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Flow-induced crystallization of high-density polyethylene : the effects of shear, uniaxial extension… Derakhshandeh, Maziar 2011

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Flow-Induced Crystallization of High-Density Polyethylene: The Effects of Shear, Uniaxial Extension & Temperature by  Maziar Derakhshandeh B.A.Sc., Shiraz University, 2009  A THESIS SUBMITTED IN PARTIAL FULLFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF  Master of Science  in  The Faculty of Graduate Studies (Chemical and Biological Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) September 2011 © Maziar Derakhshandeh, 2011  Abstract The effects of shear, uniaxial extension and temperature on the flow-induced crystallization of two different types of two high-density polyethylenes (a metallocene and a Ziegler-Natta HDPE) are examined using rheometry. Shear and uniaxial extension experiments were performed at temperatures below and well above the peak melting point of the polyethylene’s in order to characterize their flow-induced crystallization behavior at rates relevant to processing. Generally, strain and strain rate found to enhance crystallization in both shear and elongation. In particular, extensional flow was found to be a much stronger stimulus for polymer crystallization compared to shear. At temperatures well above the melting peak point (up to 25°C), polymer crystallized under elongational flow, while there was no sign of crystallization under simple shear. A modified Kolmogorov crystallization model (Kolmogorov AN (1937) On the statistics of the crystallization process on metals. Bull Akad Sci. USSR, Class Sci, Math Nat. 1:355– 359) proposed by Tanner (Tanner RI (2009) Stretching, shearing and solidification, Chem Eng Sci, 64:4576-4579) was used to describe the crystallization kinetics under both shear and elongational flow. The model was found to predict the FIC behaviour under low deformation rates and various temperatures well; however the predictions for the higher rates were not satisfactory.  ii  Table of Contents Abstract .............................................................................................................................. ii Table of Contents ............................................................................................................. iii List of Tables ..................................................................................................................... v List of Figures ................................................................................................................... vi Nomenclature .................................................................................................................... x Acknowledgments .......................................................................................................... xiii Dedication ....................................................................................................................... xiv 1. Introduction ................................................................................................................... 1 2. Rheological Measurements .......................................................................................... 4 2.1.  Shear Rheometry..................................................................................................... 4  2.2.  Extensional Rheometry ........................................................................................... 8  3. Flow Induced Crystallization ..................................................................................... 14 3.1.  Flow Induced Crystallization Studies ................................................................... 14  3.2.  Flow Induced Crystallization Models ................................................................... 16  4. Thesis Objectives ......................................................................................................... 21 5. Materials and Methodology ....................................................................................... 22 6. Results and Discussion................................................................................................ 25 6.1.  Thermal Behaviour/Quiescent Crystallization ...................................................... 25  6.2.  Flow Induced Crystallization (FIC) in Steady Shear ............................................ 26  6.3.  Flow Induced Crystallization (FIC) in Shear Stress Relaxation ........................... 35  6.4.  Flow Induced Crystallization (FIC) in Uniaxial Extension .................................. 37  6.5.  Tensile Stress Relaxation after Cessation of Steady Uniaxial Extension ............. 43  6.6.  Modeling ............................................................................................................... 46  6.6.1.  Isothermal FIC Modeling................................................................................. 46  6.6.2.  Non-Isothermal FIC Modeling ........................................................................ 50  6.6.3.  Generalization of Crystallization for Mixed Flows ......................................... 60  7. Conclusions .................................................................................................................. 62 8. Recommendation for Future Work ........................................................................... 64 Bibliography .................................................................................................................... 65  iii  Appendix A ..................................................................................................................... 75 Appendix A.1. .................................................................................................................. 75 Appendix A.2. .................................................................................................................. 77  iv  List of Tables Table 1: Schematics of different techniques used for generating shear flow .................... 6 Table 2: Schematics of different methods used to generate extensional flow ................. 10 Table 3: List of HDPEs used in this study and their molecular characteristics ............... 22 Table 4: The estimated parameters of crystallization model (Eq. 3.7) fitted to experimental results ........................................................................................... 47 Table 5: The estimated parameters of crystallization model (Eq. 3.7) fitted to experimental results ........................................................................................... 51  v  List of Figures Figure 1: Simple illustrations of two of the central concepts: (a) the historical fringed micelle crystal; (b) the folded chain crystal ....................................................... 2 Figure 2: Transmission electron micrograph showing lamellar crystals ........................... 3 Figure 3: Schematic of the Sentmanat Extensional Rheometer (Sentmanat, 2003a)....... 12 Figure 4: Schematic of two different types of crystal structure ....................................... 15 Figure 5: The DSC thermograph of ZN-HDPE. The melting crystallization under quiescent conditions was observed at 119.6°C. The melting and cooling peaks were 128.5°C and 116°C, respectively .............................................................. 25 Figure 6: The DSC thermograph of m-HDPE. The melting crystallization under quiescent conditions was observed at 117.6°C. The melting and cooling peaks were 127.3°C and 115°C, respectively .............................................................. 26 Figure 7: The shear stress growth coefficient of ZN-HDPE at temperatures 124.5°C, 125°C and 125.5°C at constant shear rate of 0.01s-1 ......................................... 27 Figure 8: The shear stress growth coefficient η+ of ZN-HDPE at different shear rates at 125.5 °C. ........................................................................................................... 28 Figure 9: The shear stress growth coefficient of ZN-HDPE at shear rates of 0.01, 0.02, 0.06 and 0.1s-1 and at 125.5°C. ......................................................................... 29 Figure 10: Comparison of the dependency of induction time with that of critical strain on the shear rate at 125.5°C.................................................................................. 30 Figure 11: The shear stress growth coefficient of m-HDPE at temperatures 123.0°C, 123.5°C and 124.0°C at constant shear rate of 0.02s-1..................................... 31 Figure 12: The shear stress growth coefficient η+ of m-HDPE at different shear rates at 123.0 °C. .......................................................................................................... 31 Figure 13: The shear stress growth coefficient of m-HDPE at shear rates of 0.01, 0.02, 0.06 and 0.1s-1 at 123.0°C ............................................................................... 32 vi  Figure 14: Comparison of the dependency of induction time and that of critical strain of m-HDPE for the onset of crystallization on the shear rate at 123.0°C. ........... 32 Figure 15: The effect of shear rate on the induction time of ZN-HDPE at various temperatures. ̇ is the critical shear rate which is needed for effective FIC . 