Extrudate Swell of High Density Polyethylene in Capillary and Slit Dies by Vinod Kumar Konaganti B.Tech., Jawaharlal Nehru Technological University (JNTU), India, 2007 M.Eng., Indian Institute of Science (IISc), India, 2009 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) December 2016 © Vinod Kumar Konaganti, 2016 ii Abstract The analysis of extrudate swell in polymer melts is of great importance in many polymer processing operations and has been the subject of interest both experimentally and numerically. The main objectives of this research work are to obtain systematic and reliable extrudate swell data of a high molecular weight HDPE, to identify a suitable constitutive model that can precisely represent extrudate swell phenomena and to predict extrudate swell accurately under various processing and operating conditions. A novel extrudate swell measuring system with an online data acquisition system is designed for the present work. This system allows one to measure extrudate swell profile under different conditions such as steady state or transient, gravity free, isothermal and non-isothermal conditions. Further, the set-up is suitable for both capillary and slit extrudates. A comprehensive analysis on the applicability and validity of various rheological (integral and differential/molecular) models in describing extrudate swell of a highly viscoelastic HDPE polymer over a broad range of shear rates (5 to 100s-1) is carried out using FEM based ANSYS POLYFLOW®. The simulation results indicated that the integral constitutive equations of K-BKZ type can account for the significant memory effects of viscoelastic polymer melts such as HDPE. Overprediction of extrudate swell by the integral K-BKZ model invoked the importance of obtaining non-linear viscoelastic properties for a broader range of deformations/deformation rates. The newly available CPP fixture from AntonPaar is used to procure such non-linear viscoelastic data and thus to determine the accurate damping function. The simulation results of extrudate swell in capillary and slit dies are in good agreement with the experimental measurements using the newly determined damping function. In addition, non-isothermal extrudate swell of the HDPE polymer is studied using the pseudo-time integral K-BKZ Wagner (i.e., the non-isothermal form) model with the differential Nakamura equation for the crystallization kinetics. The model is implemented in ANSYS POLYFLOW®. Extrudate swell measurements are obtained by extruding the polymer melt at 200ºC through long capillary and slit dies to ambient air at 25ºC and 110ºC. The numerical results are found to be in very good agreement with the experimental observations. iii Preface This thesis entitled “Extrudate swell of high density polyethylene in capillary and slit dies” presents the research the author performed during his PhD study under the supervision of Professor Savvas G. Hatzikiriakos. The extrudate swell phenomenon of a high molecular weight high density polyethylene has been extensively studied both experimentally and numerically. The results presented in Chapters 4, 5 and 6 are summarized in seven manuscripts published or to be submitted for publication: 1. Konaganti, V. K., Derakhshandeh, M., Ebrahimi, M., Mitsoulis E., and Hatzikiriakos, S. G. (2016). Non-isothermal extrudate swell. Physics of Fluids, 28, 123101-1 - 123101-18. 2. Konaganti, V. K., Ansari, M., Mitsoulis, E., and Hatzikiriakos, S. G. (2016). The effect of damping function on extrudate swell. Journal of Non-Newtonian Fluid Mechanics, 236, 73-82. 3. Konaganti, V. K., Behzadfar, E., Ansari, M., Mitsoulis, E., and Hatzikiriakos, S. G. (2016). Extrudate Swell of High Density Polyethylenes in Slit (Flat) Dies. International Polymer Processing, 31(2), 262-272. 4. Konaganti V.K., Behzadfar E., Hatzikiriakos S.G., Atsbha H., Karsch U., and Grützner R. (2016). Study on extrudate swell of high density polyethylenes in slit (flat) dies. Proceedings of the Technical Conference & Exhibition ANTEC® 2016, May 23-25, Indianapolis, IN, USA. 5. Konaganti, V. K., Ansari, M., Mitsoulis, E., and Hatzikiriakos, S. G. (2015). Extrudate swell of a high-density polyethylene melt: II. Modeling using integral and differential constitutive equations. Journal of Non-Newtonian Fluid Mechanics, 225, 94-105. 6. Behzadfar, E., Ansari, M., Konaganti, V. K., and Hatzikiriakos, S. G. (2015). Extrudate swell of HDPE melts: I. Experimental. Journal of Non-Newtonian Fluid Mechanics, 225, 86-93. 7. Konaganti, V.K., Ansari M., Ebrahimi M., Hatzikiriakos S.G., Atsbha H., and Hilgert C. (2015). Extrudate swell of high-density polyethylene using integral and differential iv constitutive equations. Proceedings of the Technical Conference & Exhibition ANTEC® 2015, March 23-25, Orlando, FL, USA. The sixth paper has written mostly by Dr. Ehsan Behzadfar, and some of the experimental set-up details presented in this paper will be used in this thesis. All other manuscripts were written mostly by the author of this PhD, and revised by Prof. Savvas G. Hatzikiriakos (research supervisor) and the other co-authors. v Table of Contents Abstract .......................................................................................................................................... ii Preface ........................................................................................................................................... iii Table of Contents ...........................................................................................................................v List of Tables ................................................................................................................................ ix List of Figures .................................................................................................................................x List of Symbols ......................................................................................................................... xviii List of Abbreviations ................................................................................................................ xxii Acknowledgements .................................................................................................................. xxiii Dedication ................................................................................................................................. xxiv Chapter 1: Introduction ................................................................................................................1 1.1 Literature review ............................................................................................................. 3 1.1.1 Experimental techniques for measuring swell ............................................................ 3 1.1.2 Numerical studies on extrudate swell ......................................................................... 4 1.1.2.1 Isothermal extrudate swell studies ...................................................................... 4 1.1.2.2 Non-isothermal extrudate swell studies .............................................................. 6 1.2 Summary/concerns .......................................................................................................... 