium or zinc)  affect their processability in extrusion? What is the highest amount of stress or pressure that can be applied on these ionomers before they lose their optical quality in production such as extrusion? How the molecular weight characteristics and number of ionic interactions affect the quality of the final products?         vi  Preface Most of this work has been published in M. Zuliki, S. Zhang, K. Nyamajaro, T. Tomkovic, S.G. Hatzikiriakos, “Rheology of sodium and zinc ionomers: Effects of neutralization and valency,” Physics of Fluids, 32, 023104 (2020) and M. Zuliki, S. Zhang, T. Tomkovic, S.G. Hatzikiriakos, “Capillary flow of sodium and zinc ionomers,” Physics of Fluids, 32, 023106 (2020). The contribution of each co-author is described below. Chapter 3. Data for 4.9-Na 0, 4.3-Na 0, 11.5-Na 0, and 19.2-Na 0 samples in Table 3.1 (used in both publications) are taken from the previous work of Tanja Tomkovic (PhD graduate in CHBE,  Department of Chemical and Biological Engineering and co-author in the first and second publications listed above) and Prof. S.G. Hatzikiriakos with authors’ permission. Degree of neutralization and mol% data of  MAA for 9.7-Zn 0, 10.8-Zn 0 and 16.8-Zn 0 samples in Table 3.1 (used in first publication) are collected and calculated by the author, Shiling Zhang (PhD visiting student from the China University of Petroleum and co-author in first publication listed above) and Kudzi Nyamajaro (PhD candidate in UBC, Department of Chemistry and co-author in the first publication listed above).  Chapter 4. Data for 4.9-Na 0, 4.3-Na 0, 11.5-Na 0, and 19.2-Na 0 samples in Figures 4.3(a), 4.4(a) (used in first publication listed above) are taken from the previous work of  T. Tomkovic and Prof. S.G. Hatzikiriakos  with authors’ permission. Micro photos in Figures 4.17 and 4.18 (used in second publication) were taken by Shiling Zhang (co-author in second publication) from extrudates that the author and Shiling produced using capillary rheometry. All other work reported in this thesis is done solely by the author with guidance from Professor S.G. Hatzikiriakos. vii  Table of Contents Abstract ................................................................................................................................... iii Lay Summary .......................................................................................................................... v Preface ..................................................................................................................................... vi Table of Contents………………………………………………………………………...………….vii List of Tables .......................................................................................................................... ix List of Figures .......................................................................................................................... x List of Symbols ..................................................................................................................... xiv List of Abbreviations ........................................................................................................... xvi Acknowledgments ............................................................................................................... xvii Chapter 1: Introduction ......................................................................................................... 1 1.1 Definition, Characteristics, and Application of Ionomers ............................................... 1 1.2 Rheological  Properties of  Ionomers .............................................................................. 4 1.2.1 Rheology ................................................................................................................... 4      1.2.1.1 Rheometers ....................................................................................................... 4           1.2.1.1.1 Rotational Rheometer ............................................................................... 5           1.2.1.1.2 Sentmanat Extensional Rheometer ........................................................... 7           1.2.1.1.3 Capillary Rheometer ................................................................................. 7 1.2.2 Review of Rheological Studies of Ionomers .......................................................... 10 Chapter 2: Thesis Objectives and Organization ................................................................ 12 2.1 Objectives ...................................................................................................................... 12 viii  2.2 Thesis Organization ....................................................................................................... 13 Chapter 3:  Materials and Experimental Methodology .................................................... 14 3.1 Materials ........................................................................................................................ 14 3.2 Rheological Experiments .............................................................................................. 17 Chapter 4: Results and Discussion ...................................................................................... 21 4.1 Linear Viscoelasticity .................................................................................................... 21 4.2 Characteristic Relaxation and Lifetimes of Association ............................................... 26 4.3 Nonlinear Rheology of Ionomers and their Parent Copolymers ................................... 29 4.3.1 The Damping Function ........................................................................................... 29 4.3.2 Uniaxial Extension ................................................................................................. 31 4.4 Experimental Results on Capillary Flow ...................................................................... 34 4.4.1 End Pressure and the Effect of Pressure on Viscosity ............................................ 34 4.4.2 Flow Curves ............................................................................................................ 36 4.4.3 Wall Slip of Ionomers............................................................................................. 39 4.5 Melt Fracture ................................................................................................................. 40 Chapter 5:  Conclusions and Recommendations ............................................................... 49 5.1 Conclusions ................................................................................................................... 49 5.2 Recommendations for Future Work .............................................................................. 51 References .............................................................................................................................. 52 Appendix: Supporting Information for Chapter 4 ............................................................ 59  ix  List of Tables Table 3.1 List of ionomers their parent copolymers studied in this work ..................................................16 Table 4.1 Estimation of characteristic times of the various ionomers ........................................................28 Table 4.2 The onset of flow instabilities (critical shear rate/stress) for the studied melts using a capillary die with L/D = 15. ........................................................................................................................47                      x  List of Figures Figure 1.1 Chemical structure of poly(ethylene-co-methacrylic acid) ionomer, where X, Y and Z represent the mole fractions of ethylene, methacrylic acid and a metal-ion methacrylate, respectively. M = Li, Na, (Zn)1/2 and (Mg)1/2 ..................................................... 1 Figure 1.2 A schematic illustrating the structure of molten ethylene-methacrylic acid sodium ionomer to show the role of sodium ion in forming ionic clusters. .......................................... 2 Figure 1.3 Schematic drawing of the ion hopping mechanism in ionomers. Circles represent ionic aggregates containing ionic groups. Curved line represents a segment of a polymer chain bearing a single ionic group, which diffuse from the left aggregate to the right aggregate. .................................................................................................................................. 3 Figure 1.4 Schematic illustration of (a) parallel-plate and (b) cone-and-plate fixtures used to generate shear flow ................................................................................................................... 5 Figure 1.5 Schematic of the cone and partitioned plate (CPP) geometry ................................ 6 Figure 1.6 Schematic of the Sentmanat extensional fixture (SER) .......................................... 7 Figure 1.7 Schematic representation of a capillary rheometer ................................................. 8 Figure 1.8 Typical Bagley correction for capillary data (ionomer example at 140 °C) .......... 9 Figure 3.1 Typical relaxation moduli after imposition of various sudden step shear strains (left) and vertically shifted relaxation moduli at various strains on the linear relaxation modulus to determine the shift factors for the damping function (right)  (ionomer example at 140 °C) .................................................................................................................................... 18 Figure 3.2 A typical tensile stress growth coefficient as a function of at three Hencky strain rates (ionomer example at 140 °C) ......................................................................................... 19 xi  Figure 4.1 The master curves of the storage (G) and loss (G) moduli (symbols) along with the Maxwell model fit (lines) for sodium ionomer samples (filled symbols) and corresponding copolymers (unfilled symbols) studied in this work at the reference temperature of 140 °C using the relaxation spectrum plotted in Figure 3. (a) 4.3-Na 69 and 4.3-Na 0 (b) 4.9-Na 63 and 4.9-Na 0 (c) 11.5-Na 65 and 11.5-Na 0 (d) 19.2-Na 65 and 19.2-Na 0 ......................................................................................................................................... 22 Figure 4.2 The master curves of the storage (G) and loss (G) moduli (symbols) along with the Maxwell model fit (lines) for zinc ionomer samples (filled symbols) and corresponding copolymers (unfilled symbols) studied in this work at the reference temperature of 140 °C using the relaxation spectrum plotted in Figure 3. (a) 9.7-Zn 40 (b) 10.8-Zn 60 and 10.8-Zn 0 (c) 16.8-Zn 33 and 16.8-Zn 0 .................................................................................................. 23 Figure 4.3 The distribution of the Maxwell relaxation strengths and times for all ionomers (a) for sodium and their copolymers (b) for zinc and their copolymers ................................. 24 Figure 4.4 The master curves of the complex viscosities ( *( )  ) of (a) sodium and (b) and zinc ionomers and their corresponding copolymers at the temperature of 140 °C ................. 26 Figure 4.5 (a) Relaxation moduli after imposition of various sudden step shear strains for the 16.8-Zn 33 melt and its copolymer 16.8-Zn 0 at the temperature of 140 °C. (b) Vertically shifted relaxation moduli at various strains on the linear relaxation modulus to determine the shift factors for the damping function ..................................................................................... 29 Figure 4.6 The damping function of ionomers and their parent copolymers obtained from step-strain relaxation experiments and its fit using the Wagner model (Eq. 4.3) ................... 31 xii  Figure 4.7 Comparison of uniaxial stress growth coefficient for zinc ionomers and their corresponding copolymers at different Hencky strain rates at temperature of 140 °C. (a) 9.7-Zn 40 (b) 10.8-Zn 60 and 10.8-Zn 0 (c) 16.8-Zn 33 and 16.8-Zn 0 ....................................... 33 Figure 4.8 Comparison of the extensional rheological properties of zinc ionomers and their corresponding copolymers using the Strain Hardening Factor (SHF) at different Hencky strain rates  temperature of 140 °C (left the zinc ionomers and right their copolymers) ........ 34 Figure 4.9 The Bagley plot for ionomers 11.5-Na 65 (left) and 9.7-Zn 40 (right)  samples at the reference temperature of 160 °C ....................................................................................... 35 Figure 4.10 The effect of apparent shear rates on the end pressure for all samples at 160 °C as determined by the Bagley method for the sodium (a) and zinc ionomers (b) .................... 