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Crystallization of polypropylene : experiments and modeling Derakhshandeh, Maziar 2015

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Crystallization of polypropylene: experiments and modeling   by Maziar Derakhshandeh  B.Sc., Shiraz University, Iran, 2009 M.A.Sc., THE UNIVERSITY OF BRITISH COLUMBIA, Canada, 2011  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES (Chemical and Biological Engineering)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver)  November 2015 © Maziar Derakhshandeh, 2015 ii  Abstract In this study, the quiescent crystallization of several polypropylenes (PPs) with different molecular characteristics was first examined using Differential Scanning Calorimetry (DSC) and Polarized Optical Microscopy (POM). The Avrami/Nakamura equation was employed to fit and predict crystallization kinetics under isothermal and non-isothermal conditions. The Avrami/Nakamura model was found to predict the non-isothermal crystallization data of the various PPs very well over a range of cooling rates. POM was used in line with a rotational rheometer to further examine the behaviour under quiescent condition at different temperatures and/or cooling rates. The growth rate of crystals was impeded exponentially with increase of temperature. To study the effect of flow on crystallization behaviour of PPs, the Anton Paar MCR-502 rotational rheometer equipped with various fixtures including parallel-plate and POM to induce shear flow was used. Generally, an increase in strain and strain rate or decrease of temperature is found to decrease the thermodynamic barrier for crystal formation and thus enhancing crystallization kinetics at temperatures between the melting and crystallization points. Popular models based on suspension theory, which are often used to relate the degree of crystallinity to normalized rheological functions are validated experimentally. It is found that the constant(s) of various suspension models should be dependent on the flow parameters in order for the suspension models to describe the effect of shear on FIC, particularly at higher shear rates.  Finally, a capillary rheometer was used to investigate flow-induced crystallization (FIC) of various resins at high shear rates relevant to polymer processing. It is found that the crystallization iii  kinetics are enhanced with increasing molecular weight indicating the importance of high-end tail of MWD on FIC. Various dies with different physics were used to investigate the effect of flow on FIC. The Cogswell analysis was applied on the capillary data to obtain the apparent extensional strain rate and strain as well as the apparent extensional viscosity. FIC was found to depend strongly on the L/D ratio of the capillary die. Finally, temperature impacted the FIC behaviour extensively since it alters the activation energy needed for the formation of macroscopic structures. iv  Preface All the presented data in this thesis was obtained by the author of this PhD thesis under the supervision of Professors Savvas G. Hatzikiriakos and Dr. Antonios Doufas (ExxonMobil) at the University of British Columbia. Several apparatuses were used in order to obtain data under various flow and thermal conditions.  The results provided in Chapter 4 were presented at The Society of Rheology (SOR) 85th annual meeting held in Montreal, QC on October 13-17, 2013; Canadian Society of Chemical Engineering Conference held in Vancouver, BC on October 14-17, 2012; and ANTEC held in Cincinnati, OH on April 22-24, 2013. Some of the results of Chapters 5 were presented at APS March Meeting held in San Antonio, TX on March 2-6, 2015; and Polyolefins Conference, SPE held in Houston, TX on February 21-24, 2015. The results provided in Chapters 3, 4 and 5 are summarized in eight manuscripts written by the author with input from Prof. Savvas G. Hatzikiriakos and Dr. Antonios Doufas and have published or have been accepted for publication: [1] M. Derakhshandeh, A.K. Doufas, and S.G. Hatzikiriakos "Quiescent and shear-induced crystallization of polypropylenes" Rheologica Acta (2014): 1-17. [2] M. Derakhshandeh, G. Mozaffari, A.K. Doufas, and S.G. Hatzikiriakos "Quiescent crystallization of polypropylene: Experiments and modeling" Journal of Polymer Science Part B: Polymer Physics 52.19 (2014): 1259-1275. v  [3] M. Derakhshandeh, B. Jazrawi, G. Hatzikiriakos, A.K. Doufas, and S.G. Hatzikiriakos "Flow induced crystallization of polypropylenes in capillary flow" Rheologica acta  (2014): 1-15. [4] A.K. Doufas, M. Derakhshandeh, and S.G. Hatzikiriakos (Invited), “Rheology and Flow-Induced Crystallization of Polyolefins” APS March Meeting. San Antonio, TX. USA, March 2-6, 2015. [5] A.K. Doufas, M. Derakhshandeh, and S.G. Hatzikiriakos, “Polypropylene Flow-Induced Crystallization Rheometry” Polyolefins Conference, SPE. Houston, TX. USA, February 21-24, 2015. [6] M. Derakhshandeh, S.G. Hatzikiriakos, and A.K. Doufas “Crystallization of polypropylene: the effect of shear rate on morphology and degree of space filling" The Society of Rheology (SOR) 85th annual meeting. Montreal, QC. Canada, October 13-17, 2013. [7] M. Derakhshandeh, S.G. Hatzikiriakos, and A.K. Doufas “Crystallization of polypropylene: the effect of shear and temperature" ANTEC. Cincinnati, OH. USA, April 22-24, 2013. [8] M. Derakhshandeh, S.G. Hatzikiriakos, and A.K. Doufas “flow induced crystallization of polyolefins" Canadian Society of Chemical Engineering Conference. Vancouver, BC. Canada, October 14-17, 2012.  vi  Table of Contents  Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iv Table of Contents ......................................................................................................................... vi List of Tables ................................................................................................................................ ix List of Figures ............................................................................................................................... xi List of Symbols .............................................................................................................................xx List of Greek Symbols ............................................................................................................. xxiv List of Abbreviations ............................................................................................................... xxvi Acknowledgements ................................................................................................................. xxvii Dedication ............................................................................................................................... xxviii Chapter 1: Introduction ................................................................................................................1 1.1 Importance of various parameters on crystallization kinetics ......................................... 3 1.1.1 The effect of flow and temperature on crystallization ................................................ 4 1.1.2 The effect of molecular characteristics on crystallization .......................................... 9 1.2 Research objectives ....................................................................................................... 10 1.3 Thesis organization ....................................................................................................... 13 Chapter 2: Materials, Experimental Equipment, and Methodology ......................................14 2.1 Materials ....................................................................................................................... 14 2.2 Differential scanning calorimetry (DSC) ...................................................................... 15 2.3 Drag flow rheometer ..................................................................................................... 20 vii  2.3.1 Cone and plate geometry .......................................................................................... 21 2.3.2 Parallel plate geometry ............................................................................................. 22 2.3.3 Polarized optical microscopy .................................................................................... 24 2.4 Capillary rheometer ...................................................................................................... 31 2.4.1 Flow complexities in capillary flow ......................................................................... 33 Chapter 3: Quiescent Crystallization of Polypropylenes .........................................................37 3.1 Differential scanning calorimetry (DSC) ...................................................................... 37 3.1.1 Thermal behaviour of the polymers .......................................................................... 37 3.1.2 Isothermal quiescent crystallization .......................................................................... 38 3.1.3 Non-isothermal quiescent crystallization .................................................................. 49 3.2 Crystal growth rate via polarized optical microscopy (POM) ...................................... 57 3.3 Conclusions ................................................................................................................... 70 Chapter 4: Flow (Shear) Induced Crystallization of Polypropylenes .....................................72 4.1 Linear viscoelasticity of the polypropylenes ................................................................ 72 4.2 Flow-induced crystallization (FIC) under shear flow (parallel plate geometry) .......... 75 4.3 Crystallization kinetics using polarized microscopy under shear ................................. 87 4.3.1 Viscosity-crystallinity suspension modeling ............................................................ 91 4.4 Conclusions ................................................................................................................... 94 Chapter 5: Flow Induced Crystallization of Polypropylenes in Capillary Flow at High Deformation Rates .......................................................................................................................96 5.1 The relaxation times of the various polypropylenes ..................................................... 96 5.2 Flow induced crystallization in capillary ...................................................................... 99 5.2.1 Transition from a melt at high temperature to a non-homogenous melt/semi-solid . 99 viii  5.2.2 The effect of contraction angle on flow induced crystallization ............................. 108 5.2.3 The effect of L/D ratio on crystallization ............................................................... 112 5.2.4 The effect of die diameter on crystallization .......................................................... 116 5.2.5 The effect of molecular weight (Mw) on crystallization ........................................ 117 5.3 Conclusions ................................................................................................................. 120 Chapter 6: Conclusions and Recommendations .....................................................................122 6.1 Conclusions ................................................................................................................. 122 6.2 Recommendations for future work ............................................................................. 124 References ...................................................................................................................................126 Appendices ..................................................................................................................................141  : Nakamura model prediction of non-isothermal crystallization of polypropylenes................................................................................................................................................. 141  : Analysing the crystal growth: definition of radius and center of spherulites ... 146  : Determination of heat rise within polymer under microscopy and corresponding data corrections ....................................................................................................................... 149  ix  List of Tables TABLE 2.1: The PP resins studied along with their molecular characteristics and thermal properties................................................................................................................................. 15 TABLE 2.2: Working equations for cone and plate geometry. .................................................... 22 TABLE 2.3: Working equations for parallel plate geometry. ...................................................... 23 TABLE 2.4: Working equations for a capillary rheometer. ......................................................... 32 TABLE 3.1: Theoretical Avrami parameters for different mechanisms of crystallization. ......... 39 TABLE 3.2: The Arrhenius activation energy, Ea, of the different resins studied ....................... 43 TABLE 3.3: Constants of WLF equation obtained for different resins ........................................ 46 TABLE 3.4: Constants of the Hoffman-Lauritzen equation obtained for different resins ........... 48 TABLE 3.5: Smallest values of crystallinity obtained by using DSC at various cooling rates for PP5. ......................................................................................................................................... 53 TABLE 3.6: Non-isothermal induction times (s) obtained by using two different functions in Equation 3.8 along with the experimentally obtained induction time for PP5 under four different cooling rates. ............................................................................................................ 56 TABLE 4.1: The PP resins studied along with their activation energy for flow, Ea. ................... 75 TABLE 4.2: The critical shear rate above which the effect of flow on crystallization become evident at 0.5T = . ................................................................................................................... 86 TABLE 4.3: Optical microscopy images before and after image processing. .............................. 90 TABLE 4.4: Models’ parameter used to predict the results in FIGURE 4.13. ............................. 93 TABLE 5.1: Relaxation time of PP5 at various temperatures. ..................................................... 97 x  TABLE 5.2: Relaxation times of various resins obtained from sinusoidal oscillation at different temperatures. ........................................................................................................................... 99 TABLE 5.3: Estimated crystalline later thickness at wall of capillary. ...................................... 115  xi  List of Figures FIGURE 1.1: Schematic of lamella (a) (adopted from Gedde 1995), spherulite crystal (b) (adopted from Elias H.G. 2008), and shish-kebabs structures (c) (adopted from Huong et al. 1992). ... 6 FIGURE 2.1: Schematic of a disk-type heat flux DSC. 1) conduction path 2) furnace 3) lid 16 FIGURE 2.2: The experimental protocol which is used in DSC to obtain the melting and crystallization peak temperatures. ........................................................................................... 19 FIGURE 2.3: Schematic of cone and plate geometry and the definition of various geometrical and operating parameters. .............................................................................................................. 21 FIGURE 2.4: Schematic of parallel plate geometry and the definition of geometrical and operating parameters. .............................................................................................................................. 22 FIGURE 2.5: (a) Crossed polarizers - the transmitting polarized light will be cut off by analyzer................................................................................................................................................. 25 FIGURE 2.6: Schematic of polarized microscope (Mitutoyo VMU-V) used in this study .......... 27 FIGURE 2.7: Morphology evolved after 3279 s at the temperature of 145oC for PP5. The test specimen was cooled down from 200oC at the rate of 10oC/min until the isothermal test temperature of 145oC was achieved. ....................................................................................... 29 FIGURE 2.8: Schematic of a capillary die including the entry region (conical zone). Extensional flow occurs in the core of the flow in the entry region, while shear dominates the flow in the die. ........................................................................................................................................... 35 FIGURE 2.9: The maximum extensional rates in the centerline of contraction capillary flow as a function of contraction angle calculated by Equations 2.4 to 2.6. .......................................... 36 xii  FIGURE 3.1: The DSC thermogram of resin PP3 listed in TABLE 2.1 for three consecutive heating/cooling cycles. (a) The second melting and crystallization peaks were observed at 160.5°C and 115°C, respectively. (b) Thermal history elimination is demonstrated by the small differences obtained between second and third melting peaks. .............................................. 38 FIGURE 3.2: The relative crystallinity (X/Xf) obtained by DSC for isothermal crystallization as a function of time for PP5 at various temperatures.  The error bars are based on four replicate experiments. ............................................................................................................................ 40 FIGURE 3.3: Morphologies observed within the melt matrix of PP5 at different temperatures. Microscope images shows co-existence of disk-like crystals (shown with a white arrow) and spherulites. .............................................................................................................................. 41 FIGURE 3.4: Analysis of crystallization data for PP5 plotted in FIGURE 3.2 based on the Avrami equation (3.1). (a) The calculated values of the Avrami index vary between 1.9 and 2.7. (b) Refit of the Avrami equation by fixing the Avrami index n to 2. ........................................... 42 FIGURE 3.5: Fits of the Arrhenius equation to the Avrami rate parameter k calculated from Equation 3.1 using n=2. .......................................................................................................... 44 FIGURE 3.6: Isothermal quiescent crystallization fits of the Avrami equation with n=2 and using the Arrhenius equation for the temperature dependence of the rate constant for PP5. ........... 45 FIGURE 3.7: Fit of the Williams-Landel-Ferry equation to the Avrami rate parameter calculated from Equation 3.1 using n=2.  The Avrami rate parameter k was fitted to Avrami Equation 3.1 as a function of temperature as shown in FIGURE 3.4b. ....................................................... 47 FIGURE 3.8: Fit of Hoffman equation to the Avrami rate constants calculated from Equation 3.1 using n=2. ................................................................................................................................ 49 xiii  FIGURE 3.9: Non-isothermal quiescent crystallization prediction of the Nakamura model [Eq. (3.7)] with n=2 and Arrhenius temperature dependence for the rate parameter k without temperature correction for high cooling rates. The smallest crystallinity amount (see TABLE 3.5) obtained by DSC was fed to Nakamura model. ................................................. 53 FIGURE 3.10: Non-isothermal quiescent crystallization prediction of the Nakamura model [Eq. (3.7)] with n=2 and Arrhenius temperature dependence [Eq. (3.2)] for the rate constants with temperature correction for high cooling rates. (a) An infinitesimally small value of crystallinity (10-6, 10-8 and 10-10) at equilibrium melting temperature (185oC) is used as initial condition independent of cooling rate to progress the numerical solution. (b) The smallest detectable crystallinity amount obtained by DSC for each cooling rate was used as initial condition. As seen, similar induction times were obtained from the use of both functions. ......................... 54 FIGURE 3.11: Non isothermal quiescent crystallization prediction of Nakamura model with n=2 and Arrhenius temperature dependence for the rate constant with temperature correction for high cooling rates. The non-isothermal induction times are estimated from Equation 3.8. ... 56 FIGURE 3.12: Non isothermal quiescent crystallization prediction with n=2 and (a) WLF /(b) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The induction time estimated from isothermal data was fed to Nakamura model to obtain upturns..................................................................................................................................... 57 FIGURE 3.13: The variation of crystal radius as a function of time elapsed for PP5 at five different crystallization temperatures. Crystal growth is the slope of the fitted lines to the data. ........ 58 FIGURE 3.14: Arrhenius fit obtained from the growth rate of resins at different temperatures. 59 FIGURE 3.15: The quiescent (DSC) half crystallization times of all PPs as a function of normalized temperature obtained from isothermal crystallization experiments. Normalized xiv  temperature is defined as Cm CT TTT T−=−, where TC and Tm are crystallization and melting peak temperatures obtained by DSC. .............................................................................................. 61 FIGURE 3.16: POM images of investigated resins at 0.5T = . As shown PP4 has the highest nucleation density among all. PP3 and PP5 show a comparable nuclei density at the studied temperature region. ................................................................................................................. 62 FIGURE 3.17: Growth rate of PP5 crystals at various temperatures. Using a non-isothermal experimental protocol allows for an efficient way to determine crystal growth rates over a wider range of temperatures.................................................................................................... 64 FIGURE 3.18: Growth rate of crystals at various temperatures for PP2, PP3, and PP5. PP2 was produced using a different catalyst system compared to PP3 and PP4 and thus exhibits the lowest growth rates possibly due to the defects in the melt matrix of this polymer. .............. 65 FIGURE 3.19: Plot of ( ) ( )( )*ln G U R T T∞+ × −  vs ( )1 T Tf∆  for PP5. .................................... 68 FIGURE 3.20: Plot of ( ) ( )( )*ln G U R T T∞+ × −  vs ( )1 T Tf∆  for PP3. .................................... 68 FIGURE 3.21: Plot of ( ) ( )( )*ln G U R T T∞+ × −  vs ( )1 T Tf∆  for PP2. .................................... 69 FIGURE 3.22: Optical micrograph of PP3 and PP5 obtained under cooling rate of 4oC/min. PP3 shows higher nucleation density at high degree of undercooling. .......................................... 70 FIGURE 4.1a-f. Master curves of the viscoelastic moduli of all PPs listed in TABLE 2.1 at the reference temperature of Tref=190°C. ..................................................................................... 73 FIGURE 4.2: The master curves of the complex viscosity of all PPs listed in TABLE 2.1 at the reference temperature of Tref=190°C. ..................................................................................... 74 xv  FIGURE 4.3: The effect of cooling rate on the shear stress growth coefficient η +  of PP1 at the shear rates of (a) 0.005 s-1 and (b) 0.1 s-1 at T=142.4°C or 0.5T = . ....................................... 76 FIGURE 4.4: The effect of cooling rate on the induction time, tind, for the onset of crystallization in start-up of steady shear experiments for the resin PP1 at  T=142.4°C or 0.5T =  for two different shear rates. ................................................................................................................ 77 FIGURE 4.5: The shear stress growth coefficient η +  of PP1 at the normalized temperatures, at T  of 0.25, 0.5, and 0.75 at the shear rate of 0.005 s-1. ................................................................ 79 FIGURE 4.6: Crystal density observed at various radial positions in the parallel plate geometry for PP1 at 0.5T = and shear rate of 2s-1 at center of the geometry (a) at r=0.75R (b) and r=R (c), with crystal alignment to increase with the radial position. .................................................... 79 FIGURE 4.7: The shear stress growth coefficient η +  of PP1 at shear rates ranging from 0.005 s-1 to 0.1 s-1 at 152.6°C or 0.75T = . ........................................................................................... 80 FIGURE 4.8: The shear stress growth coefficient η +  of PP1 at shear rates from 0.005 s-1 to 0.1 s-1 at 152.6°C or 0.75T = . .......................................................................................................... 82 FIGURE 4.9: The effect of shear rate on the induction time for the onset of crystallinity for all PPs at 0.5T = The closed symbols correspond to half time quiescent crystallization obtained from DSC. ........................................................................................................................................ 83 FIGURE 4.10: The effect of shear rate on the induction time for the onset of crystallinity for PP6 and PP2 at 0.5T = . A critical shear rate is observed at which transition from quiescent to flow induced crystallization occurs. ................................................................................................ 85 xvi  FIGURE 4.11: The induction time for the onset of crystallinity as a function of shear rate for PP1 at different normalized temperatures. The closed symbols correspond to half time quiescent crystallization obtained from DSC.......................................................................................... 87 FIGURE 4.12: The relative crystallinity of PP5 as a function of time obtained from DSC and optical microscopy. (a) Raw data without temperature correction showing the effect of different thermal history in each test (b) Avrami prediction of PP5 under microscopy set-up.................................................................................................................................................. 89 FIGURE 4.13: Normalized viscosity as a function of space filling (crystallinity) for PP5.......... 93 FIGURE 5.1: The temperature dependency of the maximum and average relaxation time of PP5 obtained using various methods. ..................................................................................... 