8 1.3 Research objectives ......................................................................................................... 8 1.4 Thesis organization ....................................................................................................... 10 Chapter 2: Material and Experimental Methods .....................................................................11 2.1 Material ......................................................................................................................... 11 2.2 Rotational rheometer ..................................................................................................... 11 2.2.1 Parallel plate (PP) geometry ..................................................................................... 12 vi 2.2.2 Cone and plate (CP) geometry .................................................................................. 13 2.2.3 Cone partitioned plate (CPP) geometry .................................................................... 14 2.2.4 SER (Sentmanat Extension Rheometer) universal testing platform ......................... 15 2.3 Capillary rheometer ...................................................................................................... 16 2.4 Extrudate swell measuring system ................................................................................ 18 2.5 Differential scanning calorimetry (DSC) ...................................................................... 20 Chapter 3: Extrudate Swell of HDPE: Modeling using Integral and Differential Constitutive Equations ................................................................................................................22 3.1 Mathematical modeling ................................................................................................ 22 3.1.1 Governing equations ................................................................................................. 22 3.1.2 Constitutive equations ............................................................................................... 23 3.1.3 Boundary conditions ................................................................................................. 26 3.1.4 Method of solution .................................................................................................... 29 3.2 Results and discussion .................................................................................................. 32 3.2.1 Rheological characterization ..................................................................................... 32 3.2.2 Flow pressure drop comparison ................................................................................ 37 3.2.3 Extrudate swell simulations ...................................................................................... 37 3.2.3.1 Integral versus differential constitutive models ................................................ 38 3.2.3.2 Swell profiles comparison................................................................................. 40 3.2.3.3 Pressure and stress profiles comparison ........................................................... 41 3.2.3.4 The effect of die length ..................................................................................... 42 3.2.3.5 The effect of temperature .................................................................................. 45 3.3 Conclusions ................................................................................................................... 46 vii Chapter 4: Extrudate Swell – The Effect of Damping Function .............................................48 4.1 Rheological characterization ......................................................................................... 49 4.1.1 Small-amplitude oscillatory shear (SAOS) ............................................................... 49 4.1.2 Step-strain stress relaxation ...................................................................................... 49 4.1.3 Steady shear .............................................................................................................. 51 4.1.4 Uniaxial extension .................................................................................................... 53 4.2 Extrudate swell simulations .......................................................................................... 54 4.2.1 Contour plots ............................................................................................................. 54 4.2.2 Damping function sensitivity analysis ...................................................................... 55 4.2.2.1 Extrudate swell ratio dependence ..................................................................... 55 4.2.2.2 Extrudate swell profile variation ....................................................................... 57 4.2.2.3 Stress profiles comparison ................................................................................ 59 4.2.3 Comparison with experimental measurements – Capillary dies ............................... 61 4.2.3.1 Effect of die length (Length/Diameter ratio) .................................................... 61 4.2.3.2 Extrudate swell profiles .................................................................................... 62 4.2.3.3 Effect of extrusion temperature ........................................................................ 63 4.2.4 Comparison with experimental measurements – Slit dies ........................................ 64 4.2.4.1 Effect of width to die gap (W/H) ...................................................................... 64 4.2.4.2 Extrudate swell profiles .................................................................................... 65 4.3 Conclusions ................................................................................................................... 66 Chapter 5: Non-Isothermal Extrudate Swell of HDPE Melt including Crystallization Kinetics..........................................................................................................................................68 5.1 Mathematical modeling ................................................................................................ 68 viii 5.1.1 Crystallization kinetics.............................................................................................. 68 5.1.2 Governing equations ................................................................................................. 70 5.1.3 Boundary conditions ................................................................................................. 73 5.1.4 Method of solution .................................................................................................... 74 5.2 Results and discussion .................................................................................................. 76 5.2.1 Rheological characterization ..................................................................................... 76 5.2.2 Crystallization kinetics.............................................................................................. 76 5.2.3 Extrudate swell simulations ...................................................................................... 83 5.2.3.1 Contour plots ..................................................................................................... 