35 Figure 4.11 The effect of wall shear stresses on the end pressure for all samples melt at 160 °C as determined by the Bagley method ................................................................................. 36 Figure 4.12 The Bagley corrected flow curves for 11.5-Na 65 (left) and 9.7-Zn 40 (right) melts at 160 °C for three dies with various L/D ratios. The solid lines represent the LVE data plotted as a flow curve at the same temperature ..................................................................... 37 Figure 4.13 The pressure-dependency coefficient of viscosity on pressure for all samples .. 38 Figure 4.14 The end-pressure corrected flow curves of 11.5-Na 65 (left) and 9.7-Zn 40 (right) samples at 160 °C ........................................................................................................ 39 Figure 4.15 The effect of the die diameter to check the possibility of slip at the wall for 19.2-Na 65 (left) and 9.7-Zn 40 (right) at T = 140 °C. The solid lines represent the LVE results at the same temperature .............................................................................................................. 40 Figure 4.16 Images of zinc extrudates from the capillary extrusion experiment at 140 °C. The apparent shear and appearance are indicated in each picture .......................................... 41 xiii  Figure 4.17 Images of sodium extrudates from the capillary extrusion experiment at 140 °C. The apparent shear and appearance are indicated in each picture .......................................... 42 Figure 4.18 The processability maps of all polymer melts as a function of apparent shear rate at 140 °C in capillary extrusions for the sodium (a) and zinc ionomers (b) ........................... 44 Figure 4.19 The flow curves of all sodium ionomers at 120-180 °C for L/D = 15. The filled and open symbols represent gross melt fracture and no melt fracture, respectively .............. 45 Figure 4.20 The flow curves of all zinc ionomers at 120-180 °C for L/D = 15. The filled and open symbols represent gross melt fracture and no melt fracture, respectively ..................... 46 Figure 4.21 The critical shear stress for the onset of melt fracture as a function of the lifetime of associations, for all ionomers as well as all temperatures, studied using capillary rheometry ................................................................................................................................ 48   xiv  List of Symbols aT                                                        horizontal shift factor in time-Temperature superposition D                                                         capillary die diameter  Db                                                         reservoir (barrel) diameter Eact                                                          activation energy G                                                         storage moduli G                                                         loss moduli G(t)                                                       linear relaxation modulus G(γ,t)                                                    nonlinear relaxation modulus Gk                                                                relaxation strength  L/D                                                        length-to-diameter ratio of die Mw                                                        molecular weight  Me                                                         entanglement molecular weight n                                                             Wagner damping function parameter Ne                                                             number of monomers per entanglement  Ns                                                           number of  monomers per ionic group   N                                                            number of monomers of the full chain P                                                            pressure Δ𝑃                                                         pressure drop 𝑃𝑒𝑛𝑑                                                       end pressure Q                                                           volumetric flow rate  R                                                            gas constant xv  T                                                            temperature Tm                                                                 melting temperature Tref                                                         reference temperature WiE                                                       Weissenberg number   ZE                                                          average number of entanglements per chain ZS                                                          average number of ionic associations per chain *( )                                                    complex viscosity η0                                                          zero shear viscosity ηE                                           tensile stress growth coefficient τe                                                            Rouse time of an entanglement strand  τs                                                           association lifetime τrep                                                         reptation time  ?̇?𝐴                                                          apparent shear rate 𝛾.𝐴,𝐶                                                            critical shear rate for the onset of melt fracture 2α                                                          contraction angle                                                              frequency   λk                                                                  relaxation time σw                                                            true wall shear stress σC                                                               critical shear stress for the onset of melt fracture βP                                                                Barus coefficient   xvi  List of Abbreviations HDPE                                           high density polyethylene LDPE                                                       low density polyethylene LVE                                               linear viscoelastic envelope MAA                                              methacrylic acid SER                                               Sentmanat extensional rheometer tTS                                                 time-temperature superposition                xvii  Acknowledgments I would like to express my sincere gratitude and appreciation to my supervisor Professor Savvas G. Hatzikiriakos for his support throughout my graduate studies. He has supported me throughout my thesis and experiments with his patience and knowledge. I really appreciate the discussions and encouragements that he gave me every week in the past 2 years. I would also to thank Eng. Ahmed Bugshan for his endless support throughout my bachelor and master studies. His financial support is the reason why I am here in the first place and now I can be a productive individual to my home country. I would also thank the great rheology group at UBC who helped throughout my research and experiments. I highly appreciate Tanja, Shiling and Nikoo for all their help during my study. Last but not least, the most profound and loving thanks to my family, to my parents and my brother who have given me the greatest love and support anyone could ever have. I am also blessed  to have a wonderful wife who supported me throughout my study.   1  Chapter 1: Introduction 1.1 Definition, Characteristics, and Application of Ionomers Ionomers are plastics in which a copolymer of ethylene or styrene with an acid, such as methacrylic, acrylic acid, or sulfonic acid, is combined with a metal.1 The term ionomer was first used by Rees and Vaughan2 to describe a copolymer of ethylene and methacrylic acid that were partially neutralized with sodium or zinc ions (Figure 1.1). Ionomers were first manufactured commercially in 1964 by DuPont. Tant and Wilkes3 defined ionomers as thermoplastic polymers that contain a small mole fraction of ionic functionalities, up to 15 mol.%.   Figure 1.1 Chemical structure of poly(ethylene-co-methacrylic acid) ionomer, where X, Y and Z represent the mole fractions of ethylene, methacrylic acid and a metal-ion methacrylate, respectively. M = Li, Na, (Zn)1/2 and (Mg)1/2  In general, ionomers contain mostly non-ionic repeat units, and a small amount of ion containing repeat units. The distribution of these ionic groups, which can be rearranged differently in the polymer backbone influence the ionomer properties.4-6 Despite that ionomers tend to form ionic multiplets and clusters (Figure 1.2). In ionomers, the nonpolar chains are grouped together and the polar ionic groups are attracted to each other but since these groups are attached to the polymer chain, they form small clusters.7 These clusters and multiplets that 2  represent the ionic interactions strongly depend on the type of polymer backbone, ionic content, type of cation and degree of neutralization (percentage of acid groups neutralized with metal ions). As a result, the properties of ionomers are greatly affected by the presence of these aggregates that act as reversible crosslinks.1  Figure 1.2 A schematic illustrating the structure of molten ethylene-methacrylic acid sodium ionomer to show the role of sodium ion in forming ionic clusters.  Due to the commercial importance of ionomers, which are processed in their melt state, the effect of ionic interactions on melt flow behavior is of particular interest (focus of the present work). As previously explained, the ionic aggregates could form reversible crosslinks that significantly affect the flow of polymer. Ionomers found to be more viscous than their non-ionic counterparts. However, they can be processed in conventional apparatuses. The flow of ionomer proceeds via mechanism of ion hoping in which the flow doesn’t necessary require the elimination of ionic bonding (Figure 1.3). However, the ionic groups tend to hop to another aggregate. This behavior represents the association lifetime, which is defined as the time 3  necessary for the ionic group to find a new partner after dissociation.8,9 During this time, polymer chains relax allowing the flow of ionomer. Another important time characteristic for entangled polymers is the reptation time. It represents the time it takes for a full chain to completely relax after the application of a stress. Hence it is important to determine these characteristic relaxation time to better understand the flow behavior of ionomers and their processability (main focus of the present work).   Figure 1.3 Schematic drawing of the ion hopping mechanism in ionomers. Circles represent ionic aggregates containing ionic groups. Curved line represents a segment of a polymer chain carrying a single ionic group, which diffuse from the left aggregate to the right aggregate.  A particular class of ionomers of commercial interest includes those obtained from semicrystalline ethylene-methacrylic acid copolymers, which their carboxylic acid groups  are neutralized with various metal ions, such as sodium, magnesium, zinc, lithium, and calcium1,6,10 forming aggregations of ionic groups generates materials with excellent properties as discussed above.10 Ionomers have also shown delayed relaxation due to the presence of reversible 4  associations and the relaxation times increase with increase of the ionic content and valency of ionic associations.11,12 They also exhibit a significant rise in modulus as the number of ions increase.2,13 Due to their unique properties (optical clarity  and improved toughness and tensile strength  compared with parent copolymer), they have found many applications such as in fabrication of membranes,14 as compatibilizers for polymer blends15 and as self-healing materials.16  1.2 Rheological  Properties of Ionomers Before presenting the objectives of the present work and the related literature review certain rheological principles that is useful in understanding the behavior of these complex structured ionomers are described in the following section. 1.2.1 Rheology Rheology is the science that studies the flow and deformation of solid-like and liquid-like materials when external forces are applied.17 In particular it studies the behavior of complex viscoelastic materials that show properties of both solid and liquid depending on the time and length scales of deformation. The term rheology is originated from the Greek words ‘rheo’ translating as ‘flow’ and ‘logia’ meaning ‘the study of’. In this section the various rheometers used to carry out the experimental part of this work are presented, discussing the type of rheological data measured. 