98 FIGURE 5.2: The flow curve of PP5 at various temperatures from 163oC to 190oC (Tm= 162.1oC). (a) The effect of flow on FIC at high shear rates is evident with the effect to be more dominant closer to Tm. (b) Applying the time-temperature principle using the shift factors determined from linear viscoelastic measurements shows that the deviation from the LVE curve (crystalline free) occurs at smaller apparent shear rate values with decrease of temperature. The increase of shear stress beyond the LVE is clearly due to the effect of flow on FIC. ... 101 FIGURE 5.3: (a) The entry pressure of PP5 at various temperatures using a die with L/D~0, D=0.51 mm and contraction angle of 2a=15o. (b) The shifted ends pressure data of PP5 at various temperatures using a die with L/D~0, D=0.51 mm and contraction angle of 2a=15o. ......... 105 FIGURE 5.4: The extensional viscosity of PP as a function of extensional deformation rate in the die entrance for PP5 at various temperatures calculated using the Cogswell analysis (Equations 2.5 and 2.8). .......................................................................................................................... 106 xvii  FIGURE 5.5: Normalized shear viscosity as a function of the apparent extensional deformation rate in the die entrance for PP5 at various temperatures. ...................................................... 107 FIGURE 5.6: The flow curves of PP5 at various temperatures using a capillary die having L/D=20 and contraction angle of (a) 02 60α =  and (b) 02 90α = . The dashed line represents the flow curve from LVE measurements. The effect of flow at high shear rates is clear from the large deviations from the LVE line which occurs at smaller apparent shear rates with decrease of temperature. Comparison of this set of data with that plotted in FIGURE 5.2b ( 02 15α = ) shows that the contraction angle (average extensional rate) has no additional effect on the crystallization behavior of PP5. ............................................................................................ 109 FIGURE 5.7: The flow curves of PP5 at 170oC using capillary dies having different contraction angles. ................................................................................................................................... 110 FIGURE 5.8: The extensional viscosity of PP5 as a function of the extensional rate using two capillary dies of different contraction angles (15o and 90o) at 170oC. The viscosity values obtained for 2α=90o agrees with 3η+ curve showing that the flow kinematics in the die entry having a contraction angle of 90o is extensional dominated whereas in that of 15o contraction is shear dominated................................................................................................................. 111 FIGURE 5.9: The flow curves of PP5 obtained by using capillary dies having different L/D ratios with all other parameters kept constant. A larger die length causes crystallization at smaller apparent shear rate values, which is due to the larger residence time (see Equation 5.3). ... 114 FIGURE 5.10: Illustration of various steps occurs in die when crystallization kicks in. ........... 114 FIGURE 5.11: The flow curve of PP5 at 170oC using dies having the same contraction angle and different diameter. As the diameter decreases, the degree of crystallinity formed increases, although a correction for viscous dissipation might counterbalance this effect. .................. 117 xviii  FIGURE 5.12: The flow curves of various resins at different temperatures shifted by applying the time-temperature superposition using the shift factors determined from LVE measurements. The dash lines represent the flow curve corresponding to LVE measurements. .................. 119 FIGURE 5.13: Shish structure formed in an extrudate of PP5 extruded at T=170oC and apparent shear rate of 250s-1 observed close to the wall of capillary die. ........................................... 120 FIGURE A.1: Non isothermal quiescent crystallization prediction of PP1 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8. ................................................................................................................ 142 FIGURE A.2: Non isothermal quiescent crystallization prediction of PP3 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8. ................................................................................................................ 143 FIGURE A.3: Non isothermal quiescent crystallization prediction of PP4 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8. ................................................................................................................ 144 FIGURE A.4: Non isothermal quiescent crystallization prediction of PP6 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8. ................................................................................................................ 145 FIGURE B.1: Circle fitted to a crystal and bisectors used to calculate the center of spherulites................................................................................................................................................ 147 xix  FIGURE B.2: Crystal radius as a function of time for PP3. Experimental data (three different crystals) were superimposed on a single line with the slope equals to the crystal growth rate................................................................................................................................................ 148 FIGURE C.1: The schematic of quartz parallel plate system used in this study. ....................... 150 FIGURE C.2: The temperature rise at r/R = 0.75 and z/h = 0.25 in a polymer sample that crystallizes at different temperatures. The temperature increases of 0.2 oC to 1.9oC at the crystallization temperatures of 131.4oC to 121.7oC respectively explain adequately the differences in FIGURE 4.12a. .............................................................................................. 151 FIGURE C.3: Fit of the Arrhenius equation to the Avrami rate parameter k using Avrami index of 2............................................................................................................................................. 152      xx  List of Symbols A  Shape constant  0A   initial cross sectional area of the sample in SER pC   heat capacity 𝑑𝑑  thickness of crystal disk D   capillary diameter bD   barrel diameter De   Deborah number enP∆   excess pressure drop in the capillary die entrance   ( ) 100% fH∆  heat of melting of 100% crystalline polymer  aE   activation energy 𝐺𝐺  growth rate of crystals ''G   loss modulus Gi  modulus of Maxwell nodes G0  pre-exponential parameter of Hoffman-Lauritzen model xxi  'G   storage modulus H   enthalpy  h  gap  between the two plates of the parallel plate geometry k  Avrami constant, thermal conductivity of polymer K   the Nakamura rate constant gK   constant of Hoffman-Lauritzen model  ok          constant of Hoffman-Lauritzen model 'k   factory defined constant for DSC L   length of capillary die 0L   intial length of test specimen in SER M   torque Mw  weight average molecular weight Mn  number average molecular weight n    Avrami index, power law exponent  𝑁𝑁  nucleation density Na  Nahme-Griffith number xxii  ?̇?𝑁  nucleation rate 1 2N N−  normal stress cP   pressure drop in capillary tube Q   volumetric flow rate q   rate of heat generation within the polymer matrix due to crystallization R   geometry and capillary radius r  radius CR   radius of circle T   temperature dt   experimental time scale Tg   glass transition temperature tind  induction time mT°  the equilibrium melt temperature of the polymer oT   reference temperature Rt   residence time xxiii  *U   universal constant in Hoffman-Lauritzen model zv   velocity of fluid in the capillary tube Wi  Weissenberg number X   degree of crystallinity CX   x-coordine of the circle center fX   the total crystallinity at the end of primary crystallization process ( )i , iX Y  coordination of three points on the circumference of a circle CY   y-coordinate of the circle center xxiv  List of Greek Symbols α   relative crystallinity, contraction angle in the die entrance β    angle of cone and plate geometry, coefficient for the pressure dependency of  viscosity Aγ   apparent shear rate γ   shear rate a wγ   apparent shear rate at the wall of the capillary die wγ   shear rate at the wall of the capillary die Rγ   shear rate at the edge of geometry 0γ   extrapolation of the limiting slope of strain-time curve to t = 0 Hε   Hencky strain rate aη   amorphous viscosity Aη   apparent viscosity η∗   complex viscosity ,u Aη   apparent extensional viscosity in the capillary die entrance  xxv  λi  relaxation time of Maxwell nodes maxλ   longest relaxation time ρ   density σ   shear stress Eσ   tensile stress ,E Aσ   apparent tensile stress in the capillary die entrance Aτ   apparent shear stress wτ    wall shear stress in capillary rheometry φ   volume fraction occupied by the spheres, angular distance traveled by fluid element φ   the rate of volume fraction increase of crystals mφ   heat flow rate in DSC maxφ   maximum volume fraction occupied by the spheres  Ω   angular velocity ω   angular frequency xxvi  List of Abbreviations DSC  differential scanning calorimetry  FIC  flow-induced crystallization MFR  melt flow ratio MWD  molecular weight distribution NMR  nuclear magnetic resonance PDI  polydispersity index POM  polarized optical microscopy PP  polypropylene    xxvii  Acknowledgements I would like to offer my sincere gratitude to my thesis supervisor Prof. Savvas G. Hatzikiriakos for his insightful guidance during my degree. His approach in solving complex problems has inspired me to further improve my interpersonal skills such as critical thinking and planning. It was an honour for me to work with such a friendly and knowledgeable leader. My special appreciations go to my co-supervisor, Dr. Antonios K. Doufas. His scientific advice was of great importance in refining my project extensively.  Working in the Rheology Group has been a privilege. I would love to thank all members, both past and present, for their hard work that has inspired so many ideas. I am also grateful for the financial support of the University of British Columbia through the prestigious 4YF program. From the bottom of my heart, I thank my good friends Hamid Rezaei, Sina Tebianian, Vinod Konaganti, Mahmoud Ansari, and my brother Babak Derakhshandeh for helping me get through the difficult times, and for all the emotional support, camaraderie, entertainment, and caring they 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.     xxviii  Dedication    Dedicated to my family:  Enaiatollah, Shahin, Babak, and Mahtab    1  Chapter 1: Introduction Polymer processing techniques such as film blowing, film casting, fibre spinning, calendaring, and injection moulding are widely used in the production of commercially important plastic products, including polypropylene the polymer under study in this PhD thesis. During these processes, the raw materials are forced to flow, thus undergoing shear and/or extensional deformation, which frequently accelerate/affect crystallization (flow-induced crystallization). Crystal formation and growth during these processing methods are crucial in deciding the final product quality in terms of mechanical and optical properties (Scelsi and Mackley 2008; Janeschitz-Kriegl 2003; Kornfield et al. 2002; Kumaraswamy et al. 1999; McHugh et al. 1995; Eder et al. 1990).   It is well known that isotactic Polypropylene (iPP) (the polymer under study in this PhD thesis) shows three types of crystals formed upon crystallization under various conditions, namely, α-form (the most common), β-form, and γ-form. The α-form often occurs in crystallization from melt or solution, while the β-form is induced by temperature gradient, molecular orientation, and β-nucleating agent (Kawai et al. 2002; Dorset et al. 1998; Lotz and Wittmann 1986; Norton and Keller 1985; Lovinger et al. 1977; Binsbergen and De Lange 1968). The γ-crystals are formed when the polymer is degraded or crystallized in the presence of pressure. The α-structure has two limiting modifications known as α1 (limit-disordered) and α2 (limit-ordered) form (Hirose et al. 2000; Auriemma et al. 2000; Immirzi and Iannelli 1998; Hikosaka and Seto 1973). The α1-crystalline form transforms into α2-form when polymer is annealed at high temperatures. Nakamura and co-workers (Nakamura et al. 2008) investigated the growth rate of α and β  2  crystalline form under a wide window of temperatures. Two crossover in growth rate of α and β crystals was observed at 133oC and 90oC. The α-form showed higher growth rate compare to β crystalline form for temperatures below 90oC or higher than 133oC. The α2 modification was evolved at temperature higher than 110oC and saturated at 140oC. Understanding these forms of crystallinity in polypropylene under various conditions is crucial in polymer processing.  An effective method of optimizing any industrial scale polymer process and/or improving the quality of the final product is by performing numerical simulations of the process. In optimizing a polymer process in terms of product formation and performance, one important element is to develop a model that predicts the degree of crystallinity as a function of time or domain position under non-isothermal crystallization conditions. It has been shown previously that the Nakamura equation can be used to predict both isothermal and non-isothermal quiescent crystallization kinetics (Chan and Isayev 1994). A correction of non-isothermal quiescent crystallization data for higher cooling rate (greater than 10oC/min) was suggested in order to obtain real temperature of the test specimens (Hammami et al. 1995; Chan and Isayev 1994). A second important aspect is to relate the degree of crystallinity to a normalized rheological function (NRF) such as normalized viscosity in order to describe the effects of flow. To put this into perspective, in the film blowing process the locked-in stresses at the frost line (location of transition from liquid to solid) dictates the tensile strength of the produced material (Kuijk et al. 1999; Han et al. 1983; Kwack et al.1983). Numerical simulation of this process and its further optimization heavily depends on the accuracy of the models and for the process under discussion a reliable model for FIC is crucial (Doufas 2013; Doufas et al. 2001; Doufas et al. 2000; Doufas et al. 1999). In such  3  models the degree of crystallinity is often related to normalized rheology functions (NRF) by various models derived from suspension theory. These models benefit from being easy to implement into simulation, however, they have their own limitations such as not being applicable to higher deformation rates. Most of the works performed before, have related normalized the rheological function which is obtained using a rheometer with relative crystallinity which is obtained by another method such as Differential Scanning calorimeter (DSC) (Lamberti et al. 2007; Acierno et al. 2003; Pantani et al. 2001; Boutahar et al. 1998; Titomanlio et al. 1997). Since thermal history can affect the crystallization kinetics significantly, it is preferred to obtain both NRF and relative crystallinity simultaneously using the same apparatus.   In the current work, the rheological properties and the degree of crystallization under various flow conditions are studied simultaneously to derive crystallization models for iPP that can be used directly in the optimization of polymer processing. Various thermo-mechanical histories are considered which as shown in this work affect the crystallization kinetics significantly. 1.1 Importance of various parameters on crystallization kinetics  Various parameters can affect the kinetics of crystallization. These parameters can be categorized as two groups, namely, the flow/thermal parameters, and molecular characteristics. The kinetics of crystallization is closely related to these parameters as discussed in the next sections.    4  1.1.1 The effect of flow and temperature on crystallization Flow stretches/orients chains in the direction of flow and disturbs their random coil configuration. Therefore, applying deformation decreases the thermodynamic barrier for crystal formation and subsequent growth (Van Meerveld et al. 2004; Keller 1968). The degree of kinetic enhancement is related to both strain and strain rate (Tiang et al. 2012; White et al. 2012; Chellamuthu et al. 2011; Tanner et al. 2009; Dai et al. 2006; Hadinata et al. 2006; Tanner et al. 2005; Stadlbauer et al. 2004; Kitoko et al. 2003; Swartjes et al. 2003; Gelfer et al. 1999; Eder et al. 1998). Extensional flow imposes an even stronger deformation on the sample compared to that of shear at comparable deformation rates and thus affecting the kinetics of crystallization more significantly. Since shear deformation is inherently a weaker flow, larger strains are necessary to observe effective Flow Induced Crystallization (FIC) (Derakhshandeh and Hatzikiriakos 2012). The main challenge facing extensional rheometers are the difficulty of generating uniform deformation throughout the sample particularly at high rates. Various apparatuses have been designed in order to investigate flow-induced crystallization (FIC) under extension. Melt-spinning technique was one such technique that has been used extensively to examine FIC (Patel et al. 2008; Paradkar et al. 2008; Ishizuka and Koyama 1997; Spruiell and White 1975; Nakamura et al. 1972; Ziabiki 1967). However, a large temperature gradient is imposed upon test specimens undergoing melt-spinning. Therefore, the flow contribution to FIC cannot be separated from that of thermal gradient.  Various other geometries were designed to induce stagnation flows, which yield strong extensional deformation. Stagnation flow was generated using opposed jets, four-roll mill, or  5  cross-slot geometry (Hassell and Mackley 2008; Swartjes et al. 2003; Mackay et al. 1995; McHugh et al. 1993; Crowley et al. 1976). Flow in these geometries are often nonhomogeneous with the maximum extension occurring at the stagnation point. A unique fixture for rotational rheometers capable of generating uniform extensional deformation at high rates which can be used to study FIC, is the Sentmanat Extensional Rheometer (SER) (Sentmanat 2004). It consists of two drums rotating in opposite direction, while stretching the sample within the chamber oven of rheometer under a precise temperature profile.  The capillary rheometry has also been used to assess FIC phenomena (Scelsi and Mackley 2008; Hadinata et al. 2007; Hadinata et al. 2006; Farah and Bretas 2004; Ness and Liang 1993; Titomanlio and Marrucci 1990; Ledbetter et al. 1984; Crater et al. 1980; Mackley et al. 1975; Southern and Porter 1970). While in the die entrance the flow is mainly extensional, in the die land, flow is purely shear. The capillary rheometer has been used extensively in an attempt to develop highly oriented crystalline structure from melt in the die entry region. Most of these studies focused on examining the properties of material in the entrance after imposing various flow and thermal histories (Ledbetter et al. 1984; Crater et al. 1980; Southern and Porter 1970). Optical observation in slit dies confirmed the formation of oriented structure at walls, which subsequently propagates toward the core. However, in the case of converging flow these structures were formed at the centerline where extensional strain and strain rate is large (Farah and Bretas 2004; Mackley et al. 1975).  The effect of flow on nucleation and growth of crystals using various optical techniques in various flow geometries was also studied previously. In particular, Coccorullo and co-workers  6  (Coccorullo et al. 2008) used optical microscopy to determine nucleation and growth rate of iPP under week shear deformation (?̇?𝛾 = 0 - 0.3s-1) using the Linkam shearing cell (CSS 450). It was reported that both nucleation rate and growth rate increase exponentially as shear deformation is increased. Increasing the deformation rate beyond a critical value yields to a transition from spherulitic nucleation to ordered structure known as shish-kebabs. Schematic of lamella (smallest units involve in crystallization), spherulite, and shish-kebabs structures are shown in FIGURE 1.1.  FIGURE 1.1: Schematic of lamella (a) (adopted from Gedde 1995), spherulite crystal (b) (adopted from Elias H.G. 2008), and shish-kebabs structures (c) (adopted from Huong et al. 1992). The transition from isotropic to oriented structures were discussed as a function of Deborah number (Baert and Van Puyvelde 2006). Shish-kebab formation at early stage of crystallization was previously monitored using Small Angle X-Ray Scattering (SAXS) and Wide Angle X-ray Diffraction (WAXD) (Somani et al. 2005). Shish structure was formed initially upon which the kebabs are grown subsequently. Shish structure found to be either mesomorphic or crystalline while kebabs are crystalline (Somani et al. 2005). When deformation stops  the shish structures grow by self-orientation of molecules or dissolve into the melt matrix due to relaxation of chains.  7   The Deborah (De) and/or the Weissenberg (Wi) numbers often used to estimate the chain orientation and stretch, which is imposed by flow (Scelsi and Mackley 2008; Baert and Van Puyvelde 2006; Hadinata et al. 2006; Acierno et al. 2003; Macosko 1994). Although the Deborah and Weissenberg numbers look similar they have different physical interpretation. The Weissenberg number is a measure of the degree of nonlinearity or orientation that is expected as a result of flow and it is defined as the ratio of the microscopic time scale of test specimen, maxλ , to the local strain rate, Aγ  (Dealy and Kim 2013) max AWi λ γ=                                (1.1) At low Weissenberg number (Wi<1) the elastic effects are negligible (random coil chain configuration), while at large values of Weissenberg numbers the flow conditions falls into Non-Newtonian region. The Weissenberg number is generally used in cases where the deformation is uniform. Uniform deformation in space and time is not often observed in industrial processes and thus Deborah number, De, is preferred. The Deborah number is defined as the ratio of material response time to experimental time scale. maxmaterial response timeexperimental time scale dDetλ= =                   (1.2)  At low values of the De number or large observation time scales, the material behaves like a Newtonian fluid with little memory effect. In the limit of De → ∞ the material resemble an elastic solid (Deshpande et al. 2010; Phan-Thien N 2002). Under the context of FIC, the Wi number indicates whether the deformation rate is large enough to enhance kinetics of crystallization and  8  impose oriented crystalline structures, while De number indicates whether the material reside within the region of deformation long enough to enhance the crystallization kinetics (Housmans et al. 2009; Elmoumni and Winter 2006; Van Meerveld et al. 2004). Therefore, at low values of the De number (i.e., smaller than 1), orientation of chains is expected if the Wi number is large; this simply means that the residence time of material under the deformation is larger than the characteristic relaxation time of the molecular chains. However, as the observation time decreases, chains orient in the melt less effectively. Chain orientation is expected to occur at Wi>1 and De<1. This orientation should be enough to impose effective shear induced crystallization if the shearing time is long enough (Acierno et al. 2003). For small De numbers (De<1) and large Wi numbers (Wi>100), chains are stretched in the direction of flow, which decreases the thermodynamic barrier for crystallization and thus enhances the crystallization kinetics. The longest relaxation time, maxλ  which correlates with the high-end tail of Molecular Weight Distribution (MWD) can be estimated from creep or sinusoidal oscillation experiments using Equations 1.3 and 1.4 (Macosko 1994). Previous studies have shown that the longest chains within the polymer matrix control the critical work which is needed to induce oriented structures (Acierno et al. 2003).  0max 0limτ γγλ→=                                                               (1.3) max 0'lim''GGω ωλ→=                                                                (1.4) Where 0γ is defined by extrapolation of the limiting slope of strain-time curve to t = 0. Under quiescent conditions, as the crystallization temperature increases, the probability of  9  obtaining stable nuclei becomes lower. The nuclei which are formed can be dissipated into the melt matrix faster at a higher temperature. Thus, a longer time is needed for crystallization to occur. Additionally, the kinetics become slower at higher temperatures amounting to a slower crystal growth.  At a given time, the degree of crystallinity is overall lower with increase of the isothermal crystallization temperature. Under the application of flow, as the temperature decreases, the longest relaxation time of various chains increases and thus more chains of smaller size are influenced by deformation effectively (higher Wi number and lower De number). Therefore, more chains having smaller De and higher Wi numbers are expected to crystallize, which in turn causes enhanced crystallization kinetics. 1.1.2 The effect of molecular characteristics on crystallization Molecular reptation theory was used to explain the differences observed in crystal growth rate of various polymers with different molecular architectures under quiescent conditions (Hoffman and Millner 1988). In the molecular reptation theory, friction of a linear chain in the matrix of polymer melt is proportional to its length (De Gennes 1971). As the molecular weight increases, the friction coefficient of chains within the melt increases, which leads to a reduced chain mobility. Therefore, the crystal growth rates are decreased with increase of molecular weight as well as the rate of crystallization (Miyata and Masuko 1998; Cheng et al. 1990; Vasanthakumari and Pennings 1983; Van Antwerpen and Van Krevelen 1972; Magill 1969; Magill 1967; Magill 1964). Crystallization from copolymers are often more complex in behaviour. The degree of co-monomer rejection by the growing face of crystals was observed to impact the crystal growth rate extensively (Lambert and Phillips 1994). These co-monomer units can decrease the probability of  10  active chains facing the growing interface of crystals and thus can be interpreted as defects. The variation in tacticity can be regarded as a factor introducing defects into the matrix and thus it should have similar effects as co-monomer insertion (Patki et al. 2007). Inclusion of small defects has tremendous effect on the growth rate of crystals (Wagner and Phillips 2001). Using various catalyst to manufacture the polymer leads to different type and degree of defects influencing the growth rate distinctly.  