33 Figure 16: The effect of shear rate on the induction time of m-HDPE at various temperatures. ̇ is the critical shear rate which is needed for effective FIC . 34 Figure 17: The effect of shear rate on the induction time at various temperatures. ̇ is the critical shear rate which is needed for effective FIC ................................ 35 Figure 18: The Relaxation behavior of ZN-HDPE under different levels of strain at 125.5°C.  shows the shearing time of specimens ......................................... 36  Figure 19: The Relaxation behavior of m-HDPE under different levels of strain at 123°C. shows the shearing time of specimens ........................................................ 37 Figure 20: The effects of temperature and Hencky strain rate on the flow-induced crystallization of ZN-HDPE ........................................................................... 38 Figure 21: The effects of temperature and Hencky strain rate on the flow-induced crystallization of m-HDPE .............................................................................. 39 Figure 22: Tensile stress growth coefficient,  of the ZN-HDPE as a function of  °  Hencky strain at 125 C.................................................................................... 40 Figure 23: Tensile stress growth coefficient,  of the m-HDPE as a function of Hencky  °  strain at 123 C ................................................................................................. 41 Figure 24: The effect of temperature and Hencky strain rate on the onset of crystallization .................................................................................................. 42 Figure 25: The effect of Hencky strain rate and temperature on the onset of crystallization of m-HDPE and ZN-HDPE ..................................................... 43 Figure 26: The effect of flow on the relaxation behavior of ZN-HDPE at 125.5°C ........ 44 Figure 27: The effect of flow on the relaxation behavior of m-HDPE at 123°C ............. 45 Figure 28: The effect of flow on the relaxation behavior of m-HDPE at 140°C ............. 46 vii  Figure 29: FIC prediction of ZN-HDPE under shear flow at 125.5°C ............................ 48 Figure 30: FIC prediction of m-HDPE under shear flow at 123°C .................................. 48 Figure 31: FIC prediction of ZN-HDPE under elongational flow at 125.5°C ................. 49 Figure 32: FIC prediction of m-HDPE under elongational flow at 125.5°C ................... 49 Figure 33: The function λ (Eq. 6.2) and its fit to experimental results ............................ 51 Figure 34: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 124.5°C ... 52 Figure 35: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 125.0°C ... 52 Figure 36: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 125.5°C ... 53 Figure 37: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 126.0°C ... 53 Figure 38: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 126.5°C ... 54 Figure 39: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 125.0°C. ......................................................................................................................... 54 Figure 40: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 126.5°C. ......................................................................................................................... 55 Figure 41: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 126.0°C. ......................................................................................................................... 55 Figure 42: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 140.0°C. ......................................................................................................................... 56 Figure 43: The function λ (Eq. 6.2) and its fit to experimental results ............................ 56 Figure 44: FIC prediction (fit) of Eq. 3.7 for m-HDPE under shear flow at 122.5°C. .... 57 Figure 45: FIC prediction (fit) of Eq. 3.7 for m-HDPE under shear flow at 123.0°C. .... 57 Figure 46: FIC prediction (fit) of Eq. 3.7 for m-HDPE under shear flow at 123.5°C. .... 58 Figure 47: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 122.5°C . 58 Figure 48: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 123.0°C . 59 Figure 49: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 140.0°C . 59 viii  Figure 50: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 150.0°C . 60 Figure A.1.1: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.02s-1 ................................................................. 75 Figure A.1.2: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.06s-1 ................................................................. 75 Figure A.1.3: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.10s-1 ................................................................. 76 Figure A.1.4: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.50s-1 ................................................................. 76 Figure A.2.1: The shear stress growth coefficient of m-HDPE at various temperatures at constant shear rate of 0.06s-1 ..................................................................... 77 Figure A.2.2: The shear stress growth coefficient of m-HDPE at various temperatures at constant shear rate of 0.10s-1 ..................................................................... 77 Figure A.2.3: The shear stress growth coefficient of m-HDPE at various temperatures at constant shear rate of 0.50s-1 ..................................................................... 78  ix  Nomenclature 2D  deformation tensor second invariant of deformation tensor  A  cross sectional area, shape constant  A0  unstretched cross sectional area  B  constant  C  shape constant  C  Cauchy-Green tensor  F  tangential force  FF  frictional force crystals growth rate  H  gap size between parallel plates  Ko, Kf  kinetic constants  Lo  sample length  M  torque  Mn  number average molecular weight  Mw  weight average molecular weight  Mz  the z - average molar mass  Mi  molar mass  m  shape constant  N  number of density of nuclei  N0  number of density of nuclei before applying the deformation  x  Nf  number of density of nuclei which is induced by flow  ̇  nucleus generation rate  Ni  number of moles of species  p  constant  PI  polydispersity index  R  radius of the plate and radius of SER windup drum  r  distance from the center of the plate  T  torque, temperature  t  time  ts, tf  stoppage time for stress relaxation experiments  Greek Letters   (t )  shear strain    shear rate, s-1   R  shear rate at the edge, s-1  H   Hencky strain rate, s-1  H   Hencky strain    viscosity, Pa*s    shear stress growth coefficient    shear stress decay coefficient  a  amorphous viscosity, Pa*s  E  tensile stress growth coefficient  E  tensile stress decay coefficient xi    time constant, s  E  principal stretching stress, Pa    angular speed, rotation rate    displacement in  direction    shear stress, Pa  a  apparent shear stress, Pa    crystallinity fraction  xii  Acknowledgments I offer my sincere gratitude and appreciation to my thesis supervisor Prof. Savvas G. Hatzikiriakos for giving me insightful scientific advice. He has supported me throughout my thesis with his patience and knowledge whist allowing me the room to work in my own way. One simply could not wish for better or friendlier supervisor. I am grateful for the financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada. From the bottom of my heart, I thank my brother Babak Derakhshandeh for helping me get through the difficult times, and for all the emotional support, camaraderie, entertainment, and caring he provided. The most profound and loving thanks to my family, to my parents who have given me the greatest love and support anyone could ever expect. I am blessed to have them and I have missed them deeply during these years. They are my true inspiration.  xiii  Dedication  This thesis is dedicated to my parents, sister and brother, for always encouraging and supporting me to achieve my goals.  xiv  1. Introduction Polymer processing techniques involve mixed shear and elongational flows and cause polymer molecules to experience flow-induced crystallization during flow and subsequent solidification (Eder et al., 1990; Dealy and Wissbrun 1990; McHugh, 1995). The mechanical properties of the final products are significantly dependent upon the degree of crystallization and types of crystals formed and therefore optimization of any polymer process requires a good understanding on how flow influences crystallization (McHugh, 1995; Eder et al., 1990; Kornfield et al, 2002; Janeschitz-Kriegl, 2003; Scelsi and Mackley, 2008; and Kumaraswamy et al., 1999). In particular, the orientation of the crystals in elongational and shear flows affect the crystallization and thus the mechanical properties (Gelfar and Winter, 1999). The type of flow deformation can also play a significant role in affecting crystallization. For example, extensional flow (which is an inherently strong flow) causes molecules to orient and stretch in the direction of extension (as in the case of fiber spinning and film blowing) facilitating the process of flow induced crystallization (Swartjes et al., 2003; Janeschitz-Kriegl, 2003; Kornfield et al., 2002: Stadlbauer et al., 2004). Although shear deformations also influence the crystallization behavior of polyethylenes (Monasse, 1995; Vleeshouwers and Meijer, 1996; Alfonso and Scardigli, 1997; Koran and Dealy, 1999; Lagasse and Maxwell, 1976; Fortenly et al., 1995; Fernandez-Ballester et al., 2009), and because these shear flow fields are not typically as strong as extensional flows, very large strains must be generated in order to observe shear flow induced crystallization.  