1.2.1.1 Rheometers Rheometry is defined as the tool used to experimentally characterize the rheological behavior of materials. It controls the environment around them, and applies and measures a wide 5  range of properties in response to the application of stress, strain, and strain rate. There are two different types of rheometers, namely stress-controlled and strain-controlled rheometers. The first measures the rotational speed, while the torque is fixed. The second type measures the torque while the rotational speed is fixed.18 The following sections discuss the main rheometers and fixtures that have been used in this study. 1.2.1.1.1 Rotational Rheometer A rotational rheometer (shear rheometer) is an instrument used to determine how a liquid or slurry flows. It applies a certain force to the polymeric sample and measures how it reacts to this force. It comes mainly with two rheological geometrical fixtures, namely a parallel plate and cone-and-plate geometries. The parallel-plate geometry has two parallel diameter alike disks and a gap between them, while the cone-and-plate geometry comes with a cone of a certain angle (4 - 7°) and a disk (see Figure 1.4). An advantage from the use of cone-and-plate over the parallel plate geometry is that it produces homogenous deformation, while the later produces non homogenous deformation. Another advantage is that it can measure the first normal stress difference.18            Figure 1.4 Schematic illustration of (a) parallel-plate and (b) cone-and-plate fixtures used to generate shear flow 6  However, those geometries are not able to reach high strains and shear rate due to edge fracture that the samples experience at high deformations and deformation rates.17,18 This fracture has a significant effect on measurements since it affects the torque and the rotational speed.19 This problem can be solved by the use of the cone-partitioned-plate that consists of a cone plate at the bottom and an upper geometry that has a small standard plate integrated into a stationary coaxial ring which directly connected to the torque transducer (Figure 1.5). This setup creates a guard ring of sample around the active measurement area, delaying the effects of edge failure, allowing for higher strains to be measured on elastic materials.19   Figure 1.5 Schematic of the cone and partitioned plate (CPP) geometry  7  1.2.1.1.2 Sentmanat Extensional Rheometer The Sentmanat Extension Rheometer (SER) system (Figure 1.6), can be used together with a rotational rheometer to measure the extensional properties of polymers. It consists mainly of two rotating drums and two clips to secure the sample in place (see Figure 1.6). To generate uniaxial extensional data, a sample with a certain area and thickness is stretched by its both ends by a certain rate (Hencky stain rate) and the tensile stress, force per unit area, needed to stretch it, is measured. This non-linear deformation provides information about molecular structure, branching, stress relaxation in extension, and brittleness that is otherwise impossible to get from linear viscoelastic data.20,21               Figure 1.6 Schematic of the Sentmanat extensional fixture (SER)  1.2.1.1.3 Capillary Rheometer Capillary rheometry is widely used in both industry and academia to assess the rheological behavior (i.e. viscosity, slip and flow instabilities) of polymer melts at high shear rates. It consists of a heated reservoir (barrel) that contains the polymer melt and a constant speed piston that drives the melt through a die having a specific diameter and length (see Figure 1.7). This die is located at the bottom end of the reservoir. Different dies with different lengths 8  and diameters are used to produce different data, mainly pressure drop-flow rate relationships.17,18   Figure 1.7 Schematic representation of a capillary rheometer  The load cell measures the force needed to move the polymer melt through the die at a specific volumetric flow rate, Q, or apparent shear rate, ?̇?A. The driving pressure at the capillary die inlet can be calculated from this force (force per unit of cross sectional area of capillary reservoir/barrel). The pressure then is used to calculate the apparent shear stress, σA, and the apparent shear rate, ?̇?A, using the following equations, ?̇?A =32QπD3                                                                    (1.1) σA =∆P4(L/D)                                                                        (1.2) 9  The simple pressure reading used to calculate the apparent shear stress, gives only the apparent viscosity value. However, the true pressure drop along the capillary is hidden by an additional pressure drop (end pressure) at the entrance of the die, where the flowing polymer melt flows from a wide reservoir to a narrow capillary die. To account for this additional pressure, the Bagley correction is applied.17,18 This correction requires at least two sets of data obtained by using the same material, same temperature, same capillary die diameter but different capillary die lengths. Hence, by plotting the pressure versus the die length-to-diameter ratios (L/D) at a fixed apparent shear rate values, the additional pressure drop can be obtained as shown in Figure 1.8. Note that different apparent shear rate values have different values of pressure drop. Eq. 1.3 gives the true shear stress, σw, by accounting the additional pressure drop from Bagley correction.  σw =∆P − Pend4(L/D)                                                                     (1.3)  Figure 1.8 Typical Bagley correction for capillary data (ionomer example at 140 °C) 10  1.2.2 Review of Rheological Studies of Ionomers  Ionomer is an advanced type of polymer with key properties that makes it a leading choice for industrial applications. As a result, extensive research on the rheological properties of ionomers having different types of ions, metal level and concentration of acid that carry ionic groups were conducted. The morphology of ionomers based on poly(ethylene-co-methacrylic acid) or EMAA are divided into three regions: amorphous phases, crystalline phases, and ionic clusters. These ionic clusters act as reversible crosslinks that improve the toughness, viscosity, optical clarity, and adhesion properties of the copolymer.22 Ward (1967)23 have demonstrated that neutralization of poly(ethylene-co-methacrylic acid) leads to the formation of ionic multiplets and clusters, which significantly affect the relaxation processes of ionomers and their thermophysical properties. There are many studies on the effect of different cations and ion content on flow, for example, Na+, Li+, Ca2+, and Mg2+ in the case of poly(ethylene-co-methacrylic acid)  ionomers. This effect was observed by conducting dynamic experiments. It was found that the valency of ionic associations has a stronger effect on the behavior of these ionomers at lower shear rates.24-27 Hinton and Alvarez28 investigated the effect of association strength on the viscoelastic properties of zinc ionomers. They found that the linear and non-linear viscoelastic rheological complexity of the melts is strongly dependent on association strength. For example, increasing the association strength directly impacts the stress and strain at break having an impact in the processability of these polymers.28 Furthermore, the influence of the pseudocross-linking structure in sodium and zinc ionomers appears in the increase of the activation energies. Compared to their parent copolymers the ionomers showed elevated activation energy, a direct consequence of the presence of ionic associations.29-31 11  While their rheology has received some attention in the literature, their flow behavior in processing has not been studied thoroughly (not adequately studied in the literature particularly for the case of zinc ionomers).32,33 A recent study showed that the end pressure of sodium ionomers is higher than those of its parent polymer and that the pressure has a significant effect on sodium ionomers viscosity.24 Moreover, extensive research on melt fracture phenomena reveals its importance in limiting the polymers production rate.34-36 It is found that capillary rheometry is an efficient experimental technique to investigate the processing instabilities (i.e. melt fracture) during the extrusion of some commercial polymers such as polyethylenes33,37 and polypropylenes.38 It has been found that processing instabilities depend strongly on the molecular characteristics of polyethylenes, a polymer used in synthesizing the carboxylated polyethylene ionomers. Characteristics such as molecular weight (Mw) and its distribution and levels of long chain branching.34,39 Since no previous studies exist on the flow instabilities and melt fracture behavior of these polymers, it is one objective to study these phenomena for the case of ionomers. As previously mentioned, the introduction of metals to the  poly(ethylene-co-methacrylic acid) backbone in the neutralization process introduces the formation of ionic multiplets and clusters. This clustering act like a physical crosslink, which affect the relaxation processes of ionomers and their thermophysical properties.7 However, some of the carboxylic groups are not neutralized and capable of forming hydrogen bonds. They allow the exchange of hydrogen ions with the metal cation which allow the polymer chains to move around giving the free carbolic groups the ability to plasticize the ionic aggregates.29,40 As a result, ionomers tend to be very sensitive to the presence of moisture and appropriate drying is needed to obtain accurate measurements.30 12  Chapter 2: Thesis Objectives and Organization 2.1 Objectives The overall objective of this work is to study the influence of the ionic groups on the rheological and processability of several zinc and sodium poly(ethylene- co -methacrylic acid) ionomers with particular focus on their nonlinear rheology (i.e. the damping function, steady shear and extensional rheology), capillary flow and melt fracture behavior. In particular, the objectives are: A. To study the nonlinear rheology (shear and extensional) of sodium and zinc ionomer melts (ionic interactions) and to compare that to the rheology of their parent polymers (no ionic interactions). B. To investigate the effect of valency of ions (Na+ vs Zn++) in these ionomers on their rheological properties. C. To analyze the rheological properties in terms of characteristic relaxation times such as reptation, lifetime of association and Rouse time in order to understand their rheological response in terms of molecular weight, number of associations and entanglements per chain, useful is rheological modeling. D. To investigate the effect of pressure on the viscosity of ionomers having different type (valency) and level of ions. E. To investigate the slip of ionomers in capillary flow. F. To study the processability of sodium and zinc ionomers in terms of their melt fracture performance and formulate a relation between the molecular characteristics (e.g. the lifetime of ionic associations, τs) of these different ionomers with their melt fracture behavior. 13  2.2 Thesis Organization  Chapter 1 presents the structure and properties of ionomers and provides a description of the  rheological methods and testing, as well as the basic rheometrical instruments used for such material characterization. The objectives and organization of this thesis are presented in chapter 2. Chapter 3 gives a full description of the specific experimental methods used to produce the copolymers from ionomers as well as how the degree of neutralization is measured. Sample preparation method prior to testing is also included in this chapter. Moreover, rheological experiments used for such material characterization, are presented as well. Chapter 4 discusses the effect of the type and level of ions on the linear and nonlinear viscoelastic properties. Moreover, the distribution of the relaxation times and lifetimes of association of all samples is also analyzed and critically discussed. Finally, the effect of the type and level of ions and temperature on the capillary flow and on the occurrence of melt fracture are presented in this chapter. Conclusions and Recommendations for future work are presented in chapter 5.       14  Chapter 3:  Materials and Experimental Methodology This chapter presents all the ionomers and copolymers used in this study and gives a full description of the specific experimental methods used to produce the copolymers from ionomers. In addition, the rheological methods used to study the ionomers and their copolymers are discussed. 