As previously stated, under the application of flow, the crystallization kinetics correlates with relaxation times of chains exist within the matrix. These relaxation times relates directly to molecular characteristics such as Molecular Weight and Molecular Weight Distribution. Starting from infinitesimal rate, flow affects the longest chains within the matrix first however as the flow rate increases smaller chain can be effectively oriented/stretch in the direction of flow contributing to a faster crystallization kinetics.  1.2 Research objectives In this research project, a capillary rheometer, a DSC, and a rotational rheometer coupled with various accessories such as Polarized Optical Microscope and parallel plates of various dimensions including optical plates were used in order to study crystallization of polypropylenes having various molecular characteristics. The research objectives are categorized into four groups: 1. To study the quiescent crystallization of several polypropylene (PP) resins with various molecular characteristics. This was investigated using Differential Scanning Calorimetry (DSC). Polarized Optical Microscopy (POM) was also used in conjunction with an Anton  11  Paar MCR 502 rheometer to observe the crystallization morphology developed during isothermal and non-isothermal conditions. The growth rate of crystals over a wide range of temperatures for the various resins were determined and compared using this technique. Modeling the isothermal and non-isothermal quiescent crystallization of PPs would be the basis for further developing a model that takes into account the effects of flow either by using a suspension theory (Kitoko et al. 2003; Kitano et al. 1981) or a frame invariant constitutive viscoelastic equation such as the two-phase microstructural model (Doufas 2014; Doufas et al. 2000; Doufas et al. 1999).  2. To study the role of temperature and shear flow on the crystallization kinetics of several polypropylenes of various molecular weights and polydispersities. In particular, the effects of important parameters for crystallization such as strain rate, strain, temperature and molecular parameters are examined under simple shear using rheometry coupled with polarised microscopy. Although some of these effects are well studied before (Bourgin & Zinet 2010; Baert et al. 2006; Kim et al. 2005; Hadinata et al. 2005; Coppola et al. 2004; Acierno et al. 2003; Duplay et al. 2000; Jay et al. 1999) including them here are beneficial. Most of the previously written papers were solely dealing with shear flow or to a less extent with extensional flow. Six different PP resins with different behavior in terms of nucleation (as observed by polarized optical microscopy), and growth rate are studied. The data provided in this section sets a basis for obtaining the model parameters which further can be validated by comparison with FIC behavior under extensional and/or mixed flow. A comprehensive model should predict crystallization behavior under quiescent, shear, extensional, and mixed flow conditions at various temperatures. Moreover, as mentioned  12  by Duplay et al. (2000), extra care should be paid when one deals with resins from different origins/manufacturing methods using different catalyst systems. 3. To study the applicability of various suspension viscosity models in modelling the crystallization under flow. These models are typically used to relate the system zero-shear-rate viscosity with the degree of crystallization evaluated experimentally. A light microscope equipped with a polarizer and an analyzer was used with a parallel-plate rheometer in order to study the degree of crystallization under various deformation rates. These data subsequently are used to evaluate the applicability of various suspension models in describing the degree of crystallinity under flow. 4. To study FIC at high shear/extensional flows relevant to polymer processing. Flow induced crystallization under mixed high shear and extensional flow rates at various temperatures was investigated in a capillary rheometer. More specifically, using many dies of different length-to-diameter ratio and entrance angles allowed to study the effect of extensional strain and rate on FIC over a wide range and also separate the relative effects of shear and extensional to a certain extent. Furthermore, the effect of shearing time was also studied using dies of identical diameter and of different length. The Cogswell analysis was applied in order to extract the extensional deformation rate and viscosity in the entrance zone. The methodology developed in this section can be used in two ways (i) assess and differentiate various polymers in terms of their crystallization behaviour at high shear and extensional rates relevant to polymer processing (ii) simulate the data to test and validate constitutive equations coupled with crystallization models.   13  1.3 Thesis organization The organization of this dissertation is as follow. Details of the apparatuses used in this study, which include a rotational rheometer, a DSC, and a capillary rheometer are described in Chapter 2, as are the thermal/molecular characteristics of different polypropylene resins used and the methodology associated with the experimental measurement techniques. Chapter 3 discusses the quiescent crystallization of all PP resins studied using first DSC and Polarized microscopy is also used in-situ in order to reveal information about the nucleation density and growth rate of crystals formed within each resin. This complements the DSC data and allows for a comprehensive comparison of the crystallizability of the various PPs (Objective 1). Chapter 4: reports the shear flow induced crystallization of resins examined in the current work. The effect of various parameters such as deformation, rate of deformation, temperature, and molecular parameters are investigated using a MCR-502 rotational rheometer (Objective 2). Furthermore, this Chapter provides the interrelation between degree of volume fraction of formed crystals and viscosity under various shear rate using an in-situ polarized microscope (Objective 3). Chapter 5 includes data obtained using a capillary rheometer. This allows, flow induced crystallization to be examined under mixed high shear and extensional flow rates at various temperatures (Objective 4).    14  Chapter 2: Materials, Experimental Equipment, and Methodology This chapter provides details of the apparatuses used in this study, which include a rotational rheometer, a Differential Scanning Calorimetry, and a capillary rheometer. Furthermore, the molecular characteristics of different polypropylene resins used in the experiments and the methodology associated with the experimental measurement techniques are discussed.  2.1 Materials Six different polypropylenes (PP) with different molecular weights and polydispersities were used in this work, all provided by the ExxonMobil Chemical Company. These resins are listed in TABLE 2.1 along with their molecular weight characteristics (molecular weight, Mw, polydispersity index (PDI=Mw/Mn) and melt flow rate, MFR). The Molecular weights (weight average molecular weight, Mw, and number average molecular weight, Mn) were determined using High-Temperature Gel-Permeation Chromatography (Polymer Laboratories PL-GPC-220) equipped with a differential refractive index detector (DRI) (Sun et al. 2001). Carbon NMR (13C) spectroscopy was used to measure the tacticity of the studied polypropylenes in terms of % molar meso pentads (mmmm). 13C NMR spectra were acquired with a 10-mm broadband probe on a Varian spectrometer having a 13C frequency of at least 100 MHz. The samples were prepared in 1,1,2,2-tetrachloroethane-d2 (TCE).  Sample preparation (polymer dissolution) was performed at 140°C where 0.25 grams of polymer was dissolved in an appropriate amount of solvent to give a final polymer solution of 3 ml.  15  TABLE 2.1: The PP resins studied along with their molecular characteristics and thermal properties. PP Resin Mw �𝑲𝑲𝑲𝑲 𝒎𝒎𝒎𝒎𝒎𝒎� � PDI MFR �𝒅𝒅𝑲𝑲 𝒎𝒎𝒎𝒎𝒎𝒎� � Tc (°C) Tm (°C) %molar meso pentads (mmmm) PP1 182 2.7 36.0 122.0 162.8 0.95 PP2 189 1.8 24.0 108.7 148.6 0.947 PP3 230 3.1 12.0 115.0 160.5 0.954 PP4 288 3.8 5.0 124.0 164.2 0.944 PP5 366 3.3 2.0 112.0 162.1 0.956 PP6 498 3.6 0.9 113.4 161.0 0.909  2.2  Differential scanning calorimetry (DSC) Two basic types of Differential Scanning Calorimeters (DSCs) are commonly used: • The heat flux DSC • The power compensation DSC They are different in the design and the theory which is used to determine the heat flow. In heat flux DSC, the temperature difference between the sample pan holder and reference pan holder is measured which correlates with the intensity of the heat exchange. The heat flow takes place via a well-defined heat conduction path with given thermal resistance and its rate is proportional to the measured signal. Heat flux DSCs can be categorized into three different groups known as disk-type measuring system, turret-type measuring system, and cylinder-type measuring system. In this section only the disk-type measuring system is explained, while detailed theoretical background about various types of DSCs are available in literature (Höhne et al. 2003).  16  In this study, the thermal behaviour of PPs was examined using a disk-type heat flux DSC calorimeter (Shimadzu DSC-60). This type of DSC is simple in design with a high sensitivity signal measurement. Small test specimens can induce sufficient signal to be measured, however the heat exchange between furnace and sample is limited and thus only medium heating and cooling rates are obtained. In disk-type heat flux DSCs the heat conducts from the furnace to the samples symmetrically through a disk of medium thermal conductivity in which the thermocouples are installed (FIGURE 2.1).       << PCCalibration KRecorderϕmT(t)S RΔT321 FIGURE 2.1: Schematic of a disk-type heat flux DSC. 1) conduction path 2) furnace 3) lid The sample pans are placed on this disk symmetrical to the center. Each temperature sensor covers the area of disk, therefore calibration can be carried out independent of the sample position inside the pan. If a sample transition occurs, a differential temperature signal is induced. The heat flow rate ϕm is internally assigned to this signal ΔT by factory-installed provisional calibration:   17  ' Tm kφ = − × ∆                                                           (2.1) The total uncertainty of the heat measurement is about 5 % (Höhne et al. 2003). In our study, dried nitrogen gas was chosen as heating flow when necessary in order to induce convection without any degradation of test specimens. The DSC samples weighted typically 1-2 mg were cut from PP pellets and sealed in aluminum pans.  Several heating and cooling cycles were applied on PP samples and the crystallization and melting peak temperatures were obtained by using the experimental protocol shown in FIGURE 2.2. The test specimen was heated at the rate of 10oC/min from 50oC to 200oC. Each sample was equilibrated at 200°C for 15 minutes in order to eliminate the thermal and flow histories that the polymer experienced during manufacturing. Subsequently, the test specimen was cooled to 50°C at a rate of 10oC/min and then reheated to 200°C with the same rate in order to obtain the crystallization and melting peak temperatures. History elimination is validated by performing a second cooling and a third heating cycles. The crystallization and melting peak which are observed during these cycles were almost identical to those found in the previous ones. The crystallization and melting peak temperature (𝑇𝑇𝐶𝐶 and 𝑇𝑇𝑚𝑚 respectively) are summarized in TABLE 2.1 for all resins. The non-isothermal crystallization kinetics were investigated by selecting four different cooling rates of 2, 5, 10, and 20oC/min via DSC. Identical protocol for history elimination was implemented as described previously, that is each sample was equilibrated at 200oC for 15 minutes before cooling at the desired rate. For a given cooling rate, the relative crystallinity (degree of transformation) X(T)/Xf, where Xf is the total crystallinity at the end of primary crystallization  18  process, as a function of temperature was calculated from the DSC exotherm by integrating the area under the crystallization peak of the DSC thermogram from the equilibrium melting temperature ( mT°  ) up to the desired temperature as follows:  ' '' '( )( )( )mfmTTTfTH T dTX TXH T dT°°=∫∫     (2.2) where H is the enthalpy (heat flow), Tf is the temperature corresponding to the intercept of the exotherm with the baseline in the region of the glass transition temperature and mT°   is the equilibrium melting temperature. The integrations were performed using a generalized trapezoidal rule.  To obtain the degree of crystallinity under isothermal conditions each test specimen was soaked at the temperature of 200oC for 15 min and then cooled at the cooling rate of 10oC/min to the desired crystallization temperature. The thermal behavior of the sample was recorded and the degree of crystallinity was determined using Equation 2.3.  ( ) oottftdH dtdtX tdHX dtdt∞   =   ∫∫     (2.3) where H is the enthalpy (heat flow), and to is the time at which the desired crystallization  19  temperature is reached. The half-time of isothermal crystallization was also determined at different temperatures between the determined crystallization, Tc and melting, Tm, DSC peaks. These were defined in DSC thermograms as the time required for a sample to reach 50% of its final crystallinity at the designated crystallization temperature.   Since the investigated PP resins exhibit different melting and crystallization peaks, the isothermal crystallization for the various resins was studied at fixed values of a normalized temperature defined by ( ) ( )/C m CT T T T T= − − . This enables also a consistent and fair comparison of the half times of crystallization of all resins at specific values of 0.25, 0.5, and 0.75T =  on the same graph. 0 20 40 60 80 100 120 140501001502002nd Cooling cycle 3rd  Heating cyclePeak Melting Temperature Determination  Temperature, T (oC)Time, t (s) DSC protocolHistoryEliminationPeak CrystallizationTemperature DeterminationHistoryEliminationValidation FIGURE 2.2: The experimental protocol which is used in DSC to obtain the melting and crystallization peak temperatures.  20  2.3 Drag flow rheometer In drag flow rheometers, measured quantities (such as torque and angular velocity) are converted to stress and strain on the sample. Such data are used to determine rheological material parameters needed in a particular constitutive equation. Two distinct type of drag flow rheometers are commercially available, namely, controlled strain rheometer and controlled stress rheometer. In a controlled strain rheometer the angular velocity is fixed while monitoring the torque signal. The most versatile controlled strain design is direct coupling of a DC motor to the rotating shaft. A tachometer is used to control the angular velocity and a capacitance transducer is used to control the angular position. In a controlled stress rheometer, the torque is fixed while the angular motion is measured. In a common design, magnetic field rotates around a copper cup which is supported by an air bearing. The field generates eddy currents in the cup and thus it follows the rotating field. The advantages and disadvantages of each these systems are discussed in detail elsewhere (Macosko 1994). In this work, a controlled stress Anton Paar rotational rheometer (MCR 502) was used to study the rheology of samples. Polypropylene pellets were melted at the temperature of 200oC and then were pressed in a compression molding apparatus using a pressure of 500 kPa to produce sheets of various thickness. The thickness varied from 0.1 mm to 1 mm depending on the accessories used in conjunction with the rheometer.  21  2.3.1 Cone and plate geometry Cone and plate is one of the most popular geometries to study the rheology of non-Newtonian fluids since it generates homogenous deformation and allows the direct measurement of first normal force difference. A sketch of the cone and plate geometry is shown in FIGURE 2.3. Spherical coordinates are used to drive the underlying equations used to determine shear stress and rate. Detailed discussion of the background theory and the derivation of underlying equations summarized in TABLE 2.2 can be found elsewhere (Macosko 1994).  FZ MΩ,φ θ β rR FIGURE 2.3: Schematic of cone and plate geometry and the definition of various geometrical and operating parameters. The linear viscoelastic properties of all PP resins, namely the storage modulus, 'G , the loss modulus, ''G , and the complex viscosity, ( )η ω∗  is measured using this cone and plate geometry. These data are useful to compare resin behaviour in terms of their zero-shear viscosity. In addition, the flow curve of each resin can be computed as ( )σ η ω ω∗= ×  and plotted together with the corresponding capillary data (flow curve) in order to identify deviations due to FIC effects.   22  TABLE 2.2: Working equations for cone and plate geometry. Shear stress  12 332MRφθσ σπ= =  Shear strain φγβ=  Shear rate γ βΩ=  2.3.2 Parallel plate geometry Parallel plate geometry is used in flow induced crystallization extensively since it allows highly viscous material to be examined. The flow in parallel plate is similar to that of cone and plate although non-homogenous. The parallel disks geometry is sketched in FIGURE 2.4. Detail discussion of the background theory and the derivation of underlying equations which are summarized in TABLE 2.3 can be found elsewhere (Macosko 1994).   FZ MΩ,φ Rh FIGURE 2.4: Schematic of parallel plate geometry and the definition of geometrical and operating parameters. In palate-plate geometry the shear rate increases linearly from the center to the edge of the plate. If the shear rate is reported at radius of 0.75R as suggested by Macosko 1994, the difference  23  between the true and the measured viscosity is often less than 2% (Macosko 1994). Crystallization starts from the edge due to the higher shear deformation experiences by the test specimen. For modeling purposes the induction times for the onset of crystallization are usually defined at very small relative crystallinity (~1-5%) and therefore the crystallization phenomenon which starts from the edge has a minimal effect on the rheological behavior captured at induction times. When crystallization begins, polymer shrinks and thus the underling equations used for rheological characterization are not applicable without modifications regardless of the geometry used. The importance of induction times are also discussed elsewhere in detail (Godara et al. 2006).  TABLE 2.3: Working equations for parallel plate geometry. Shear stress 12 3ln32 lnz RM d MR dθσ σπ γ = = +   Apparent or Newtonian shear stress 32AMRσπ=  Representative shear stress ( ) ( ) 2%A Aη τ η σ= ±  ln0.76 1.4lnA Rd Mfor anddσ σγ= < Shear rate RRhγ Ω=  Shear strain rhθγ =   The shear induced crystallization behaviour of polymers was examined in simple shear flow using the parallel plate geometry with plates of 25mm in diameter. For this protocol, the test specimens had thickness of about 1 mm. These experiments are confined to the temperature region between the crystallization and the melting peaks found in the DSC thermograms under the heating and cooling modes. This temperature range is of great importance in FIC due to the tendency of  24  molecules to crystallize and reorient. Since higher temperatures impede the crystallization kinetics excessively, FIC at higher temperatures is extremely time-consuming given the low achievable shear rates which are imposed by apparatus limitations. Of note, the crystallization behaviour of resins at shear rates larger than 2 s-1 couldn’t be investigated considering edge failure effects. The thermal and flow histories of the samples were eliminated by heating up each test specimen to 200°C (around 40°C above melting peak temperature) for 15 minutes prior to cooling at the rate of 10°C/min to the desired temperature and experimental testing. Fast and accurate cooling is crucial to reach the desired crystallization temperature without any temperature undercooling which would lead into premature crystallization. The Peltier system of the rheometer was used for this study since it can produce higher cooling rates more precisely compared to those of the convection oven system. The influence of several parameters such as temperature, deformation and deformation rates (shear) on the crystallization kinetics of the PPs were studied at shear rates up to 2 s-1.  2.3.3 Polarized optical microscopy Natural light waves vibrate in random directions and thus is called non-polarized light. Linear polarizers are used to filter all the light waves except the ones which are parallel to the polarizer. Therefore linearly polarized light possess a unique direction. When a second polarizer (analyzer) is used the relative direction of polarizer and analyzer dictates how much light is transferred. To put this into perspective, if the polarizer and analyzer are in the orthogonal directions, the transmitting polarized light will be cut off by the analyzer. Therefore, an isotropic sample will appear as black in the microscope. If anisotropy exists within  25  the matrix, the image appears with a better contrast compared to the non-polarized microscope. Such state is known as crossed nicols or cross polarizers. Parallel nicols or parallel polarizers is the state in which the analyzer direction matches the direction of polarizer and thus the amount of light transmittance is maximized. Cross polarizers and parallel polarizers states are shown schematically in FIGURE 2.5. Light source Polarizer AnalyzerLight source Polarizer Analyzera)b) FIGURE 2.5: (a) Crossed polarizers - the transmitting polarized light will be cut off by analyzer  (b) parallel polarizers - the amount of transmitted light is maximized. Polarized optical microscopy is used in this work under both quiescent and flow conditions. Optical microscopy under quiescent condition is often used to extract the growth rate and nucleation density of crystals, which are formed within the matrix. These data are used along with DSC results to shed light on the differences exist between crystallization kinetics of resins studied.  Upon the application of flow, in-situ optical microscopy is used to obtain interrelation between viscosity and volume fraction of the formed structures. This relationship is of great importance for model validation purposes.  26  a) Optical microscopy under quiescent condition Polarized optical microscopy was implemented to obtain growth rate of crystals using two different protocols. In the first approach, the radius of a desired crystal is monitored as a function of time elapsed under isothermal conditions. The growth rate was then obtained from the slope of the line that best fits the data. This approach was used for low degree of undercooling (i.e. high temperatures). At high degree of undercooling, the heat of crystallization imposes significant temperature gradients within the sample, thus jeopardizing isothermal crystallization. The increase in temperature of test specimen depends on the thickness and weight of sample. Larger samples generate more heat, while thicker samples hinder the heat transfer rate from the core to the surface.  In the second approach, the radius of crystals are determined using a non-isothermal protocol under various cooling rates (Chen and Chung 1998; Martins et al. 1993; Chung and Chen 1992). The slope of radius versus time elapsed curve at each temperature defines the growth rate of crystals. As the temperature decreases, larger heat is generated per unit of time, which should be transferred efficiently. This problem is more pronounced at higher cooling rates (Ding and Spruiell 1997; Ding and Spruiell 1996; Martins et al. 1993). Therefore, the samples which are used in this method should be very small in both weight and thickness. The crystal growth rates obtained using a non-isothermal experimental protocol, are often validated by the rates obtained under isothermal conditions. Self-nucleation process was suggested to extend the non-isothermal approach to the low undercooling region (Di Lorenzo et al. 2000). It consists of creating crystals at lower temperatures, then soaking the test specimen at a higher temperature (160oC for iPP) for a certain period of time after which the cooling rate is applied while monitoring crystal growth.  27  The temperature at which the sample is maintained should be high enough in order to ensure relaxation of any imposed ordering in the melt while the nuclei remain intact (Di Lorenzo et al. 2000). Half-mirrorLensC-mount cameraFiber-optic cablePolarizerAnalyzer FIGURE 2.6: Schematic of polarized microscope (Mitutoyo VMU-V) used in this study In this work, both approaches described above have been implemented in Chapter 3. A Mitutoyo VMU-V polarized microscope (Schematic of which is shown in FIGURE 2.6) was equipped with a Lumenera LU 165 color CCD camera and used with an Anton Paar MCR-502 in order to study the evolution of crystalline structure. In a typical experiment, the test specimen is  28  prepared by pressure molding of a desired resin at the temperature of 200oC and the pressure of about 500 kPa.  Using the first approach, the test specimen is placed in the space between two glass parallel plates (43 mm in diameter). After reaching the desired gap size of 0.1 mm, the test specimen is equilibrated at 200oC for 15 min in order to eliminate the thermo-mechanical histories before cooling at the rate of 10oC/min to the desired crystallization temperature. The crystalline microstructure, which is formed within the polymer melt matrix over time, is observed in situ with the microscope. The gap size of about 0.1 mm is necessary to allow light transmission through the sample thickness in order to observe clearly the crystals formed during crystallization. The morphology developed during crystallization as a function of time at the specified temperature is captured as a video at 15 frames/second (fps) and then is decompiled to frames using Matlab. One such frame is shown in FIGURE 2.7 for PP5 after cooling at the rate of 10oC/min from 200oC to 145oC and exposing the sample at 145oC for 3279s. The isothermal crystal growth rate under different crystallization conditions for various polypropylenes are determined by choosing twenty frames equally spaced in time. At least three distinct crystals are analyzed by a numerical code written in C# in order to check reproducibility. Details of the method used to obtain the radius of spherulites are described in the Appendix B. It is noted that in the case of isothermal crystallization experiments, “time” is defined as the time elapsed from the moment at which the isothermal crystallization temperature is reached.  