83 5.2.3.2 Effect of shear rate on quiescent crystallization ............................................... 84 5.2.3.3 Effect of ambient temperature on quiescent crystallization .............................. 87 5.2.3.4 Comparison with experiments .......................................................................... 88 5.3 Conclusions ................................................................................................................... 90 Chapter 6: Conclusions and Recommendations .......................................................................92 6.1 Conclusions ................................................................................................................... 92 6.2 Recommendations for future work ............................................................................... 93 Bibliography .................................................................................................................................95 Appendices ..................................................................................................................................113 Appendix A : Bagley and slip (Mooney analysis) of the capillary data ................................. 113 Appendix B : More details on extrudate swell measuring system .......................................... 116 Appendix C : Evaluation of material functions ...................................................................... 118 Appendix D : Contour plots and stress profiles in long slit dies ............................................ 122 ix List of Tables TABLE 3.1: Relaxation spectrum at temperatures 160ºC, 180ºC, and 200ºC for HDPE melt with K-BKZ model parameters (Wagner and PSM damping functions, θ=0)……….............................34 TABLE 3.2: Relaxation spectrum at temperatures 160oC, 180 oC, and 200oC for HDPE melt with PTT model parameters……………………………………………………………………...……34 TABLE 3.3: The relaxation spectrum along with Giesekus model parameters at 200ºC……..…..36 TABLE 3.4: The relaxation spectrum along with DCPP model parameters at 200ºC………….....36 TABLE 4.1: Relaxation spectrum at temperatures 160ºC, 170ºC, 180ºC, and 200ºC for HDPE melt with K-BKZ Wagner model parameters (n=0.35 and β=0.4)………………………..……………49 TABLE 5.1: Half-times and Avrami indices at the four different crystallization temperatures for the studied HDPE resin…………………………………………………………………………...79 TABLE 5.2: The necessary physical and thermal properties of the present HDPE resin. Also, includes Ziabicki model parameters………………………………………………………..….....79 x List of Figures FIGURE 1.1: Schematic representation of twin sheet extrusion process (first and key step in NGFS process) where the polymer sheets are extruded with a desired thickness profile…………….........2 FIGURE 1.2: Extrudate swell observed in Newtonian fluid (water) (on the left) and Viscoelastic fluid (mineral oil with metallic soaps) (on the right)……………………………………….……....2 FIGURE 2.1: Schematic representation of parallel plate rheometer………………………….......12 FIGURE 2.2: Schematic representation of cone and plate rheometer…………………………….13 FIGURE 2.3: Schematic of cone and partitioned plate (CPP) fixture. The flow is generated by the lower part cone while the resulting torque is measured by the upper partitioned plate..…………..15 FIGURE 2.4: Schematic of SER (Sentmanat Extension Rheometer) testing platform…………...16 FIGURE 2.5: Schematic representation of capillary rheometer………………………..................18 FIGURE 2.6: The Instron capillary rheometer along with the novel extrudate swell measurement set-up placed under the die exit (on the left) and a schematic (on the right) representing the ability of the present set-up to measure both isothermal and non-isothermal measurements………….....19 FIGURE 2.7: Schematic representation of relative crystallinity fraction, α computation.…….…21 FIGURE 3.1: Schematic representation of flow domain for a typical capillary extrudate swell simulation problem along with all the necessary boundary conditions……………………….......28 FIGURE 3.2: A typical finite element grid for the simulation of the 12:1 abrupt circular contraction with L /D = 16, Lext /D = 31.64, D=0.79 mm and 2a = 180º…………………………………….…30 FIGURE 3.3. Computational variable residual with the number of iterations at different apparent shear rates ranging from 5 to 100s-1……………………………………..………………………..31 xi FIGURE 3.4: The master curves of storage (G') and loss (G") moduli and the corresponding fitting using a six-mode Maxwell spectrum at the reference temperature of Tref = 200oC…......................33 FIGURE 3.5: Experimental data (symbols) and model fits (lines) of the shear stress growth coefficient, ηS+, using both integral (K-BKZ Wagner and K-BKZ PSM) and differential (PTT, Giesekus and DCPP) constitutive models……………………………………..………………….35 FIGURE 3.6: Experimental data (symbols) and model fits (lines) of uniaxial stress growth coefficient, ηE+, using both integral (K-BKZ/Wagner and K-BKZ/PSM) and differential (PTT, Giesekus and DCPP) constitutive models……………………………..………………………….35 FIGURE 3.7: Experimental data (solid symbols) and model predictions of the total pressure drop for flow in a capillary having L/D=16 and D=0.79 mm, using the K-BKZ Wagner, K-BKZ PSM, PTT, Giesekus, DCPP models for the HDPE polymer melt at 200ºC………………………….…37 FIGURE 3.8: Experimental data (solid symbols) and model predictions of extrudate swell ratios predicted by using integral K-BKZ models with Wagner and PSM damping functions and differential PTT, Giesekus and DCPP models with reservoir…………………………………….39 FIGURE 3.9: Experimental data (solid symbols) and model predictions of extrudate swell ratios predicted by using K-BKZ models with Wagner and PSM damping functions and the PTT model with and without reservoir…………………………………………………………………….….39 FIGURES 3.10: Comparison of experimental extrudate profiles with simulations for the capillary die having L/D=16, D=0.79 mm at (a) a low shear rate, 5 s-1 and (b) a high shear rate, 100s-1….....40 FIGURE 3.11: Comparison of (a) Pressure, P (b) Shear stress, τ12 (c) First normal stress differences, N1,W along the wall and free-surface and (d) First normal stress differences, N1,sym, along the axis of symmetry, using integral K-BKZ PSM and differential PTT models, at an apparent shear rate, γ̇A= 26 s-1……………………………………………….....…………...........41 xii FIGURE 3.12: Extrudate swell simulations using the K-BKZ PSM model for three different dies with L/D=5, 16, and 33 and their comparison with experimental results (a) including the reservoir in the flow simulation (b) excluding the reservoir in the flow simulation………………………...43 FIGURE 3.13: Extrudate swell simulations using the 6-mode PTT model for three different dies with L/D=5, 16, and 33 and their comparison with experimental results. Similar simulation results with or without the reservoir………………………………………………….…………………..43 FIGURE 3.14: Variation of extrudate swell with length to diameter ratio, L/D, using integral K-BKZ PSM model with reservoir at three different shear rates 11, 26 and 64s-1…………..…..…...44 FIGURE 3.15: Variation of extrudate swell with temperature simulated using (a) K-BKZ PSM and (b) PTT models compared with experimental data (solid symbols)………….………...