3.1 Materials Partially neutralized poly(ethylene-co-methacrylic acid) with zinc and sodium ions, in the shape of pellets, was kindly provided by DuPont (Experimental station, Wilmington, DE, USA). Table 3.1 lists all the studied ionomers along their molecular weight, zero shear viscosity at 140 °C as well the energy of activation determined from the time-Temperature superposition (tTS) principle. It was determined by titration41,42 that these ionomers contain 4.3-19.2 wt% (1.5-7.2 mol%) of methacrylic acid (MAA), which is statistically distributed along the polymer chain. These are listed in Table 3.1. PerkinElmer Frontier FTIR spectrometer was used to determine two characteristic peaks of the carboxylic acid dimer (1700 cm-1), and polyethylene band (1465 cm-1), which were then used to calculate the neutralization level from the ratio of the integrated peak areas as explained previously.41-43 In the case of zinc ionomers, the unneutralized polymers (no ionic interactions) were prepared by refluxing the ionomer beads in a solution of tetrahydrofuran: xylene (95:5 % V/V) until the polymer has completely dissolved. The free polymer was obtained by precipitating from a 1:1 mixture of ethanol-water, washed several times with cold methanol; and then dried under 15  vacuum for 48 h.41,42 The total removal of the zinc was confirmed using Fourier-transform infrared spectroscopy.44 The acid content of the polymer was determined by titration. These unneutralized copolymers still contain MAA groups at amounts in the range of 3.64-6.31 mol% that can potentially form hydrogen bonding. It has been reported that MAA groups form relatively weak hydrogen bonds and these hydrogen bonds in polymers containing MAA content up to 6 mol.% have little to no effect on material properties.44 For sodium ionomers, the procedure and details have been reported elsewhere.30 The samples studied in this work are labeled as X – Zn Y and X – Na Y, where X and Y indicate the weight percent content of MAA group and the degree of neutralization by sodium or zinc, respectively. For example, 16.8–Zn 33 indicates that this sample contains 16.8 wt.% of MAA and that 33% of the MAA groups are neutralized with zinc ions. Unneutralized polymeric samples are labeled as X-Zn 0 or  X–Na 0. All polymers used in this work (both neutralized and unneutralized) are listed in Table 3.1 along with various thermophysical and rheological properties described below. Finally, the grades of the original ionomers as received and labeled by DuPont Co., are also provided in Table 3.1 indicated as Surlyn®. Sample preparation. In order to eliminate the influence of moisture, the ionomer pellets were annealed for one week at 75 °C and then allowed to cool down slowly to room temperature by turning off the vacuum oven heater.30 After drying, the sample was fabricated using compression molding, where the ionomer pellets were sandwiched between two poly(tetrafluoroethylene) sheets. First, the press was preheated to 130 °C in the case of sodium ionomer and to 170 °C for zinc ionomer since the later show lower melting rate comparing to the sodium ionomers. Consequently, the ionomers were allowed to melt for 15 min without applying pressure. A pressure of about 3 MPa was applied for 5 min, and the sample was allowed to cool 16  down to room temperature. The thickness of the prepared samples was about 1 mm suitable for rheological measurements.30 They were stored in a desiccator over CaSO4 at room temperature before testing. It should be pointed out that incomplete drying significantly affects the rheological properties as previously reported.30,45 For example, the viscosity at low frequencies may decrease by up to 3 times in the absence of drying.30 For capillary experiments, the dried pellets were directly loaded into the barrel of the capillary rheometer while minimizing their exposure with ambient air. Table 3.1 List of ionomers their parent copolymers studied in this work Sample mol% -MAA- Degree of neutralization Mw (g/mole) η0 (140 °C)  (kPa∙s) Eact  (kJ/mol) Tm (°C) 4.3 – Na 0 1.5 - 72,400 7 75.5 98 4.3 – Na 69 (Surlyn® 1601) 69 72,500 86.4 87.3 99 4.9 – Na 0 1.6 - 114,200 33 69 94 4.9 – Na 63 (Surlyn® 1605) 63 114,500 55.4 81.9 96 11.5 – Na 0 4.1 - 70,500 6.4 76 94 11.5 – Na 65 (Surlyn® 1707) 65 71,000 176 90.3 94 19.2 – Na 0 7.2 - 64,400 4.7 64 97 19.2 – Na 65 (Surlyn® 1802) 65 65,200 33.9 79.3 99 9.7 – Zn 0 3.4 - 61,400 4 - 96 9.7 – Zn 40 (Surlyn® 9650) 40 62,400 38 91.6 97 10.8 – Zn 0 3.8 - 80,400 10 80.1 92 10.8 – Zn 60 (Surlyn® 1650) 60 81,900 58 87.4 93 16.8 – Zn 0 5.9 - 113,300 32.2 67.7 94 16.8 – Zn 33 (Surlyn® 9120) 33 114,200 322 126.6 96 17  3.2 Rheological Experiments Small Amplitude Oscillatory Shear. Rheological measurements were conducted using the Anton Paar MCR 502 device (Anton Paar, Graz, Austria) with the parallel-plate geometry. The thermal stability of the ionomer was studied for 3 h isothermally at the temperature range from 140 and 180 °C. It was found that the ionomer samples are thermally stable at these temperatures and over these periods.30,31,46 Frequency sweep experiments were performed in the linear viscoelastic region at different temperatures, namely, 120, 140, 160, and 180 °C for ionomers and 110, 120, 130, and 140 °C for copolymers. Lower temperatures were used for the copolymers since they are thermally unstable at higher ones. The resulting storage, G’, and loss moduli, G”, were shifted horizontally to the reference temperature of 140 °C using the time-Temperature superposition (tTS) principle in order to produce master curves of the linear viscoelastic moduli that cover a wide range of frequencies. These shift factors are then used to calculate the energy of activation, Eact, of all samples (discussed below). Stress Relaxation. To study the nonlinear viscoelastic properties of ionomers and their copolymers the Anton Paar MCR702 (Anton Paar, Graz, Austria) rotational rheometer was used, equipped with a cone-partitioned-plate geometry. In comparison with conventional cone-and-plate geometries, the cone-partitioned-plate allows reliable measurements at high shear strains and rates overcoming edge fracture effects.17-19 Stress relaxation after imposition of sudden step-strain experiments were conducted in the range of shear strains from 0.05 to 10 to determine the damping function. The stress relaxation data for each strain imposed are plotted as relaxation modulus, G(t), versus t in a log-log plot (Figure 3.1 (left)). For strains usually 0.2 or less, the relaxation modulus values are almost the same indicating that the material response is independent of the strain and lies in their linear region. In order to obtain the damping function, 18  the curves at higher strains, where the response depends on the strain, are vertically  shifted by different factors, h(γ), to superpose on the linear relaxation modulus (Figure 3.1 (right)).17,18 These shift factors are then plotted against the step strains to get the damping function which is used to construct constitutive equations to predict the shear and extensional flow of many polymers in polymer processing.17,18,47  Figure 3.1 Typical relaxation moduli after imposition of various sudden step shear strains (left) and vertically shifted relaxation moduli at various strains on the linear relaxation modulus to determine the shift factors for the damping function (right)  (ionomer example at 140 °C)  Uniaxial Extension. The elongational viscoelastic properties were studied using the second generation Sentmanat Extensional Rheometer (SER2) fixture.20,21 Specimens were prepared as films with a specific width and thickness. In this experiment, a sample of initial length, L0, is stretched by its both sides at a constant Hencky starin rate, ɛ̇ = dɛ / dt, where ɛ is the Hencky strain represented by Eq. 3.1. ɛ (t) = lnL(t)L0                                                                     (3.1) Time, t (s)10-1 100 101 102 103Relaxation modulus, G (Pa)10-1100101102103104105106g = 0.05g = 0.1g = 0.2g = 0.5g = 1g = 2g = 5g = 10T = 140 oCTime, t (s)10-1 100 101 102 103Relaxation modulus, G (Pa)10-1100101102103104105106g = 0.05g = 0.1g = 0.2g = 0.5g = 1g = 2g = 5g = 10T = 140 oC19  Figure 3.2 plots the tensile stress growth coefficient, ηE, versus time at different Hencky strains for an ionomer as an example. The solid curve represents the linear viscoelastic envelop (LVE) determined by the frequency sweep tests (discussed above). ηE (t) = 3η+(t)                                                    (3.2)  Figure 3.2 A typical tensile stress growth coefficient as a function of at three Hencky strain rates (ionomer example at 140 °C)  Capillary Flow. Capillary flow tests were performed by using a constant piston-speed pressure-driven capillary rheometer (Instron) with a reservoir (barrel) of diameter, Db = 0.9525 cm. The end pressure (Bagley correction)32 and the viscosity as a function of the wall shear stress, σw, and the apparent shear rate, ?̇?A = 32Q / πD3, were determined at the reference temperature of 160 °C, where Q is the volumetric flow rate and D is the capillary diameter. For the evaluation of the end-pressure effects (Bagley correction), and the effect of pressure on the T = 140 oCTime, t (s)10-2 10-1 100 101Uniaxial Stress Growth Coefficient, hE (Pa.s)1031041051061071083h+e = 0.5 s-1e = 2 s-1e = 5 s-1...20  viscosity of the ionomer, several dies of the same diameter (D = 0.762 mm) and various length-to-diameter ratios (L/D = 4, 15, 33), all having a contraction angle (die entry region) of 2α = 180° were used. To study possible slip effects, three capillary dies with the same length-to-diameter (L/D) ratio, and different diameters, namely 0.432 mm, 0.889 mm, and 1.22 mm, having a contraction angle of 2α = 180°  were used. Extrudates from the capillary experiments were collected and examined to assess the onset of flow instabilities such as extrudate distortions (melt fracture). An Olympus MIC-D digital microscope was used for obtaining various images which are presented below.          21  Chapter 4: Results and Discussion  In this section the rheological data of all samples listed in Table 3.1 are presented including linear and nonlinear viscoelasticity as well as capillary flow studies and their processability in terms of melt fracture. 4.1 Linear Viscoelasticity As mentioned above, the rheological properties of the ionomers were determined using  parallel-plate geometry. The results (master curves) are plotted in Figures 4.1(a)-(d) for sodium ionomers and 4.2(a)-(c) for zinc ionomers. In each of these Figures the linear viscoelastic properties of each of the ionomers are compared with the properties of their corresponding parent copolymer. The effect of ionic associations is evident. In most cases an increase of these properties (G, G) by one order of magnitude is observed (particularly at the terminal relaxation zone). In fact, horizontal shifting, aT, of the ionomer data superposes well with that of its copolymer indicating a similar thermal dependence of relaxation with simply longer relaxations times for the ionomers. To this end the negligible effect of the present hydrogen bonds in both ionomers and their corresponding copolymers simplifies the picture. It is expected that ionomers with a lower degree of neutralization and higher MAA content would exhibit more complex dynamics possibly leading to failure of tTS. The differences in G and G between ionomers and corresponding parent copolymers appear to decrease at higher frequencies as the inverse of these frequencies (time scales) are closer to the lifetime of ionic associations. In other words, if the lifetime of ionic associations defined here as τs, is less than the inverse of a particular frequency (1/), its effect on the linear viscoelastic measurements is negligible or slowly diminishes at frequencies  < s. The experimental data of Figures 4.1 and 4.2 shows that the lifetimes of 22  ionic associations are small, practically of the order of 10-4 - 10-3 s in which the values of G and G of ionomers and corresponding copolymers converge. As shown previously the lifetime of hydrogen bond associations are even lower, well separated from those of ionic associations.48   Figure 4.1 The master curves of the storage (G) and loss (G) moduli (symbols) along with the Maxwell model fit (lines) for sodium ionomer samples (filled symbols) and corresponding copolymers (unfilled symbols) studied in this work at the reference temperature of 140 °C using the relaxation spectrum plotted in Figure 3. (a) 4.3-Na 69 and 4.3-Na 0 (b) 4.9-Na 63 and 4.9-Na 0 (c) 11.5-Na 65 and 11.5-Na 0 (d) 19.2-Na 65 and 19.2-Na 0 4.3 - Na 69 (filled symbols)4.3 - Na 0 (unfilled symbols)Tref = 140 oCw.aT (rad/s)10-3 10-2 10-1 100 101 102 103 104G'/bT & G''/bT (Pa)100101102103104105106107G'G''Maxwell Model(a)4.9 - Na 63 (filled symbols)4.9 - Na 0 (unfilled symbols)Tref = 140 oCw.aT (rad/s)10-3 10-2 10-1 100 101 102 103 104G'/bT & G''/bT (Pa)100101102103104105106107G'G''Maxwell Model(b)11.5 - Na 65 (filled symbols)11.5 - Na 0 (unfilled symbols)Tref = 140 oCw.aT (rad/s)10-3 10-2 10-1 100 101 102 103 104G'/bT & G''/bT (Pa)100101102103104105106107G'G''Maxwell Model(c)19.2 - Na 65 (filled symbols)19.2 - Na 0 (unfilled symbols)Tref = 140 oCw.aT (rad/s)10-3 10-2 10-1 100 101 102 103 104G'/bT & G''/bT (Pa)100101102103104105106107G'G''Maxwell Model(d)23   Figure 4.2 The master curves of the storage (G) and loss (G) moduli (symbols) along with the Maxwell model fit (lines) for zinc ionomer samples (filled symbols) and corresponding copolymers (unfilled symbols) studied in this work at the reference temperature of 140 °C using the relaxation spectrum plotted in Figure 3. (a) 9.7-Zn 40 (b) 10.8-Zn 60 and 10.8-Zn 0 (c) 16.8-Zn 33 and 16.8-Zn 0  The continuous lines in Figures 4.1 and 4.2 represent fits of the multi-mode Maxwell model: ( ) ( )1'exp ' 't Nij k ijk kt tt G t dt =− −= −                                  (4.1) Where G𝑘  are the relaxation strengths and λ𝑘  are the relaxation times. The obtained parameters of λ𝑘 and G𝑘 found from the fitting are plotted in Figure 4.3(a) and 4.3(b) for sodium and zinc ionomers respectively (see Appendix Tables A.1 and A.2 for more details). From the superposition of G and G the horizontal shift factors (𝑎𝑇) were calculated, and they were found 9.7 - Zn 40Tref = 140 oCw.aT (rad/s)10-3 10-2 10-1 100 101 102 103 104G'/bT & G''/bT (Pa)10-1100101102103104105106107G'G''Maxwell Model(a)10.8 - Zn 60 (filled symbols)10.8 - Zn 0 (unfilled symbols)Tref = 140 oCw.aT (rad/s)10-3 10-2 10-1 100 101 102 103 104G'/bT & G''/bT (Pa)10-1100101102103104105106107G'G''Maxwell Model(b)16.8 - Zn 33 (filled symbols)16.8 - Zn 0 (unfilled symbols)Tref = 140 oCw.aT (rad/s)10-3 10-2 10-1 100 101 102 103 104G'/bT & G''/bT (Pa)10-1100101102103104105106107G'G''Maxwell Model(c)24  to follow the Arrhenius equation aT = exp [Eact / R(1/T - 1/Tref)], where 𝑇𝑟𝑒𝑓 = 140 °C. The Eact of ionomers (Table 3.1) are higher than that of their parent copolymer.30,40 Moreover, Eact of zinc ionomers are in general higher than those of sodium ones. This is  possibly due to the energy of zinc ionic interactions (form of physical cross-linking) are stronger than that of the sodium ions due to the +2 valency of Zn compared to +1 valency of Na. Six relaxation times are enough for all ionomers and their corresponding copolymers to provide optimum rheological representation (plotted in Figure 4.3). The relaxation time distribution of ionomers compared to that of copolymers is similar in shape, although is shifted to higher values of the relaxation modulus and higher relaxation times, obviously due to the effect of ionic interactions. The moduli distribution of ionomers and corresponding copolymers start roughly together at low relaxation times and they both start decreasing at higher time scales with the shift of the distribution to increase with a higher content of MAA groups. The relaxation time distributions of ionomers and copolymers converge at small relaxation times and extrapolation of these distributions indicate that at time scales in the range of 10-4-10-3, the effects of ionic interactions are negligible.  Figure 4.3 The distribution of the Maxwell relaxation strengths and times for all ionomers (a) for sodium and their copolymers (b) for zinc and their copolymers T = 140 oCRelaxation time, lk (s)10-4 10-3 10-2 10-1 100 101 102 103Relaxation modulus, Gk (Pa)1011021031041051064.3 - Na 694.3 - Na 04.9 - Na 634.9 - Na 011.5 - Na 6511.5 - Na 019.2 - Na 6519.2 - Na 0(a)T = 140 oCRelaxation time, lk (s)10-4 10-3 10-2 10-1 100 101 102 103Relaxation modulus, Gk (Pa)1011021031041051069.7 - Zn 4010.8 - Zn 6010.8 - Zn 016.8 - Zn 3316.8 - Zn 0(b)25  Figures 4.4(a) and 4(b) plot the complex viscosity of all sodium (Figure 4.4(a)) and zinc (Figure 4.4(b)) ionomers and their parent copolymers studied, exhibiting a similar shape and terminal zone relaxation. In spite of the presence of ionic clusters (strong associations), terminal zone (zero-shear viscosity) is reached mainly due to the short lifetime of these associations.30,40 In general, zinc ionomers possess a higher viscosity compared to their sodium counterparts at comparable molecular weights and considering the MAA content and degree of neutralization mainly due to their +2-valency that possibly form stronger ionic associations. The zero-shear viscosity is an indication of the molecular weight and it is used here to estimate the molecular weights of the various polymers used. The molecular weight of sample 11.5–Na 65 is 71,000 g/mole as its viscosity material function matches the one studied by Tierney and Register.30,40 Its corresponding copolymer 11.5–Na 0 should have slightly smaller molecular weight due to the small content of metal ions. The zero-shear viscosities of the other copolymers can be used together with the scaling law of  η0 ∝ Mw 3.4  to provide an estimate of their molecular weights and corresponding ionomers once the number of ionic groups are estimated (discussed below). It should be mentioned that although the scaling η0 ∝ k Mw 3.4 (η0 in Pa.s and Mw in kg/mol with the equation constant equals to 0.00333 mol3.4.m-1.s-1.kg-2.4) is applicable mostly for linear polymers, Janzen and Colby49 found that this works for branched polymer of similar simple structure. In fact, the parent copolymers have a simple, similar branched structure (“quasilinear” referred to in28,49) since the terminal zone (zero-shear viscosity) is reached at frequencies of about 10-1 rad/s. The values of the calculated molecular weights (Mw) of all polymers used are listed in Table 3.1. 26   Figure 4.4 The master curves of the complex viscosities (*( )  ) of (a) sodium and (b) and zinc ionomers and their corresponding copolymers at the temperature of 140 °C   4.2 Characteristic Relaxation and Lifetimes of Association Following the scaling analysis of Leibler et al.,8 Tomkovic et al.,30 analysed the viscoelastic moduli plotted in Figure 4.1 for sodium ionomers. A similar procedure was followed to perform a similar analysis for the zinc ionomers. Starting first with the reptation time that can be written as8,30 ( )3 3 1 2/rep e e s e sN N N N N  − −= = , where τe is the Rouse time of an entanglement strand  (2 2e s e sN N −= ) and τs is the association lifetime, Ne is the number of segments per entanglement, Ns is the number of segments per ionic group and N is the number of segments of the full chain. These scaling laws can be expressed in terms of the average number of ionic associations per chain, ZS and the average number of entanglements per chain, ZE as, 2/ (Z )S rep E SZ =  and 2 2/ Ze s S EZ =  respectively. It is noted that the last three parameters specify the structure of the polymer.8  Tref = 140 oCAngular frequency, w (rad/s)10-3 10-2 10-1 100 101 102 103 104Complex viscosity, | h*| (Pa.s)10-110010110210310410510611.5 - Na 654.3 - Na 694.9 - Na 6319.2 - Na 654.9 - Na 04.3 - Na 011.5 - Na 019.2 - Na 0Maxwell Model(a)Tref = 140 oCAngular frequency, w (rad/s)10-3 10-2 10-1 100 101 102 103 104Complex viscosity, | h*| (Pa.s)10-110010110210310410510616.8 - Zn 3310.8 - Zn 609.7 - Zn 4016.8 - Zn 010.8 - Zn 0Maxwell Model(b)27  The reptation time for each ionomer and corresponding copolymer can be calculated from LVE by finding the interception of the two limited behaviours (G ∼ ω and G ∼ ω2) at repand inverting this to obtain 1/rep rep  . For the sake of simplicity, we take as reptation time, the longest relaxation time (not much different than τrep) determined from the fit of the linear viscoelastic moduli (plotted in Figure 4.3). The determined values of the lifetime of ionic associations are listed in Table 4.1. The calculations for the lifetimes of ionic associations proceed as follows. Considering a molecular weight of 114,200 g/mol and the entanglement molecular weight of polyethylene, Me=1,200 g/mol50 the number of entanglements per chain is ZE=114,200/1200 ≈ 95. The value of the number of ionic associations per chain, ZS, can be estimated from our titration analysis (mole fraction of MAA and degree of neutralization), that is ZS = [(mol% of MAA)]×(degree-of-neutralization)×( Mw/86), where 86 is the molecular weight of the MAA group. For sample 16.8–Zn 33, ZS=(0.059)×(0.33)× (11420086) ≈26. The corresponding value of the molecular weight of each ionomer can be obtained by adding the molecular weight of the number ions in each macromolecule (sodium or zinc) to the corresponding  Mw of the parent copolymer. All estimated lifetimes of ionic associations are listed in Table 4.1. The values of the lifetimes of associations practically are of the order of 10-4-10-2 with most to be of the order of 10-3 or lower which was also concluded from Figures 4.1-4.3, where the viscoelastic moduli of ionomers and those of their corresponding copolymers seem to converge at these time scales. A value of 1.3×10-3 was reported for the lifetime of associations (referred to as “ion hoping”) by Tierney and Register40 for an ionomer similar to 11.5-Na 65, a value to be compared with the value of 3.0×10-3, reported in Table 4.1. Although the values 28  listed in Table 4.1 are estimates, they are certainly of the correct magnitude. Note also that for all ionomers studied here ZE>ZS , that is the number of entanglements per chain exceed the number of ionic interactions per chain. It is noted that the Rouse time of segments between entanglements, τe (listed in Table 4.1) for all ionomers are similar of the order of 10-4 s (sign of consistency of these estimates) and as such its effect is not evident in Figures 4.1 and 4.2.30,51 Table 4.1 Estimation of characteristic times of the various ionomers Sample mol% MAA Mw g/mol Degree of neutralization ZE ZS τrep (s) τ𝒔 (s) τe (s) 4.3 – Na 69 1.5 72,570 69 60 8 69.2 1.6  10-2 2.8  10-4 4.9 – Na 63 1.6 114,490 63 95 14 75.5 4.3  10-3 0.93  10-4 11.5 – Na 65 4.1 71,000 65 59 22 86.9 3.0  10-3 4.2  10-4 19.2 – Na 65 7.2 65,190 65 54 35 44.6 6.5  10-4 2.7  10-4 9.7 – Zn 40 3.4 62,400 40 52 10 22.1 4.6  10-3 1.7  10-4 10.8 – Zn 60 3.8 81,900 60 68 21 86.4 2.8  10-3 2.7  10-4 16.8 – Zn 33 5.9 114,200 33 95 26 211 3.4  10-3 2.5  10-4 29  4.3 Nonlinear Rheology of Ionomers and their Parent Copolymers 4.3.1 The Damping Function Stress relaxation experiments after the imposition of step strains were performed using the cone-partitioned-plate geometry. The relaxation modulus was measured at various strains from 0.05 to 10 at the reference temperature of 140 °C as shown in Figure 4.5. The threshold between the linear and nonlinear region was found to be at the strain of 0.2 with the relaxation curves obtained at lower values of strain to be independent of strain. Figure 4.5(a) depicts the relaxation modulus of ionomer 16.8-Zn 33 (taken as an example) and its parent copolymer, 16.8–Zn 0, at different shear strains. Figure 4.5(b) plots the reduced relaxation modulus of these two samples obtained by shifting the data of nonlinear (relaxation modulus at strain of 0.2 or higher) to the linear viscoelastic regime (relaxation modulus at strain of 0.1) to obtain the shift factors, h(γ).30 Excellent superposition is obtained for both ionomer and parent copolymer which indicates that the time-strain separability holds in the relaxation process and that the networks are not yet fractured in this strain region but significantly deformed.   Figure 4.5 (a) Relaxation moduli after imposition of various sudden step shear strains for the 16.8-Zn 33 melt and its copolymer 16.8-Zn 0 at the temperature of 140 °C. (b) Vertically shifted relaxation moduli at various strains on the linear relaxation modulus to determine the shift factors for the damping function 16.8 - Zn 33Time, t (s)10-2 10-1 100 101 102 103Relaxation modulus, G (Pa)10-1100101102103104105106g = 0.05g = 0.1g = 0.2g = 0.5g = 1g = 2g = 5g = 10T = 140 oC16.8 - Zn 0(a)16.8 - Zn 33Time, t (s)10-2 10-1 100 101 102 103Relaxation modulus, G (Pa)100101102103104105106g = 0.05g = 0.1g = 0.2g = 0.5g = 1g = 2g = 5g = 1016.8 - Zn 0T = 140 oC(b)30  Similar results were obtained for all other samples not plotted here (their shift factors, h(γ), are plotted in Figure 4.6). The shift factors essentially define the damping function, h(γ), that is,17,30 h(γ) =G(γ, t)G(t)                                                                     (4.2) where G(t) and G(γ,t) are the linear and nonlinear relaxation modulus, respectively. The obtained values of the shift factors that make up the damping function (Figure 4.6) are fitted using the Wagner damping function,47 which for simple shear is reduced to h(γ) = exp(−nγ)                                                       (4.3) The Wagner damping function is used to construct constitutive equations to predict the shear and extensional flow of many polymers in polymer processing such as blow modeling.32,47 Good fit of the experimental results was obtained with the Wagner parameter with values n = 0.13–0.18, as depicted in Figure 4.6. Overall, the corresponding damping function values of ionomers are higher than those of their parent copolymers. In other words, the ionic interactions slow down the relaxation of the polymers, the more so for the lower molecular weight polymers. Note that the damping function for copolymers are the same here as in case of sodium parent copolymers. The values of the parameter “n” are listed in the legend of Figure 4.6. The two lines in Figure 4.6 correspond to predictions of Eq. 4.3 with the highest and lowest values of parameter n needed to describe all the experimental data of the various polymers.  