In the second approach, after eliminating the thermal and flow histories described above, each test specimen is cooled to 100oC at different cooling rates of 0.5, 1, 4, 8, and 13oC/min while  29  video frames are captured. These videos are then decompiled frame by frame as previously mentioned and analyzed subsequently.    FIGURE 2.7: Morphology evolved after 3279 s at the temperature of 145oC for PP5. The test specimen was cooled down from 200oC at the rate of 10oC/min until the isothermal test temperature of 145oC was achieved.       30  b) Optical microscopy under flow deformation Polarized light microscopy was also used in-situ with an Anton Paar MCR502 in order to examine the structure evolution and volume fraction of formed crystals under different shearing conditions. In a typical experiment, the sample (disk) is placed in the space between two glass plates of the rheometer (parallel-plate geometry of 43mm in diameter) and the microstructure is observed with the microscope as a function of time.  The space filling of crystals in the polarised images is calculated through image analysis as a function of time and then converted to degree of crystallinity by using the densities of crystal and polymer melt (0.94 and 0.85 𝑔𝑔 𝑐𝑐𝑐𝑐3� ) respectively (Natta et al. 1955).  The microscopy videos are decompiled to frames which are analyzed by an image processing software (Matlab and ImageJ) in order to calculate the amount of space filling. Two different image processing methods are utilized to validate the obtained results. In the first approach, the grayscale filter is first applied to all extracted images to set each pixel value to a single number that represents the brightness of the pixel. The pixel format is typically the byte image which gives a range of possible values from 0 to 255. Frequently, zero is taken to be black while 255 are taken as white. Values in between 0 and 255 construct different shades of gray. Each pixel value is normalized between 0 and 1 in Matlab in order to define a threshold. All pixels within the microscopy images are converted into black or white based on the specified threshold, which was estimated by Matlab and further tuned by user. Eventually, the space filling is estimated as the ratio of the number of white pixels to the total number of pixels within the image.   31  In the second approach, the area of crystals is estimated by the area of circles with the same center and radios as that of the crystals. In this method, any overlap between the circles is considered once. Circles were drawn manually and then processed by ImageJ software.  Image quality plays a significant role in acquiring accurate results in any image processing technique. Therefore, the errors contributing to this technique is minimized by obtaining clear images at gap size of about 0.5mm. Although a smaller gap size is preferable for a better light transmission, gap size of smaller than 0.5mm could affect the shear stress data obtained by the rheometer. Most of the works which are performed before relate a normalized rheological function obtained using a rheometer with the relative crystallinity which is obtained by another method such as DSC (Acierno et al. 2003; Lamberti et al. 2007; Pantani et al. 2001; Boutahar et al. 1998; Titomanlio et al. 1997). Since the relative crystallinity and NRF are obtained using the same experiment in this study, the temperature and shear histories imposed on the samples are identical and thus the experimental data are expected to be more reliable.  2.4 Capillary rheometer Capillary rheometers are among the oldest methods used for measuring viscosity. A capillary rheometer is a pressure-driven flow and thus the flow is nonhomogeneous. Therefore, capillary rheometers are only used to measure steady shear functions. These pressure-driven rheometers are simple in operation and give accurate viscosity data. Pressure is often generated using electric motors upon the material in a reservoir. A capillary die of radius R and length L is connected using a die holder to the bottom of the reservoir. Pressure drop over the capillary tube is used to determine shear stress. The working equations for the capillary is shown in TABLE 2.4.  32  To study flow induced crystallization (FIC) at high shear rates and temperatures, a capillary rheometer (Instron 4465) fitted with several capillary dies of various lengths, diameters and entrance angles was used. This was necessary in order to assess a number of effects, including the effects of length-to-diameter (L/D) ratio, die diameter (D) and entrance angle, 2α. Also in many cases was necessary to determine the capillary entry pressure loss (Bagley correction) in order to determine the true flow curves (entry pressure corrected) of polypropylenes to be compared with the corresponding ones obtained from linear viscoelastic measurements.  TABLE 2.4: Working equations for a capillary rheometer. Wall shear stress 2cwPRLσ =  Wall shear rate , 34A wQRγπ=  ,c1 lnQ34 lnPw A wddγ γ = +     ( ),1 13 model4w A wfor power lawnγ γ  = +     Representative shear rate ( ) ( ) 2%A Aη γ η γ= ±   ln0.83 0.2 1.3lnA cd Qfor andd Pγ γ= < <    The capillary experiments were performed at temperatures above the melting peak, Tm, as follows. The resins were placed inside the capillary barrel and melted at the temperature of 200oC for at least 15 minutes. After that, the temperature was decreased to the desired experimental temperatures; this step required approximately 15 minutes according to the test temperature.  Since the temperature window used in the experiments was above the Tm of all resins, premature  33  crystallization due to undercooling was not expected before the application of flow. Quiescent crystallization experiments using the DSC demonstrated no sign of crystallization for several hours at temperatures above Tm. Therefore, the crystallization behaviour observed was solely linked to the deformation effects. The barrel (reservoir) of the capillary rheometer is equipped with 4 individual heating zones from the barrel top to the die land in order to maintain the temperature uniform along the barrel. 2.4.1 Flow complexities in capillary flow Consider flow in a capillary rheometer, where the barrel diameter is Db, the die diameter is D, the die length is L and the entry contraction angle to the die is 2α . Initially the polymer melt flows in the entry region of the capillary, where the flow is mixed extensional and shear as shown in FIGURE 2.8. In the core region of the entry, the flow is largely extensional with the highest extensional rate to be attained at the centerline. The flow in the entry becomes shear close to the wall. The extensional as strong flow is the dominant flow that induces crystallization at times at least one order of magnitude less than those seen in shear. In addition for temperatures above Tm, extensional has been observed to induce crystallization, which is much more dominant compared to shear (Derakhshandeh and Hatzikiriakos 2012; Hadinata et al. 2007). Two different approaches were suggested to estimate the extensional properties in the die region. The first approach is based on continuum mechanics and was developed for laminar flow of an incompressible Newtonian fluid in a conical geometry similar to this depicted in FIGURE 2.8 (Everage and Ballman 1974).   34  ( ) ( )max 233 1 cos 28cot 1 cos 1 2cosA αα α αγε − =  − +         (2.4) where 332A Q Dπγ =  is the apparent shear rate at the wall and Q is the volume flow rate. The maximum extensional rate, as previously mentioned, occurs at the centerline of the entry (conical zone). Equation 2.4 indicates that as the contraction angle increases, the maximum extensional rate also increases. The second approach which is based on geometrical analysis and it was developed by Cogswell (Cogswell 1972). According to Cogswell the maximum extensional rate, maxε , as a function of contraction angle, 2α , apparent shear rate, Aγ , and the local power-law exponent, n, is as follows: max(3 1) tan2( 1)Annγ αε+=+      (2.5) where the local power-law exponent, n, is given by the slope of a double logarithmic plot of the shear stress at the wall versus the apparent shear rate. For a Newtonian fluid, n is equal to one and the apparent shear rate becomes equal to the wall shear rate, Wγ . Thus for Newtonian fluids Equation 2.5 is reduced to max tanW αγε =  . For a Newtonian fluid Equations 2.4 and 2.5 yield similar results for contraction angles, 2α , of less than 90o as shown in FIGURE 2.9. An alternative model to the Cogswell formula has been proposed (Metzner and Metzner 1970):  35  max1 cossin2Aε αγ α+=      (2.6) All these models show comparable behavior for small contraction angles (as shown in FIGURE 2.9).  DbDLshear flowExtensional flowShear flow FIGURE 2.8: Schematic of a capillary die including the entry region (conical zone). Extensional flow occurs in the core of the flow in the entry region, while shear dominates the flow in the die.  Using the Cogswell analysis, the apparent extensional stress, ,E Aσ  is defined as , 11 22E Aσ τ τ≡ −, and the apparent extensional viscosity are given as:   , 11 223 ( 1)8E A enn Pσ τ τ= − = + ∆      (2.7)   ,,maxE Au Aσηε=      (2.8)  36  The total Hencky strain experienced by a fluid element moving along the centreline of the contraction region from far upstream to the die exit is given by (Rothstein and McKinley 1999):   0 2( )max max0 ( )lnzzt v z LbzzvDdvdtv D=−∞  ε ≡ ε = =      ∫ ∫     (2.9) where to is the time spent by the polymer melt in the centreline and Db/D is the contraction ratio. These equations will be used to analyse the experimental data in an attempt to infer the relative effects of shear and extensional flows at high rates relevant to polymer processing. 0 10 20 30 40 50 60 70 80 90 10001234567           γw=1 s-1          n = 1 Equation 1 Equation 2 Equation 3Maximum extensional rate, εmax (s-1)Half of contraction angle, α.. FIGURE 2.9: The maximum extensional rates in the centerline of contraction capillary flow as a function of contraction angle calculated by Equations 2.4 to 2.6.   37  Chapter 3: Quiescent Crystallization of Polypropylenes In this chapter the quiescent crystallization of all PP resins listed in TABLE 2.1 is studied using first DSC. Polarized microscopy is also used in-situ in order to reveal information about the nucleation density and growth rate of crystals formed within each resin. This complements the DSC data and allows for a comprehensive comparison of the crystallizability of the various PPs.    3.1 Differential scanning calorimetry (DSC) 3.1.1 Thermal behaviour of the polymers  FIGURE 3.1a depicts a typical thermogram of one of the PPs (PP3 in TABLE 2.1) using the experimental protocol discussed in section 2.2. The melting and the crystallization peak temperatures are observed to occur at 160.5°C and 115°C, respectively. As shown in FIGURE 3.1b, the thermo-mechanical history causes a difference of 2.5°C between the first melting peak (163.0°C) and second melting peak temperature (160.5°C). The difference between second and third heating peaks is only 0.2°C which is within experimental error. This further indicates that the sample soaking time of 15 minutes at 200°C is long enough to eliminate thermal history (FIGURE 3.1b). Similar comments apply to the crystallization/cooling peaks as well as to the DSC results (second melting and cooling peaks) for all other samples listed in TABLE 2.1.    38  100 200-50510Tm=160.5oCTC=115oCHeat, Q (mW)Temperature,T (oC)     PP3 DSC  a150 155 160 165 170 175-50     PP3 DSC  First heatingpeakSecond and thirdheating peakHeat, Q (mW)Temperature,T (oC)b FIGURE 3.1: The DSC thermogram of resin PP3 listed in TABLE 2.1 for three consecutive heating/cooling cycles. (a) The second melting and crystallization peaks were observed at 160.5°C and 115°C, respectively. (b) Thermal history elimination is demonstrated by the small differences obtained between second and third melting peaks.  3.1.2 Isothermal quiescent crystallization FIGURE 3.2 depicts the relative crystallinity (from DSC) as a function of time for PP5 at four different quiescent crystallization temperatures, namely 120oC, 125oC, 130oC, and 135oC. As the crystallization temperature increases, the probability of obtaining stable nuclei becomes lower. The formed nuclei can be dissipated into the melt matrix faster at a higher temperature. Thus, a longer time is needed for crystallization to occur. Additionally, the kinetics become slower at higher temperatures amounting to a slower crystal growth.  At a given time, the degree of crystallinity is overall lower with increase of the isothermal crystallization temperature as shown in FIGURE 3.2. The isothermal Avrami formula (Equation 3.1) is often used to model/represent quiescent crystallization data obtained by DSC. This equation is valid within the primary growth region,  39  which corresponds to relative crystallinity of 30-70%. It is noted that the Avrami equation does not account for secondary growth of crystals, crystal perfecting process, and the shrinkage of sample due to crystallization (Patki et al. 2007; Mandelkern 2002). 1nkteα −= −       (3.1) where “n” is the Avrami index and “k” is the Avrami rate parameter. The Avrami parameter k contains the temperature dependence of the nucleation and crystal growth processes, while the exponent “n” depends on the geometry and dimensionality of the growth as well as the nature of the nucleation process (McHugh et al. 1995; Mandelkern 2002).    TABLE 3.1: Theoretical Avrami parameters for different mechanisms of crystallization. Crystal growth shape Nucleation mechanism Avrami index n Avrami rate constant k Crystal Dimensionality Sphere Sporadic (homogeneous) 4.0 313 NGπ   3-dimensional Sphere Predetermined (heterogeneous) 3.0 343 NGπ* 3-dimensional Disc Sporadic (homogeneous) 3.0 21  3 NG dπ  ** 2-dimensional Disc Predetermined (heterogeneous) 2.0 2  NG dπ   2-dimensional Rods Sporadic (homogeneous) 2.0 21  4 Nd Gπ   1-dimensional Rods Predetermined (heterogeneous) 1.0 21  2 Nd Gπ   1-dimensional        * N is nucleation density        ** d is thickness of the disk   40  While the nucleation and crystal growth rate are lumped into the rate parameter k, one cannot determine the two effects unless the crystal growth rate G is measured independently e.g. via optical microscopy as will discussed later (crystal growth analysis). Theoretical values of the Avrami parameters for different mechanisms of crystallization is listed in TABLE 3.1 (Patki et al. 2007). The Avrami equation is fitted to isothermal crystallization data obtained by DSC for all PPs at four different quiescent crystallization temperatures. Here the details of the analysis will be presented only for PP5. FIGURE 3.2 shows the average data over four replicate measurements for PP5 with error bars obtained with a confidence intervals of 95%.  FIGURE 3.2: The relative crystallinity (X/Xf) obtained by DSC for isothermal crystallization as a function of time for PP5 at various temperatures.  The error bars are based on four replicate experiments. The Avrami index varies around the theoretical value of 2 (1.9-2.7) for PP5 over the range of the investigated temperatures. This implies that the crystals resemble like disks rather than  41  spherulites. In fact as shown in FIGURE 3.3 often mixed spherulites and disk like crystals are formed. Ellipsoids shown in FIGURE 3.3 are disk-like crystals with their normal axis not aligned in the view direction. Previous studies showed that polypropylene with long chain branching form disk-like structures with Avrami index of approximately 2 (Tian et al. 2006).   FIGURE 3.3: Morphologies observed within the melt matrix of PP5 at different temperatures. Microscope images shows co-existence of disk-like crystals (shown with a white arrow) and spherulites.  42  The Avrami indexes which are obtained from the data plotted in FIGURE 3.4a, did not correlate with temperature. Therefore, the theoretical Avrami index of 2 was chosen to re-evaluate the Avrami rate constants (“k”) of PP5 at these temperatures. The Avrami fits for PP5 at various temperatures with a fixed Avrami index of 2 is shown FIGURE 3.4b. A similar procedure is implemented by various authors to estimate Avrami rate constants (Chan and Isayev 1994). The dependency of the obtained Avrami rate parameter on the crystallization temperature is subsequently modelled by three different methods, namely, using the Arrhenius equation, the WLF equation, and the Hoffman-Lauritzen formula, discussed below in detail.  FIGURE 3.4: Analysis of crystallization data for PP5 plotted in FIGURE 3.2 based on the Avrami equation (3.1). (a) The calculated values of the Avrami index vary between 1.9 and 2.7. (b) Refit of the Avrami equation by fixing the Avrami index n to 2. The Arrhenius Equation: The Arrhenius equation (Equation 3.2) was suggested in 1889 to describe the temperature dependence of the reaction rates and it has been extended to analyse many other thermally-induced processes (Arrhenius 1889).   43  1 1( ) / ( ) exp aooEk T k T R T T  = −        (3.2) where, "𝑅𝑅" is the universal gas constant, 𝐸𝐸𝑎𝑎 is the activation energy, and 𝑇𝑇0 is a reference temperature. The Arrhenius equation is used in this study to represent the dependency of the Avrami rate parameter on temperature. The calculated Avrami rate parameter is plotted as a function of temperature in FIGURE 3.5 along with the Arrhenius fit (Equation 3.2). The activation energy obtained by this method is used to compare the temperature dependency of the crystallization kinetics of different resins. In general, the higher the activation energy, the less sensitive are the crystallization kinetics on the temperature variation. TABLE 3.2 lists the activation energies of the different resins examined. As shown in TABLE 3.2, PP3 and PP5 have the lowest activation energy, which indicates that their crystallization kinetics are most sensitive to temperature variation. Calculated activation energies are later used to obtain the Avrami rate parameter under non-isothermal conditions at various cooling rates. TABLE 3.2: The Arrhenius activation energy, Ea, of the different resins studied Resin Temperature range (K) Activation energy Ea (kcal/mol) T0(K) k (T0) ( ns− ) PP1 403-418 210.3 403 051.51 10−×  PP2 383-403 202.7 383 061.54 10−×  PP3 391-413 156.1 391 043.42 10−×  PP4 403-418 239.4 403 063.47 10−×  PP5 393-408 138.5 393 057.49 10−×  PP6 398-413 205.6 398 069.26 10−×    44   FIGURE 3.5: Fits of the Arrhenius equation to the Avrami rate parameter k calculated from Equation 3.1 using n=2. FIGURE 3.6 depicts the Avrami equation fits for isothermal quiescent crystallization of PP5 using the fixed Avrami index of 2. Arrhenius equation was used in order to include the effect of temperature on the rate constant. At all studied temperatures, the Avrami equation with the Arrhenius model for the kinetic parameter k can represent the isothermal crystallization of PP5 well. The model shows a small deviation from the experimental data for relative crystallinity > 70% due to secondary crystal growth particularly at low temperatures. Similar behaviour was observed for all other resins studied and the results are summarised in Appendix A.  45   FIGURE 3.6: Isothermal quiescent crystallization fits of the Avrami equation with n=2 and using the Arrhenius equation for the temperature dependence of the rate constant for PP5. WLF (Williams-Landel-Ferry) Equation: The WLF Equation (Equation 3.3) is a model suggested for time-temperature superposition of rheological data for polymers.  12( )( )ln( ) ( )oo oC T Tk Tk T C T T  −=  + −      (3.3) where T and oT  are the temperature and reference temperature respectively. The kinetic parameter ratio, ( ) / ( )ok T k T , should be available for at least three different temperatures. The dependency of the Avrami rate parameter k on temperature was studied using the WLF equation. When the WLF parameters are obtained using experimental data above the glass transition (Tg) of the polymer, the extrapolation to a temperature close to Tg is usually erroneous. While the WLF  46  behaviour well above Tg is similar to that obtained from the Arrhenius model, the behaviour close to Tg differs. Often, negative values for C1 and C2 are obtained for temperatures below Tg. The WLF fit to the data for PP5 is shown in FIGURE 3.7. Overall the data representation is excellent. Similar results were obtained for all other polymers (see Appendix A). The constants C1 and C2 for resins studied have been tabulated in TABLE 3.3. TABLE 3.3: Constants of WLF equation obtained for different resins Resin Temperature range (K) C1 C2 (K) T0 (K) k(T0) ( 10 ns −× ) PP1 403-418 -60.2 81.32 403 051.51 10−×  PP2 383-403 -88.63 101 383 061.54 10−×  PP3 391-413 -55.33 95 391 043.42 10−×  PP4 403-418 -49.93 65.91 403 063.47 10−×  PP5 393-408 -26.58 51 393 057.49 10−×  PP6 398-413 -115.5 163 398 069.26 10−×    47   FIGURE 3.7: Fit of the Williams-Landel-Ferry equation to the Avrami rate parameter calculated from Equation 3.1 using n=2.  The Avrami rate parameter k was fitted to Avrami Equation 3.1 as a function of temperature as shown in FIGURE 3.4b. The Hoffman-Lauritzen equation: The Hoffman-Lauritzen theory demonstrates the crystallization kinetics of linear flexible macromolecules in molecular terms (Hoffman et al. 1976). These macromolecules are crystallized from the melt into chain folded lamellae. This theory has been used to interpret and model the crystallization behavior of a large number of polymers. The Hoffman-Lauritzen formula is best suited to describe the chain folded crystallization of polyethylene. This is written as:  ( )ln( ) * / ( ) ln( ) /o gk U R T T k K T T∞+ − = − ∆     (3.4)  48  where 30gT T C∞ = − ° , *U  is a universal constant having a value of 1500 Cal/mol, T∆  is degree of undercooling defined as mT T° −  , and mT°  is the equilibrium melt temperature of the polymer. In order to estimate the unknown material parameters (i.e., gK  and 0k ), the equilibrium melt temperature of the investigated polymers should be known. For polypropylene, a value of 185oC was reported as 𝑇𝑇𝑚𝑚°  (Petraccone et al. 1985). FIGURE 3.8 shows the Hoffman-Lauritzen fit of experimental data which is used to estimate the Avrami rate parameter k for non-isothermal crystallization experiments. Similar results were obtained for all other polymers and the constants have been tabulated in TABLE 3.4. TABLE 3.4: Constants of the Hoffman-Lauritzen equation obtained for different resins Resin Temperature range (K) k0 ( 10 ns −× ) Kg  (210K −× ) PP1 403-418 75.4298 10×  56.074 10×  PP2 383-403 172.4010 10×  61.488 10×  PP3 391-413 111.2356 10×  57.573 10×  PP4 403-418 101.4540 10×  56.943 10×  PP5 393-408 91.2930 10×  56.665 10×  PP6 398-413 122.6620 10×  59.135 10×   49   FIGURE 3.8: Fit of Hoffman equation to the Avrami rate constants calculated from Equation 3.1 using n=2. 3.1.3 Non-isothermal quiescent crystallization As described in the classical monograph of Mandelkern (Mandelkern et al. 2002), most of the theories of isothermal and non-isothermal quiescent crystallization are based on a derived Avrami expression with a variety of modifications. A number of different forms of Kolmogoroff-Avrami-Evans theory (simply referred to as “Avrami”) are reviewed in several articles (Mubarak et al. 2001; Di Lorenzo and Silvestre 1999).   The Nakamura model (Equation 3.5) is a modified form of the Avrami model with an isokinetic approximation and is often used in numerical simulations of processes that involve non-isothermal crystallization. This model can be written as (Nakamura et al. 1972):   50  / 1 exp ( )ntfoX X K T dt   = − −    ∫      (3.5) where fX  is the final degree of crystallinity, and ( )K T  is the crystallization rate parameter which can be obtained readily from the temperature dependence of Avrami rate parameter k as: 1/n( ) ( )K T k T=       (3.6) Once, a comprehensive set of data for crystallinity and growth rate is given, all parameters of models proposed for crystallinity (for instance the Schneider rate parameters) can be extracted. Nonetheless, in the current study, we have implemented the Nakamura equation to capture the crystallization behaviour. It has been reported that the original Nakamura equation makes no allowance for the induction time for nucleation (Chan and Isayev 1994). An induction temperature (onset) to crystallization for non-isothermal crystallization can be actually estimated by the Nakamura model, provided one considers an infinitesimally small amount of crystallinity (e.g. on the order of 10-10) for n ≠ 1 at the equilibrium melting point or melt temperature as an initial condition in order to progress the numerical solution.  The value of this initial condition does not generally affect the prediction of crystallinity profile and onset temperature for crystallization as a function of cooling rate, as long as the initial crystallinity is infinitesimally small (e.g. < 10-6), see also FIGURE 3.10a.  As shown in FIGURE 3.10a a critical initial crystallinity (X0=10-15) is needed to observe the onset of crystallization in the Nakamura model. Therefore, estimating the onset of crystallization in a more rigorous fashion, as described below, on the basis of the differential form of Nakamura equation [Eq. (3.7)] and the smallest detectable DSC crystallinity, is highly desirable  51  for improving model predictive capability in simulation of polymer processes. Differentiating Equation 3.5 with respect to time yields the derivative form of the Nakamura model (Patel and Spruiell 1991). ( ) ( )1/ dt ( ) 1 / ln 1 /nnf fdX nK T X X X X− = − − −     (3.7) Polymer processes frequently deal with non-isothermal crystallization and therefore a form such as Equation 3.7 is convenient for numerical simulations due to its differential form compared to the integral one (Equation 3.5). Four different cooling rates of 2, 5, 10, and 20oC/min were chosen to study the non-isothermal crystallization behavior of the studied polypropylenes. Higher cooling rates could not be achieved due to the apparatus limitations. Increasing the cooling rate causes the induction time to occur at a lower temperature.  