1  X-ray diffraction of polymers (including high-density polyethylenes the subject of the present work) reveals its semi-crystallinity. It was first assumed that fringed micelles, schematic of which is shown in the Figure 1, cause the polymers to crystallize. Later investigation showed that this is not a good representation. Thence, chain folding was suggested by Storks (1938) as the route to the formation of crystals. Transmission electron microscopy (TEM) revealed that crystal lamellae are thinner than the chain length which supports the Storks idea. In general, crystal morphology can be divided into three stages. At the first step lamellae is formed by stretching and folding the polymer chains. Afterward, stacks are formed in which lamellae found to be parallel to each other while having amorphous phase in between (GEDDE 1995).  Figure 1: (a) The fringed micelle crystal concept; (b) The folded chain crystal theory (GEDDE 1995).  Figure 2 shows a typical picture of polymer stack which is obtained by transmission electron microscopy (TEM). Polymers with low to intermediate molecular weight were found to form stacks which have lots of lamellae. High molecular weight or highly branched polymers showed a higher tendency to create stacks with fewer lamellae. 2  Finally, supermolecular structure is formed which includes lamellae and stacks. Polarized microscopy is the standard technique to study the supermolecular structures, while TEM is used to study the lamellae crystals. The major experimental problem with semicrystalline polymer is the low density difference between the crystals and the amorphous phase.  Figure 2: Transmission electron micrograph (TEM) showing lamellar crystals (GEDDE 1995).  In this thesis, the effects of shear, uniaxial extension and temperature on the flowinduced crystallization of two different types of two high-density polyethylenes (a metallocene and a Ziegler-Natta HDPE) are examined using rheometry. Shear and uniaxial extension experiments are performed at temperatures below and well above the peak melting point of the polyethylene’s in order to characterize their flow-induced crystallization behavior at rates relevant to processing. Finally a crystallization model is proposed that can be used in mixed flow simulations to predict crystal formation.  3  2. Rheological Measurements Rheology studies the flow and deformation of matter when external forces are imposed (Dealy and Wissbrun, 1990). In other words, it attempts to understand why a material behaves in a certain way when a force is applied. Different techniques have been developed in rheometry. Rheometry is defined as the tool used to experimentally characterize the rheological behavior of materials. Some of these methods which are used in the present study are discussed below. The equations which highlight the operation of the rheological equipment and data analysis gained from these rheological techniques are also discussed to a certain extent.  2.1.  Shear Rheometry  The rheometers used to investigate material’s behaviour in shear can be categorized into two groups based on the flow driving force. First a pressure difference between two points in a channel may cause a molten polymer to shear deform and flow. The capillary rheometer can generate a pressure driven flow by means of a piston driving the molten polymer through a contraction from a reservoir into a capillary die of wellknown length, L and diameter, D. The flow in capillary rheometer and its governing equations are discussed in details in several books of rheology and since it is not utilized in the present work, it will not be discussed either (Dealy and Wissbrun, 1990; Macosko, 1994). Another method to generate shear flow includes drag force. The apparatuses which use this technique are among others the sliding plate and the rotational rheometers.  4  Sliding parallel plate is the simplest way by which the shear flow can be generated (see Table 1). The main experimental difficulty with this rheometry method is the edge effect (edge fracture) which becomes worse as the strain increases. Use of large plates can decrease the edge effect, however it can jeopardize the parallelism of the plates. Rotational rheometers are more suitable for measuring the viscoelastic behavior of low viscous fluids compared to sliding plates. Rotational rheometers are currently being used extensively due to their ability to use different geometries and to measure the material properties precisely (see Table 1). These geometries include concentric cylinder, cone-and-plate, and parallel plates. Concentric cylinder was the first practical way of measuring rheological properties based on drag flow. Cone-and-plate is the most common geometry, which is used to study non-Newtonian flow behaviour. Homogenous flow can be developed by this fixture, however it is not suitable for highly viscous melts since loading of samples becomes almost impossible (Macosko, 1994). The third geometry which can be mounted on the rheometer is the parallel plate one. The flow which is generated by this fixture is not homogenous; nonetheless it offers more flexibility since the gap size can be changed easily. As seen in Table 1, in the parallel plate rheometer, two circular plates are mounted on a common axis of symmetry, and the sample is placed between them. Shear deformation is imposed upon the test specimen by rotation of the upper plate at a predefined angular velocity (t). Generally, the movement of the upper plate can be programmed so as to create different type of deformations, and the corresponding torque, M, is measured (strain controlled rheometer). An alternative mode of operation is fixing the torque and measuring the resulting displacement (stress  5  controlled rheometer). Table 1 shows schematic of different techniques which are used in generating purely shear flows.  Table 1. Schematics of different techniques used for generating purely shear flow.  Rheometer  Fixture  Simple schematic  Sliding plate  Concentric Rotational cylinders  Rotational  Cone-and-plate  Rotational  Parallel plates  6  The simplest type of deformation that can be generated with the parallel plate geometry is simple shear. In this experiment the upper plate is rotated with a constant rotational speed,  (t ) and the torque is measured. Certain assumptions are made to simplify the underlying equations (Macosko, 1994):   Steady, laminar, isothermal flow    Cylindrical edge    Negligible body and inertia forces    No velocity components in z and r directions    No slip boundary condition at solid walls  Under these assumptions the shear rate can be easily shown to be (Macosko, 1994): ̇( ) Where r is the distance from the center of the plate,  (2.1). is the angular velocity, and H is the  gap size (distance between the two plates). Therefore, the shear rate at the edge (  )  is: ̇  (2.2).  Shear strain is found to be dependent on position (non-homogenous flow) and is given by the following equation: ( )  (2.3).  Shear stress is derived from the torque balance in conjunction with the equation of motion and it has the following form (Macosko, 1994): 7  ( )  [ ̇  ]  (2.4).  Where M is the torque which sensed by the apparatus. Therefore, shear stress can be obtained precisely when  is plotted versus  ̇  and the corresponding slope is  inserted in the Equation 2.4. For the Newtonian fluid this slope is found to be one, and thus the apparent shear stress,  a is defined as: ( )  2.2.  (2.5).  Extensional Rheometry  Another important rheological characterization which gives additional important information on the structure of polymeric materials is extensional flow. In this experiment, a polymeric sample (i.e cylindrical or rectangular) is stretched by its ends by a certain rate and the tensile stress (force per unit of cross sectional area) needed for this is measured (uniaxial extension – see Table 2). Therefore, the polymer is subjected to typically non-linear deformation, which generates structural information that is otherwise impossible to obtain from linear viscoelastic data. For instance, the molecular weight distribution and the degree of long chain branching can have the same influence on the linear viscoelastic properties of a polymer. However, these effects can be separated by studying the sample under extensional flow (Münstedt, 1980). Different techniques have been developed in order to study the behavior of polymers under extensional flow. The main problem with extensional flow is its difficulty to generate a homogeneous deformation. The streamlines converge as the  8  extensional flow is developed which leads to thinner samples. Therefore, usually a lower limit of strain is needed for extension compared to that with shear. The various extensional deformations include extension, compression, and various stagnation flows (Table 2) (Macosko, 1994). One approach of generating extensional flow is compression. Polymer specimen is squeezed between two plates which creates biaxial extensional flow (see Table 2). Since, shearing of test specimen can happen at the surface of the plates, lubricant is used to help the polymer squeeze easier. An alternative way of developing multi axial extension is sheet stretching in which clamps are used to hold the sample edges. These clamps stretch the specimen and the force on clamps is monitored. The underlying formula for each of these methods can be found in Macosko (1994). Using oil in conjunction with the above methods helps the specimens stretch uniformly. Oil penetration is checked by monitoring the viscosity of the specimen over a period of time. Obviously, the oil should not penetrate into the sample. Holding the sample edges is a difficult task for lower viscous melts. In these cases, fiber spinning and stagnation flows are used to study the rheology of low viscous melts (typically less than 1000 Pa.s). In fiber spinning the melt is extruded out of a pipe which follows by stretching (Table 2). Rotating wheel or vacuum suction is used to stretch the polymer and thus no holding of the sample ends is required. Finally, stagnation flow is created by impinging streams of polymeric melt. This method is capable of obtaining high extensional strain particularly at the stagnation point where the strain is infinity (Macosko, 1994). Therefore, highly oriented structures can be induced by using this method. Table 2 shows a simple schematic of all methods discussed above.  