31   Figure 4.6 The damping function of ionomers and their parent copolymers obtained from step-strain relaxation experiments and its fit using the Wagner model (Eq. 4.3) Relatively smaller values have been reported for the corresponding sodium ionomers (n = 0.11-0.17).30 Zinc ionomers form hydrogen bonding between two –COOHs, ionic bonding between Zn2+ and two –COO–, but no coordinate bonding (between Zn and –COOH). The higher damping functions in zinc ionomers compared to the sodium ionomers ones reported by Tomkovic et al.30  can be explained by the suggestion that the ionic bonding between Zn2+ and – two COO– acts as pseudo-long-chain branching point (physical crosslink), which also can be seen in the increase of the activation energies in zinc ionomers compared to the sodium ones (Table 3.1). This might be related with ion structures, where Zn2+ has two sites ( –2COO–, no acid cation exchange) and Na+ gives three sites (COO– and –2COOH, acid cation exchange).29  4.3.2 Uniaxial Extension As discussed above, using the SER2 fixture, uniaxial extension experiments were conducted at the reference temperature of 140°C at several Hencky strain rates, namely, 0.5, 2, T = 140 oCShear strain, g10-1 100 101Damping function, h10-110016.8 - Zn 33   0.1816.8 - Zn 0     0.139.7 - Zn 40     0.1310.8 - Zn 60   0.1710.8 - Zn 0     0.13Wagner modeln = 0.18n = 0.13n 32  and 5 s−1. The results for all zinc ionomers and their corresponding parent copolymers are presented in Figure 4.7. The results for the sodium ionomers have been reported elsewhere with the same Hencky strain rates used here.30 All ionomers and their corresponding parent copolymers exhibit strain hardening. The Weissenberg number in extension based on the reptation times τrep (listed in Table 4.1), is defined as WiE  ≡  ɛ̇H τrep. The values of WiE well exceed the value of 1 in all cases (note the large reptation times of ionomers and copolymers from Table 4.1), which justifies the strain hardening behavior in all cases. While the strain hardening of ionomers originates from the strong ionic associations, for the copolymers it is due to a combination of the effects of branching and hydrogen bonding. It is clear from Figure 4.7 that the strain hardening effects are stronger in the case of ionomers and these differences depend on the number of associations per chain, ZS. For example, compare the differences for samples 16.8-Zn 33 and 16.8-Zn 0 in Figure 4.7(c) (ZS = 26) and those of samples 10.8-Zn 60 and 10.8-Zn 0 in Figure 4.7(b) (ZS = 21). In addition, the uniaxial stress growth coefficient, 𝜂𝐸, is more abrupt in the case of ionomers compared to that of copolymers, which is due to the extra relaxation mode of the lifetime of associations at short time scales. Overall, zinc ionomers have a similar growth coefficient compared to sodium ionomers which were reported elsewhere.30 33   Figure 4.7 Comparison of uniaxial stress growth coefficient for zinc ionomers and their corresponding copolymers at different Hencky strain rates at temperature of 140 °C. (a) 9.7-Zn 40 (b) 10.8-Zn 60 and 10.8-Zn 0 (c) 16.8-Zn 33 and 16.8-Zn 0 To compare more conclusively the uniaxial behaviour of ionomers and their copolymers, the Strain Hardening Factor (SHF) can be used which is defined as ( , )( )3E tStrain Hardening Factor SHF  +=    (4.4) where 𝜂0+ is the shear stress growth coefficient. The SHF shows the fractional strain hardening compared to the linear viscoelastic envelope. It can clearly be seen now from Figure 4.8 that the strain hardening of zinc ionomers are stronger than that of their corresponding copolymers. 9.7 - Zn 40T = 140 oCTime, t (s)10-2 10-1 100 101Uniaxial Stress Growth Coefficient, hE (Pa.s)1031041051061071083h+e = 0.5 s-1e = 2 s-1e = 5 s-1...(a)10.8 - Zn 60 (filled symbols)10.8 - Zn 0 (unfilled symbols)T = 140 oCTime, t (s)10-2 10-1 100 101Uniaxial Stress Growth Coefficient, hE (Pa.s)1031041051061071083h+e = 0.5 s-1e = 2 s-1e = 5 s-1....(b)16.8 - Zn 33 (filled symbols)16.8 - Zn 0 (unfilled symbols)T = 140 oCTime, t (s)10-2 10-1 100 101Uniaxial Stress Growth Coefficient, hE (Pa.s)1031041051061071083h+e = 0.5 s-1e = 2 s-1e = 5 s-1...(c)34   Figure 4.8 Comparison of the extensional rheological properties of zinc ionomers and their corresponding copolymers using the Strain Hardening Factor (SHF) at different Hencky strain rates  temperature of 140 °C (left the zinc ionomers and right their copolymers)  4.4 Experimental Results on Capillary Flow  4.4.1 End Pressure and the Effect of Pressure on Viscosity The capillary flow of all ionomers at the temperature of 120-180 °C as a function of apparent shear rate in the range from 5 s-1 to 1000 s-1 for several dies having various L/D ratios was studied. To determine the viscosity from capillary experiments first the entry pressure effects should be determined to extract them from the total pressure drop. The Bagley plots (capillary flow pressure versus L/D ratio at several apparent shear rates) of a representative sodium and zinc polymers are shown in Figure 4.9 (see Appendix Figure A.1 for the corresponding Bagley plots for the other ionomers). The results show a slight upward curvature at higher apparent shear rates, which is consistent with the assumption of pressure-dependent viscosity (discussed below). Second-order polynomial was used to fit the data for each apparent shear rate and extrapolated to zero die length in order to determine the end pressure values. These are plotted in Figures 4.10 (a) and (b) as functions of the apparent shear rate and in Figure T = 140 oCTime, t (s)10-2 10-1 100 101Strain hardening factor10-11001011029.7 - Zn 4010.8 - Zn 6016.8 - Zn 33e = 0.5 s-1e = 2 s-1e = 5 s-1...T = 140 oCTime, t (s)10-2 10-1 100 101Strain hardening factor10-110010110210.8 - Zn 016.8 - Zn 0e = 5 s-1e = 2 s-1 e = 0.5 s-1...35  4.11 as functions of the wall shear stress. A first observation is that these entry pressures are much higher than those determined before for non-associating polymers.32,33,52,53 A second observation is that zinc and sodium ionomers have similar end pressures (Figure 4.10), which can better be seen if the data of Figure 4.10 is plotted as a function of the wall shear stress (Figure 4.11).   Figure 4.9 The Bagley plot for ionomers 11.5-Na 65 (left) and 9.7-Zn 40 (right)  samples at the reference temperature of 160 °C  Figure 4.10 The effect of apparent shear rates on the end pressure for all samples at 160 °C as determined by the Bagley method for the sodium (a) and zinc ionomers (b) 11.5 - Na 65T = 160 oC  gA (s-1)L/D5 10 15 20 25 30 35Pressure (MPa)02040608010012014016018051126641704001000.9.7 - Zn 40T = 160 oC  gA (s-1)L/D5 10 15 20 25 30 35Pressure (MPa)02040608051126641704001000.T = 160 oCD = 0.762 mmApparent shear rate, γA (s-1)100 101 102 103 104End pressure, Pend (MPa)10-11001011024.3 - Na 694.9 - Na 6311.5 - Na 6519.2 - Na 65(a).T = 160 oCD = 0.762 mmApparent shear rate, γA (s-1)100 101 102 103 104End pressure, Pend (MPa)10-11001011029.7 - Zn 4010.8 - Zn 6016.8 - Zn 33(b).36   Figure 4.11 The effect of wall shear stresses on the end pressure for all samples melt at 160 °C as determined by the Bagley method  4.4.2 Flow Curves  The entry pressures determined above were used to correct the pressure drop for the entry effects in order to calculate the true wall shear stress using Eq. 4.5. ( )4endwP PL D −=                                                             (4.5)                                                       Figure 4.12 plots the flow curves (true wall shear stress versus apparent shear rate) of the 11.5-Na 65 and 9.7-Zn 40 ionomers at 160 °C obtained with capillary dies of the same diameter and three different L/D ratios, namely 4, 15 and 33 having a contraction angle (die entry region) of 2α = 180°. The flow curves of both samples superposed well implying that Figures 4.10 and 4.11 give the correct end pressure values.  T = 160 oCD = 0.762 mmL/D = 15Wall shear stress, σW (MPa)10-2 10-1 100 101End pressure, Pend (MPa)10-11001011024.3 - Na 694.9 - Na 6311.5 - Na 6519.2 - Na 659.7 - Zn 4010.8 - Zn 6016.8 - Zn 3337   Figure 4.12 The Bagley corrected flow curves for 11.5-Na 65 (left) and 9.7-Zn 40 (right) melts at 160 °C for three dies with various L/D ratios. The solid lines represent the LVE data plotted as a flow curve at the same temperature  Essentially the capillary data are above the LVE curve due to effect of pressure of viscosity. The viscosity dependence on pressure is defined by an exponential function, known as the Barus equation, which is: ( )0exppppa p= =     (4.6) where βP is the pressure coefficient of viscosity, 𝜂P0 is the viscosity at ambient pressure, and p the absolute pressure. The values of the absolute pressures were assumed to be Δp / 2 based on the assumption that the pressure is decreasing linearly along with the die, where Δp (plotted in Figure 4.9) is the pressure drop along the whole length of the die including the die entrance. The experimental data obtained with capillary dies with high values of L/D were shifted vertically to superpose with the LVE data, and as a result, the shear-rate-dependent pressure coefficient βP at 11.5 -  Na 65T  = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Wall shear stress or Complex modulus, σW or G* (MPa)10-310-210-1100101L/D = 4L/D = 15L/D = 33LVE.9.7 - Zn 40T = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Wall shear stress or Complex modulus, σW or G* (MPa)10-310-210-1100101L/D = 4L/D = 15L/D = 33LVE.38  various levels of pressure was determined for all polymers (Figure 4.13). As previously reported52,54 βP depends on pressure, following a power-law equation: pnp mp−=          (4.7) The data and the fit of Eq. 4.7 to the data are plotted in Figure 4.13. As pressure increases, the βP decreases showing that the βP is more significant at lower pressures. Eq. 4.7 captures the coefficient βP and the parameters fitting the data are presented in the Figure (m=0.33 kPa0.067, np=-0.93) showing a weak dependence of viscosity on pressure similar to that reported previously.30 Figure 4.14 depicts the flow curves at various levels of L/D after applying the end pressure correction. The data superposes well, and the capillary data agrees well with the LVE data showing that the calculated coefficient βP is correct. The flow curves of other ionomers are plotted in Figure A.2 (Appendix).  Figure 4.13 The pressure-dependency coefficient of viscosity on pressure for all samples T = 160 oCPressure, P (kPa)102 103 104 105 106Pressure coefficient of viscosity, bP (kPa-1)10-610-510-410-3L/D = 4L/D = 15L/D = 33bP(kPa-1) = 0.33p-0.9339   Figure 4.14 The end-pressure corrected flow curves of 11.5-Na 65 (left) and 9.7-Zn 40 (right) samples at 160 °C  4.4.3 Wall Slip of Ionomers   The die diameter dependence has been studied in order to assess the slip behavior of ionomer. Figure 4.15 plots the flow curves of 9.7-Zn 40 and 19.2-Na 65 at the temperature of 140 °C determined by using three capillary dies with significantly different diameters and the same L/D ratio. The experimental results show that the flow curves superpose well, indicating that this polymer does not slip due to strong interactions between the chains at the interface and the wall (possible strong adsorption effects).32 Similar results observed for all samples showing that ionomers do not slip. 11.5 -  Na 65T  = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Pressure-corrected wall shear stress or complex modulus, σW/exp( bP) or G* (MPa)10-310-210-1100101L/D = 15L/D = 33LVE.9.7 -  Zn 40T = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Pressure-corrected wall shear stress or complex modulus, σW/exp( bP) or G* (MPa)10-310-210-1100101L/D = 15L/D = 33LVE.40   Figure 4.15 The effect of the die diameter to check the possibility of slip at the wall for 19.2-Na 65 (left) and 9.7-Zn 40 (right) at T = 140 °C. The solid lines represent the LVE results at the same temperature  4.5 Melt Fracture The melt fracture behavior of all ionomers was studied over a wide range of temperatures 120-180 °C. For a given ionomer there is a critical shear rate/shear rate at which extrudate distortions were observed. It is well known that there are two different types of melt fracture, namely surface melt fracture (sharkskin) and gross melt fracture with different origin.55 Typically polymers exhibiting strain hardening (high melt strength polymers) exhibit only gross melt fracture that originates from the capillary entry due to the high extensional stresses/strains developed in the entry region. Sharkskin originates at the capillary exits and involves small strains that melts of high strength can withstand.56-58 All ionomers exhibit strong strain hardening effects25 and thus do not have sharkskin melt fracture. However, they exhibit gross melt fracture at certain temperatures particularly in the range of 120-160 °C. Typical images for the zinc and sodium ionomers are shown in Figures 4.16 and 4.17, respectively, at 140 °C. It can be clearly 19.2 - Na 65T = 140 oCL/D = 15Apparent shear rate or Angular frequency, γA or w (s-1)100 101 102 103Wall shear stress or Complex modulus, σA or G* (MPa)10-310-210-1100101D = 0.432 mmD = 0.763 mmD = 1.22 mmLVE.9.7 - Zn 40T = 140 oCL/D = 15Apparent shear rate or Angular frequency, γA or w (s-1)100 101 102 103Wall shear stress or Complex modulus, σA or G* (MPa)10-310-210-1100101D = 0.432 mmD = 0.763 mmD = 1.22 mmLVE.41  seen that the transition from smooth extrudate to gross melt fracture occurs at a critical shear rate without the occurrence of sharkskin melt fracture. 