…………45 FIGURE 4.1: Step-strain relaxation data after imposition of sudden step strains of different levels of the present HDPE melt at 200ºC……………………………………………………………….49 FIGURE 4.2: The resulting damping function obtained from step stain relaxation experiments plotted in FIGURE 4.1 (solid symbols) along with the Wagner model predictions using n=0.35 (best fit)………………………………………..………………………………………………....50 FIGURE 4.3: Experimental data (symbols) and the K-BKZ Wagner model predictions (lines) of the shear stress growth coefficient, ηS+……………………………………………………………51 FIGURE 4.4: The slip corrected flow curve of the HDPE at 200oC compared with the LVE data along with the steady-state values of start-up of shear experiments. Good agreement is shown, demonstrating the validity of the Cox-Merz rule. The plot also includes the K-BKZ Wagner model predictions for three different values of n=0.2, 0.35 and 0.5………………………..…………….51 FIGURE 4.5: Experimental data (symbols) and the K-BKZ Wagner model predictions (lines) for three different values of β=0.2, 0.4 and 0.9 of the uniaxial stress growth coefficient,ηE+. The value of 0.4 is the optimum value that describes the experimental data well……………...…………….52 xiii FIGURE 4.6: Contours of field variables obtained using the integral K-BKZ Wagner model at an apparent shear rate of 26s-1: (a) Axial velocity, Uz, (b) Pressure, P, (c) Shear stress, τrz, (d) First normal stress difference, N1………………………………………………………………..……..54 FIGURE 4.7: The dependence of extrudate swell ratio predicted using the integral K-BKZ model (Wagner damping function) on the Wagner exponent n at different apparent shear rates ranging from 5 to 100s-1 and using the optimum value of Wagner parameter, β=0.4……………...………55 FIGURE 4.8. The dependence of extrudate swell ratio predicted using the integral K-BKZ model (Wagner damping function) on the Wagner parameter β at different apparent shear rates ranging from 5 to 100s-1 and using the optimum value of Wagner exponent, n=0.35…………………..…56 FIGURE 4.9: Comparison of simulated swell profiles for the capillary die having L/D=16 and D=0.79mm at a moderate apparent shear rate of 26s-1 for the Wagner exponent n ranging from 0.15 to 0.5 and using the optimum value of Wagner parameter, β=0.4. Solid symbols represent the experimental data in agreement with the predictions for the optimum value of n=0.35…………..57 FIGURE 4.10 Comparison of simulated swell profiles for the capillary die having L/D=16 and D=0.79 mm at a moderate apparent shear rate of 26s-1 for the Wagner parameter β ranging from 0.2 to 0.9 and using the optimum value of Wagner exponent, n=0.35. Solid symbols represent the experimental data in agreement with the predictions for the optimum value β=0.4………………58 FIGURE 4.11 Comparison of (a) Pressure P (b) Shear stress τ12 (c) First normal stress differences N1,W along the wall and free-surface and (d) First normal stress differences N1,sym, along the axis of symmetry, using the integral K-BKZ model with three different Wagner exponents n=0.2, 0.35 and 0.5 and using Wagner parameter β=0.4, at an apparent shear rate, γ̇A=26s-1…........................59 FIGURE 4.12: Extrudate swell simulations (solid lines) with the presence of the reservoir using integral K-BKZ Wagner model with n = 0.35 and β=0.4, for three different dies with L/D=5, 16, and 33 and their comparison with experimental results (solid symbols)………………………….61 xiv FIGURE 4.13: Comparison of experimental extrudate profiles (solid symbols) with simulations (solid lines) for the capillary die having L/D=16, D=0.79 mm at shear rates, 5, 26 and 100s-1……62 FIGURE 4.14: Variation of extrudate swell with temperature: simulation results using integral K-BKZ Wagner model (lines) compared with experimental measurements (solid symbols) for a capillary die having L/D=16 and D=0.79 mm at shear rates, 11, 26 and 64s-1………….…………63 FIGURE 4.15: Extrudate swell simulations (lines) using the K-BKZ Wagner model with n= 0.35 and β=0.4, for two different dies with W/H=18 and 36 (L/H=40, W=18mm) and their comparison with experimental results (solid symbols)……………………………………………….……….64 FIGURE 4.16: Comparison of experimental thickness swell (BT) profiles (solid symbols) with simulations (solid lines) for the slit die L/H=40, W/H=18, and H=1 mm at low and high shear rates, 5 and 100s-1, respectively……………………………………………...…………………………65 FIGURE 5.1: Relevant boundary conditions of non-isothermal extrusion flow through a long capillary die……………………………………………………..……………………………......73 FIGURE 5.2: The DSC thermograph of the HDPE polymer of present study. The melting, Tm and cooling, Tc peaks were observed at 132.3ºC and 113.5ºC, respectively. The onset of crystallization temperature, Tonset=118.6ºC is also marked in the plot……………………………..……………..75 FIGURE 5.3: Isothermal crystallization data for HDPE used in this study at different crystallization temperatures…………………………………………………..……………….....77 FIGURE 5.4: Avrami analysis of HDE isothermal crystallinity data……………..………………77 FIGURE 5.5: Haffman-Lauritzen fit of isothermal crystallization data……………..……………80 FIGURE 5.6: Inverse half time data extrapolated using Haffman-Lauritzen theory along and the corresponding Ziabicki model fit………………………………………………………….....…..80 xv FIGURE 5.7: Non-isothermal crystallization data obtained at various cooling rates and the corresponding model fittings using the differential form of Nakamura model………..………….81 FIGURE 5.8: Contours of field/flow variables obtained flow simulations using integral K-BKZ Wagner model with Nakamura crystallization kinetics at an apparent shear rate of 26s-1 and ambient air at 70ºC (a) Axial velocity, Uz, (b) Pressure, P, (c) Shear stress, τrz, (d) First normal stress difference, N1 (e) Temperature, T, and (f) relative crystallinity, α………………..……...…83 FIGURE 5.9: The flow curves of the present HDPE polymer at different temperatures shifted by using TTS using the shift factors determined from LVE measurements (T0=200°C). The solid line represents the flow curve corresponding to LVE measurements…………………………….…...84 FIGURE 5.10: Contour plots of temperature in non-isothermal extrudate swell simulations at three apparent shear rates 5, 26, and 100s-1 where the polymer melt at 200ºC is extruded through a long capillary die to ambient air at 70ºC……………………………………………………….………85 FIGURE 5.11: Relative crystallinity profiles along the free surface boundary at apparent shear rates 5, 26, and 100s-1 obtained from non-isothermal swell simulations where the polymer melt at 200ºC is extruded through a long capillary die to ambient air at 70ºC……………………..……...86 FIGURE 5.12: The contour plots of relative crystallinity fraction, α over the extrudate at different ambient temperatures from 40-100ºC obtained at an apparent shear rate of 26s-1 using a long capillary die of L/D=33, D=0.79mm……………………………………………………………..87 FIGURE 5.13: The extrudate swell as a function of apparent shear rate when the polymer melt at 200°C extruded to two different ambient temperatures 110ºC and 25°C using (a) a long capillary die of L/D=33, D=0.79mm and (b) a slit die long slit die of L/H=40, W/H=18, H=1mm. Solid symbols represent the experimental data and solid lines are corresponding simulation predictions……………………………………………………………………………………..…88 FIGURE 5.14: Non-isothermal extrudate swell profile data compared with simulation results using the integral K-BKZ (Wagner) model with Nakamura crystallization kinetics at an apparent shear rate of 26s-1 in long (a) Capillary and (b) Slit dies……………………………………..