The most accurate method to predict the onset of crystallization appears to be obtained by employing the Nakamura equation with the lowest detectable value of crystallinity in DSC under non-isothermal conditions. The smallest non-zero value of crystallinity obtained by using the Shimadzu DSC-60 calorimeter is shown in TABLE 3.5. Different time intervals was chosen for various cooling rates in order to capture the crystallization peak using DSC effectively. Smaller time interval is needed for higher cooling rates since non-isothermal crystallization happens within a smaller temperature window. These distinct intervals contribute to dissimilar smallest detectable value of crystallinity in DSC. While these values are of the same order of magnitude, their effect on the predicted profile is negligible. The prediction of the Nakamura model using the smallest detectable value of crystallinity under non-isothermal conditions in DSC is shown in FIGURE 3.9 for PP5. In these predictions, the Arrhenius  52  equation was used to capture the dependency of the Avrami rate parameter k on temperature. As shown in FIGURE 3.9, the Nakamura model captures the behavior well for cooling rate of less than 10oC/min. However, for the cooling rate of 10 and 20oC/min, there is a discrepancy due to a temperature lag experienced by the sample. At these high cooling rates, the sample cannot be cooled according to the desired profile since the resistance to heat transfer between test specimen and furnace becomes important. This resistance includes the resistance to heat transfer in between the pan and furnace, the resistance in between pan and sample, and the resistance within the sample. The energy equation should be used in order to correct the temperature lag provided that the resistance in between pan and furnace is the controlling resistance (Chan and Isayev 1994). The details on temperature correction can be found elsewhere (Chan and Isayev 1994). The prediction of the Nakamura model after applying the temperature correction for the data under high cooling rates is shown in FIGURE 3.10. As also shown in FIGURE 3.10a, prediction of the onset temperature of crystallization and relative crystallinity profiles as a function of cooling rate using an infinitesimally small amount of crystallinity (10-6 to 10-10) as initial condition is good overall, however using the lowest detectable value of crystallinity for DSC for each cooling rate as initial condition seems to give slightly more accurate fits of the crystallinity data.       53  TABLE 3.5: Smallest values of crystallinity obtained by using DSC at various cooling rates for PP5. Cooling rate (oC/min) Temperature at which the smallest crystallinity value is detected (oC) Smallest detected crystallinity in DSC 2 126.9 32.53 10−×  5 123.2 31.89 10−×  10 121.8 31.35 10−×  20 119.2 31.07 10−×    FIGURE 3.9: Non-isothermal quiescent crystallization prediction of the Nakamura model [Eq. (3.7)] with n=2 and Arrhenius temperature dependence for the rate parameter k without temperature correction for high cooling rates. The smallest crystallinity amount (see TABLE 3.5) obtained by DSC was fed to Nakamura model.    54  As previously mentioned, the smallest detectable value of crystallinity under non-isothermal conditions in DSC should be used in the Nakamura equation in order to be able to integrate the differential equation (3.7) for n > 1 and generate crystallization kinetics predictions. This method requires additional DSC data at the desired cooling rate at which the prediction is attempted. Therefore, it is not practically the best method to model non-isothermal crystallization under processing conditions, where cooling rates could be highly variable.   FIGURE 3.10: Non-isothermal quiescent crystallization prediction of the Nakamura model [Eq. (3.7)] with n=2 and Arrhenius temperature dependence [Eq. (3.2)] for the rate constants with temperature correction for high cooling rates. (a) An infinitesimally small value of crystallinity (10-6, 10-8 and 10-10) at equilibrium melting temperature (185oC) is used as initial condition independent of cooling rate to progress the numerical solution. (b) The smallest detectable crystallinity amount obtained by DSC for each cooling rate was used as initial condition. As seen, similar induction times were obtained from the use of both functions.  The onset of crystallization under different cooling rates can be estimated using the induction times obtained by isothermal quiescent crystallization data as shown by Equation 3.8 (Sifleet et al. 1973). Thus, all predictions are performed based on isothermal data to determine the  55  induction time. Induction time in this study is defined as the elapsed time (t-t0) at which the relative crystallinity of 1% is obtained. 0/ ( ) 1indtdt f T =∫       (3.8) The non-isothermal induction times are then estimated using a function ( ( )f T ) fitted to induction times at different isothermal quiescent crystallization temperatures. The time at which this ratio approaches unity is defined as the time for the onset of crystallization. Equation 3.8 holds true for both isothermal and non-isothermal crystallization experiments. Since ( )f T  can be any arbitrary function, the sensitivity of induction time prediction to the proposed model is checked by using two different functions; namely, a power law and an exponential function. These functions along with their non-isothermal induction time predictions under various cooling rates are summarized in Table 3.6.  The exponential function was used in this study, since it was found more convenient to be applied numerically. FIGURE 3.11 depicts the prediction of the Nakamura model (Equation 3.7) coupled with the estimated induction times obtained by Equation 3.8. It noted that the solid lines in FIGURE 3.11 are obtained by using solely the quiescent isothermal crystallization data as explained in detail.  56  TABLE 3.6: Non-isothermal induction times (s) obtained by using two different functions in Equation 3.8 along with the experimentally obtained induction time for PP5 under four different cooling rates. PP5          Cooling rate ( ) ( )71 34.51.36 10f T T−= ×  ( )( )1301.135 5136.3Tf T e− ×  = ×  Experimental tind (s) 2oC/min 2182 2079 2208 5oC/min 926 916 945 10oC/min 485 493 502   FIGURE 3.11: Non isothermal quiescent crystallization prediction of Nakamura model with n=2 and Arrhenius temperature dependence for the rate constant with temperature correction for high cooling rates. The non-isothermal induction times are estimated from Equation 3.8.   57  The WLF Equation 3.3 can be used as an alternative to Arrhenius Equation 3.2 to capture the Avrami rate parameter dependency on temperature. FIGURE 3.12a shows that the obtained crystallinity profile predictions from the WLF and Arrhenius equations are very similar. However, using the Hoffman-Lauritzen theory for capturing the rate parameter behavior under different cooling rates yields erroneous prediction for this resin (FIGURE 3.12b). For other resins studied all three time temperature superposition methods gave comparable results with good agreements with the experimental data. The results obtained using these methods for other resins are shown in Appendix A.    FIGURE 3.12: Non isothermal quiescent crystallization prediction with n=2 and (a) WLF /(b) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The induction time estimated from isothermal data was fed to Nakamura model to obtain upturns. 3.2 Crystal growth rate via polarized optical microscopy (POM) As discussed above, polarized optical microscopy was used to determine the crystal growth rates for all studied polypropylenes. Polymers with high nucleation density cannot be easily  58  studied using this microscopic technique due to the large amount of nuclei within the bulk making it intractable to “isolate” distinct spherulites and measure their growth rate. Therefore, the growth rate was studied only for three of the polypropylenes, namely PP2, PP3, and PP5. Since these resins have high isotacticity and Mw, the effect of cross-hatched lamellae on crystallization kinetics can be neglected (Janimak et al. 1991). Using the isothermal experimental protocol described above, twenty different frames at various times over regular time intervals were analyzed at each temperature in order to determine the crystal radius as a function of time. At each temperature three distinct crystals were analysed in detail as described above. The crystal radii, R, as a function of time are shown in FIGURE 3.13 for PP5. The crystal growth rate, G, can be calculated from the radius by /G R t≡ ∂ ∂ .  0 1000 2000 3000 4000 5000 6000 7000 8000 900020304050607080 T=135.0oC T=137.5oC T=140.0oC T=142.5oC T=145.0oCG = 0.0110 µm/sG = 0.0497 µm/sG = 0.0170 µm/sG = 0.0297 µm/sG = 0.0067 µm/s  Crystal radius, R (µm)Time, t-t0 (s)PP5 FIGURE 3.13: The variation of crystal radius as a function of time elapsed for PP5 at five different crystallization temperatures. Crystal growth is the slope of the fitted lines to the data.  59  A linear increase of spherulite radius with time is observed implying a constant crystal growth rate at a given temperature. Increase of the quiescent crystallization temperature, causes a significant decrease in the growth rate of crystals. The obtained data are in a good agreement with previous studies (Boyer and Haudin 2010; Janeschitz-Kriegl 2006).  The Arrhenius time-temperature superposition can be applied to the obtained growth rates at different temperatures in order to represent the dependency of crystal growth rate on temperature. These activation energies along with those obtained using DSC can be used to compare different resins with each other. FIGURE 3.14 depicts the crystal growth, G, as a function of temperature for three of the resins. PP3 and PP5 show very similar activation energy within the temperature range studied. Their activation energy Ea,g is much lower compared to that of PP2.   FIGURE 3.14: Arrhenius fit obtained from the growth rate of resins at different temperatures.   60  Therefore, the crystal growth rate of PP3 and PP5 are the most sensitive to the temperature variations. As shown in FIGURE 3.16a, d, f, resins PP1, PP4, and PP6 had high nucleation density and thus optical microscopy could not be used to study them. PP1 and PP4 have higher nucleation density compared to that of PP2 and thus they showed lower half time of crystallization at higher temperatures as shown in FIGURE 3.15. Half time of crystallization is defined as the time at which 50% of final crystallinity is obtained. The effect of nucleation density on crystallization kinetics for PP4 is more pronounced than the other resins as observed in FIGURE 3.16d. Although, PP6 was observed to have higher nucleation density than PP2, they revealed comparable behavior in FIGURE 3.15. Therefore, PP2 should have larger growth rate at comparable normalized temperature in order to balance the effect of nuclei formation on the total crystallization kinetics.  The molecular reptation theory (De Gennes 1971) is used now to explain the differences observed in the crystal growth rate of various polymers with different molecular architectures (Hoffman and Miller 1988). As the molecular weight increases the friction coefficient of chains within the melt increases which leads to a reduced chain mobility. Therefore, the crystal growth rates are decreased with increase of molecular weight, which is supported by the present and previous studies (Miyata and Masuko1998; Cheng et al. 1990; Vasanthakumariand and Pennings 1983; Van Antwerpen and Van Krevelen 1972; Magill 1969; Magill 1967; Magill 1964).   61   FIGURE 3.15: The quiescent (DSC) half crystallization times of all PPs as a function of normalized temperature obtained from isothermal crystallization experiments. Normalized temperature is defined as Cm CT TTT T−=−, where TC and Tm are crystallization and melting peak temperatures obtained by DSC. Crystallization from copolymers are often more complex in behaviour. The degree of co-monomer rejection by the growing face of crystals was observed to impact the crystal growth rate extensively (Lambert and Phillips 1994). These co-monomer units can decrease the probability of active chains facing the growing interface of crystals and thus can be interpreted as defects. The variation in tacticity can be regarded as a factor introducing defects into the matrix and thus it should have similar effects as co-monomer insertion (Patki et al. 2007). Inclusion of small defects has tremendous effect on the growth rate of crystals (Wagner and Phillips 2001).  62   FIGURE 3.16: POM images of investigated resins at 0.5T = . As shown PP4 has the highest nucleation density among all. PP3 and PP5 show a comparable nuclei density at the studied temperature region.  63  Using various catalyst to manufacture the polymer leads to different type and degree of defects influencing the growth rate distinctly. Therefore the difference in the growth rate observed for PP2 compared with other resins are mainly related to defects imposed by different catalyst used for its production. To put this into perspective, it is known that metallocene catalyst yields head to head defects within the matrix, which can impede the growth rate. Polarized optical microscopy was also used under non-isothermal protocol with cooling rates of 0.5, 1, 4, 8, 13oC/min. High cooling rates often impose a lag in between the polymer temperature and the temperature measured by the rheometer. A fast-response (0.15 ms) K-type thermocouple was inserted in each sample to monitor the real temperature of test specimen during the cooling period. Twenty to thirty frames equally spaced in time were extracted from the obtained videos and analyzed further to obtain the radius of each crystal. The growth rates were subsequently determined using the obtained radius data points. FIGURE 3.17 shows the growth rate as a function of temperature for PP5. The results obtained using this non-isothermal experimental protocol is in agreement with those obtained by the isothermal approach indicating the good consistency of the non-isothermal growth rate determination.   64   FIGURE 3.17: Growth rate of PP5 crystals at various temperatures. Using a non-isothermal experimental protocol allows for an efficient way to determine crystal growth rates over a wider range of temperatures. FIGURE 3.18 compares the growth rate of crystals for various resins under a wide range of temperature. PP3 and PP5 reveal comparable growth rates at lower degrees of undercooling (higher temperatures, 128-145oC). As the temperature decreases, the controlling mechanism of crystal growth changes from lateral growth to nuclei deposition on the surface. Based on FIGURE 3.18, PP5 has a higher degree of nuclei deposition on the surface of growing crystals compare to that of PP3 at lower temperatures. PP2 used a different catalyst for its production, which yields different degree of defects within the polymer. These defects can impede the growth of crystals immensely throughout the temperature region studied as they decrease the probability of active chain/nuclei to confront with the growing surface of crystals.    65   FIGURE 3.18: Growth rate of crystals at various temperatures for PP2, PP3, and PP5. PP2 was produced using a different catalyst system compared to PP3 and PP4 and thus exhibits the lowest growth rates possibly due to the defects in the melt matrix of this polymer. The Lauritzen-Hoffman analysis was applied on the experimental data shown in FIGURE 3.18 to reveal the transition temperature between the crystallization regimes. According to this theory the growth rate kinetics can be modeled as: *0C CU( ) exp exp(T T ) f T TgKG T GR ∞   = − −   − × ∆                                                 (3.9) where G0 is a pre-exponential parameter nearly independent of temperature, U* is a universal constant characteristic of the activation energy for transport of polymer segments to the growing surface of crystals which has a value of 1,500 cal/mol, R is the gas constant; T∞  represents the temperature below which all motion associated with viscous flow or reptation ceases. T∞  is  66  usually assumed to be equal to T 30( )g C− °  in which Tg  is the glass transition temperature of polymer. The correction term ( )f 2 C C mT T T °= +  accounts for the temperature dependence of the heat of crystallization.   FIGURE 3.19-FIGURE 3.21 depict the results of Lauritzen-Hoffman analysis for various resins. The Lauritzen-Hoffman theory analysis of materials studied has shown transition from regime II to regime III at ~130, 127, and 121oC for PP5, PP3, and PP2, respectively. Regime I to regime II transition was only observed for low Mw PP2 at ~124oC. These transition temperatures for PP5 and PP3 are in good agreement with the previous studies done on iPP (Patki et al. 2007; Mandelkern 2002). In Lauritzen-Hoffman formulation, regimes are categorized according to the competition between the rate of lateral surface spreading and the rate of secondary deposition. Regime I occurs at high temperatures (low degree of undercooling) where rate of surface spreading dominates the growth process. In regime II, the rate of lateral surface spreading is comparable to the rate of deposition of secondary nuclei, while in the regime III the rate of deposition of secondary nuclei dominates the process.  It has been shown that as the bulk viscosity is decreased, the temperature at which transition to regime III occurs at a lower temperature (Patki et al. 2007; Silvestre et al. 1999). In regime II, the rate of lateral growth is comparable with the rate of deposition of a nuclei on the growing surface due to lower energy barrier of lateral growth. Transition to regime III occurs when the mobility of chains become less (at lower temperature) and the rate of nuclei deposition competes with the rate of lateral growth. PP5 has the highest molecular weight and lowest MFR compared  67  with PP3 and PP2, therefore the chain segments are less mobile (higher friction) which causes the transition from regime II to regime III to occur at a higher temperature. PP2 has the highest MFR and different degree/type of defects, thus has a lower temperature for transition from regime II to regime III as expected. These results are in agreement with the observed behaviour in the iPP/poly(α-pinene) blend system (Patki et al. 2007; Silvestre et al. 1999). KgIII/ KgII ratio is 2.7 and 1.7 for PP5 and PP3, respectively which is fairly close to the value of 2 predicted by the theory (Clark and Hoffman 1984; Hoffman 1983). Of note, KgII and KgIII obtained here for PP3 and PP5 are in good agreement with previous studies (Xu et al. 1998; Cheng et al. 1990; Martuscelli et al. 1982; Wlochowicz and Eder 1981; Goldfarb 1978; Lovinger et al. 1977; Binsbergen and Delange 1970; Keith and Padden 1964; Falkai and Stuart 1960; Falkai and Stuart 1959). The KgIII/ KgII ratio for PP2 is 1.2 which is not close to the predicted value. It is noted that the II-III transition for PP2 was hardly observed under the conditions studied. Higher cooling rates could not be applied on the resin due to apparatus limitation. PP2 shows transition from regime I to regime II at the unusual temperature of 124oC. It is worth noting that PP2 has different melting peak temperature (~148oC) compared to other resins (~161oC) and thus its melting and transition temperatures should be different (Magill 1964). These sets of data have shown that the presence of defects affects the transition behaviour immensely. Smooth transition from regime II to III which is observed for PP3 and PP2 is also reported in other studies (Xu et al. 1998; Point and Janimak 1993).   68   FIGURE 3.19: Plot of ( ) ( )( )*ln G U R T T∞+ × −  vs ( )1 T Tf∆  for PP5.  FIGURE 3.20: Plot of ( ) ( )( )*ln G U R T T∞+ × −  vs ( )1 T Tf∆  for PP3.  69   FIGURE 3.21: Plot of ( ) ( )( )*ln G U R T T∞+ × −  vs ( )1 T Tf∆  for PP2. PP5 and PP3 have very similar KgII implying that they possess comparable growth kinetics. This is also previously shown in FIGURE 3.18. However, in regime III, PP5 has a larger Kg value, indicating higher growth rate than that of PP3. FIGURE 3.15 reveals almost identical behaviour in half time of crystallization for both resins under the investigated temperature range. In regime II where growth kinetics are comparable, the nucleation density of both resin should be of the same magnitude as shown in FIGURE 3.16c and FIGURE 3.16e in order to contribute to the same bulk crystallization kinetics. Although in regime III, PP5 have larger growth kinetics it behaves similar in terms of total crystallization kinetics as PP3 due to lower nucleation density. The difference in nucleation density is clear in their micrograph as depicted in Figure 3.22. Higher nucleation density in regime III was observed in previous studies (Celli et al. 2003).    70   FIGURE 3.22: Optical micrograph of PP3 and PP5 obtained under cooling rate of 4oC/min. PP3 shows higher nucleation density at high degree of undercooling. 3.3 Conclusions The Avrami rate parameter reflects the total crystallization kinetics, however it does not provide insight/information into the crystal microscopic differences existing between different resins such as the nucleation density and crystals growth rates. Optical microscopy should be used in conjunction with the Avrami analysis of DSC crystallization data in order to capture both crystal macroscopic and microscopic differences of the investigated resins. Polypropylene resins with different molecular characteristics were found to have different nucleation density and growth rate. In particular, different catalytic system used in polymer production yield different degree of defects within the polymer matrix contributing to a much slower growth kinetics and smoother regime transition.   Data obtained using DSC under high cooling rates should be corrected for thermal lag effects, which is bounded to the test specimens in order to observe the true crystallization material behavior. Furthermore, it was shown that induction times under non-isothermal conditions can be  71  estimated using the induction times obtained under isothermal conditions.  The obtained induction times under non-isothermal condition were found to be nearly insensitive to the proposed model as long as the model fits the isothermal induction times well.  An Avrami/Nakamura model in its differential form was found to be able to fit and predict DSC non-isothermal crystallization kinetics data very well over a range of cooling rates of 2-20oC/min for all studied PP resins.  Such a crystallization model is a useful tool for numerical simulations of polymer processes including injection molding, fiber spinning and film blowing as such it is of great industrial applicability for product and process development.  72  Chapter 4: Flow (Shear) Induced Crystallization of Polypropylenes  4.1 Linear viscoelasticity of the polypropylenes The master curves of the linear viscoelastic moduli of all PPs studied at the reference temperature of 190°C produced by the application of the time-temperature superposition (TTS) technique are depicted in FIGURE 4.1a-f (Ferry 1980). For the low molecular weight materials, the Newtonian region (e.g. PP1, PP2) at low frequencies was reached. For the high molecular weight materials (PP4, PP5 and PP6) stress relaxation experiments after imposing a sudden strain were also performed to obtain data over a wide range of frequencies. This is possible by converting the relaxation modulus to dynamic data at very low frequencies in order to determine the zero-shear viscosity (Schwarzl method) (Ferry 1980). This procedure was used just for three of the PPs in this study i.e. the high molecular weight resins. The continuous lines in FIGURE 4.1a-f represent fits of the multi-mode Maxwell model {Gi, λi} which shows that this model is capable of representing the data very well.  FIGURE 4.2 plots the complex viscosity of all PPs as a function of frequency at the reference temperature of 190oC. This Figure is in agreement with the MFR values of the resins listed in TABLE 2.1. The resin with the larger viscosity has higher resistance to flow and thus it has a lower MFR.  73  10-2 10-1 100 101 102 10310-210-1100101102103104105106  G' G'' Maxwell modelLoss and Storage Modulus, G'' &G' (Pa)Shifted Frequency, aTω (rad/s)     PP1Tref=190oCa100101102103 |η∗|Complex viscosity, |η∗| (Pa.s)10-2 10-1 100 101 102 10310-210-1100101102103104105106  G' G'' Maxwell modelLoss and Storage Modulus, G'' &G' (Pa)Shifted Frequency, aTω (rad/s)     PP2Tref=190oC102103104 |η∗|Complex viscosity, |η∗ | (Pa.s)b10-2 10-1 100 101 102 103100101102103104105  G' G'' Maxwell modelLoss and Storage Modulus, G'' &G' (Pa)Shifted Frequency, aTω (rad/s)     PP3Tref=190oCc102103104 |η∗|Complex viscosity, |η∗ | (Pa.s)10-3 10-2 10-1 100 101 102 10310-1100101102103104105106  G' G'' Maxwell modelLoss and Storage Modulus, G'' &G' (Pa)Shifted Frequency, aTω (rad/s)     PP4Tref=190oCd102103104105 |η∗|Complex viscosity, |η∗| (Pa.s)10-3 10-2 10-1 100 101 102 10310-1100101102103104105106  G' G'' Maxwell modelLoss and Storage Modulus, G'' &G' (Pa)Shifted Frequency, aTω (rad/s)     PP5Tref=190oCe102103104105 |η∗|Complex viscosity, |η∗| (Pa.s)10-3 10-2 10-1 100 101 102 103100101102103104105106  G' G'' Maxwell modelLoss and Storage Modulus, G'' &G' (Pa)Shifted Frequency, aTω (rad/s)     PP6Tref=190oC102103104105106 |η∗|Complex viscosity, |η∗ | (Pa.s)f FIGURE 4.1a-f. Master curves of the viscoelastic moduli of all PPs listed in TABLE 2.1 at the reference temperature of Tref=190°C.  74          10-4 10-3 10-2 10-1 100 101 102 103 104101102103104105Shifted Frequency, aTω (rad/s)   PP1 PP2 PP3 PP4 PP5 PP6Complex viscosity, |η∗ | (Pa.s) FIGURE 4.2: The master curves of the complex viscosity of all PPs listed in TABLE 2.1 at the reference temperature of Tref=190°C. TABLE 4.1 lists the activation energy for flow, Ea, using the shift factors obtained from the construction of the master curves depicted in FIGURE 4.1a-f and FIGURE 4.2. The Arrhenius model is used: 01 1aTEa expR T T  = − −          (4.1) Where, Ea is the activation energy for flow, a measure of the sensitivity of the viscoelastic properties to temperature.  It can be seen that PP5 has the highest energy of activation, Ea.   75   TABLE 4.1: The PP resins studied along with their activation energy for flow, Ea. PP Resin Ea (kcal/mol) PP1 9.55 PP2 7.63 PP3 9.63 PP4 10.56 PP5 11.27 PP6 9.91  4.2 Flow-induced crystallization (FIC) under shear flow (parallel plate geometry) FIGURE 4.3a and FIGURE 4.3b demonstrate the effect of cooling rate from T = 200°C to the test temperature of 142.4°C or 𝑇𝑇� = 0.5 on the shear stress growth coefficient 𝜂𝜂+ = 𝜏𝜏 ?̇?