9  Table 2. Schematics of different methods used to generate extensional flows.  Method name  Simple schematic  Uniaxial Extension  Compression  a) before deformation  b) after deformation  Sheet stretching  10  Fiber spinning  Stagnation flow  In the present study to examine flow-induced crystallization in extension the Sentmanat Extensional Rheometer (SER) is used. A schematic of the SER (Sentmanat Extensional Rheometer) fixture is shown in Figure 3. This extensional rheometer is designed for use as a detachable fixture on commercially available rotational rheometer host systems whose main function was described in the previous section. As such, this fixture is able to fit within the host oven chamber allowing precise temperature controlled testing.  11  Figure 3: Schematic of the Sentmanat Extensional Rheometer (Sentmanat, 2003a).  As shown in Figure 3, the extensional fixture is made of master and slave windup drums. The shaft of the rheometer rotates the master drum with an angular velocity of ω which in turn leads to an equal but opposite rotation of the slave drum. This makes the end sections of the attached polymer wound up onto the drums resulting in the specimen being stretched over an unsupported length (Lo). For a constant rotation rate of ω, the Hencky strain rate imposed on the sample specimen can be expressed as (Sentmanat, 2003b): ̇  (2.6).  Where R is the radius of the drums, and Lo is the unsupported length of the test specimen being stretched which is equal to the centerline distance between the windup drums. The polymer resistance to extensional flow is revealed as a tangential force of magnitude F acting upon the drums. The force is transferred as a torque through the 12  chassis housing the assembly to the torque shaft. Torque is determined from the summation of moments about the axis of the torque shaft. This yields (Sentmanat, 2003b): (  )  (2.7).  Where T is the final torque measured by the torque transducer, and FF is the frictional force which is appeared due to the bearings and intermeshing gears. Precision bearings and gears cause the frictional term to become quite small (typically less than 2% of the measured torque signal) and thus it can be neglected such that Equation (2.7) further simplifies to (Sentmanat, 2003b): (2.8). If the nominal and actual strain rates are the same for polymeric melts, the instantaneous cross-sectional area, ( ), of the stretched material changes exponentially with time for a constant Hencky strain rate experiment and can be expressed as (Sentmanat, 2003b): ( )  [  ̇ ]  (2.9).  Where, Ao indicates the unstretched cross-sectional area of the sample. The tensile stress growth coefficient,  ( ), of the stretched sample under constant Henky strain rate  can then be expressed as (Sentmanat, 2003b): ( )  Where,  ( ) ̇ ( )  (2.10).  ( ) is the instantaneous extensional force at time t imposed by the  sample due to its resistance to flow. 13  3. Flow Induced Crystallization 3.1.  Flow Induced Crystallization Studies  The exact route to the crystal formation still remains unknown although several explanations were suggested (Janeschitz-Kriegl, 1997; Janeschitz-Kriegl et al., 1997; 2003). It was assumed that polymeric melts contain precursors which are not activated at high temperature under quiescent (no flow) conditions. These precursors can be excited when their favourite condition is reached by decreasing the temperature or applying deformation. Saturation of the number of nuclei density is reached as the temperature is decreased and/or the deformation rate is increased. However, this theory cannot explain the gelation phenomenon in a crystallizing polymer (Steenbakkers et al., 2010). Different crystal morphologies can be evolved depending upon the flow conditions. Spherulites are induced under no flow condition or when the deformation rate is diminishingly low (i.e. next to quiescent condition) (Monasse, 1995; Pagodina et al., 2001). Since, impurities exist within the polymer bulk, quiescent crystallization is found to be heterogeneous. Furthermore, the spherulite growth is found to be isotropic and the growth rate is reported to have an exponential relationship with the temperature (Wunderlich, 1977; Mandelkern, 1964; Patel et al., 1991). Row-like nucleation and anisotropic growth occur as the deformation rate increases; at more extreme conditions oriented layer (known as shish-kebab) can be formed (Pogodina, 2001). The shish-kebab morphology was first reported in the work of Pennings (Pennings et al., 1965) who observed these structures in a polymeric solution. An explanation to the pathway of  14  shish-kebab formation was recently reported by Hashimoto et al. (2010). A schematic of shish kebab and spherulite structure is shown in Figure 4.  (a)  (b)  Figure 4: (a) Schematic of spherulite crystals (Elias HG., 2008), (b) Schematic of shish kebab structure (Huong et al., 1992)  A factor that can affect the crystallization kinetics is the presence of nucleating agents such as organoclays (Lagasse et al., 1976; Jerschow et al., 1997; Spruiell et al., 1996; Nakajima et al. 1996; Huo et al., 2004; Somwangthanaroj et al., 2003). Moreover, the kinetics of crystallization significantly changes under flow conditions at temperatures near the melt peak temperature, and is a function of both strain and strain rate (Kitoto et al., 2003; Hadinata et al., 2006; Dai and Tanner, 2006; Tanner and Qi, 2009). Therefore, to manufacture products of enhanced mechanical properties, it is of significant importance to control the polymer morphology by controlling the flow-induced crystallization phenomena.  15  3.2.  Flow Induced Crystallization Models  Modeling of flow-induced crystallization is a useful tool to optimize polymer processes. Most of these models are phenomenological based on experimental data. The ultimate goal in modeling the flow-induced crystallization is to predict both shear and elongational behaviour using a single model; in this respect one may find the study by Tanner et al. (2009) useful in which a formula was developed based on the simple suspension theory (Tanner et al., 2009). To the best of our knowledge, there is no such a model that can map both elongational and shear data well. This is partly due to fact that there are very few elongational data available in the literature (Stadlbauer et al., 2004 Part I; Bove et al., 2002; Sentmanat and Hatzikiriakos, 2010; Sentmanat et al., 2010) and most of them are limited to small Hencky strain rates. Different authors have used different basic hypothesis so as to be able to correlate the flow induced crystallization phenomenon. Polymers facing crystallization is assumed as molecular mixtures (Bushman and McHugh, 1996, Doufas et al., 1999; 2000a; 2000b, Doufas and McHugh, 2001a; 2001b; 2001c, Van Meerveld, 2005, Van Meerveld et al., 2008). The crystals are also assumed to be dispersed through the polymer bulk and thus it was concluded that suspension theory is suitable to be used (Boutahar et al., 1996; 1998, Carrotet al., 1993, Steenbakkers and Peters, 2008, Tanner, 2002; 2003). Pictures which are obtained by microscopy support the second idea more convincingly. The other assumptions which have drawn attention in the literature are physically cross-linking networks (Horst and Winter, 2000a; 2000b, Pogodina and Winter, 1998, Pogodina et al. 1999a; 1999b; 2001, Winter and Mours, 1997) and locally physically cross-linking melts  16  (Custódio et al., 2009, Peters et al., 2002, Peters, 2003, Swartjes et al., 2003, Zuidema, 2000, Zuidema et al., 2001). In this section, some of the suspension theory formulas are presented and discussed. According to this theory, the crystallizing melt is assumed to be a mixture of solid particles (crystals) in a Newtonian fluid. For such a system the following formula can be used to relate the viscosity of the crystallizing melt to crystallinity fraction (Metzner, 1985): (  Where  )  (3.1).  