………....89 xvi FIGURE A.1: Bagley plot of the HDPE capillary data at 200ºC……..…………………..……..112 FIGURE A.2: The Bagley-corrected flow curves using three different capillary dies of diameters D=0.43, 0.79, and 2.11 mm (with L/D=16) along with the LVE plotted as a flow curve (no-slip). The diameter dependence of the flow curves implies the presence of slip, and the Mooney method can be used to determine slip as a function of the wall shear stress…………………………..….113 FIGURE A.3: The slip velocity (Vs) as a function of wall shear stress (τw) for the HDPE melt at 200oC. The continuous line is a fit to the experimental data indicating a slope of 3.4…...............114 FIGURE B.1: Keyence® LS-7030Moptical micrometers used for extrudate swell measurements………………………………………………………………………..………….115 FIGURE B.2: Monitor image depicting tilt correction using Keyence® LS-7030M optical micrometer………………………………………………………………………………..…….116 FIGURE B.3. Set-up to hold and move optical micrometers for reliable extrudate swell data.….116 FIGURE C.1: Variation of (a) shear viscosity, η and (b) first normal stress difference coefficient ψ1 with shear rate predicted by using the K-BKZ constitutive model with Wagner and PSM damping functions and the PTT constitutive model………………………………..……………117 FIGURE C.2: Predictions of several material functions of the HDPE polyethylene melt at 200oC using the K-BKZ models with Wagner and PSM damping functions and the multi-mode PTT model (a) shear stress growth coefficient, ηS+(b) shear stress decay coefficient, ηS- (c) first normal stress difference coefficient, ψ1+ and (d) first normal stress difference decay coefficient, ψ1- …...118 FIGURE C.3: Predictions of (a) steady recoverable shear strain, γr,s and (b) steady recoverable elongational strain, εr,s using the K-BKZ model with Wagner and PSM damping functions and the multi-mode PTT model…………………………………………………………………...…….120 xvii FIGURE D.1: Contours of filed variables obtained using the integral K-BKZ Wagner model at an apparent shear rate of 5 s-1: (a) Stream function, STF, (b) Axial velocity, Uy, (c) Pressure, P, (d) Shear stress, τrz, (e) First normal stress difference, N1…………………………………………...122 FIGURE D.2: Comparison of (a) Thickness swell, BT (b) Pressure, P (c) shear stress, τyz (d) First normal stress differences, N1 profiles, along the wall and free-surface predicted using integral K-BKZ Wagner model at three different apparent shear rates 5, 26 and 100s-1………………...123 xviii List of Symbols 2a entrance angle b slip exponent of Navier’s slip law aT time-temperature shift factor d diameter of the extrudate t thickness of the extrudate from slit flow h distance between parallel disks hT local heat transfer coefficient gi relaxation moduli h(γ) damping function ∆Hf heat of fusion k thermal conductivity m Avrami index n Wagner exponent ?̂? unit outward normal vector p pressure Δp applied pressure or pressure drop q heat flow qn number of arms of pom-pom t observer’s time t1/2 half time of crystallization (1/t1/2)0 intercept of Hoffman-Lauritzen fit u velocity vector A parameter of non-sphericity BD extrudate swell ratio in capillary die BT thickness swell ratio in slit die C Hoffman-Lauritzen theory constant CP heat capacity Ct Cauchy-Green tensor Ct-1 Finger strain tensor xix D diameter of the die Db diameter of the reservoir (barrel) D rate of deformation tensor Ea activation energy F normal force Ft deformation gradient tensor Gi shear modulus in DCPP model G* complex modulus G'(ω) storage modulus G"(ω) loss modulus G(t) relaxation modulus H thickness of a slit die IC first invariant of Ct IC-1 first invariant of Ct-1 I unit tensor Kmax kinetic parameter at the maximum crystal growth rate KN crystallization kinetic parameter KN(T) temperature dependent crystallization rate parameter KN(fa) flow dependent crystallization rate parameter L length of the die Lext extrudate length Lt velocity gradient tensor M torque MW Weight average molecular weight N number of relaxation modes N1 (≡τ11-τ22) first normal stress difference N2 (≡τ22-τ33) second normal stress difference N1,w first normal stress difference along the wall and free-surface N1,sym first normal stress difference along the axis of symmetry P pressure xx Q volumetric flow rate R universal gas constant ?̇? quiescent crystallization rate α relative crystallinity fraction αi material constant of Giesekus model for each mode αp shear parameter of PSM damping function β extensional parameter of Wagner or PSM damping function βsl slip coefficient of Navier’s slip law τ12 shear stress τW wall shear stress τrZ or τZ shear stress τa apparent or Newtonian shear stress τi individual contribution of extra stress tensor τ extra stress tensor γ shear strain ?̇? shear rate ?̇?𝐴 apparent shear rate εH Hencky strain rate ξi non-dimensional shear material parameters of PTT model εi non-dimensional elongational material parameters of PTT model ζi nonlinear material constants for each model in DCPP model λi relaxation time ?̅? mean relaxation time λsi relaxation time related to the stretching mechanism of DCPP model λci relaxation time of the crystallizing polymer melt η0 zero shear viscosity ηi partial viscosity ηci partial viscosity of the crystallizing polymer melt ηs+ shear stress growth coefficient ηs- shear stress decay coefficient xxi ηE+ uniaxial stress growth coefficient ηE- uniaxial stress decay coefficient 1+ first normal stress difference coefficient 1− first normal stress difference coefficient Λ stretching scalar ω angular frequency ρ density ξ elapsed time xxii List of Abbreviations ALE Arbitrary Lagrangian-Eulerian CCD Charge Coupled Camera CLIPS C Language Integrated Programming System CP Cone and Plate CPP Cone Partitioned Plate DSC Differential Scanning Calorimetry DEVSS Discontinuous Elastic-Viscous Stress Split DCPP Double Convected Pom-Pom EVSS Elastic-Viscous Stress Split XPP Extended Pom-Pom FEM Finite Element Method HDPE High Density Polyethylene HWNP High Weissenberg Number Problem K-BKZ Kaye-Bernstein-Kearsley-Zapas LLDPE Linear Low-Density Polyethylene LVE Linear Viscoelastic Elasticity LDPE Low-Density Polyethylene NGFS Next Generation Fuel System PSM Papanastasiou-Scriven-Macosko PP Parallel Plate PTT Phan-Thien-Tanner SER Sentmanat Extensional Rheometer STA Simultaneous Thermal Analyzer SAOS Small Amplitude Oscillatory Shear SFEM Streamline Finite Element Method SUPG Streamline-Upwind/Petrov-Galerkin TTS Time-Temperature superposition xxiii Acknowledgements I would like take this opportunity to thank all the people who have supported me with this project over the past few years. Firstly, I would like to express sincere gratitude and appreciation to my supervisor Prof. Savvas G. Hatzikiriakos for his advice, assistance and encouragement throughout my PhD research. His constant support and thoughtful insight helped me to address several challenges I faced during this research work. I would also like to thank Prof. Evan Mitsoulis for his constant support, valuable suggestions and comments throughout this project. I would also like to thank my colleagues Dr. Mahmoud Ansari, Dr. Ehsan Behzadfar and Ms. Marzieh Ebrahimi from Polymer Processing and Rheology Lab at UBC for giving me valuable experimental data at times and for having some thoughtful discussions on flow modeling and