𝛾⁄  as a function of time for PP1 at two different shear rates. These tests are performed to identify the optimum cooling rate that would best satisfy conditions of isothermal flow-induced crystallization. Cooling rates smaller than a value of about 5°C/min allow the polymer to crystallize non-isothermally as it takes significant time for the quenching. For example, this is reflected in an earlier upturn of the shear stress growth coefficient when the cooling rate of 1oC/min is used at both shear rates (FIGURE 4.3a and FIGURE 4.3b). This earlier upturns are clearly due to premature crystallization before reaching the test temperature. Cooling rates greater than about 10°C/min, allow nearly isothermal crystallization by minimizing premature crystallization. The results become independent of cooling rate for values greater than 10oC/min.   76  To show this effect more clearly, the time required to observe the onset of crystallization (induction time, tind) is defined as the time at which 𝜂𝜂+ becomes 20% (arbitrary value) greater than its steady state value (Derakhshandeh and Hatzikiriakos, 2012). The induction times as functions of cooling rates are plotted in FIGURE 4.4 for two different shear rates for PP1 at 𝑇𝑇� = 0.5. High cooling rates (greater than 20oC/min) may also impose undercooling (undershooting in the temperature profile due to limitations in the temperature control of the rheometer in fast transient tests). Therefore in this study, the cooling rate used is 10°C/min in order to minimize the effect of undercooling and on the same time to eliminate the effect of premature crystallization due to long transients except at temperatures close to the crystallization peak as discussed below.   102 103 104110             PP1       γ = 0.005 s-1           T=0.5   1oC/min    5oC/min    10oC/min    20oC/min    Shear stress growth coefficient, η+ (kPa.s)Time, t (s).30a102 103 104110               PP1       γ = 0.1000 s-1             T=0.5  1oC/min   5oC/min   10oC/min   20oC/min   30oC/min Shear stress growth coefficient, η+ (kPa.s)Time, t (s).30b FIGURE 4.3: The effect of cooling rate on the shear stress growth coefficient η +  of PP1 at the shear rates of (a) 0.005 s-1 and (b) 0.1 s-1 at T=142.4°C or 0.5T = .  77  0 5 10 15 20 25 305006007008009001000              PP1         T = 0.5 γ = 0.005 s-1  γ = 0.100 s-1Induction time, t ind (s)Cooling rate, h (oC/min).. FIGURE 4.4: The effect of cooling rate on the induction time, tind, for the onset of crystallization in start-up of steady shear experiments for the resin PP1 at  T=142.4°C or 0.5T =  for two different shear rates. The effect of temperature on flow-induced crystallization for PP1 is demonstrated in FIGURE 4.5. Initially, imposition of shear flow causes the shear stress growth coefficient to increase with time reaching its steady-state value after a certain time. The dashed horizontal lines in the graph show the steady-state viscosity of crystal-free melt which can be used as a basis to determine the induction time. The normalized temperatures of 0.5 and 0.75, in combination with the cooling rate (10oC/min), are high enough to minimize premature crystallization during the cooling period before shearing as well as during the initial ascending part of the shear stress transient. However, crystallization occurs after a certain time that causes a sudden increase of the shear stress growth coefficient. At the lower normalized temperature of 0.25 the shear stress growth coefficient, η+ increases before reaching its steady-state. The times for the onset of  78  crystallization and transient to reach its steady-state are comparable as shown in the graph. In addition, nucleation evolved during the cooling period (due to the low temperature) also contributes to this deviation. As seen from FIGURE 4.5, the temperature has a noticeable effect on the induction time. A decrease of roughly 10°C (for example from 𝑇𝑇� = 0.5 𝑡𝑡𝑡𝑡 0.25) causes the onset of crystallization to occur at times of roughly one order of magnitude earlier. In general, crystallization kinetics is significantly enhanced as temperature decreases.  The gradual increase of viscosity as the crystallization begins can be used to determine the induction time which is arbitrarily defined as the time at which a 20% of increase of the viscosity is obtained. These induction times correspond to a small degree of crystallinity (typically 1-2%) and if the true viscosity is measured in parallel plate (i.e. ?̇?𝛾 reported at r = 0.75R), the increase in torque caused by crystallization does not influence the induction time measurements significantly. In the parallel plate geometry the shear deformation rate varies linearly with radial position, being zero at the center and attains its highest value at the edge. The effect of this non-homogenous shearing on the nucleation density of PP1 is shown in FIGURE 4.6. The nucleation density increases significantly from the center (FIGURE 4.6a) to the edge (FIGURE 4.6c). Moreover shear also induces crystal alignment with the highest alignment and nuclei density close to the edge of the parallel-plate geometry.  79  0.1 1 10 100 1000 1000002468101214161820T = 0.25T = 0.50T = 0.75          PP1 T = 0.25 T = 0.50 T = 0.75Shear stress growth coefficient, η+  (kPa.s)Time, t (s) FIGURE 4.5: The shear stress growth coefficient η +  of PP1 at the normalized temperatures, at T  of 0.25, 0.5, and 0.75 at the shear rate of 0.005 s-1.  FIGURE 4.6: Crystal density observed at various radial positions in the parallel plate geometry for PP1 at 0.5T = and shear rate of 2s-1 at center of the geometry (a) at r=0.75R (b) and r=R (c), with crystal alignment to increase with the radial position.  FIGURE 4.7 depicts the shear stress growth coefficient as a function of the time for various imposed shear rates at 𝑇𝑇� = 0.5 for PP1. A larger shear rate orients molecules more effectively after a certain strain. This activates more nuclei which is also supported by more spherulites of smaller size in the microscopy images (Huo et al. 2004). Therefore, a larger deformation rate decreases  80  the induction time for the onset of crystallization by enhancing the nucleation density and possibly crystal growth rates as mentioned in previous studies (Duplay et al. 2000).  103 1041032x1033x1034x1035x1036x103       PP1Tc = 122.0oCTm = 162.8oCT = 0.75   γ = 0.005 s-1 γ = 0.010 s-1 γ = 0.050 s-1 γ = 0.100 s-1....1032Time, t (s)Shear stress growth coefficient, η+  (Pa.s) FIGURE 4.7: The shear stress growth coefficient η +  of PP1 at shear rates ranging from 0.005 s-1 to 0.1 s-1 at 152.6°C or 0.75T = . FIGURE 4.8 plots the shear stress growth coefficient data of FIGURE 4.7, this time as function of strain. Typically, the required shear strain for the onset of crystallization increases with increase of the applied deformation rate. In other words, more strain is needed at higher shear rates to initiate crystallization. In particular, at the shear rates of 0.01s-1 a strain of 70 is required to initiate crystallization, while at the shear rate of 0.1s-1 crystallization occurs after 600 strain units. This is in agreement with experimental results reported by other authors (D’Haese et al. 2010; Eder et al. 1998). The increase of the amount of strain with increase of shear rate for the onset of crystallization (see FIGURE 4.8) is a counterintuitive observation.  81  FIGURE 4.9 shows the effect of shear rate on the induction time for all PPs studied.  Open symbols in FIGURE 4.9 are obtained from rheometry as described above, while closed symbols are obtained by DSC. As seen flow enhances and speeds up the onset of crystallization mainly at rates greater than a certain critical value. At deformation rates smaller than this value, flow does not disturb the molecules from their equilibrium configurations and thus no effect on their crystallization behavior occurs. Therefore, over this region (?̇?𝛾 < ?̇?𝛾𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑐𝑐) the crystallization behavior is essentially similar to that of quiescent condition. Larger shear rates (?̇?𝛾 > ?̇?𝛾𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑎𝑎𝑐𝑐) activate more nuclei causing a dramatic reduction of the induction time. Similar observations were reported for other polymers (Paradkar et al. 2008; Coppola et al. 2004). Differences in the molecular parameters of the investigated resins such as molecular weight, molecular weight distribution and tacticity likely contribute to differences in the induction times observed in FIGURE 4.9. Among the resins, PP5 and PP3 have the slowest crystallization kinetics at small shear rates. However, flow enhances crystallization kinetics of PP5 and causes dramatic reductions of its induction time so that flow-induced crystallization is faster compared to other resins. Similar induction time under very low deformation rate was observed between PP6 and PP2, however increasing shear rate enhances crystallization kinetics of PP6 more than that of PP2.  82  100 101 102 1031032x1033x1034x1035x1036x103  γ = 0.005 s-1 γ = 0.010 s-1 γ = 0.050 s-1 γ = 0.100 s-1Shear stress growth coefficient, η+  (Pa.s)Strain, γ1032       PP1Tc = 122.0oCTm = 162.8oCT = 0.75.... FIGURE 4.8: The shear stress growth coefficient η +  of PP1 at shear rates from 0.005 s-1 to 0.1 s-1 at 152.6°C or 0.75T = . PP2 does not have as long chain tail as PP6 due to its narrower MWD (TABLE 2.1) and thus requires higher shear rate before significant reduction of the induction time occurs.  PP6 has higher Mw and broader MWD than that of PP2, which likely cause the induction time to drop at a lower shear rate. It is noted from FIGURE 4.9 that the half-times of crystallization do not agree quantitatively with the induction times from simple shear experiments at very small shear rates. The former are greater than the latter for all resins. The half-times of crystallization correspond to the times where 50% relative crystallinity is formed. Polypropylenes studied here have crystallinity of ~46% and therefore 50% relative crystallinity corresponds to roughly 23%. On the other hand, the induction time determined from the flow experiments is defined as the time at which the  83  viscosity increases by 20% above its steady-state value for samples free of crystallinity. Therefore these two times are different and therefore the half-times of crystallization are expected to be somehow higher than those obtained from shear testing at very small shear rates where the effect of flow is minimal.  0.00000 0.01 0.1 1 100200040006000      T=0.5 PP1 PP2 PP3 PP4 PP5 PP6Induction time, t ind(s)Shear rate, γ (s-1).  FIGURE 4.9: The effect of shear rate on the induction time for the onset of crystallinity for all PPs at 0.5T = The closed symbols correspond to half time quiescent crystallization obtained from DSC. To see this more clearly, we consider the suspension model η/ηa=(1- 𝛼𝛼 /A)-2, where ‘A’ is a shape constant, 𝜂𝜂𝑎𝑎 is amorphous viscosity, and 𝛼𝛼 is relative crystallinity. Setting the ratio η/ηa =1.2 (induction time) with A=0.6 (frequently assumed), the crystallinity 𝑋𝑋 is calculated to be about 0.024 (2.4%), which is the time determined from shear experiments. This time is expected to be lower than the half-time of crystallization which corresponds to roughly 23% crystallinity. For the  84  two times to be comparable, the induction time for crystallization in shear should have been defined as the time at which the ratio, η/ηa, becomes equal to η/ηa =(1-0.3/0.6)-2=4, that is when the viscosity increases to 4 times its steady state value, instead of 1.2 times. In spite of these qualitative differences, these two times agree qualitatively. In FIGURE 4.9 two different regimes of crystallization can be defined. At very low deformation rates, the crystallization kinetics essentially resemble that of quiescent condition where shear rate does not change the induction time (Coppola et al. 2004). Increase of deformation rate causes chain orientation, lowers the thermodynamic barrier for crystallization, and thus enhances the kinetics of crystallization. In this region, shear rate significantly decreases the induction time as mentioned by other authors (Coppola et al. 2004). These two regions are more clearly defined for PP2 and PP6 in FIGURE 4.10 by means of two lines; namely a horizontal where the induction time is independent of the rate of shear and a line having a negative slope showing the decrease of induction time with increase of shear rate. The intersection of these two lines define the critical shear rate required to start observing the effect of flow. These critical shear rates for all resins studied are summarized in TABLE 4.2. As mentioned by Acierno (Acierno et al. 2003) orientation of chains under shearing (low Weissenberg number) are sufficient to enhance the crystallization phenomenon by inducing higher nucleation rate compared to quiescent conditions (Godara et al. 2006; Devaux et al. 2004; Koscher & Fulchiron 2002). Large deformation rates (i.e. large Weissenberg numbers) stretch chains in the direction of flow changing the mechanism of crystallization from spherulitic to rodlike nucleation which further facilitates the kinetics of  85  crystallization. POM images in our case suggest spherulitic crystallization at low Weissenberg numbers (typically of the order of 1). 0.00000 0.01 0.1 1 1002000      T=0.5 PP2 PP6Induction time, t ind(s)Shear rate, γ (s-1).γc = 0.057s-1.γc = 0.31s-1. FIGURE 4.10: The effect of shear rate on the induction time for the onset of crystallinity for PP6 and PP2 at 0.5T = . A critical shear rate is observed at which transition from quiescent to flow induced crystallization occurs. It was hypothesised earlier that dormant nuclei exists within polymer matrix. These precursors are activated by imposing undercooling and/or applying deformation rates (Janeschitz-Kriegl and Ratajski 2005; Janeschitz-Kriegl et al. 2003). Higher deformation rates and higher degree of undercooling activate more dormant nuclei. It was assumed and supported by experimental data that at very high deformation the nuclei density saturates and no more nuclei are formed (Janeschitz-Kriegl et al. 2003). These behaviours were also supported by various works done previously (D’Haese et al. 2011; D’Haese et al. 2010). PP4 has the highest nucleation density  86  among all of the investigated resins (i.e. closer to the saturated state) and thus a higher critical shear rate is needed to enhance the quiescent crystallization kinetics as seen in TABLE 4.2. As previously mentioned applying flow usually cause the polymer to induce higher nuclei density. PP2 uses different catalyst for its production and thus cannot be compared with the other resins. The different catalyst system for production can introduce different growth rate and kinetics since it imposes different molecular structures (Wunderlich 1976). PP5 and PP3 have very similar tacticity and PDI. PP5 has higher molecular weight in comparison with PP3 and thus shows lower critical shear rate for flow enhanced crystallization. Although PP6 has the highest molecular weight and PDI, it exhibits a higher critical shear rate than PP5, PP3, and PP1 due to its lower tacticity. Tacticity seems to play a major role in FIC.  FIGURE 4.11 depicts the effect of shear rate on the induction time at various normalized temperatures for PP1. DSC results for 𝑇𝑇� = 0.75 could not be obtained since the heat of crystallization which is released at this high temperature is not detectable by the DSC apparatus used. The induction time increases by one order of magnitude for a change of roughly of 10oC. On the same time, increase of shear rate from 0 to 0.5 decreases the induction time by a factor of 2. TABLE 4.2: The critical shear rate above which the effect of flow on crystallization become evident at 0.5T = . Resin Critical shear rate (s-1) PP1 0.02 PP2 0.31 PP3 0.03 PP4 0.10 PP5 0.02 PP6 0.06  87  0.00000 0.01 0.1 1110100100010000        PP1T = 0.25T = 0.50T = 0.75Induction time, t ind (s)  Shear rate, γ (s-1). FIGURE 4.11: The induction time for the onset of crystallinity as a function of shear rate for PP1 at different normalized temperatures. The closed symbols correspond to half time quiescent crystallization obtained from DSC. 4.3 Crystallization kinetics using polarized microscopy under shear In addition to rheometry, polarized light microscopy was used in conjunction with the Anton Paar MCR-502 parallel plate rheometer in order to study the crystallization kinetics of PPs under shear. The microscope is equipped with a polarizer and an analyzer which are always crossed during experimental testing. Therefore, only crystals which are present within the polymer matrix can be observed due to their birefringence (Wang et al. 2008). Similar heating and cooling protocol as described in the experimental section were used so as to eliminate flow and thermal history.  Quiescent crystallization of resins at desired temperatures was investigated by both optical microscopy and DSC in order to experimentally validate the accuracy of this setup. As a first step,  88  the relative crystallinity of PP5 as a function of time was calculated by integrating the area underneath the peak of the DSC thermogram and then normalizing it by the total area. In a second step, the degree of space filling of crystals in the polarized images was calculated from optical microscopy by using the image processing techniques described above. The degree of space filling was converted to degree of crystallization using the density of crystals (0.94 g/cm3) and that of the melt (0.85 g/cm3) (Natta et al. 1955). The relative crystallinity obtained from the optical microscopy was compared to that obtained from DSC to check for accuracy of the optical microscopic technique.  TABLE 4.3 shows typical microscopy images before and after image processing with methods 1 and 2 described in the experimental section (section 2.3.3).  Satisfactory agreement between the DSC and image processing results was found as can be seen from the comparisons in FIGURE 4.12a. The differences are small and can be attributed to experimental errors originating from slightly different thermal history in DSC and rheometer as well from errors in the resolution of the image processing technique. Samples used in DSC are relatively small (1-2 mg) sealed in an aluminum pan and thus the crystallization heat released can be carried out of the system by convection effectively. However, in the rheometer the polymer is sandwiched in between two quartz plates of 43mm in diameter with the upper plate having thickness of 6.4mm. In the microscopy setup, the heat of crystallization cannot be convected out rapidly. This can impede crystal growth rate in the POM measurements and contributes to the lower degree of crystallinity as also shown in FIGURE 4.12a. Therefore, the differences observed in FIGURE 4.12a are clearly caused by the different thermal histories in each case.   89  Heat transfer calculations explained in Appendix C demonstrate that the temperature rises in the POM samples in the range from 0.2oC to 1.9oC at the crystallization temperatures of 131.4oC to 121.7oC. Using the Avrami equation (see details in Appendix C) and the temperature rise calculated from the heat transfer equations, the relative crystallinity as a function of time was calculated for the case of POM results indicating excellent agreement in FIGURE 4.12b. This clearly explains the origin of the differences observed in FIGURE 4.12a.  101 102 1030.00.20.40.60.81.01.2                PP5 T=121.7oC-DSC T=121.7oC-OM T=126.6oC-DSC T=126.6oC-OM T=131.4oC-DSC T=131.4oC-OMRelative crystallinity,X/Xf Time, t (s)5 103a101 102 1030.00.20.40.60.8               PP5 T=131.4oC-OM T=126.6oC-OM T=121.7oC-OM Avrami predictionTime, t (s)Relative crystallinity,X/Xf 1035b FIGURE 4.12: The relative crystallinity of PP5 as a function of time obtained from DSC and optical microscopy. (a) Raw data without temperature correction showing the effect of different thermal history in each test (b) Avrami prediction of PP5 under microscopy set-up. The effect of shear on flow induced crystallization was also examined by the optical microscopy technique. Different shear rates in the range of 0.0 to 2s-1 were chosen to investigate the effect of shear. The captured images were undergone image processing to obtain the degree of space filling and relate this to degree of crystallization as discussed above. The obtained shear stress growth coefficient (η + ) from the rheometry was normalized by using the value of steady- 90  state viscosity that corresponds to crystal-free melt at the corresponding rate and temperature, indicated as aη . The normalized viscosity as a function of space filling is plotted in FIGURE 4.13.  TABLE 4.3: Optical microscopy images before and after image processing. Material Before image processing After image processing -method 1 After image processing -method 2 PP5    Morphology structure of PP5 after 685s at 138.8°C    Morphology structure of PP5 after 685s at 138.8°C    Morphology structure of PP5 after 685s at 138.8°C   Morphology structure of PP5 after 985s at 138.8°C   Morphology structure of PP5 after 985s at 138.8°C    Morphology structure of PP5 after 985s at 138.8°C Both space filling images and normalized viscosity were obtained from the same experimental test (identical temperature history). This is essential in order to capture the true crystallization behavior of resins. Previous works typically measured the relative crystallinity from DSC and then relate it with rheological measurements (Lamberti et al. 2007; Acierno et al. 2003; Pantani et al. 2001; Boutahar et al. 1998; Titomanlio et al. 1997). As depicted in FIGURE 4.13,  91  the data at very low shear rates (?̇?𝛾 < 0.5 𝑠𝑠−1) shows comparable crystallization behavior. Increasing the shear rates alters the crystal-melt interaction and thus causes the normalized viscosity to show upturns at lower degrees of space fillings (crystallizations).  4.3.1 Viscosity-crystallinity suspension modeling Different models based on suspension theory are used to model the crystallization behavior of the PPs. These models are using the data generated from coupling of degree of crystallinity obtained by DSC and normalized rheological function obtained by rotational rheometer (Lamberti et al. 2007). The first model examined was the one proposed by Graham for concentrated suspension of rigid interacting spheres (Graham et al. 1981). Graham proposed an additional term into equation 4.2 which modifies the behavior for concentrated suspension of rigid interacting spheres (Graham et al. 1981). ( )( )25 9 112 4 1 0.5 1aη φη φ φ φ = + +  + +   3312 maxmaxφφϕφφ  −    =    (4.2) ∅𝑚𝑚𝑎𝑎𝑚𝑚 is the maximum volume fraction occupied by the spheres. For the same system Mooney proposed the following equation while Frankel and Acrivos suggested equation 4.4 (Frankel et al. 1967; Mooney 1952):  92  5  2 1amaxexpη φφηφ  = × −   (4.3) 13139 81maxamaxφφηη φφ   = × −    (4.4)   The last two formulae which were studied in this work have similar form and were proposed by Ball and Richmond in 1980 (Eq. 4.5) and Kitano et al. in 1981 (Eq. 4.6) (Kitano et al. 1981; Ball et al. 1980; Kataoka et al. 1978). The latter is useful for concentrated suspension of particles of any shape with “A” being a shape coefficient (for instance A = 0.68 for smooth spheres): 52 1maxmaxaφη φφη− × = −   (4.5) ( ) 2 1aAη φη−= −  (4.6) To compare the capabilities of these equations, we have set the adjustable parameters in such a way that all models result the same induction time at low shear rates. This induction time corresponds to the space filling required to obtain normalized viscosity ratio, η/ηa of 1.2. The parameters, which were used in models are listed in TABLE 4.4. Frankel and Acrivos formulation (Eq. 4.4) predicts NRF of zero instead of one at zero degree of space filling as shown in FIGURE 4.13. Equation 4.3 and 4.5 exhibit the strongest NRF dependence on the degree of space  93  filling. Equations 4.2 and 4.6 show a similar trend and seem to predict the relation between space filling and NRF better. Experimental data under higher shear rate of 0.5 and 2 s-1 could not be predicted by using the same model constants, therefore “A” constant in equation 4.6 was reduced by a factor of 50 to mimic the behavior properly (see dash line curve in FIGURE 4.13).  TABLE 4.4: Models’ parameter used to predict the results in FIGURE 4.13. Equation Parameter Value Eq. 4.6,a A 0.2 Eq. 4.6,b A 0.004 Eq. 4.5 maxφ  0.2313 Eq. 4.4 maxφ  0.4233 Eq. 4.3 maxφ  0.2882 Eq. 4.2 maxφ  0.5  10-5 10-4 10-3 10-2 10-1 1000.00.20.40.60.81.01.21.41.61.82.0PP5 γ = 2.00 γ = 0.50 γ = 0.10 γ = 0.05 γ = 0.00 Eq. 28,a  Eq. 28,b Eq. 37 Eq. 36 Eq. 35 Eq. 24  Normalized viscosity, η/ηaDegree of space filling, φ.....tind FIGURE 4.13: Normalized viscosity as a function of space filling (crystallinity) for PP5.  94  The suspension models are applicable at low deformation rates, since there is only dependence of the viscosity on the degree of filling but there is no explicit dependence on the deformation rate (Newtonian-like models). A proper constitutive equation for suspensions, however, should include both the degree of filling (crystallinity) and the deformation rate dependencies. For relatively small shear rates (e.g. ≤ 0.1 s-1 in our case), the regime of viscosity Newtonian plateau at the testing temperature is reached and the NRF vs. degree of space filling experimental curves more or less collapse as shown in FIGURE 4.13.  However, above a critical shear rate (e.g. 0.5 s-1) we likely deviate from the Newtonian plateau and the experimental NRF vs. degree of filling curves deviate from the ones corresponding to the low deformation conditions (≤0.1 s-1).  Therefore, at the high shear rate conditions (i.e., ≥0.5 s-1), the true NRF which is affected by the shear rate and deviates from the low shear rate curve, irrespective of the suspension model used.  The dependency of suspension model parameter on the shear rate indicates that the viscosity of the semi-crystalline system is a function of both crystallinity and shear rate.  Therefore, we suggest that a proper suspension/rheology model for semi-crystalline systems should be a frame invariant constitutive equation with viscoelastic basis and both crystallinity and deformation rate dependences (Doufas 2013; Tanner and Qi 2009; Tanner and Qi 2005; Doufas et al. 2000). 4.4 Conclusions Flow-induced crystallization of several PP samples under simple shear was studied in this chapter in order to elucidate the effects of flow deformation relative to quiescent conditions. In simple shear, there is a critical strain required for the onset of crystallization at given shear rates and temperatures. In shear experiments the critical strain needed for polymer to crystallize  95  decreases with decreasing rate. Temperature was found to be an important variable in crystallization as expected and change of temperature by a few degrees changes the induction times for crystallization by one order of magnitude. The suspension models are applicable only to low deformation rates, since there is only dependence of the viscosity on the degree of filling. A proper model should also include viscoelastic effects in order to be able to capture the crystallization behaviour well (e.g. the two-phase constitutive/microstructural model for FIC of Doufas (2013) and Doufas et al. (2000).   