is the amorphous viscosity of polymeric melt,  is the crystallinity fraction, A  is a constant which depends on the crystals shape and its value is usually between 0.44 and 0.6, and  is the viscosity of the crystallizing melt. The crystallinity fraction is  defined by the following formula (Kolmogorov, 1937): ( Where  )  (3.2).  is: ∫  ̇ ( ) [∫  ( )  ]  (3.3).  In Equation 3.3, the m parameter is reported to have a value between 1 to 5; for instance, its value for spherulitic nucleation and rod like nucleation is 3 and 2, respectively. ̇ ( ) is the nucleus generation rate at time s, C is a shape constant, and ( ) is the crystals growth rate.  17  The nucleus density can be divided into two parts. One part includes the number of nucleus which is presented before the deformation and the second term is imposed by applying the flow. Thus, the following formula can be used (Tanner and Qi, 2009): (3.4). Combining Equations 3.2, 3.3, and 3.4 the following formula for crystallinity can be derived (Tanner and Qi, 2009): ∫ ̇ (  { [  )  ]}  (3.5).  Where Ko and Kf are kinetic constants for the crystallization. Following the work by Tanner and Qi (2009) one can define the nucleation rate as follows: ̇ Where  |  | (√  )  (3.6).  is the second invariant of deformation tensor, a and p are constants, and  is the trace of the Cauchy-Green tensor (finite deformation tensor) (Tanner and Qi, 2009). Combining the last two equations and performing the integration, the following formula can be derived: { ( ) ( Where,  |  | √  )}  (3.7).  is a time constant and m depends on the crystalline structure. Note that under  no flow (    ) Equation 3.7. simplifies into:   1  exp   t    m   , an equation  known as the Kolmogorov or Avrami for quiescent crystallization. The Cauchy-Green tensor for simple shear flow has the following form (Macosko, 1994)  18  [ (t 2 )   (t1 )]  1   Cij (t1 , t 2 )  [ (t 2 )   (t1 )]  0   Where  1  [ (t  )   (t1 )]  2  2  0  0 1    0  (3.8).  is the time at which a material element is in its reference configuration, and  is the time at which the strain is elevated, relative to the configuration at time  . Setting  to zero (beginning of flow) and the current time t2=t, the Cauchy-Green tensor becomes:   (t2 ) 1  Cij (t1 , t2 )   (t2 ) 1   (t2 ) 2  0 0    (3.9).  is equal to the shear strain ( ( )). Moreover, the rate of  Therefore, the √ deformation tensor (    0  0  1  ) for simple shear has the following form:  2D =  0  (t ) 0  (t ) 0 0 0  (3.10).  0 0  Hence, the second invariant of rate of deformation tensor (  ) is the shear rate ( ̇ ( )).  The corresponding expression of Cauchy-Green tensor for the uniaxial extensional flow is (Macosko, 1994):  e 2[ (t2 ) (t1 )]  Cij (t1 , t 2 )   0  0   0 e [ (t2 ) (t1 )] 0    0  [ ( t2 ) ( t1 )]  e  0  (3.11).  Assuming the reference time, t1 as zero simplifies the above tensor into:  19  [ ( )]  ( )  [  ]  [ ( )]  (3.12).  [ ( )]  Therefore, the √ deformation tensor (  [ ( )]  [ ( )]  . The rate of  ) for uniaxial extension is:  2D =  Thus,  is equal to √  is equal to  2 (t ) 0  0 -  (t )  0 0  0  0  -  (t )  ̇ . Substituting  and √  (3.13).  into the Equation 3.7  and fitting the resulting model into the shear and elongational data the four constants, namely,  and  can be calculated at a given temperature. Since λ and m can be  obtained from quiescent crystallization (Kolmogorov constants), they should be the same for both types of deformation (shear and extensional). The ideal is to have all four parameters be the same for both shear and extensional flows as well as for mixed flows. As will be discussed later this turns out not to be the case and therefore a modification to the Tanner model is proposed as part of the developmental work of this thesis.  20  4. Thesis Objectives The overall objective of this research work was to first study experimentally the effects of shear and extensional flows on the crystallization of two different types of high-density polyethylenes (a metallocene and a Ziegler-Natta HDPE) at various temperatures around and well above the melt peak temperature. Secondly, using the crystallization data, a model is to be developed that describes flow induced crystallization under both shear and extensional flows. In particular, the objectives are: 1. To study the crystallizability of two different types of high-density polyethylenes (HDPEs), namely a Ziegler-Natta and a Metallocene HDPE. 2. To study the effect of the important parameters on the crystallization of HDPEs such as strain rate, strain and temperature under both shear and uniaxial deformations. 3. To formulate a simple crystallization model based on the experimental findings and the proposed models developed by Kolmogorov (1937) and Tanner and Qi (2009). The developed model should be useable for both shear and extensional as well as for mixed flows.  21  5. Materials and Methodology The materials used in this study include two different types of HDPE, namely, a metallocene HDPE (m-HDPE) and a Ziegler-Natta HDPE (ZN-HDPE) both provided by Chevron Phillips Company LP. The ZN-HDPE has a molecular weight of 328 kg/mole and polydispersity index of 14, while m-HDPE has a molecular weight of 230 kg/mole and polydispersity index of 42. Polydispersity index is defined as the ratio of weight average molecular weight over the number average molecular weight ( PI  Mw/Mn ). It shows how broad the molecular weight distribution is. Mw and Mn can be calculated by using the following formulae:  M i Ni Mn   Ni   M i Ni Mw   M i Ni 2  Where Ni is the number of moles of each polymer species and Mi is the molar mass of that species. Table 3 summarises some of their molecular characteristics. Their rheological and processing behaviour has been studied in detail by Ansari et al. (2010) and Ansari et al. (2011) respectively.  Table 3. List of HDPEs used in this study and their molecular characteristics.  Mn  Mw  (Kg/mole)  (Kg/mole)  ZN-HDPE  23.0  328.4  14.3  1541.0  4.7  m-HDPE  5.3  222.5  42.0  876.7  3.9  Resin  Mz PI=Mw/Mn  Mz/Mw (kg/mole)  22  In Table 3, Mz is the z average molar mass and it is obtained using the following expression:   M i Ni 3  Mz    M i Ni 2  A Shimadzu DSC-60 calorimeter with air as heating flow was used to study the thermal behaviour of the high-density polyethylene samples. To obtain the polymer melting and cooling peaks, 1-2 mg samples sealed in an aluminum pan were heated at the rate of 10°C/min from 50°C to 180°C. The sample was kept at 180°C for 5 minutes before cooling it down to 50°C at the same rate. An Anton Paar rotational rheometer (MCR 501) was used to study the crystallization behaviour under flow using two types of flow. Simple shear experiments were performed with circular parallel plates of 25mm in diameter in order to investigate flow-induced crystallization phenomenon in shear. In addition, the SER2 (Sentmanat Extension Rheometer) Universal Testing Platform was used in conjunction with the MCR-501 rotational rheometer in order to study flow-induced crystallization behaviour in extension. Most of the experiments were carried out at temperatures between the peaks found in the DSC thermographs in the heating and cooling modes. This temperature range is the most important in flow-induced crystallization due to the tendency of molecules to crystallize and reorient. To eliminate thermal history of the samples each test specimen was heated up to 180°C (around 50°C above melting peak temperature) and remained within the thermo chamber oven (CTD450) of MCR 501 for 5 minutes prior to cooling down to experimental testing. The effects of various parameters such as temperature, deformation (strain) and deformation (strain) rate on the crystallization of 23  the two HDPE polymers were studied in both shear and extensional flows. Moreover, the relaxation behaviour of samples under both shear and elongation was monitored after applying flow for different times. Additional experiments at temperatures well above the heating melting peak temperatures were carried out particularly for uniaxial extension and as will be discussed below these were found to be quite significant. Reproducibility of the experimental results was examined. The error for the times which correspond to the onset of crystallization was found to be small (±10%).  24  6. Results and Discussion 6.1. Thermal Behaviour/Quiescent Crystallization Figures 5 and 6 depict the thermographs of the two HDPE samples. The onset of crystallization for ZN-HDPE and m-HDPE under quiescent conditions was observed at 119.6°C and 117.6°C, respectively, a difference of only 2°C. The peak heating and cooling temperatures differ by 1.2°C and 1°C respectively. A temperature difference of 2°C is set as the base for comparing the crystallization behaviour of the two polymers i.e. isothermal crystallization under flow is compared for the two HDPEs at corresponding temperatures differing by 2°C. Based on these thermographs it can be concluded that for all practical purposed the ZN-HDPE and m-HDPE exhibit similar quiescent crystallization behaviour.  Figure 5: The DSC thermograph of ZN-HDPE. The melting crystallization under quiescent conditions was observed at 119.6°C. The melting and cooling peaks were 128.5°C and 116°C, respectively.  25  Figure 6: The DSC thermograph of m-HDPE. The melting crystallization under quiescent conditions was observed at 117.6°C. The melting and cooling peaks were 127.3°C and 115°C, respectively.  