9.7 – Zn 40, L/D = 15  A  = 11 s-1, Smooth  A  = 400 s-1, Smooth  A  = 1000 s-1, Gross 10.8 – Zn 60, L/D = 15  A  = 11 s-1, Smooth  A = 100 s-1, Gross  A  = 1000 s-1, Gross 16.8 – Zn 33, L/D = 15  A  = 11 s-1, Smooth  A  = 26 s-1, Gross  A  = 1000 s-1, Gross  Figure 4.16 Images of zinc extrudates from the capillary extrusion experiment at 140 °C. The apparent shear and appearance are indicated in each picture   42  4.3 – Na 69, L/D = 15  A  = 100 s-1, Smooth  A  = 260 s-1, Gross  A  = 1000 s-1, Gross 4.9 – Na 63, L/D = 15  A  = 100 s-1, Smooth  A  = 170 s-1, Gross  A  = 1000 s-1, Gross 11.5 – Na 65, L/D = 15  A  = 100 s-1, Smooth  A  = 170 s-1, Gross  A  = 1000 s-1, Gross 19.2 – Na 65, L/D = 15  A  = 100 s-1, Smooth  A  = 260 s-1, Gross  A  = 1000 s-1, Gross  Figure 4.17 Images of sodium extrudates from the capillary extrusion experiment at 140 °C. The apparent shear and appearance are indicated in each picture 43  Another way to summarize the results is by presenting the processability maps. Figures 4.18 (a) and (b) depict the processability maps of all ionomers at the temperature of 140 °C. Gross melt fracture (GMF) occurs at lower shear rates with decrease in temperature, i.e. see the filled symbols indicating the conditions that gross melt fracture occurs. Gross melt fracture is phenomenon that makes it impossible to produce extrudate having a smooth surface at a high extrusion rate in high-speed processes.59 It is clear from the processability maps that sodium ionomers exhibit superior processability in capillary extrusions compared to zinc ionomers as they fracture at significantly higher shear rates. For example, both 16.8-Zn 33 and 11.5-Na 65 have almost the same viscosity curve, but the latter exhibits a higher critical shear rate of 170 s-1 compared to 26 s-1 for 16.8-Zn 33 sample. This is due partly to the highest Mw of 16.8-Zn 33 compared to that of 11.5-Na 65. Although previous studies showed that molecular weight and molecular characteristics play important roles in the flow instabilities of polymer melts, in the case of associating polymers the number of associations and lifetime of associations which both affect the reputation time should also be considered as critical parameters (discussed below).60 It was found for all samples that as the reptation time ( τrep) decreases; therefore samples exhibit better processability. For instance, sample 9.7-Zn 40 has a smaller reptation time of 22 s, and its critical shear rate is 640 s-1 compared to 26 s-1 and 100 s-1 for 10.8-Zn 60 and 16.8-Zn 33, respectively.  44   Figure 4.18 The processability maps of all polymer melts as a function of apparent shear rate at 140 °C in capillary extrusions for the sodium (a) and zinc ionomers (b) Another way to represent the melt fracture data is by plotting the flow curves at different temperatures. Figures 4.19 and 4.20 plot flow curves of all sodium and zinc ionomers at various temperatures. Open symbols represent smooth extrudates, whereas filled represent sample with gross melt fracture. While the dependence of the critical shear rate for the onset of gross melt fracture depends strongly on temperature increases, the dependence of the critical shear stress on temperature is less sensitive, and in some cases it is indeed independent.     Die: L/D=15, D = 0.762 mm, 2a=180o m T = 140 oCApparent shear rate, γA (s-1)100 101 102 10319.2 -  Na 654.3 -  Na 694.9 -  Na 6311.5 -  Na 65SmoothGross.Die: L/D=15, D = 0.762 mm, 2a=180o m T = 140 oCApparent shear rate, γA (s-1)100 101 102 1039.7 -  Zn 4010.8 -  Zn 6016.8 -  Zn 33SmoothGross.45   Figure 4.19 The flow curves of all sodium ionomers at 120-180 °C for L/D = 15. The filled and open symbols represent gross melt fracture and no melt fracture, respectively 4.3 -  Na 69Die: L/D=15, D = 0.762 mm, 2a=180oApparent shear rate, γA (s-1)101 102 103Apperant shear stress, σA (MPa)10-210-1100101T = 120 oCT = 140 oCT = 160 oCT = 180 oC(a).4.9 -  Na 63Die: L/D=15, D = 0.762 mm, 2a=180oApparent shear rate, γA (s-1)101 102 103Apperant shear stress, σA (MPa)10-210-1100101T = 120 oCT = 140 oCT = 160 oCT = 180 oC(b).11.5 -  Na 65Die: L/D=15, D = 0.762 mm, 2a=180oApparent shear rate, γA (s-1)101 102 103Apperant shear stress, σA (MPa)10-210-1100101T = 120 oCT = 140 oCT = 160 oCT = 180 oC(c).19.2 -  Na 65Die: L/D=15, D = 0.762 mm, 2a=180oApparent shear rate, γA (s-1)101 102 103Apperant shear stress, σA (MPa)10-210-1100101T = 120 oCT = 140 oCT = 160 oCT = 180 oC(d).46   Figure 4.20 The flow curves of all zinc ionomers at 120-180 °C for L/D = 15. The filled and open symbols represent gross melt fracture and no melt fracture, respectively Table 4.2 lists the critical shear rates, ?̇?A,C, and critical shear stresses, σ𝐶 , for the onset of gross melt fracture at 120 °C, 140 °C and 160 °C. At the higher temperature of 160 °C, ionomers 4.3–Na 69, 9.7–Zn 40, and 10.8–Zn 60 did not exhibit melt fracture at all shear rates tested. At the highest temperature of 180 °C, no melt fracture was observed for neither of the ionomers, although this is a high temperature for ionomers amenable to degradation.   9.7 - Zn 40Die: L/D=15, D = 0.762 mm, 2a=180oApparent shear rate, γA (s-1)101 102 103Apperant shear stress, σA (MPa)10-210-1100101T = 120 oCT = 140 oCT = 160 oCT = 180 oC(a).10.8 - Zn 60Die: L/D=15, D = 0.762 mm, 2a=180oApparent shear rate, γA (s-1)101 102 103Apperant shear stress, σA (MPa)10-210-1100101T = 120 oCT = 140 oCT = 160 oCT = 180 oC(b).16.8 - Zn 33Die: L/D=15, D = 0.762 mm, 2a=180oApparent shear rate, γA (s-1)101 102 103Apperant shear stress, σA (MPa)10-210-1100101T = 120 oCT = 140 oCT = 160 oCT = 180 oC(c).47  Table 4.2 The onset of flow instabilities (critical shear rate/stress) for the studied melts using a capillary die with L/D = 15 Sample Critical shear rate, ?̇?𝐀,𝐂 (s-1) Critical shear stress, σ𝑪 (MPa) 120 °C 140 °C 160 °C 120 °C 140 °C 160 °C 4.3 – Na 69 40 260 - 0.91 0.85 - 4.9 – Na 63 40 170 400 0.42 0.38 0.36 11.5 – Na 65 40 170 640 1.38 0.82 0.75 19.2 – Na 65 64 260 640 0.43 0.41 0.33 9.7 – Zn 40 170 640 - 1.11 0.97 - 10.8 – Zn 60 26 100 - 1.02 0.86 - 16.8 – Zn 33 11 26 1000 0.85 0.83 0.80  Figure 4.21 examines the existence of a possible correlation between the critical shear stress for the onset of melt fracture, σ𝐶 , (listed in Table 4.2) and the lifetime of associations, τs, (listed in Table 4.2).  It was noted that the lifetime of associations scales with 2/ ( )S rep E SZ Z =and thus takes into account the terminal relaxation time as well as the molecular parameters of the number of entanglements and associations.28,30 Within the context of this correlation it seems that the number of associations and the number of entanglements dominate the melt fracture behavior of these polymers. The higher the ZE and ZS the lower the critical shear stress for the onset of melt fracture (easier to fracture). It was noted that melt fracture is a characteristic of 48  linear polymers of low entanglement molecular weight (high ZE). Therefore, the correlation (continuous line in Figure 4.21) seems to result the correct trend, σ𝐶 = 2.12 τs1/5, ( τs in s and σ𝐶  in MPa).     Figure 4.21 The critical shear stress for the onset of melt fracture as a function of the lifetime of associations, for all ionomers as well as all temperatures, studied using capillary rheometry          D = 0.762 mmThe association lifetime, tS (s)10-4 10-3 10-2 10-1Critical shear stress, σC (MPa)10-1100101T = 120 oCT = 140 oCT = 160 oC49  Chapter 5:  Conclusions and Recommendations 5.1 Conclusions The rheology of entangled polydisperse polyethylene ionomers was studied using a parallel-plate rheometer equipped with a partitioned plate, and the Sentmanat extensional fixture (SER). The ionomers were unneutralised by removing the ions to produce their parent copolymers in order to examine the relative effects of ionic interactions. Particular emphasis has been placed on the distribution of the relaxation times to identify the characteristics times such as reptation, Rouse and lifetime of the ionic associations which are related to the number of entanglements and reversible ionic associations. As such scaling laws have been used to calculate the order of magnitude of these time scales that are important parameters for their rheological modeling. The lifetime of associations, τs, was found to scale with the number of associations per chain, ZS, τs ∝ Zs -2 and to be of the order of 10-4 to 10-2 s using three different methods showing that their effects at time scales greater than 10-2 s are significant. It noted that for all ionomers the number of reversible associations is smaller than the number of entanglements per chain, that is segments between associations are larger than those between entanglements. As the lifetimes of association are significant in the case of ionomers, they can be used in assessing the processability of these polymers. To study the effects of ionic associations in more detail, the commercial ionomers were completely unneutralized and their rheological behavior was compared directly with their associative counterparts (parent copolymers). The rheological comparison included the linear viscoelastic moduli, the damping function, and extensional rheology showing the effects of ionic interactions.  Overall due to the presence of associations the relaxation processes slow down, and 50  the modulus shifts to significantly higher values, the more so as the number of ionic associations increase. An increase of one order of magnitude was observed. However, both sodium and zinc ionomers almost have the same relaxation modulus. As a result of these ionic associations, extensional experiments have shown that the tensile stress growth coefficients of two zinc ionomers samples are 1 to 1.5 orders of magnitude higher than the parent copolymer. The latter only exhibit a small degree of strain hardening. At short times, the extensional results follow the linear viscoelastic envelope (LVE), whereas at long times, there is a significant increase in extensional viscosity (strain hardening effect), which clearly originates from the strong effect of hydrogen bonding interactions (in case of copolymers) and from the effect of ionic associations (in case of zinc ionomers). These commercial ionomers have been also studied in capillary rheometry to assess their processability in terms of flow instabilities, such as the occurrence wall slip and extrudate distortion known as melt fracture. It was found that their entry pressure is high, compared to non-associating polymers, indicating the significant effect of the ionic interactions on the entry pressure. Also, the capillary experiments have shown a distinct, relatively small pressure-dependency of viscosity with a pressure coefficient to be a power-law function of pressure similar to all ionomers, both zinc and sodium ones. Using capillary dies of different diameters, it was shown that these polymers do not slip over solid boundaries possibly due to strong interactions of ionic associations with solid boundaries.  Capillary extrusion experiments have shown that these ionomers exhibit only gross melt fracture and no surface (sharkskin) melt fracture. This is a characteristic of polymers exhibiting strong extensional strain hardening effects, which is the case for strong associating polymers 51  such as the present ones. The gross melt fracture was found to occur at a critical shear stress value independent of temperature. Finally, the critical shear stress, σ𝐶 , was found to depend on the lifetime of associations, σ𝐶 = 2.12 τs1/5, independent of the molecular weight and type of ion (zinc or sodium). This correlation indicates that entanglements and, more importantly, the ionic associations play a significant role in melt fracture. In other words, highly entangled and highly associated polymers fracture at lower shear stresses and therefore lower shear rates.  5.2 Recommendations for Future Work In this section recommendations for future work are listed to help address some issues related to the rheology of poly(ethylene- co -methacrylic acid) ionomers. First, ions such as Li1+ , Ca2+ and Mg2+ that can be associated with the MAA groups can be further examined with a particular focus on their capillary flow and processability since each ion act rheologically differently. Additionally, controlling the number of reversible associations can give a better insight into the relaxation processes. This could be achieved by synthesizing copolymers or making blends having more than one ionomer type with different valency.  