experimental analysis. Financial assistance from NSERC (Automotive Partnership Canada), the federal funding agency of Canada, is greatly acknowledged. xxiv Dedication To my parents1 Chapter 1: Introduction The relative increase in the dimensions of extrudate compared to those of the die due to the sudden release of stresses upon exiting the die is known as extrudate swell. This phenomenon plays a crucial role in several polymer processing operations (Mitsoulis 2010) such as blow molding, fiber spinning, film blowing, and profile extrusion. This project is related to the optimization of a novel manufacturing process referred to as Next Generation Fuel System (NGFS) developed by Kautex Textron GmbH & Co. One of the main steps of this process involves the sheet extrusion of two polymer sheets (refer FIGURE 1.1). Consequently, all the necessary fuel system components like pumps, valves and fuel gauges are held between these two sheets before the two sheets are welded together to form the fuel tank. With the help of the cavity tools on both sides of the sheets these parts are incorporated into the fuel tank in a simple step. Thus, the main challenge in this process is to control the thickness profile/distribution of the extruded sheets as it governs the thickness uniformity and thus the mechanical strength of the final product. In this thesis, the axisymmetric and flat sheet extrudate swell of a highly viscoelastic polymer melt such as High Density Polyethylene (HDPE) is studied in detail in order to devise a model that can be used for the optimization of NGFS process. Extrudate swell in polymers occurs when long, entangled and coiled chains of molecules are forced to flow through a channel which converges from broad entrance to a narrow die, as usually happens in extrusion processes. Due to the “memory effect”, the polymer melt/fluid attempts to revert back to the original, un-oriented state as it exits the die. Such expansion of the extrudate is called extrudate swell. FIGURE 1.2 depicts the significant difference between extrudate swell observed for a viscoelastic fluid to that of Newtonian fluid. For capillary dies, the extrudate swell is defined as BD=d/D, where D is the die diameter and d is the diameter of the extrudate. In slit dies, the thickness swell is defined as BT=t/H, where H is thickness of a slit die and t is the thickness of the extrudate. This depends on the location of its measurement with a tendency of increasing in the downstream direction (Garcia-Rejon and Dealy 1982). In addition, extrudate swell in polymer melts depends on many parameters such as the past deformation history (memory effects), die design characteristics, viscoelastic properties of the material (extended) and process conditions (Tanner 1970, 1973, 2000, 2005; Koopmans 1988; Mitsoulis 2010). 2 FIGURE 1.1: Schematic representation of twin sheet extrusion process (first and key step in NGFS process) where the polymer sheets are extruded with a desired thickness profile in order to produce a fuel tank of uniform thickness when they are expanded to form the tank FIGURE 1.2: Extrudate swell observed in Newtonian fluid (water) (on the left) and Viscoelastic fluid (polystyrene standard) (on the right) d 3 Early works on extrudate swell of polymer fluids in capillary and slit dies were presented in the 1970s (Tanner, 1970; Graessley et al., 1970; Vlachopoulos et al., 1972; Utracki et al., 1975; Han, 1973; Vlachopoulos, 1977; Huang and White, 1979), where the phenomenon was studied both experimentally and theoretically. The early numerical studies on the extrudate swell problem were mainly concerned with Newtonian fluids (Tanner 1973; Nickell et al. 1974). The advent of advanced numerical techniques and constitutive models made extrudate swell simulation possible for viscoelastic fluids, such as polymer melts and/or solutions (Crochet and Keunings 1982; Luo and Tanner 1986a, 1986b; Kiriakidis and Mitsoulis 1993). Although the extrudate swell mechanism has been studied extensively both experimentally and theoretically, it is not fully understood (Ansari et al. 2013; Ganvir et al. 2011; Kalyon et al. 1980; Béreaux et al. 2012; Konaganti et al. 2015). There are no studies to show quantitative agreement between experimental observations and flow model simulations. Such a model would be invaluable in optimization of this complex process. One of the main objectives of this work is to identify a constitutive model that can quantitatively predict the extrudate swell of the present HDPE resin over a wide range of parameters discussed above. 1.1 Literature review As mentioned before, extrudate swell in polymer melts or solutions has been the subject of interest both experimentally and numerically for many researchers. Some of the important studies are reviewed below. 1.1.1 Experimental techniques for measuring swell Although extrudate swell (which is significant under sufficiently high levels of stress) is usually considered as an instability for extrudates (Denn 2001; Georgiou 2003; Georgiou and Crochet 1994), it can be controlled to manufacture or produce products with desired dimensions and thus precise swell measurement is essential in polymer processing applications. However, accurate measurement of extrudate swell dimensions is a challenging task due to its sensitivity to processing, operating and measuring conditions. One of the early methods of extrudate swell measurement in blow molding is the use of a pinch-off method combined with experimental photography where the extrusion rate, weight and 4 size of the solidified extrudate are very important for calculating the time-dependent swell. The inefficiency of this method is very apparent as it involves shrinkage of polymer extrudate upon direct extrusion to ambient air, thereby it does not consider the free relaxation and swelling of the extrudate (Kalyon et al. 1980; Dutta and Ryan 1982). Other methods include infrared transmission (Glascock 2007), filming method (Huang and Li 2006), charge coupled camera (CCD) (Maziers and Pestiaux 2009), and optical techniques (Swan et al. 1996; Béreaux et al. 2012), which also suffer from several setbacks including inability of detecting transparent objects (i.e., for many polymer melts), flat extrudates, transient measurements and complete determination of the extrudate profile. The effects of gravity (i.e., sagging) and ambient temperature also need to be controlled for accurate extrudate swell measurements. In addition, the measurement system should be noninvasive and capable of operating at high temperatures (i.e., in the range of polymer process temperatures). 1.1.2 Numerical studies on extrudate swell Many researchers have attempted to predict the extrudate swell behavior in polymer melts or solutions using various constitutive equations including integral, differential, and molecular models and compared the results with experiments (Reddy and Tanner 1978; Luo and Tanner 1986a, 1986b, 1988; Goublomme et al. 1992; Housiadas and Tsamopoulos 2000; Béraudo 1998; Russo and Phillips 2010). The major difficulty in accurately assessing the extrudate swell in polymer melts is the choice of a constitutive equation that is appropriate for a wide range of deformations and deformation rates, both in shear and elongation. 