6.2. Flow Induced Crystallization (FIC) in Steady Shear     (t )  Figure 7 depicts the shear stress growth coefficient, η+(t)  (t )     of ZN-  HDPE at the constant shear rate of   0.01s 1 at three different temperatures between the peak heating and cooling temperatures, namely 124.5°C, 125°C and 125.5°C. This coefficient increases with time under the application of constant shear rate till eventually reaches a plateau (steady-state) as normally expected for a polymer melt. At these temperatures (all above the quiescent crystallization temperature), the samples are all crystal-free and in the molten state; hence they exhibit almost the same behaviour over this region. However, after a certain time, the shear stress growth coefficient, η+(t),  26  exponentially increases. This is clearly due to the occurrence of crystallization which becomes gradually more and more dominant. An induction time for the onset of crystallization can be defined. It is defined as the time at which η+(t) increases by 20% from its steady-state. Some authors define this time at 100% increase from its steady-state (Tanner et al., 2009a). It should be stressed, however, that the final conclusions do not depend on the defined cut-off point. As seen from Figure 7, temperature appears to have a significant influence on the induction time of crystallization. A decrease in temperature, even as small as by 0.5°C decreases significantly the induction time. In general, crystallization tends to occur more quickly as temperature decreases. This was found to be the case at other shear rates (see Figure A.1.1, A.1.2, A.1.3, A.1.4 in Appendix A.1.). This is in agreement with the exponential relationship of crystallites growth rate with temperature (Patel et al., 1991).  Figure 7: The shear stress growth coefficient of ZN-HDPE at temperatures 124.5°C, 125°C and 125.5°C at constant shear rate of 0.01s-1.  27  Figure 8 shows the effect of shear rate on crystallization of ZN-HDPE. It plots the stress growth coefficient η+(t), as a function of time for various imposed levels of shear rates, namely 0.01, 0.02, 0.06 and 0.1s-1 at 125.5°C. The data at short times follow closely the linear viscoelastic envelope (LVE) before deviation due to nonlinearity. It appears that the higher deformation rates stimulate the melt more efficiently, and thus cause crystallization faster. Therefore, the induction time for the onset of crystallization decreases with increase of shear rate due to both the increase of the number of nuclei and the crystals growth rate (Janeschitz-Kriegl et al., 2003; Monasse, 1995).  Figure 8: The shear stress growth coefficient η+ of ZN-HDPE at different shear rates at 125.5 °C.  Figure 9 depicts the results of Figure 8 as a function of the shear strain. One intuitively expects that at higher shear rates, the required shear strain to observe 28  crystallization should decrease, in other words less strain is needed for the onset of crystallization at higher shear rates. However, as seen from Figure 9, the critical shear strain for the onset of crystallization increases with increase of shear rate.  Figure 9: The shear stress growth coefficient of ZN-HDPE at shear rates of 0.01, 0.02, 0.06 and 0.1s -1 at 125.5°C.  The effects of shear rate on the induction time and on the critical strain for the onset of crystallization are summarized in Figure 10. The induction time and critical strain are found to depend nonlinearly with deformation rate. The induction time decreases with increase of shear rate, while more strain is needed to observe the occurrence of crystallization.  29  Figure 10: Comparison of the dependency of induction time and that of critical strain of ZN-HDPE for the onset of crystallization on the shear rate at 125.5°C.  The same behaviour was also observed for the m-HDPE. Figures 11, 12, 13 and 14 plot the shear stress growth coefficient for m-HDPE as a function of t and strain for different temperatures, and shear rates. The induction time for crystallization increases with increase of temperature, decrease of shear rate and more strain is needed for the onset of crystallization at a higher strain rate. Similar results to those plotted in Figure 11 at other shear rates are plotted in Figures A.2.1, A.2.2, and A.2.3 in Appendix A.2.  30  Figure 11: The shear stress growth coefficient of m-HDPE at temperatures 123.0°C, 123.5°C and 124.0°C at constant shear rate of 0.02s-1.  Figure 12: The shear stress growth coefficient η+ of m-HDPE at different shear rates at 123.0 °C.  31  Figure 13: The shear stress growth coefficient of m-HDPE at shear rates of 0.01, 0.02, 0.06 and 0.1s -1 at 123.0°C.  Figure 14: Comparison of the dependency of induction time and that of critical strain of m-HDPE for the onset of crystallization on the shear rate at 123.0°C.  32  Increasing the rate of deformation from zero (quiescent condition), a critical shear rate is observed after which effective flow-induced crystallization evolve (Coppola et al., 2004). Since, very low shear rates cannot stimulate the melt efficiently, chain orientation does not occur extensively and the crystallization kinetics will be practically the same as that in quiescent crystallization. Figures 15 and 16 show this effect on the induction time at various temperatures for the ZN-HDPE and m-HDPE respectively. At a given temperature, a critical shear rate is needed to enhance crystallization compared to that under no-flow conditions (quiescent). It appears also that the critical shear rate increases with increase of temperature in agreement with Coppola et al. (2004). Furthermore, it seems that all data at various temperatures converge showing that the effect of temperature on the induction time becomes less important compared to that due to flow (increased shear rate).  Figure 15: The effect of shear rate on the induction time of ZN-HDPE at various temperatures. ̇ is the critical shear rate which is needed for effective FIC.  33  Figure 16: The effect of shear rate on the induction time of m-HDPE at various temperatures. ̇ is the critical shear rate which is needed for effective FIC.  Figure 17 compares its behaviour with that of ZN-HDPE in terms of the induction time for the onset of crystallization as a function of shear rate at several temperatures. More specifically, the data plotted in Figures 15 and 16 for ZN-HDPE (125°C-126°C) and m-HDPE (122.5°C-124°C) are plotted together in order to compare the crystallization behaviour of the two HDPEs in shear. Clearly the crystallization behaviours are similar showing that (i) a critical shear rate is needed for flow to enhance crystallization and (ii) at high rates the effect of temperature diminishes as the effect of flow becomes more dominant.  34  Figure 17: The effect of shear rate on the induction time at various temperatures. ̇ is the critical shear rate which is needed for effective FIC.  6.3. Flow Induced Crystallization (FIC) in Shear Stress Relaxation The relaxation behaviour of samples in steady shear flow cessation at different times (levels of strain) at a constant temperature was also examined. The corresponding results for ZN-HDPE and m-HDPE are shown in Figures 18 and 19, respectively. First for the ZN-HDPE (Figure 18), cessation of flow at times where the polymer is believed to be in the melt state (i.e. at times 252s and 1650s) did not prevent the crystallization of the material at later times during relaxation. Note the leveling off of the shear stress at significant levels of shear stress or at least the significant delay in stress relaxation. The later the cessation of flow it happens, the slower the relaxation is. Therefore, while the onset of crystallization can only be detected once the shear stress deviates significantly from its steady-state value, the flow effects have initiated the process of crystallization at much earlier times not evident from the macroscopic measurements. For the case of m35  HDPE, the stress relaxation curves exhibit an upturn after shearing at long times of 902s and 1307s (i.e. strain of 90.2 and 130.7) due to crystal formation. This observation is in agreement with the short time shearing protocol of Janeschitz-Kriegl and coworkers (Eder et al., 1997; Liedauer et al., 1993; Liedauer et al., 1995; Jerschow et al., 1996; Jerschow et al., 1997). Finally, when the flow is stopped once the crystallization has started (shear stress increases significantly), the crystals are formed fast within the melt and the stress is relaxed much less. At these shearing times (i.e. 2000s for ZN-HDPE in Figure 18 and 1803s for m-HDPE in Figure 19), the relaxation curves drop rapidly after a certain time due to the detachment of the solidified melt off the shearing plate. Such experimental data would be extremely useful to test crystallization models which should be able to predict the shear stress evolution in more complex tests other than steady shear.  Figure 18: The relaxation behaviour of ZN-HDPE under different levels of strain at 125.5 °C. shearing time of specimen or the cessation time of the uniaxial extension.  shows the  36  Figure 19: The relaxation behaviour of m-HDPE under different levels of strain at 123 °C. shearing time of specimen or the cessation time of the uniaxial extension.  