Second, the degree of neutralization or  molecular weight of all samples in this work were not fixed. A more systematic study of the degree of neutralization at fixed molecular weight can be interesting to examine the effect of acid level on the rheological properties.  Finally, the study of normal stress differences and how it is affected by ionic interactions would be another important. This would definitely contribute to the understanding of the nonlinear rheology of different ionomer samples.  52  References  (1) Eisenberg, A.; Kim, J.-S. “Introduction to Ionomers,” John Wiley & Sons, New York,     USA (1998).  (2) Rees, R. W.; Vaughan, D. J. “Physical Structure of Ionomers.” Polym. Prepr. Am. Chem. Soc. Div. Polym. Chem. (1965), 6, 287–295. (3) Tant, M.R.; Wilkes, G.L. “An overview of the viscous and viscoelastic behavior of ionomers in bulk and solution,” J. Macromol. Sci., Rev. Macromol. Chem. Phys., C28, 1–63 (1988). (4) Zhong, X. F.; Eisenberg, A. “Aggregation and Critical Micellization Behavior of Carboxylate-Terminated Monochelic Polystyrene.” Macromolecules (1994), 27, 1751–1758. (5) Ro, A. J.; Huang, S. J.; Weiss, R. A. “Synthesis and Thermal Properties of Telechelic Poly(Lactic Acid) Ionomers.” Polymer (2008), 49, 422–431. (6) Tant, M.R.; Mauritz, K.A.; Wilkes, G.L. “Ionomers Synthesis, Structure, Properties   and Application,” Blackie Academic Press, London, UK (1997). (7) Otocka, E.P.; Kwei T.K. “Properties of Ethylene-Metal Acrylate Copolymers,” Macromolecules, 1, 401–405 (1968).  (8) Leibler, L.; Rubinstein M.; and Colby, R.H. “Dynamics of Reversible Networks,” Macromolecules, 24, 4701-4707 (1991). (9) Rubinstein, M.; Semenov, A. N. “Dynamics of Entangled Solutions of Associating Polymers.” Macromolecules (2001), 34, 1058–1068. (10) Holliday, L. Ionic Polymers, (John Wiley & Sons, New York, USA 1975). 53  (11) Chen, Q.; Tudryn, G.J.; Colby, R.H. “Ionomer dynamics and the sticky Rouse model,” J. Rheology, 57, 1441-1462 (2013). (12) Zhang, Z., Huang, C.; Weiss, R.A.; Chen, Q. “Association energy in strongly associative polymers,” J. Rheology, 61, 1199-1207 (2017). (13) Statz, R. “Ethylene Copolymer Ionomers.” In History of Polyolefins: The World’s Most Widely Used Polymers; Seymour, R. B., Cheng, T., Eds.; D. Reidel Publishing Company: Dordrecht, (1986); pp 177–192. (14) Yeager, H.L.; Gronowski, A.A. “Membrane applications”. In Ionomers: Synthesis, Structure, Properties and Applications; Tant, M.R., Mauritz, K.A., Wikes, G.L., Eds., (Blackie Academic Press, London, UK 1997). (15) Gao, Z.; Molnar, A.; Eisenberg, A. “Blend Compatibilization”. In Ionomers: Synthesis, Structure, Properties and Applications. Tant, M.R., Mauritz, K.A., Wikes, G.L., Eds., (Blackie Academic Press, London, UK 1997) (16) Zhang, L.; Brostowitz, N.R.; Cavicchi, K.A.; Weiss, R.A. “Perspective: Ionomer Research and Applications,”  Macromol. React. Eng., 8, 81-99 (2014). (17) Dealy, J.M.; Wissbrun, K.F. Melt rheology and its role in plastics processing - Theory and applications, Van Nostrand Reinhold, New York, 1990. (18) Macosko C.W. “Rheology: Principles, Measurements and Applications,” Wiley-VCH, New York, 1994. (19) Snijkers, F.; Vlassopoulos D. “Cone-partitioned-plate geometry for the ARES rheometer with temperature control,” J. Rheol., 55, 1167–1186 (2011). (20) Sentmanat, M.L. “Dual windup extensional rheometer,” US Patent No. 6,578,413 B2, 2003. 54  (21) Sentmanat, M.L. “Miniature universal testing platform: From extensional melt rheology to solid-state deformation behavior,” Rheol. Acta, 43,  657–669 (2004). (22) Register, R.A.; Cooper, S.L. Macromolecules 1990; 23(1): 318 e23. (23) Ward, T.C.; Tobolsky, A. V. “Viscoelastic study of Ionomers,” J. Appl. Polym. Sci., 11,  2403–2415 (1967). (24) Bonner, F.; Bonotto, S.; Bonner, E. F. “Effect of Ion Valency on the Bulk Physical Properties of Salts of Ethylene-Acrylic Acid Copolymers.” Macromolecules (1968), 1, 510–515. (25) Hirasawa, E.; Yamamoto, Y.; Tadano, K.; Yano, S. “Effect of Metal Cation Type on the Structure and Properties of Ethylene Ionomers.” J. Appl. Polym. Sci. (1991), 42, 351–362. (26) Spencer, M. W.; Wetzel, M. D.; Troeltzsch, C.; Paul, D. R. “Effects of Acid Neutralization on the Properties of K+ and Na+ Poly(Ethylene-Co-Methacrylic Acid) Ionomers.” Polymer (2012), 53, 569–580. (27) Winey, K. I.; Laurer, J. H.; Kirkmeyer, B. P. “Ionic Aggregates in Partially Zn-Neutralized Poly (Ethylene-Ran-Methacrylic Acid) Ionomers : Shape, Size, and Size Distribution.” 176 Macromolecules (2000), 33, 507–513. (28) Hinton, Z.R.; Alvarez, N.J. “The trade-off between processability and performance in commercial ionomers,” Rheologica Acta, 58 499–511 (2019). (29) Vanhoorne, P.; Register, R. A. “Low-Shear Melt Rheology of Partially-Neutralized Ethylene−Methacrylic Acid Ionomers,” Macromolecules,  29, 598–604 (1996). (30) Tomkovic, T.; Hatzikiriakos, S.G. “Nonlinear Rheology of Poly(ethylene-co-methacrylic acid) Ionomers,” J. Rheology, 62, 1319-1329 (2018) 55  (31) Nishioka, A.; Takahashi, T.; Masubuchi, Y.; Takimoto, J.; Koyama, K. “Rheological characterization of ionic bonding in ethylene-ionomer melts with low neutralization degree” J. of Rheology 46  1325 (2002)  doi: 10.1122/1.1516787 (32) Tomkovic, T.; Mitsoulis, E.; Hatzikiriakos, S.G. “Contraction Flow of Ionomers,” J. Non-Newtonain Fluid Mech., 262, 131-141 (2018) (33) Tomkovic, T.; Mitsoulis, E.; Hatzikiriakos, S.G. “Contraction Flow of Ionomers and their Corresponding Copolymers: Ionic and Hydrogen Bonding Effects,” Physics of Fluids, 31, 033102 (2019) (34) Ramamurthy, A. V. “Wall slip in viscous fluids and influence of materials of construction,” J. Rheol. 30, 337-357 (1986). (35) Hatzikiriakos, S. G; Dealy, J. M. “Role of slip and fracture in the oscillating flow of HDPE in a capillary,” J. Rheol. 36, 845-884 (1992a). (36) Vlachopoulos, J.; Lidorikis, S. “Melt fracture of polysturene,” Polymer Eng. Sci. 11, 1-5 (1971). (37) Wang, S. G; Drda, P. A. “Stick-slip transition in capillary flow of linear polyethylene:3. Surface conditions,” Rheol. Acta 36, 128-134 (1997). (38) Kazatchkov, I. B.; Hatzikiriakos, S. G.; Stewart, C. W. “ Extrudates distortion in the capillary/slit extrusion of a molten polypropylene,”  Polymer Eng. Sci. 35, 1864-1871 (1995). (39) Allal, A.; Vergnes, B. “Molecular design to eliminate sharkskin defect for linear polymers,” J. Non-Newtonian Fluid Mech. 146, 45-50 (2007). 56  (40) Tierney, N.K.; Register, R.A. “Ion hopping in ethylene-methacrylic acid ionomer melts as probed by rheometry and cation diffusion measurements,” Macromolecules, 35, 2358–2364 (2002).  (41) Han K.; Williams, H.L. “Ionomers: The sodium salt of poly(ethylene-co-methacrylic acid),” J. Appl. Polym. Sci. 38 73–86 (1989). (42) Haslam, J.; Willis, H.A.; Squirrell, D. C. M. “Identification and Analysis of Plastics,” 2nd ed. (Liffe, London, 1972). (43) Earnest, T.R.; MacKnight, W.J. “Effect of Hydrogen bonding and ionic aggregation on the melt rheology of an Ethylene - Methacrylic Acid copolymer and its sodium salt,” J. Polym. Sci. Polym. Phys. Ed., 16, 143–157 (1978). (44) Münstedt, H.; Schmidt, M.; Wassner, E. “Stick and slip phenomena during extrusion of polyethylene melts as investigated by laser-Doppler velocimetry,” Journal of Rheology (1978-present), 44 413-427 (2000). (45) Shabbir, A.; Huang, Q.; Baeza, G.P.; Vlassopoulos, D.; Chen, Q.; Colby, R.H.; Alvarez, N.J.; Hassager, O. “Nonlinear shear and uniaxial extensional rheology of polyether-ester-sulfonate copolymer ionomer melts,” J. Rheology, 61, 1279-1289 (2017)  (46) Sakamoto, K.; MacKnight, W.J.; Porter, R.S. “Dynamic and steady shear melt rheology of an ethylene-methacrylic acid copolymer and its salts,” J. Polym. Sci. Part A-2., 8, 277–287 (1970). (47) Wagner, M. H. “Analysis of stress-growth data for simple extension of a low-density branched polyethylene melt,” Rheol. Acta 15, 133–135 (1976).  (48) Chen, Q.; Zhang, Z.; Colby, R.H. “Viscoelasticity of Entangled random Polystyrene 57  Ionomers,” J. Rheology, 60, 1031-1040 (2016)  (49) Janzen J.; Colby, R.H. “Diagnosing long-chain branching in polyethylenes,” J. Molecular Structure, 485-486, 569-584 (1999). (50) Vega, J. F.; Rastogi, S.; Peters, G. W. M; and Meijers, H. E. H. “Rheology and Reptation of Linear Polymers: Ultrahigh Molecular Weight Chain Dynamics in the Melt,” J. Rheol., 48, 663-678 (2004). (51) Kurian T.; Nishio, M., Nishioka, A.; Takahashi, T.; Koyama, K. “Dynamic Melt Rheological Properties of Ionomers Based on Poly(Ethylene-co-Acrylic Acid) and Poly(Ethylene-co-Methacrylic Acid),” Intern. J. Polym. Materials, 56, 135-145 (2007) (52) Ansari, M.; Zisis, T.; Hatzikiriakos, S.G.; Mitsoulis, E. “Capillary flow of low‐density polyethylene,” Polymer Engineering & Science, 52, 649-662 (2012).  (53) Ansari, M.; Zisis, T.; Hatzikiriakos, S.G.; Mitsoulis, E. “Slip effects in HDPE flows,” J. Nonnewton. Fluid Mech., 167–168, 18–29 (2012).doi:10.1016/j.jnnfm.2011.09.007.  (54) Liang, J.Z. “Pressure effect of viscosity for polymer fluids in die flow,” Polymer (Guildf). 42, 3709–3712 (2001). doi:10.1016/S0032-3861(00)00507-3.  (55) Verges, B., “Extrusion Defects and Flow Instabilities of Molten Polymers,” International Polymer Processing, 30, 3-28 (2015) .  (56) Delgadillo-Velázquez, O.; Georgiou, G.; Sentmanat, M.; Hatzikiriakos, S.G. “Sharkskin and Oscillating Melt Fracture: Why in slit and capillary dies and not in annular dies?” Polymer Eng Sci., 48, 405-414 (2008). (57) Sentmanat, M.; Hatzikiriakos, S.G. “Mechanism of Gross Melt Fracture Elimination in the Extrusion of Polyethylenes in the Presence of Boron Nitride,” Rheologica Acta., 43, 624-633 (2004). 58  (58) Sentmanat, M.; Muliawan, E.B.; Hatzikiriakos, S.G. “Fingerprinting the Processing Behavior of Polyethylenes from Transient Extensional Flow and Peel Experiments in the Melt State,” Rheologica Acta, 44, 1-15 (2005). (59) Kim, S.; and Dealy, J.M. “Gross melt fracture of polyethylene. I: A criterion based on tensile stress,” 42, 482-494 (2002).  (60) Ebrahimi, M.; Tomkovic, T.; Liu, G.; Doufas, A.K.; Hatzikiriakos, S.G. “Melt Fracture of Linear Low-Density Polyethylenes: Die Geometry and Molecular Weight Characteristics,” Physics of Fluids, 30, 053103 (2018).                59  Appendix: Supporting Information for Chapter 4      Table A.1 Maxwell model parameters for zinc ionomers Sample 𝝀𝒊, 𝒔 𝑮𝒊, Pa 9.7 – Zn 40 1.0 x 10-3 3.77 x 105 9.14 x 10-3 1.54 x 105 0.068 6.39 x 104 0.475 2.12 x 104 3.26 4.16 x 103 22.1 4.30 x 102 10.8 – Zn 60 1.5 x 10-3 4.25 x 105 0.015 1.45 x 105 0.115 4.97 x 104 0.896 1.39 x 104 6.75 3.01 x 103 50.1 3.87 x 102 16.8 – Zn 33 1.68 x 10-3 5.41 x 105 0.019 2.64 x 105 0.176 1.19 x 105 1.46 3.92 x 104 12.58 8.44 x 103 133.8 1.14 x 103       60  Table A.2 Maxwell model parameters for zinc copolymers Sample 𝝀𝒊, 𝒔 𝑮𝒊, Pa 10.8 – Zn 0 2.46 x 10-4 3.72 x 105 2.58 x 10-3 1.03 x 105 0.021 3.78 x 104 0.17 1.19 x 104 1.33 2.95 x 103 11.5 4.59 x 102 16.8 – Zn 0 3.55 x 10-4 1.14 x 106 3.0 x 10-3 3.47 x 105 0.02 1.19 x 105 0.146 3.57 x 104 1.0 8.56 x 103 14.43 1.35 x 103         61      Figure A.1 The Bagley plot for 4.3-Na 69, 4.9-Na 63, 19.2-Na 65, 10.8-Zn 60 and 16.8-Zn 33 samples at the reference temperature of 160 °C  4.3 - Na 69Tref = 160 oC  gA (s-1)L/D5 10 15 20 25 30 35Pressure (MPa)020406080100120511266417040010004.9 - Na 63Tref = 160 oC  gA (s-1)L/D5 10 15 20 25 30 35Pressure (MPa)0204060805112664170400100019.2 - Na 65Tref = 160 oC  gA (s-1)L/D5 10 15 20 25 30 35Pressure (MPa)02040605112664170400100010.8 - Zn 60Tref = 160 oC  gA (s-1)L/D5 10 15 20 25 30 35Pressure (MPa)0204060801001201405112664170400100016.8 - Zn 33Tref = 160 oC  gA (s-1)L/D5 10 15 20 25 30 35Pressure (MPa)02040608010012014016051126641704001000 62    Figure A.2 The end-pressure corrected flow curves of 4.3-Na 69, 4.9-Na 63, 19.2-Na 65, 10.8-Zn 60 and 16.8-Zn 33 samples at 160 °C  4.3 -  Na 69Tref = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Pressure-corrected wall shear stress or complex modulus, σW/exp( bP) or G* (kPa)101102103L/D = 4L/D = 15L/D = 33LVE.4.9 -  Na 63Tref = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Pressure-corrected wall shear stress or complex modulus, σW/exp( bP) or G* (kPa)101102103L/D = 4L/D = 15L/D = 33LVE..19.2 -  Na 65Tref = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Pressure-corrected wall shear stress or complex modulus, σW/exp( bP) or G* (kPa)101102103L/D = 4L/D = 15L/D = 33LVE.10.8 -  Zn 60Tref = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Pressure-corrected wall shear stress or complex modulus, σW/exp( bP) or G* (kPa)101102103L/D = 4L/D = 15L/D = 33LVE..16.8 -  Zn 33Tref = 160 oCD = 0.762 mmApparent shear rate or Angular frequency, γA or w (s-1)101 102 103Pressure-corrected wall shear stress or complex modulus, σW/exp( bP) or G* (kPa)101102103L/D = 4L/D = 15L/D = 33LVE.

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