1.1.2.1 Isothermal extrudate swell studies There is a plethora of numerical studies reported on the simulation of extrudate swell of polymer melts or solutions. Interestingly, the majority of these studies focus on steady state isothermal swell/flow simulations and the numerical results are subsequently compared with experimental observations. Most of the early computational works study the extrudate swell of highly viscoelastic polymer melts, such as linear low-density polyethylene (LLDPE), low-density polyethylene (LDPE) and high-density polyethylene (HDPE) melts, using integral constitutive equations (Luo 5 and Tanner 1986a, 1986b; Luo and Mitsoulis 1989). Luo and Tanner (1986a, 1986b) have simulated the extrudate swell of a highly viscoelastic IUPAC-LDPE melt using the integral K-BKZ model for both long and short capillary dies using the streamline finite element method (SFEM). Luo and Mitsoulis (1990) have later modified this method to decouple the computation of the free surface shape and position from that of velocity and stress fields. This new finite element technique was successful in obtaining reliable numerical results for viscoelastic flow simulations with abrupt contraction that can simulate recirculating regions using integral constitutive equations. Goublomme et al. (1992) and Barakos and Mitsoulis (1995) have further modified different characteristics of Luo and Mitsoulis’s method to address the numerical difficulties associated with SFEM and to account for the effects of upstream contraction and recirculation on extrudate swell at higher shear rates. In all the aforementioned studies, researchers have used integral K-BKZ model with a spectrum of relaxation times and a damping function either of the Wagner (Wagner 1976, 1979) or of the Papanastasiou (Papanastasiou et al. 1983) type. The swell predictions observed using memory integral constitutive equations are comparable with the experimental results for long dies. However, these models largely overpredict the swell in short dies with a length L and a diameter D (0 ≤ L/D ≤ 5), especially at high apparent shear rates (Kiriakidis and Mitsoulis 1993; Barakos and Mitsoulis 1995; Konaganti et al. 2015). Ansari et al. (2013) have also observed that the integral K-BKZ PSM model highly overestimates the extrudate swell of a HDPE polymer melt even after considering a non-zero second normal stress difference and significant amount of slip at the wall. On the other hand, several authors have successfully demonstrated the use of differential constitutive equations such as the PTT, Giesekus and XPP models for the simulation of extrudate swell in polymer melts or solutions (Goublomme et al. 1992; Crochet and Keunings 1982; Crochet 1989; Rajagopalan et al. 1992; Guénette and Fortin 1995; Otsuki et al. 1999; Oishi et al. 2011). Most of the early efforts were focused on the well-known high Weissenberg number problem (HWNP). However, this problem was solved by using numerical schemes best suited for hyperbolic equations (Goublomme and Crochet 1993; Crochet 1989). Subsequently, Elastic-Viscous Stress Split (EVSS) (Rajagopalan et al. 1992) and/or Discontinuous Elastic-Viscous Stress Split (DEVSS) (Guénette and Fortin 1995) formulations on extra-stress tensor were 6 introduced to obtain convergent and numerically accurate solutions. In a recent study, Ganvir et al. (2009, 2011) have used an arbitrary Lagrangian-Eulerian (ALE) based finite element method and simulated transient extrudate swell of LLDPE and HDPE polymer melts. The numerical results showed good agreement with reported experimental results only corresponding to polymer melts of moderate elasticity (swell ratio of up to 1.4). Further, apparent shear rates as high as those used in practical applications could not be reached. Limited studies exist on extrudate swell using both integral and differential rheological models on the same polymer melt of high elasticity, typically used in blow molding applications. Moreover, there is no systematic study of the extrudate swell of viscoelastic polymer melts where flow model predictions are compared with experimental results for various operating conditions (temperature and flow rate/apparent shear rate) relevant to polymer processing and for various geometrical characteristics such as length-to-diameter ratio, L/D (in capillary dies), width-to-thickness ratio, W/H (in slit dies), and the presence of the reservoir. The latter plays a significant role in the numerical predictions and their consistency with experimental results on extrudate swell as this thesis attempts to demonstrate. 1.1.2.2 Non-isothermal extrudate swell studies The effects of temperature are very crucial in many polymer processing applications as the melts start crystallizing upon emergence from the die exits (Tanner 1985). These effects are more significant close to the phase change (transition from the melt to the solid state) where the temperature dependence is extremely high. Controlling the extrudate dimensions through the adjustment of local die temperature, melt temperature and cooling rates has been extensively employed in extrusion processes (Chang and Yang 1994). Given the significance of non-isothermal conditions in extrusion processes to control the dimensions and/or properties of the extrudate, understanding such processes has become the subject of interest both experimentally and numerically (Phuoc and Tanner 1980; Luo and Tanner 1985, 1987; Barakos and Mitsoulis 1996; Patel et al. 1991; Huynh 1998, 2000; Doufas and McHugh 2001; Ziabicki et al. 2004; Cherukupalli et al. 2005; Tanner and Qi 2005; Henrichsen and McHugh 2007; Kolarik and Zatloukal 2009; Wo et al. 2012; Doufas 2014). 7 Luo and Tanner have developed a temperature dependent integral constitutive equation of the K-BKZ type for simulating non-isothermal viscoelastic flows (Luo and Tanner 1987). Since then several researchers have used modified/improvised versions of this model to simulate non-isothermal viscoelastic flows such as capillary/slit extrusion (Barakos and Mitsoulis 1996; Marín and Rasmussen 2009), annular extrusion (Barakos and Mitsoulis 1996), and film blowing (Alaie and Papanastasiou 1993). On the other hand, several authors have also demonstrated the use of differential constitutive equations such Upper Convected Maxwell (UCM) (Cain and Denn 1988; Luo and Tanner 1985), PTT (Peters and Baaijens 1997; Muslet and Kamal 2004) and eXtended Pom-Pom (XPP) (Sarafrazi and Sharif 2008) models for non-isothermal polymer melt flow simulations. Moreover, these simulation studies were aimed mainly to account for the thermally induced extrudate swell and extrudate bending phenomena (Karagiannis et al. 1989; Phuoc and Tanner 1980; Barakos and Mitsoulis 1996; Chang and Yang 1994). However, the self-heating is not significant for the present HDPE extrusion experiments as the average velocity is very small due to the small radius or thickness of the dies used even at the highest shear rates studied. Similar behavior was observed for IUPAC extrusion measurements as reported by Luo and Tanner (1987). Moreover, many of these non-isothermal swell simulation studies lack accurate experimental validation and are limited only to fiber spinning and film blowing, where extrudate swell is relatively unimportant. In addition, several researchers developed and used macroscopic continuum models that combine rheological constitutive