shows the  6.4. Flow Induced Crystallization (FIC) in Uniaxial Extension The effects of temperature and Hencky strain rate on the crystallization of ZNHDPE and m-HDPE are shown in Figures 20 and 21, respectively. The tensile stress growth coefficient,  E of the HDPE samples as a function of time at several Hencky strain rates ranging from 0.01s-1 to 30s-1 and three temperatures. First, for the ZN-HDPE, Figure 20 shows that there is no sign of crystallization at 150°C, which is 21.5°C higher than the melting peak point (see insert in Figure 20). Decreasing the temperature to 140°C (11.5°C above the melting peak point), it causes the melt to crystallize under the application of only high Hencky strain rates (3s-1 and 10s-1). The onset of crystallization is observed from strain hardening, that is deviation from the linear viscoelastic envelope 37  indicated by  , where   (t ) is the shear stress growth coefficient obtained in steady  shear at small enough shear rates (linear regime). In most cases, before strain hardening, the data exhibit a small dip. This is due to necking which is obtained due to transition from a liquidlike to a solidlike behaviour. Further decrease of temperature to 125.5°C, causes flow induced crystallization at all applied Hencky strain rates examined, namely from 0.03s-1 to 30s-1.  Figure 20: The effects of temperature and Hencky strain rate on the flow-induced crystallization of ZNHDPE.  As mentioned, elongational flow enhances crystallization kinetics more than shear at comparable deformation rates. Comparing the induction times under elongational flow with those found in steady-shear flow at 125.5°C (Figures 8 and 20) reveals that the induction times in shear are at least 2 orders of magnitude longer than those seen in extension. In addition, flow-induced crystallization occurred in uniaxial extension at 38  temperatures several degrees above the melting peak point, whereas in shear was never observed, although experimental tests were persisted for several hours (in some cases up to 9 hrs). The m-HDPE exhibited a stronger crystallizability in uniaxial extension compared to ZN-HDPE. While there was no sign of crystallization at 150°C for ZN-HDPE, mHDPE shows the ability to crystallize under stretching at 150°C (22.3°C above the melting peak point) as shown in Figure 21. Furthermore, m-HDPE has shown strain hardening due to crystallization at the Hencky strain rates greater than 0.3s-1 at 140°C not seen in ZN-HDPE. It is noted that m-HDPE has a lower molecular weight but much higher polydispersity index than the ZN-HDPE; thus it seems that the polydispersity index dominates the molecular weight effect in these polymers under extension (Jay et al., 1999).  Figure 21: The effects of temperature and Hencky strain rate on the flow-induced crystallization of mHDPE.  39  Figure 22 shows the tensile stress growth coefficient,  of the ZN-HDPE as a  function of the applied Hencky strain at various Hencky strain rates at 125°C. A critical strain of approximately 2-3 is required to observe strain hardening due to crystallization independent of the Hencky strain rate. Similar results have been found for the m-HDPE as can be seen in Figure 23.  Figure 22: Tensile stress growth coefficient, of the ZN-HDPE as a function of Hencky strain at 125.5°C, showing that crystallization commences at a Hencky strain rate of 2-3.  40  Figure 23: Tensile stress growth coefficient, of the m-HDPE as a function of Hencky strain at 123°C, showing that crystallization commences at a Hencky strain rate of 1.5-2.3.  The effect of temperature and Hencky strain rate on the induction time for the onset of crystallization is shown in Figure 24 for ZN-HDPE at various temperatures. The induction time decreases significantly with increase of the Hencky strain rate and decreases mildly with decrease of the temperature. Furthermore, the temperature influence becomes less as the rate increases and this is in agreement with precursor theory which was suggested by Eder and co-worker (Janeschitz-Kriegl, 1997; JaneschitzKriegl et al., 1997; 2003). In general, the induction time is affected by the number of density of nuclei and also the crystal growth rate. Thus, increasing the rate makes more of the precursors become excited causing the number of density of nuclei to saturate (Janeschitz-Kriegl et al., 2003).  41  Figure 24: The effect of temperature and Hencky strain rate on the onset of crystallization for the ZNHDPE.  Figure 25 compares the effect of Hencky strain rate and temperature on the induction time for the onset of crystallization for both m-HDPE and ZN-HDPE at several temperatures. Despite the difference in the molecular weight and polydispersity index, for all practical purposes there is no difference between the induction times of the two polymers. All points fall on the same line with a slope of approximately 1. Although, the temperature window for the induction time of m-HDPE is wider (~17°C) than the ZNHDPE, the effect of flow is still dominating the temperature effect, particularly for the Hencky strain rate of more than 10s-1.  42  Figure 25: The effect of Hencky strain rate and temperature on induction time for crystallization for mHDPE and ZN-HDPE.  6.5. Tensile Stress Relaxation after Cessation of Steady Uniaxial Extension As discussed before and witnessed by the data in Figures 20 and 21, at the early stages of stretching the polymer appears to remain in a molten state displaying little deviation from the LVE envelope. However, at longer times and/or strains typically greater than 1, FIC crystallization appears evident from the strain hardening behaviour and sudden deviation from the LVE. In order to characterize the stress relaxation behaviour of the polymers after the onset of FIC, cessation of extensional experiments (in which the stretching deformation is suddenly stopped at a pre-determined time/strain) were performed on samples after a pre-determined strain imposition corresponding to various points along the  curve. Results for ZN-HDPE are plotted in Figure 26 for a  constant Hencky strain rate of 1s-1 and 125.5°C. The uniaxial extension has been stopped 43  after 1 and 2s when the polymer exhibits near-LVE behaviour, and after 3.5, 4.5, and 5s during the FIC process. When the stretching is stopped at the early stages of the experiment (after 1s and 2s) the polymer is still in a molten state and stress relaxation occurs normally as shown by the approximate exponential decay in the tensile stress growth coefficient. However, this is not the case upon the onset of FIC at cessation times greater than 2s when the FIC process has started. As seen from Figure 26, the tensile stress growth coefficient shows no such exponential decay but rather tends to approach an equilibrium stress as in the case of a rubber-like material. The relaxation has been significantly delayed due to transition to a solidlike behaviour. Similar relaxation behavior is observed for m-HDPE as shown in Figure 27. This was also reported in the literature for a linear low density polyethylene (LLDPE) by Sentamant and Hatzikiriakos (2010) and Sentmanat et al (2010).  Figure 26: The effect of flow induced crystallization (FIC) on the relaxation behaviour of ZN-HDPE at 125.5°C.  44  Figure 27: The effect of flow induced crystallization (FIC) on the relaxation behaviour of m-HDPE at 123°C.  The behaviour plotted above in Figures 26 and 27 has been observed at higher temperatures as well as above the melting peak point. An example is shown in Figure 28 for the m-HDPE for stress relaxation after cessation of uniaxial extension at 1s-1 at 140°C. At this high temperature (T=140.0°C) the initial relaxation behavior resembles that of a melt which is suddenly changes to that of solidlike material. The coefficient approaches a steady-value. At higher temperatures, although crystals are formed, initially are not interconnected due to the lower tendency of polymer to crystallize; thus when the flow is stopped discrete crystals can exist within the polymer bulk. Once these crystals form a network, solidlike relaxation becomes evident and the relaxation curve reaches a plateau.  45  Figure 28: The effect of flow on the relaxation behaviour of m-HDPE at 140°C.  6.6. Modeling 6.6.1. Isothermal FIC Modeling To model the crystallization data under shear and extension, the suspension model is used as developed by Tanner and Qi (2005) and Eder and Janeschitz-Kriegl (1998). The suspension theory and the relevant formulae have been discussed in section 3.2. In this section the main crystallization Equation 3.7 using the corresponding forms of the Cauchy-Green tensor and  for shear and elongational, is fitted to crystallization data  and the parameters of the model are calculated. Constants of the model were calculated by using the experimental data at different rates, at T=125.5oC for ZN-HDPE and 123oC for m-HDPE. This allowed the parameters to be averaged over the entire shear rate region and thus the best fitted parameters are obtained. An alternative way of fitting the model to data is to first fit the quiescent curve to the quiescent model (without the effect of flow) 46  in order to detemine m and λ. The other constants can be estimated by fitting Equation 3.7 to one of the higher rate curve (for instance for shear using the deformation rate of 0.1s-1 and for extensional the Hecky strain rate of 10s-1). The fitted model parameters are listed in Table 4. As seen, the parameters B and p parameters are quite different for shear and extensional. It was found impossible to fit simultaneously both shear and extensional crystallization data with single values and therefore the fitting was performed independently for the two types of deformation.  