96  Chapter 5: Flow Induced Crystallization of Polypropylenes in Capillary Flow at High Deformation Rates  5.1 The relaxation times of the various polypropylenes  It is well understood that temperature affects the crystallization kinetics of polymers under both quiescent and flow conditions (Ness and Liang 1993; Crater et al. 1980; Tan and Gogos 1976; Southern and Porter 1970). As the crystallization temperature increases, the probability of obtaining stable crystalline structures becomes lower. The crystalline parts formed can dissipate into the melt matrix faster at a higher temperature since the relaxation times of formed structures are smaller. Thus, a longer time or a larger deformation and deformation rate is needed for crystals to evolve.   Rheometric data obtained using capillary are often in good agreement with the results obtained using a conventional rheometer if all the corrections are applied precisely (Macosko 1994). These corrections are: (i) those associated with the errors linked to the experimental testing such as correction for barrel resistance and (ii) those corrections applicable to obtained data for revealing the true material behaviour such as Bagley and Rabinowitsch corrections (Mitsoulis et al. 1998). However, when crystallization occurs within the system of study, a discrepancy is expected between the results obtained using capillary and LVE data due to formation of crystals in the former case.   As mentioned in section 1.1.1, longest relaxation time of samples can be obtained using  97  Equations 1.3 and 1.4. Alternatively, one can use the average of Maxwell relaxation times, Mλ  extracted from the Linear Viscoelastic Master (LVE) curve at the desired temperature to estimate the relaxation time of chains within the matrix (Scelsi and Mackley 2008). The Maxwell model with 5 relaxation modes was found to describe the LVE data of PP5. TABLE 5.1 compares the longest relaxation times for PP5 obtained at various temperatures with the average of relaxation times obtained using Maxwell model.  TABLE 5.1: Relaxation time of PP5 at various temperatures. Temperature Longest relaxation time in s (Creep) Average Maxwell relaxation time in s Longest relaxation time in s (SAOS) 190oC 5.0 6.7 17.0 180oC 7.1 9.2 22.3 170oC 10.0 14.0 29.6 165oC 12.0 16.5 34.3  The Arrhenius equation can be used to capture the dependency of the relaxation time on temperature as depicted in FIGURE 5.1: max max,01 1exp aoER T Tλ λ  = −         (5.1) where, " "R  is the universal gas constant, aE  is the activation energy, 𝑇𝑇0 is the reference temperature (190oC in our case), and max,0λ is the longest relaxation time at the reference temperature of 190oC. Since the longest relaxation times were found to correlate better with temperature using the Arrhenius equation (FIGURE 5.1), they were preferred to be used here over  98  the average relaxation times.  The longest relaxation times obtained using small amplitude oscillation shear (SAOS) experiments for all polypropylenes shown in TABLE 5.2, i.e. the maximum time needed to describe the LVE master curve. It should be mentioned though that the use of different methods to calculate the longest relaxation times will not affect the conclusions, since the performance of the various PPs in FIC is compared qualitatively. 0.0 5.0x10-5 1.0x10-4-0.10.00.10.20.30.40.50.60.70.80.91.0Ea-λmax-oscillation=11.27 kcal/molEa-λM=14.98 kcal/mol λmax - Creep experiment λM - Maxwell model λmax - Oscillationln(λmax/λmax,0) or ln(λ M/λM,0) (1/T-1/T0)Ea-λmax-creep=14.07 kcal/molPP5T0=463.15 K FIGURE 5.1: The temperature dependency of the maximum and average relaxation time of PP5 obtained using various methods.     99  TABLE 5.2: Relaxation times of various resins obtained from sinusoidal oscillation at different temperatures. Resin 190oC (s) 180oC (s) 170oC (s) 165oC (s) Ea-oscillation (kcal/mol) PP1 0.16 0.20 0.26 0.29 11.27 PP2 0.10 0.12 0.15 0.16 10.56 PP3 0.52 0.66 0.84 0.95 9.63 PP4 5.00 6.45 8.41 9.65 7.63 PP5 17.00 22.30 29.61 34.29 11.27  5.2 Flow induced crystallization in capillary 5.2.1  Transition from a melt at high temperature to a non-homogenous melt/semi-solid The flow curve of resin PP5 over a wide temperatures (163oC -190oC) is shown in FIGURE 5.2a,b. For all studied shear rates, the Wi numbers were larger than 200 indicating strong flows with tendency to stretch the high end tail of MWD in the flow direction. Each point represent a steady-state value i.e. pressure drop was stabilised over an extended period of time. FIGURE 5.2a shows significant changes in the shape of the flow curve with decrease of the temperature. This observation clearly indicates breakdown of the time-temperature superposition and a transition from a melt at high temperature to a semi-solid (partly crystalline) state at lower temperatures. Since the die geometrical characteristics remain the same and only temperature is varied at each shear rate (given flow rate), the average residence time (ratio of volume of the flow field over the volume flow rate) is the same. Therefore, the De number for a given chain length should increase as the temperature is decreased and thus the longest chain may not stretch due to a lower residence time. This may not apparently agree with the data at first glance, however, as the temperature decreases the longest relaxation time of various chains increases and thus more chains of smaller  100  size are influenced by deformation effectively (higher Wi number and lower De number). Therefore, more chains having smaller De and higher Wi numbers are expected to crystallize, which in turn causes thicker shish structures at capillary wall as shown by Farah and Bretas (2004). Of note, Kimata et al. (2007) found the shish to be consisted of chains with various length which is in agreement with our discussion.   To see more clearly the failure of time-temperature superposition (the more so as temperature decreases), FIGURE 5.2b plots the data (shown in FIGURE 5.2a) after applying the time-temperature superposition principle at the reference temperature of 190oC by using the shift factors determined from linear viscoelastic measurements using a rotational rheometer (Anton Paar MCR-502). These have been reported earlier in terms of the Arrhenius equation (see Chapter 4, section 4.1). The LVE curve plotted here is obtained using a rotational rheometer and thus the range of shear rates/frequencies for which the data could be obtained is limited. The agreement between the capillary results and the LVE master curve plotted as a flow curve is excellent at 180oC and 190oC (at least in the shear range of LVE plotted). These temperatures are well above the melting peak temperature of resins obtained in DSC, and therefore no crystalline structure evolves using small amplitude oscillatory shear in the rotational rheometer. Thus, at these temperatures the polymer remains mainly crystalline-free at the entry and through the capillary die even at very high shear rates. Although very high deformation rates at the entry can stretch chains along the flow direction, the strain (shearing time) imposed on the melt is not enough to induce any ordered structures. However, as the temperature decreases, the relaxation time of the formed structures increases and thus the oriented/stretched chains can evolve as crystalline structures.  101  Furthermore, at lower temperatures smaller chains can be stretched in the direction of deformation (due to larger relaxation time/Wi number) which lowers the energy required for transformation into solid. Therefore, these oriented chains can evolve into a crystalline structure. For temperatures below 180oC a deviation from the master curve is observed due to FIC. As the deformation rate increases the deviation from the LVE curve (crystalline free) becomes larger. Since the crystalline layer which is formed at the die wall increases with the increase of shear rate (as will be shown and discussed in detail later), this increase in deviation is attributed to reduction of the effective die diameter. Higher deformation rates stretch chains more effectively contributing to a lower barrier for crystal formation. As the temperature decreases more, the deviation from the LVE curve occurs at smaller apparent shear rate values. 101 102 103 1040.112  PP5MFR 2L/D=20D=0.51mm2α=15ο T=163oC T=165oC T=167oC T=170oC T=180oC T=190oCApparent shear rate, γA (s-1)Shear stress, σw(MPa).0.06a 101 102 103 104 1050.010.11   T=163oC, aT=2.25 T=165oC, aT=2.12 T=167oC, aT=1.99 T=170oC, aT=1.83 T=180oC, aT=1.34 T=190oC, aT=1.00    LVEShear stress, σw(MPa)aT.Apparent shear rate, γA (s-1)2.PP5MFR 2L/D=20D=0.51mm2α=15οb FIGURE 5.2: The flow curve of PP5 at various temperatures from 163oC to 190oC (Tm= 162.1oC). (a) The effect of flow on FIC at high shear rates is evident with the effect to be more dominant closer to Tm. (b) Applying the time-temperature principle using the shift factors determined from linear viscoelastic measurements shows that the deviation from the LVE curve (crystalline free) occurs at smaller apparent shear rate values with decrease of temperature. The increase of shear stress beyond the LVE is clearly due to the effect of flow on FIC.   102   Synergistic effects of pressure on the crystallization temperature and thus on FIC in capillary flow is reported to be negligible, when pressure is increased using a downstream reservoir to apply a back pressure (Titomanlio and Marrucci 1990). In addition the effect of pressure on shear viscosity can be calculated by using the Barus equation is (Barus, 1981): ( )0 peβη η=                              (5.2) Where η(0) is the viscosity at atmospheric conditions and β is the coefficient for the pressure dependency of viscosity.  Using a pressure coefficient of 6.4 (GPa-1) for polypropylene (Bindings et al. 1998), the small increase of shear stress due to pressure cannot explain the results plotted in FIGURE 5.2b. The length-to diameter ratio of the capillary die is small (L/D=20) and the good agreement between the capillary data and LVE indicate that this effect is indeed negligible. Therefore the behaviour observed in FIGURE 5.2a,b is mainly due to the combined effect of temperature and flow deformation on FIC.  The FIC observed in FIGURE 5.2a,b can be linked to both shear and extensional deformation. Extensional deformation is imposed on melt in the contraction region of dies close to the center, while shear deformation occurs close to the walls in the contraction region and in the capillary die with the effects more dominant close to the wall, where the shear rate and residence time are higher. The extensional flow in the entry orients molecules effectively, possibly creates significant amount of precursors and perhaps small crystals, which can materialize into crystals in the die land if the relaxation time of formed structure is larger than the residence time in the die (for instance see Figure 9 in Faras and Bretas (2004) which shows large degree crystallinity in the  103  core). As the deformation rate increases the melt/crystal interaction changes, which leads to a higher viscosity at a comparable degree of crystallinity (see Chapter 4, section 4.3.1). The growth rate of quiescent crystallization for PP5 is estimated to be around 0.23 nm/s at T=163oC (see Chapter 3, section 3.2). This growth rate would be increased with a factor 5-10 due to orientation induced by flow (Duplay et al. 2000). Thus, a growth rate of 1.15-2.3 nm/s is reasonable to be assumed. The average residence time, Rτ , can be estimated as: 24 81 13 tanb b bRA AD D DVolume V LVolumetric flowrate Q D D D Dτγ α γ      = = = + + − +                       (5.3) Where Db is the barrel diameter and D is the die diameter. The first term is the average residence time in the contraction region and the second term is the average residence time in the die land region. Using Equation 5.3 for L/D=20 and D=0.58 mm (smallest diameter die), 2α=90, and the range of apparent shear rates used in the experiments, the average residence times are falling between 0.015s and 3.7s. Using the crystal growth estimated above, the expected size of crystals at the exit should be in the range of 0.000034-0.009 µm. The nucleation density can be estimated as 1.8E11 nuclei/m3 (see Chapter 3, section 3.2). Thus for the smallest die diameter of 0.58 mm, and it turns out that this is ~1066 times larger than the crystalline structure formed. These structures are carried out of the die readily with no effect on the effective die diameter. However, in the capillary die, chains which exist close to the wall are exposed to larger shear rates and possess much higher residence times (small De number) and thus formed oriented structures are often observed close to the wall in the form of layers parallel to the flow direction (Farah and Bretas 2004). As a result, the effective diameter for flow decreases leading to the apparent increase  104  of the wall shear stress. However, once the process of deposition of crystals on the capillary wall kicks in, why this process does not continue until the capillary die blocks off completely?  However, instead a steady-state is obtained at a given shear rate as discussed above. The steady-state occurs due to slip at the interface of melt/crystallization layer interface that stops the continuous crystal deposition establishing a dynamic equilibrium. Molten polypropylenes are known to slip frequently significantly at high shear rates (Hatzikiriakos 2012; Mitsoulis et al. 2005; Rosenbaum et al. 1997; Kazatchkov et al. 1995). In summary, the deviation of shear stress from LVE (FIGURE 5.2b) is contributed to the reduced effective radius of capillary as well as crystals which are present in the core. The relative importance of each parameter will be investigated later. As discussed in Chapter 2, several models have been proposed to facilitate the approximation of extensional properties such as extensional viscosity and stress. The Cogswell analysis (Cogswell 1972) is utilized here to extract the extensional viscosities and rates from the data shown in FIGURE 5.2a and FIGURE 5.2b. The local power-law exponent, n, is found to be equal to 0.33 at 190oC, where no crystallization occurred which is in good agreement with the previous studies on polymer melts (Farah and Bretas 2004). Using an orifice die of negligible L/D (<<1) with the diameter and contraction angle of 0.51mm and 15o, respectively, the entrance pressure drop data needed for Cogswell analysis is obtained. FIGURE 5.3a depicts the end pressure data as a function of apparent shear rate at various temperatures, while FIGURE 5.3b shows the same set of data after applying the time temperature superposition. These are implemented along with the local power law exponent of 0.33 to obtain the extensional viscosities and rates using Equations 2.5 and 2.8. The parallelicity of the data at high temperatures plotted in FIGURE 5.3a,b  105  shows that no significant crystals are formed in the entry although highly oriented molecules create precursors or tiny crystals as discussed above. As shown in FIGURE 5.3b the experimental data obtained for temperatures lower than 167oC deviate from the data obtained at higher temperatures after the application of time temperature superposition. Structure formation within the die entrance can contribute to this observation. For small contraction angle, the length of contraction region is significant and the flow is shear dominated (extensional is minor due to small acceleration due to small contraction angle). In addition, the residence time increases significantly compared to a die with a 90o entrance (from Equation 5.3, ( ) 1tanRτ α−∝ ). For such cases, structures formed in the entry region may affect the pressure drop contributing to the deviation observed in FIGURE 5.3b (Cogswell 1981). 102 103 1040.82.03.24.45.66.88.020.0 T=163oC T=165oC T=167oC T=170oC T=180oC T=190oCEntrance pressure drop, ∆P (MPa)Apparent shear rate, γA (s-1).0.62  101 2  104PP5MFR 2Orifice dieD= 0.51mm2α = 15oa102 103 1040.82.03.24.45.66.88.020.0PP5MFR 2Orifice dieDdie = 0.51mm2α = 15o   T=163oC, aT=2.25 T=165oC, aT=2.12 T=167oC, aT=1.99 T=170oC, aT=1.83 T=180oC, aT=1.34 T=190oC, aT=1.000.62  101 2  104.aT.Apparent shear rate, γA (s-1)Entrance pressure drop, ∆P (MPa)b FIGURE 5.3: (a) The entry pressure of PP5 at various temperatures using a die with L/D~0, D=0.51 mm and contraction angle of 2a=15o. (b) The shifted ends pressure data of PP5 at various temperatures using a die with L/D~0, D=0.51 mm and contraction angle of 2a=15o. The extensional viscosity is plotted as a function of extensional rate in FIGURE 5.4. The zero extensional viscosity at low rates was only observed at 190oC (dotted-line). The ratio of  106  extensional to shear viscosity is determined as about 3, which is the theoretical value of Trouton’s ratio at small shear rates, thus showing the consistency of the experimental data. No sign of extensional thickening was observed for the studied deformation rates. In the converging zone of the die, the residence time of a polymer element is limited by the contraction angle, die diameter, and deformation rate (see Equation 5.3). Therefore, often transient extensional behaviour is observed and the viscosities obtained should be used with caution (Padmanabhan and Macosko 1997). However, the good agreement of the data at various temperatures shows that no significant crystal formation is seen in the entry.   3 10 40 70 100 400104105 T=163oC T=165oC T=167oC T=170oC T=180oC T=190oCApparent extensional  viscosity, η E,A (Pa.s)Apparent extensional rate, εH,A (S-1).6005  1032  105PP5MFR 2L/D = 20D = 0.51mm2α = 15o FIGURE 5.4: The extensional viscosity of PP as a function of extensional deformation rate in the die entrance for PP5 at various temperatures calculated using the Cogswell analysis (Equations 2.5 and 2.8).  107  Crystallization models in FIC simulations include an enhancement factor which shows the apparent increase of viscosity of the melt with respect to its viscosity value in the absence of crystallization (amorphous state). The normalized shear viscosity as a function of extensional deformation rate is plotted in FIGURE 5.5a,b. The FIC behaviour at temperatures lower than 180oC is clearly observed as significant deviation from the base line at normalized shear viscosity of 1. This set of data is useful to validate models proposed for FIC since it combines both shear (which yields oriented structures at wall) and extensional (which yields crystals in the core) deformations simultaneously. Most of these models capture the behaviour well only under shear or extensional deformation separately and to the best of our knowledge they were never tested under mixed flow conditions. 101 102 1031PP5MFR 2L/D = 20Ddie = 0.51mm2α = 15o T=163oC T=165oC T=167oC T=170oC T=180oC T=190oCNormalized shearviscosity, η/ηaApparent extensional rate, εH,A (S-1).a101 102 10312345 T=165oC T=170oC T=180oC T=190oCPP5MFR 2L/D = 20D = 0.51mm2α = 90oNormalized shearviscosity, η/ηaApparent extensional rate, εH,A (S-1).b FIGURE 5.5: Normalized shear viscosity as a function of the apparent extensional deformation rate in the die entrance for PP5 at various temperatures.   108  5.2.2 The effect of contraction angle on flow induced crystallization  Extensional parameters such as extensional strain and strain rate are dependent on the die contraction angle and diameter ratio, Db/D. The maximum extensional deformation rate is dependent on the entrance angle as shown by Equations 2.4-2.6, while the maximum extensional strain on the diameter ratio Db/D (Equation 2.9). Therefore, the effect of extensional rate on FIC in the capillary die is studied using dies of same diameter (D=0.51 mm) but various contraction angles from 15o-90o with Db=9.525 mm. Since these dies have the same diameter, the average strain imposed on the polymer melt is identical for all of them (Equation 2.9). The maximum extensional strain experienced by the matrix in the die entrance is ( ) ( )22 2max ln ln 9.525 0.51 5.9bD Dε = = = . It is suggested that the average strain is approximately half of the maximum strain determined (Cogswell 1978). It was shown that the extensional viscosity determined in the die entrance for maximum extensional strain of 6 (or average strain of 3) compares to the data obtained using melt spinning experiments at strain of 3 (Padmanabhan and Macosko 1997). This observation suggests that the residence time in the mentioned die in its die entrance is enough to obtain steady state.  109  101 102 103 104 1050.010.11   T=165oC, aT=2.12 T=170oC, aT=1.83 T=190oC, aT=1.00    LVEShear stress, σw(MPa)aT.Apparent shear rate, γA (s-1)2.PP5MFR 2L/D=20D=0.51mm2α=60οa101 102 103 104 1050.010.11   T=165oC, aT=2.12 T=170oC, aT=1.83 T=180oC, aT=1.34 T=190oC, aT=1.00    LVEShear stress, σw(MPa)aT.Apparent shear rate, γA (s-1)2.PP5MFR 2L/D=20D=0.51mm2α=90οb FIGURE 5.6: The flow curves of PP5 at various temperatures using a capillary die having L/D=20 and contraction angle of (a) 02 60α =  and (b) 02 90α = . The dashed line represents the flow curve from LVE measurements. The effect of flow at high shear rates is clear from the large deviations from the LVE line which occurs at smaller apparent shear rates with decrease of temperature. Comparison of this set of data with that plotted in FIGURE 5.2b ( 02 15α = ) shows that the contraction angle (average extensional rate) has no additional effect on the crystallization behavior of PP5. FIGURE 5.6 plots the flow curves of PP5 at various temperatures using dies with contraction angles of 2a=90o and 60o. Comparison of this set of data with that plotted in FIGURE 5.2b (2a=90o) shows that the contraction angle (apparent extensional rate) does not have an additional effect on the crystallization behaviour. Thus this set of data suggests that regardless of the extensional rate employed in the die entry region the crystallization behaviour in the die entry is comparable. It is noted that the average Hencky strain is independent of the contraction angle. This last conclusion can also be drawn by the considering the data depicted in FIGURE 5.7 where the flow curves of PP5 at 170oC are plotted for dies having various contraction angles. The results are similar showing that the contraction angle has no additional effect on the crystallization behaviour of PP5. It is worth noting that extensional studies using the SER fixture have shown that  110  the extensional strain is a more important parameter than the extensional deformation rate (Derakhshandeh and Hatzikiriakos 2012; Hadinata et al. 2007). The Wi number is large (>100) for the data shown in FIGURE 5.7. Therefore practically it is possible to obtain stretched chains ready to crystallize, however the corresponding De numbers are different for the various dies. In fact the De numbers are smaller than 1 for 2α=15o and in the range of 0.27-7 for the die with 2α=90o, that is the latter is eight times more than that of 2α=15o. In spite of this, the strength of the flow might not be enough to stretch the longest chains for crystallization.  30 100 10000.070.080.090.10.20.30.40.50.60.70.8   2α = 15o 2α = 30o 2α = 60o 2α = 90oShear stress, σw (MPa)Apparent shear rate, γA (s-1)PP5MFR 2L/D = 20D = 0.51 mmT=170oC. FIGURE 5.7: The flow curves of PP5 at 170oC using capillary dies having different contraction angles. However, while the extensional flow is more dominant in the die with 2α=90o, the residence time in the die with 2α=15o is about 8 times more (calculated from Equation 5.3) and this balances the combined effects of extensional and shear (Cogswell 1981). The effect of smaller  111  contraction angles on the crystallization were noted in previous works using a temperature gradient imposed on the die (Crater et al. 1980). The difference in the behaviour was explained in terms of residence time within the entrance zone since a temperature gradient was implemented. However, we suspect that this higher apparent shear stress at very low contraction angles (FIGURE 5.7) correlates with the residence time only. 101 102 103103104105   2α=90o 2α=15o 3η+Extensional viscosity, ηE or 3η+Extensional rate, ε (s-1) or angular frequency, ω (rad/s).    PP5T=170oC FIGURE 5.8: The extensional viscosity of PP5 as a function of the extensional rate using two capillary dies of different contraction angles (15o and 90o) at 170oC. The viscosity values obtained for 2α=90o agrees with 3η+ curve showing that the flow kinematics in the die entry having a contraction angle of 90o is extensional dominated whereas in that of 15o contraction is shear dominated. To show this effect more clearly, FIGURE 5.8 plots the extensional viscosity as a function of extensional deformation rate for dies with contraction angle of 15o and 90o. The line labelled as 3η+ was obtained from LVE measurements and also plotted as the master curve at 170oC in  112  FIGURE 5.8. As shown, the extensional viscosities determined for the contraction angle of 90o agrees with the 3η+. The data for the 15o are shear dominated and therefore do not agree with the extensional data.  5.2.3 The effect of L/D ratio on crystallization Highly oriented structures in the vicinity of slit wall were reported in various works which shows the importance of shear rate/shear strain (Farah and Bretas 2004). The effect of shearing time (or length) was studied using dies with identical entrance angle and diameter and different L/D ratios. Since all samples have undergone the same thermal and extensional deformation history within the die entrance region, any difference in the behaviour is linked to shearing time or shear strain in the die land region.  Dies having a larger L/D allow for larger shearing time (which leads to a smaller De), thus contributing to a stronger stretching of chains. This also allows more time for the crystalline structures to evolve. Therefore, as L/D increases the FIC phenomenon (deviation from LVE) is expected to occur at lower shear rates. This is shown clearly in FIGURE 5.9, where the flow curves of PP5 are plotted obtained by using dies having various L/D ratios with all other geometrical parameters kept constant. Increasing the L/D ratio causes deviation from the LVE curve (absence of any crystals) at smaller apparent shear rate values. This observation has also been reported previously (Farah and Bretas 2004, Titomanlio and Marrucci 1990). Of note, FIC studies in slit dies have shown an increase in the thickness of oriented structure with the increase in the length of capillary (Farah and Bretas 2004).  