Table 4. The estimated parameters of crystallization model (Eq. 3.7) fitted to experimental results  Parameter  o  o  ZN-HDPE (T = 125.5 C)  m-HDPE (T = 123.0 C)  shear  Extensional  Shear  Extensional  m  3.25  3.25  3.25  3.25   (s)  15189  15189  14753  14753  B  10.5  12.3  0.34  0.46  p  3.3  11.0  2.70  13.00  The fitted model predictions to shear and extensional data at the given temperature are shown in Figures 29 and 31 for ZN-HDPE and Figures 30 and 32 for mHDPE, respectively. Reasonable fit is obtained for most cases. The quiescent crystallization curves (Kolmogorov expression) are also shown at diminishingly small deformation rates.  47  Figure 29: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 125.5°C.  Figure 30: FIC prediction (fit) of Eq. 3.7 for m-HDPE under shear flow at 123°C.  48  Figure 31: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under elongational flow at 125.5°C.  Figure 32: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 123.0°C.  49  6.6.2. Non-Isothermal FIC Modeling The effect of temperature on FIC was introduced by a power law function into Equation 3.7 Under quiescent condition Equation 3.7 is simplified as: { ( ) }  (6.1).  Spherulitic crystals are formed when the polymer is crystalized quiescently which means theoretically m should be equal to 3 (at this work was set equal to 3.25). Therefore, the only constant in Equation 6.1 which can vary to model the effect of temperature is . A simple functional form is proposed for : (  )  (6.2).  Where λ0 and b are constants and Tm is the peak melting temperature. The values of λ that best fit the experimental results for each temperature are first calculated. These values are plotted as a function of temperature in Figures 33 and 43 for ZN-HDPE and mHDPE, respectively. The corresponding fits of Equation 6.2 are included in these Figures. The effect of temperature was integrated by inserting this function (Eq. 6.2.) in Equation 3.7. The parameters m, p, and B are set constant and independent of temperature, essentially equal to values listed in Table 4. All parameters are listed in Table 5. Model predictions are shown in Figures 34-42 for various rates at various temperatures for ZNHDPE. The corresponding predictions for m-HDPE are plotted in Figures 43-50. The predictions of FIC in shear flow are found to be adequate, while those in extensional flow are only qualitatively good. Further modifications are needed in order to improve models performance in the extensional flows.  50  Table 5. The parameters of crystallization model (Eq. 3.7 and Eq. 6.2) fitted to experimental results  Parameter  ZN-HDPE  m-HDPE  shear  Extensional  shear  Extensional  m  3.25  3.25  3.25  3.25  λ0  807893  807893  400823400  400823400  b  174.5  174.5  283.6  283.6  B  10.5  12.3  0.34  0.46  p  3.3  11.0  2.70  13.00  Figure 33: The function λ (Eq. 6.2) and its dependence to temperature.  51  Figure 34: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 124.5°C.  Figure 35: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 125.0°C.  52  Figure 36: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 125.5°C.  Figure 37: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 126.0°C.  53  Figure 38: FIC prediction (fit) of Eq. 3.7 for ZN-HDPE under shear flow at 126.5°C.  Figure 39: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 125.0°C.  54  Figure 40: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 126.5°C.  Figure 41: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 126.0°C.  55  Figure 42: FIC prediction of Eq. 3.7 for ZN-HDPE under elongational flow at 140.0°C.  Figure 43: The function λ (Eq. 6.2) and its dependence to temperature.  56  Figure 44: FIC prediction (fit) of Eq. 3.7 for m-HDPE under shear flow at 122.5°C.  Figure 45: FIC prediction (fit) of Eq. 3.7 for m-HDPE under shear flow at 123.0°C.  57  Figure 46: FIC prediction (fit) of Eq. 3.7 for m-HDPE under shear flow at 123.5°C.  Figure 47: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 122.5°C.  58  Figure 48: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 123.0°C.  Figure 49: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 140.0°C.  59  Figure 50: FIC prediction of Eq. 3.7 for m-HDPE under elongational flow at 150.0°C.  6.6.3. Generalization of Crystallization for Mixed Flows To modify the crystallization model in order to make it usable for cases of mixed shear and extensional flows, the frame invariant flow parameter,  can be used. This was  introduced by Larson (1985) and its definition is as follows: (  )  (6.1).  ̇ This takes a value of 1 for pure shear and high positive values for elongational. Based on  the objective flow type parameter can be defined: (6.2).  This parameter  is zero under simple shear flow, one under elongational flow,  and between zero and one for mixed flows. Derivation of the objective flow type parameter for simple shear and uniaxial extension can be found in Mitsoulis and 60  Hatzikiriakos (2009). The model constants B and p can now be written as functions of the parameter and thus the crystallization model can be used for mixed flows in flow simulations through complex flows. This is as follows:  Where,  and  crystallization data and  (  )  (6.3).  (  )  (6.4).  are the value of B and and  are the B and  obtained by fitting the shear values obtained by fitting the  elongational crystallization data. As discussed above the other constants (m and λ) are set the same as those calculated from quiescent crystallization data.  61  7. Conclusions The flow-induced crystallization of a Ziegler-Natta and a metallocene HDPEs under simple shear and uniaxial extension was studied. First in shear, it was found that flow enhances crystallization for shear rates greater than a critical value. This enhancement increases with increase of shear rate and decrease of temperature. While the induction time for the onset of crystallization decreases with increase of shear rate, a higher strain is needed, which is a counterintuitive observation. Elongational flow was found to cause flow-induced crystallization more strongly than shear as the induction times were found to be at least two orders of magnitudes smaller. As a strong flow, extension aligns molecule that facilitates crystallization. Moreover crystallization was observed in extension even at temperatures well above the DSC melting point in heating mode, while no crystallization was observed in simple shear at similar conditions. It was shown that ZN-HDPE crystallizes faster than m-HDPE under quiescent condition because of its higher molecular weight (Figures 5 and 6). However, under flow the m-HDPE was found to show a higher tendency to crystallize. This higher tendency was shown by means of steady shear and stress relaxation experiments after cessation of steady shear as well as with extensional experiments at elevated temperatures. The behavior of a modified Kolmogorov-Avrami equation was examined under shear and elongation in modeling flow-induced crystallization. The model predicted the shear behaviour reasonably well which is in agreement with previously published work (Tanner and Qi, 2005; Tanner and Qi, 2009). A simple modification was proposed to generalize the equation for use in mixed flows. The model with the calculated fitted 62  parameters was used to predict the behaviour of the materials for several temperatures and the results were found reasonable. However, more experimental data are needed to fine tune the model and its capabilities in predicting the FIC of other polymers as well.  63  8. Recommendation for Future Work The suspension theory formula which was used to relate the viscosity to the crystallinity was developed based on a Newtonian fluid. However, polymers are nonNewtonian and therefore to achieve a better fit under extensional flow and higher shear rate a new suspension theory formula based on non-Newtonian fluid properties should be derived. The SER fixture in its existed form is not capable of obtaining data at very low Hencky strain rates. Perhaps using a liquid bath to offset buoyancy may overcome this difficulty. Moreover, the conventional rotational rheometers cannot apply high shear rate for high viscous fluids due to the torque limitation. Therefore, shear data for crystallization are limited to small shear rates. Using other techniques (such as capillary, sliding plate rheometer etc.) is highly recommended to study FIC behaviour of HDPE at high deformation rates. 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Ph.D. thesis,Technische Universiteit Eindhoven.  74  Appendix A Appendix A.1.:  Figure A.1.1: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.02s-1.  Figure A.1.2: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.06s-1.  75  Figure A.1.3: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.10s-1.  Figure A.1.4: The shear stress growth coefficient of ZN-HDPE at various temperatures at constant shear rate of 0.50s-1. The dip in the data is caused by slip between the shaft and the shaft holder due to high amount of torque at this high rate.  76  Appendix A.2.:  Figure A.2.1: The shear stress growth coefficient of m-HDPE at various temperatures at constant shear rate of 0.06s-1.  Figure A.2.2: The shear stress growth coefficient of m-HDPE at various temperatures at constant shear rate of 0.10s-1.  77  Figure A.2.3: The shear stress growth coefficient of m-HDPE at various temperatures at constant shear rate of 0.50s-1.  78  

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