The thickness of crystalline layer at the wall of capillary can be estimated roughly using the LVE curve along with the obtained capillary data. For example  113  using the data of FIGURE 5.6b, the apparent shear rate of 732s-1 at 170oC corresponds to a steady shear stress value of 0.28 MPa. If no crystallization occurred in the system, to obtain shear stress of 0.28 MPa one should have applied an apparent shear rate of ~2695s-1. If we assume that the increase in shear stress comes only from the reduction in effective radius due to the presence of the crystallization layer, it can be easily shown that a crystallization layer of about 90 µm should be present. These calculations are done based on no-slip assumption. As discussed above PP exhibit slip (Mitsoulis et al. 2005; Rosenbaum and Hatzikiriakos 1997; Kazatchkov et al. 1995) and this complicates the analysis further and a flow simulation should be performed to deconvolute the various effects (beyond the scope of the present work). TABLE 5.3 shows typical crudely estimated crystalline layer thickness under various conditions using the assumptions of no-slip. An increase of the crystalline layer thickness with apparent shear rate and L/D ratio can be observed as expected.  When crystallization kicks in, one may expect that a steady-state condition will never be reached. However, as discussed before the reported experimental data represent steady-state, also reported elsewhere (Farah and Bretas 2004). During the time which is needed to reach the steady state a crystalline layer slowly build up on the capillary wall thus reducing the effective radius of die (see schematic of FIGURE 5.10). This crystalline layer will be thicker at the bottom of the die as shown in FIGURE 5.10. While the effective diameter decreases, the shear rate increases and its value become dependent on the z position along the die length. Eventually due to increased levels of shear stress, slip at the interface occurs which reduces accordingly the residence time of the  114  molecules at the interface thus preventing further growth of the crystalline layer. This yields steady state conditions as observed in this study. 100 10000.1120.065000   L/D=5 L/D=10 L/D=20 L/D=40Shear stress, σw (MPa)Apparent shear rate, γA (s-1)PP5MFR 2Ddie=0.51mm2α=90oT=170oC30. FIGURE 5.9: The flow curves of PP5 obtained by using capillary dies having different L/D ratios with all other parameters kept constant. A larger die length causes crystallization at smaller apparent shear rate values, which is due to the larger residence time (see Equation 5.3).   FIGURE 5.10: Illustration of various steps occurs in die when crystallization kicks in.  115   TABLE 5.3: Estimated crystalline later thickness at wall of capillary. Die Temperature (oC) Apparent shear rate (s-1) Thickness(µm) L/D=20 D=0.51mm 2α=90o 170 250 42 400 89 650 125 1000 ~157 165 40 66 65 89 100 129 160 138 250 ~198 L/D=40 D=0.51mm 2α=90o 170 100 43 120 85   The above analysis and discussion has neglected the effect of viscous dissipation which might be significant. The Nahme-Griffith number, Na, is the critical parameter for estimating the importance of shear heating in rheometry. It measures how much the temperature rise will affect the viscosity. For capillary the Na number can be written as: 2γ4NakRβτ=                             (5.4) where, k is thermal conductivity of polymer, R is die radius, τ is shear stress, γ is shear rate, and β  is temperature sensitivity of viscosity. The viscosity as a function of temperature can be written as ( )00T Teβη η −= . The Na numbers calculated for most experimental data in this work are less than 1 (except for FIGURE 5.9 for the data for L/D=5 and L/D=10). These are shorter dies  116  with insufficient time for developing steady temperature distribution and thus the effect of viscous heating is delayed to a higher Na number) (Bair 2007). Viscous dissipation becomes important for shear rates higher than about 500 s-1 (for die with larger diameters and L/D=ratios) in our study. Although viscous dissipation was not considered, the conclusions remain the same. In summary, a flow model should be considered to fully assess the viscous heating effects together with the crystalline layer formation and slip at the interface as discussed above. This can test the validity of crystallization models by simulating the results of the resent work. 5.2.4 The effect of die diameter on crystallization Previous studies on FIC under a well-defined extensional flow emphasized on the effect of extensional strain (Derakhshandeh and Hatzikiriakos 2012, Hadinata et al. 2007). A critical strain for the onset of crystallization was observed, regardless of the extensional rates used. Dies having the same contraction angle and L/D and different diameters were used to examine the effect of extensional strain on crystallization. Three different die diameters were used namely of 0.51, 0.76, and 1.3mm. These result maximum extensional strains of 5.9, 5.1, and 4.0 respectively. The Deborah number increases as the die diameter increases and thus for the smaller die diameter one may expect more oriented/stretched chains to be evolved in the entrance. The data seem to show this, that a smaller diameter should promote crystallization more. However, it is worth noting that the data obtained at the higher apparent shear rates of 400, 650 and 1000s-1 are subjected to viscous heating and this effect is for the largest die diameter. Therefore, the small difference observed in FIGURE 5.11 can be attributed to viscous dissipation rather than to differences in crystallization initiated at the entry region due to different strains in each die.  117  100 10000.10.20.30.40.50.6PP5MFR 2L/D = 202α=90oT=170oC   D=0.51mm D=0.76mm D=1.30mmShear stress, σw (MPa)Apparent shear rate, γA (s-1). FIGURE 5.11: The flow curve of PP5 at 170oC using dies having the same contraction angle and different diameter. As the diameter decreases, the degree of crystallinity formed increases, although a correction for viscous dissipation might counterbalance this effect.  5.2.5 The effect of molecular weight (Mw) on crystallization Molecular Weight has a strong effect on crystallization kinetics. Under quiescent conditions at temperature close to melting temperature, crystallization is controlled mainly by diffusion of chains onto the crystal growing surface. Reptation time increases with viscosity (or Mw) increase since stronger friction is imposed on the chains. Therefore, crystallization kinetic under quiescent condition impedes as Mw increases (Van Antwerpen and Van Krevelen 1972; Magill 1969; Magill 1967; Magill 1964). Flow orients or stretches (under strong deformation/large Wi number) chains in the flow direction which lowers the thermodynamic barrier for crystal formation. These oriented/stretched chains can relax into spherical coil configuration. Relaxation  118  time depends on the deformation temperature as well as the molecular weight characteristics of resins. Generally, the relaxation time is shorter at higher temperatures as shown in TABLE 5.2.   Long chains within the matrix relax slower. These long chains stretch in the flow direction and subsequently form oriented crystalline structures known as shish (under low De number). The degree of orientation can be qualitatively estimated using Wi and De number, which correlate with high-end tail of MWD in the system. Longer chains relax slower leading to a Larger Wi number. FIGURE 5.12 shows the flow curves of various resins at different temperatures. As shown in FIGURE 5.12a and FIGURE 5.12b, PP1 and PP2 which have the lowest viscosity and longest relaxation time values (TABLE 5.2) shows no tendency to form crystals in the capillary. At all investigated temperatures above the corresponding melting points, the experimental data follow the flow curve obtained using sinusoidal oscillation protocol in the MCR-502 rheometer. However, as molecular weight is increased (or in other word longest relaxation time/viscosity is increased) the deviation from the LVE occurs at lower temperatures as shown for the case of PP3 in FIGURE 5.12c. At 170oC, the relaxation time of PP3 (0.84 s) is not long enough to cause deviation except at very high deformation rates. Deviation due to crystallization is observed even at higher temperatures in the case of resins with larger zero shear viscosity (PP4 and PP5 in FIGURE 5.12d and FIGURE 5.12e respectively).   119  101 102 103 104 1050.010.11   T=165oC, aT=2.02 T=170oC, aT=1.79 T=180oC, aT=1.41 T=190oC, aT=1.00    LVEShear stress, σw (MPa)aT.Apparent shear rate, γA (s-1)PP1MFR=36L/D=16Ddie=0.79mm2α=180o.a101 102 103 104 1050.010.11   T=165oC, aT=2.34 T=170oC, aT=2.11 T=180oC, aT=1.23 T=190oC, aT=1.00    LVEShear stress, σw (MPa)aT.Apparent shear rate, γA (s-1)PP2MFR=24L/D=16D=0.79mm2α=180o.b101 102 103 1040.010.11 T=165oC, aT=1.82 T=170oC, aT=1.62 T=180oC, aT=1.26 T=190oC, aT=1.00    LVEShear stress, σw (MPa)aT.Apparent shear rate, γA (S-1).cPP3MFR=12L/D=16D=0.79mm2α=180o101 102 103 104 1050.11   T=165oC, aT=1.87 T=170oC, aT=1.63 T=180oC, aT=1.25 T=190oC, aT=1.00    LVEShear stress, σw (MPa)aT.Apparent shear rate, γA (s-1)PP4MFR=5L/D=16D=0.79mm2α=180o.d101 102 103 104 1050.11   T=163oC, aT=2.25 T=165oC, aT=2.12 T=167oC, aT=1.99 T=170oC, aT=1.83 T=180oC, aT=1.34 T=190oC, aT=1.00    LVEShear stress, σw (MPa)aT.Apparent shear rate, γA (s-1)PP5MFR=2L/D=16D=0.79mm2α=180o.e FIGURE 5.12: The flow curves of various resins at different temperatures shifted by applying the time-temperature superposition using the shift factors determined from LVE measurements. The dash lines represent the flow curve corresponding to LVE measurements.   120  Shish structures which are formed in the extrudate in the vicinity of capillary wall (or in the vicinity of crystalline structure formed at wall) can be captured by using a polarized microscope (Mitutoyo microscope set up equipped with Lumenera LU 165 color CCD camera and two polarizers) as shown in FIGURE 5.13. Shish formation close to capillary wall was also reported elsewhere (Farah and Bretas 2004; Liedauer et al. 1993).  FIGURE 5.13: Shish structure formed in an extrudate of PP5 extruded at T=170oC and apparent shear rate of 250s-1 observed close to the wall of capillary die.  5.3 Conclusions Flow induced crystallization of various polypropylenes with different molecular weight and molecular weight distribution was studied using capillary rheometry. All investigated temperatures were above melting peak temperature of polymers obtained using DSC. In this temperature region, only resins having high Mw (or low MFI) were found to undergo crystallization within the capillary die which shows the effect of high-end tail of MWD on FIC. Decreasing temperature and/or increasing deformation rates enhanced the kinetics of crystallization immensely. A smaller deformation rate was necessary to induce crystallization using dies with larger L/D (larger shear strain or shearing time). Dies with different entry regions (contraction angles) was also examined to illustrate the effect of extensional flow within the  121  contraction zone. Extensional flow parameters did not affect the obtained data which shows that shear deformation in the die is the main parameter contributing to enhance FIC. This rather comprehensive set of data reported in this Chapter can be used in flow simulations to test crystallization models that describe FIC in mixed shear and extensional flows that mimic real polymer processing operations.  122  Chapter 6: Conclusions and Recommendations This work has generated a comprehensive and reliable set of data on quiescent and flow induced crystallization of various polypropylenes of different molecular characteristics. This study can be used by (1) experimentalists employing similar apparatuses as a guideline to study other polymeric systems; and (2) modellers in order to extract model parameters and compare their predictions with experimental data as a benchmark for model validation and process optimization. This chapter summarizes the conclusions of the work and provides recommendations for future work.  6.1 Conclusions Key conclusions and new contributions to knowledge are discussed and summarized: • The Avrami rate parameter reflects the total crystallization kinetics, however it does not give insight into the crystal microscopic differences existing between different resins such as the nucleation density and crystals growth rates. Optical microscopy should be used in conjunction with the Avrami analysis of DSC crystallization data in order to capture both crystal macroscopic and microscopic differences of the investigated resins. • Polypropylene resins with different molecular characteristics were found to have different nucleation density and growth rate. In particular, different catalytic system used polymer production yield to different degree of defects within the polymer matrix contributing to a much slower growth kinetics and smoother regime transition.   123  • Data obtained using DSC under high cooling rates should be corrected for thermal lag effects which is bounded to the test specimens in order to observe the true crystallization material behavior. • It was shown that induction times under non-isothermal conditions can be estimated using the induction times obtained under isothermal conditions. The obtained induction times under non-isothermal condition were found to be insensitive to the proposed model as long as the model fits the isothermal induction times well. • An Avrami/Nakamura model in its differential form was found to be able to fit and predict DSC non-isothermal crystallization kinetics data very well over a range of cooling rates of 2-20oC/min for all studied PP resins.  Such a crystallization model is a useful tool for numerical simulations of polymer processes including injection molding, fiber spinning and film blowing as such it is of great industrial applicability for product and process development.  • In simple shear, there is a critical strain required for the onset of crystallization at given shear rates and temperatures. In shear experiments the critical strain needed for polymer to crystallize decreases with decreasing rate. Temperature was found to be an important variable in crystallization as expected and change of temperature by a few degrees changes the induction times for crystallization by one order of magnitude. These observations can be used in model development to predict flow induced crystallization phenomena in shear. • The suspension models are applicable only to low deformation rates, since there is only dependence of the viscosity on the degree of filling. A proper model should include  124  viscoelastic effects in order to be able to capture the crystallization behaviour well (e.g. the two-phase constitutive/microstructural model for FIC of Doufas 2013; Doufas et al. 2000). • All investigated temperatures were above melting peak temperature of polymers obtained using DSC. In this temperature region, only resins having high Mw (or MFI) were found to undergo crystallization within the capillary die which shows the effect of high-end tail of MWD on FIC. Decreasing temperature and/or increasing deformation rates enhanced the kinetics of crystallization immensely. A smaller deformation rate was necessary to induce crystallization using dies with larger L/D (larger shear strain or shearing time). • Dies with different entry regions (contraction angles) was also examined to illustrate the effect of extensional flow within the contraction zone. Extensional flow parameters did not affect the obtained data which shows that shear deformation in the die is the main parameter contributing to enhance FIC.   6.2 Recommendations for future work The results of this work may be useful in developing new models for flow simulation which includes flow induced crystallization under various deformation/thermal conditions. The following points represent general recommendations for future studies: • Using scattering techniques such as Wide Angle X-ray Scattering (WAXS), Small Angle X-ray Scattering (SAXS), and Light scattering in-situ with rheological measurements is recommended in order to elucidate FIC at the early stages of crystallization. This can  125  clarify further how shish-kebab structures are formed and also show its relationship to various parameters such as Mw, MWD, and deformation/thermal conditions.    • Rheo-optical data at high deformation rates are still not available due to apparatus limitations. 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Ziabicki A (1967) Kinetics of polymer crystallization and molecular orientation in the course of melt spinning Applied polymer symposia 6: 1–18.  ¶ 141  Appendices : Nakamura model prediction of non-isothermal crystallization of polypropylenes  The prediction of the Nakamura model (Equation 3.7) using the estimated induction times obtained by Equation 3.8 are shown here for the PP resins. In FIGUREs A.1 to A.4 the Arrhenius equation was used to capture the dependency of Avrami rate parameter, k, on temperature. Experimental data obtained under high cooling rates (>10oC/min) were corrected for the temperature lag, which exists in between test specimens and furnace as describe previously (section 3.1.3). The Nakamura model prediction using the Arrhenius equation to capture the temperature dependency of Avrami rate parameter was found to be satisfactory. FIGUREs A.1b to A.4b and A.1c to A.4c reveal the Nakamura prediction coupled with WLF and Hoffman-Lauritzen theories, respectively. Comparable results were obtained by using all three different theories to capture the effect of temperature on the Avrami rate parameter as shown here.       142  110 115 120 125 130 1350.00.51.02 oC/min5 oC/min10oC/min20oC/min Nakamura prediction   aPP1  Relative crystallinity, X/XfTemperature, T (oC)110 115 120 125 130 1350.00.51.02 oC/min5 oC/min10oC/min20oC/min Nakamura prediction   bPP1  Relative crystallinity, X/XfTemperature, T (oC)  110 115 120 125 130 1350.00.51.02 oC/min5 oC/min10 oC/min20 oC/min Nakamura prediction   cPP1  Relative crystallinity, X/XfTemperature, T (oC)  FIGURE A.1: Non isothermal quiescent crystallization prediction of PP1 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8.    143  100 105 110 115 120 125 1300.00.51.0 2 oC/min5 oC/min10 oC/min20 oC/min Nakamura prediction   aPP3 Relative crystallinity, X/XfTemperature, T (oC)100 105 110 115 120 125 1300.00.51.0 2 oC/min5 oC/min10 oC/min20 oC/min Nakamura prediction   bPP3 Relative crystallinity, X/XfTemperature, T (oC)  100 105 110 115 120 125 1300.00.51.0 2 oC/min5 oC/min10 oC/min20 oC/min Nakamura prediction   cPP3 Relative crystallinity, X/XfTemperature, T (oC)  FIGURE A.2: Non isothermal quiescent crystallization prediction of PP3 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8.   144  115 120 125 130 135 1400.00.51.02oC/min5oC/min10oC/min20oC/min     aPP4 Nakamura predictionRelative crystallinity, X/XfTemperature, T (oC)115 120 125 130 135 1400.00.51.02oC/min5oC/min10oC/min20oC/min     bPP4 Nakamura predictionRelative crystallinity, X/XfTemperature, T (oC)115 120 125 130 135 1400.00.51.02oC/min5oC/min10 oC/min20 oC/min     cPP4 Nakamura predictionRelative crystallinity, X/XfTemperature, T (oC)  FIGURE A.3: Non isothermal quiescent crystallization prediction of PP4 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8.     145  105 110 115 120 125 130 1350.00.51.02 oC/min5 oC/min10oC/min20oC/min Nakamura prediction   aPP6  Relative crystallinity, X/XfTemperature, T (oC)105 110 115 120 125 130 1350.00.51.02oC/min5 oC/min10 oC/min20 oC/min Nakamura prediction   bPP6  Relative crystallinity, X/XfTemperature, T (oC)  105 110 115 120 125 130 1350.00.51.02oC/min5 oC/min10 oC/min20 oC/min Nakamura prediction   cPP6  Relative crystallinity, X/XfTemperature, T (oC)  FIGURE A.4: Non isothermal quiescent crystallization prediction of PP6 with n=3 and (a) Arrhenius (b) WLF (c) Hoffman theory estimation for the rate constant with temperature correction for high cooling rates. The non-isothermal induction temperatures are estimated from Equation 3.8.    146  : Analysing the crystal growth: definition of radius and center of spherulites The center of a given crystal ( CX , CY ) can be determined by specifying three points1 on the circumference. As shown in FIGURE B.1 these three points form a unique triangle with three perpendicular bisectors. The intersection of these bisectors defines the crystal center. The mathematical equations to obtain the radius and the coordinates of the center of the spherulite are easily obtained from simple geometrical equations and relations: ( ) ( ) ( ) ( )2 1 1 0 1 0 2 1X X Y Y X X Y Yδ = − × − − − × −   B.1 ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 22 1 0 0 0 2 1 1 1 0 2 22CY Y X Y Y Y X Y Y Y X YX δ − × + + − × + + − × + = ×   B.2 ( ) ( ) ( ) ( ) ( ) ( )2 2 2 2 2 22 1 0 0 0 2 1 1 1 0 2 22CX X X Y X X X Y X X X YY δ − − × + + − × + + − × + = ×  B.3 ( ) ( )2 2C C i C iR X X Y Y = − + −    B.4                             Where Xi and Yi are the coordinates of any of the three points on the edge of crystal. When user clicks on the image, the written C# code defines the coordination of mouse pointer as the                                                  1  ( ) ( ) ( )0 0 1 1 2 2, , , , ,X Y X Y and X Y   147  points on the circumference. It calculates the center and radius of circle (spherulite) using Equations B.1-B.4. In order to verify that the obtained circle fits the shape of the crystal, a red circle is sketched on top of the crystal as indicated in FIGURE B.1. Since the microscopic technique used with the Anton Paar MCR-502 has 20X magnification and the CCD camera used has 1.4 MP resolution, each pixel corresponds to 0.3225 µm in length. Thus, the calculated radius is converted to microns by multiplying the radius of circle obtained in pixels by 0.3225 µm/pixel.  Using this procedure at least three crystals is analysed as a function of time. FIGURE B.2 summarises the radius of crystals as a function of time at the temperature of 145oC for the PP3 sample. As shown, the data for all three analyzed crystals define a single line with the slope defined as the growth rate (µm/s).    FIGURE B.1: Circle fitted to a crystal and bisectors used to calculate the center of spherulites.  148  This procedure was repeated for most resins amenable to microscopy analysis. The analysis has been performed for five different temperatures in order to study the effect of quiescent crystallization temperature on the growth rate of crystals formed within the bulk of the polymer. 5.0x104 1.0x105 1.5x10550100150200250Crystal Radius, R (µm)Time, t (s)              PP3           T=145oC  Crystal radiusSlope=0.023 (µm/s) FIGURE B.2: Crystal radius as a function of time for PP3. Experimental data (three different crystals) were superimposed on a single line with the slope equals to the crystal growth rate.    149   : Determination of heat rise within polymer under microscopy and corresponding data corrections The coupled energy equations for the system shown in FIGURE C.1, where the polymer in the space between two quartz plates is crystallized are developed. This is necessary to calculate the increase of temperature in the optical microscopy results due to the heat of crystallization which is accumulated in the sample. This has an effect on the degree of crystallinity which needs to be quantified in order to explain the differences between the POM and DSC results with reference to FIGURE 4.12a (see discussion in section 3.2). Crystal formation releases heat within the polymer at a rate:  ( ) 100%fq H ρ φ= ∆ × ×                                (C.1) where ( ) 100%fH∆  is the heat of melting if the polymer was 100% crystalline, 𝜌𝜌 is the density of crystals, and φ  is the rate of volume fraction increase of crystals. The heat transfer equations for the polymer and the quartz plate located on top of the polymer (FIGURE C.1) are:        ,  ,  11polymer p polymer polymer polymerquartz p quartz quartz quartzT T TC k r k qt r r r z zT T TC k r kt r r r z zρρ∂ ∂ ∂ ∂ ∂   = + +   ∂ ∂ ∂ ∂ ∂   ∂ ∂ ∂ ∂ ∂   = +   ∂ ∂ ∂ ∂ ∂                            (C.2)  150  The equations are solved using POLYFLOW with the following boundary conditions. Convection boundary condition is imposed on the plate in contact with the nitrogen gas which is purged into the oven chamber of rheometer as the heating/cooling gas. The heat transfer coefficient used in these calculations was h = 100W/(m2•K). The quartz plate underneath the polymer imposes an adiabatic boundary condition at the lower side of the polymer. Continuity of T and heat rate at the polymer/quartz interface completes the required BC.s. N2 CirculationN2 CirculationQuartz D = 43mmPolymerHoodInsulated0.5mm6.4mmHT-Glass plate FIGURE C.1: The schematic of quartz parallel plate system used in this study. After solving the system of equations (C.2), the temperature elevation within the polymer sample can be obtained as a function of T. The temperature rise as a function of time at r = 0.75R and z = 0.25mm is shown in FIGURE C.2. The temperature rise in the polymer matrix decreased as the temperature is increased. At higher temperatures the kinetics of crystallization slows down and thus less heat is released per unit of time. This essentially explains the larger differences between OM and DSC results depicted in FIGURE 4.12a at the lower temperatures (section 4.3).   151  100 101 102 103120122124126128130132134  T0= 121.7oC   T0= 126.6oC   T0= 131.4oCTemperature, T (oC)Time, t (s)PP5 FIGURE C.2: The temperature rise at r/R = 0.75 and z/h = 0.25 in a polymer sample that crystallizes at different temperatures. The temperature increases of 0.2 oC to 1.9oC at the crystallization temperatures of 131.4oC to 121.7oC respectively explain adequately the differences in FIGURE 4.12a.  To correct the obtained data shown in FIGURE 4.12a, The Avrami equation is used (Eq. 3.1). The Arrhenius equation (Eq. 3.2) is implemented here to represent the dependency of the Avrami rate parameter on temperature as shown in FIGURE C.3. The Arrhenius equation is then used in combination with the Avrami equation to predict the behavior of the sample in the microscopy set up.   152  -7x10-5 -6x10-5 -5x10-5 -4x10-5 -3x10-5 -2x10-5 -1x10-5 00.00.20.40.60.81.0       PP5 Arrhenius fitEa = 140.8 Kcal/molT0 = 394.85 Kk(T0) = 4E-5 K/K(T0)1/T-1/T0  FIGURE C.3: Fit of the Arrhenius equation to the Avrami rate parameter k using Avrami index of 2. The value of Avrami rate parameter obtained using the temperature history in FIGURE C.2 along with Arrhenius equation is used in the Avrami equation. The prediction of the Avrami model is shown in FIGURE 4.12b and compared with the experimental data from the microscopy setup. The agreement was found to be excellent which shows that the difference in FIGURE 4.12a is related to different thermal histories during crystallization.  

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