equations with crystallization kinetics to simulate the non-isothermal extrusion processes like fiber spinning and film blowing. Most of these reported studies so far used simple shear thinning models (Patel et al. 1991; Ziabicki et al. 2004; Tanner 2003) or differential constitutive equations (Doufas and McHugh 2001; Doufas 2014) and the effects of crystallization on the fluid behavior are accounted by incorporating the Nakamura model (Nakamura et al. 1973) along with the Ziabicki (1976) or Hoffman-Lauritzen (Hoffman et al. 1976) equation. These studies have shown some successes in predicting the velocity and temperature distributions along the melt/film at least when advanced rheological models such as Giesekus, XPP and PTT are used in combination with appropriate crystallization kinetics. However, it was found that the differential/molecular constitutive equations (such as Giesekus, PTT, and DCPP) cannot properly account for the memory effects associated with highly 8 viscoelastic polymer melts and fail to predict extrudate swell accurately (Béraudo et al. 1998; Konaganti et al. 2015, 2016a). This points that further work is needed for the development of a non-isothermal extrudate swell model with crystallization kinetics that can account for the memory effects associated with viscoelastic polymer melts accurately, while describing the ongoing crystallization across the extrudate. 1.2 Summary/concerns Despite many efforts on the measurement and prediction of extrudate swell of viscoelastic polymer melts, the following gaps/concerns still remain:  Limited experimental data are available on extrudate swell of viscoelastic polymer melts considering the effects of various parameters such as geometrical characteristics of the die, wall slip, and temperature (barrel and/or extrudate).  Identification of a rheological model that can represent the extrudate swell behavior observed in highly viscoelastic polymer melts such as HDPE  There are very few systematic studies on the effect of temperature (of the barrel or extrudate environment) on extrudate swell experimentally and numerically.  There are limited studies on the effects of rheological properties/parameters on the extrudate swell of polymer melts It is important to study the flow behavior of polymer melts and the associated swell phenomenon, considering parameters such as deformation history (i.e., memory effects), wall slip in the die, non-isothermal effects for the optimization of die design and process conditions and thus better prediction of extrudate swell. 1.3 Research objectives The main objective of this research project is to obtain systematic and reliable extrudate swell data of a high molecular weight industrial grade HDPE and to accurately predict extrudate swell by identifying or developing suitable constitutive model. The particulars of the main objective are presented below: 9 1. To develop an efficient experimental technique with an on-line data acquisition system that can record reliable experimental data on extrudate swell. This was done by using a novel extrudate swell measurement system installed at the die exit. This system consists of a pair of movable optical micrometers to measure the extrudate dimensions and a pair of infrared heaters to maintain the ambient environment at a desired temperature. The new set-up is designed to eliminate the shortcomings of existing measuring techniques and can obtain gravity free and isothermal/non-isothermal swell readings. 2. To identify a suitable rheological/constitutive model that can accurately represent the extrudate swell phenomenon especially in a highly viscoelastic polymer melt such as HDPE. This was addressed by performing a detailed analysis on the applicability and validity of various rheological (integral and differential/molecular) models in describing extrudate swell of the studied HDPE melt in capillary dies at high shear rates (in the range of 5 to 100s-1) (Konaganti et al. 2015). A series of extrudate swell experiments considering the effects of temperature, slip and die geometrical characteristics in capillary dies are carried out to gather the experimental data necessary for comparison with simulation results. 3. To study the non-linear viscoelastic flow properties of the polymer resin for a broad range of deformations/deformation rates in an attempt to determine the damping function of the material. The newly available CPP fixture mounted onto AntonPaar MCR 702 is used to procure such data. Obtaining the accurate damping function is found to be the most important in simulating extrudate swell in polymer melts (Konaganti et al. 2016a). The sensitivity analysis of extrudate swell predications to damping function is also examined in detail. The simulation results of extrudate swell in capillary and slit dies obtained using the newly determined damping function parameters are compared with their corresponding experimental observations (Konaganti 2016a, 2016b). 4. To develop a non-isothermal extrudate swell model that is appropriate for highly viscoelastic polymer melts while accounting for the ongoing crystallization throughout the extrudate volume. Non-isothermal extrudate swell measurements were carried out at a constant extrusion temperature inside the die and at different ambient air 10 temperatures (below the equilibrium melting point of the polymer) surrounding the extrudate. An integral constitutive model of K-BKZ type along with Nakamura crystallization kinetics is used to predict the non-isothermal extrudate swell measurements in capillary and slit dies. 1.4 Thesis organization The organization of this dissertation is as follows. Details of the apparatuses used in this study, which include a rotational rheometer (AntonPaar MCR 502 or AntonPaar MCR 702) with parallel plate geometry, cone and plate (CP) geometry, cone partitioned plate fixture (CPP), Sentmanat Extensional Rheometer (SER) fixture, a capillary rheometer (Instron tester 4465), a novel extrudate swell measuring system (with a pair of optical micrometers (Keyence® LS-7030M) and a pair of infrared heaters), and Differential Scanning Calorimeter (DSC) (PerkinElmer STA 6000) and the methodology associated with these experimental techniques are described in Chapter 2 (Objective 1). The complete rheological characterization of the present resin using various integral, differential and molecular constitutive equations is presented in Chapter 3. Chapter 3 further includes the 2D FEM viscoelastic flow simulations in capillary dies using various constitutive models to identify an appropriate rheological model that can accurately represent the extrudate swell behavior of the present HDPE polymer. This allows a comprehensive comparison of the simulation results thus obtained using various models with the experimental observations (Objective 2). Chapter 4 reports the use of CPP fixture to obtain non-linear viscoelastic properties of the preset resin over a broad range of shear strains / shear rates. This was found to be very crucial for accurately determining the damping function parameters of the constitutive model and thus the predictions of extrudate swell. Furthermore, this chapter includes the sensitivity analysis of the damping function parameters on extrudate swell (Objective 3). A newly developed extrudate swell model that combines viscoelasticity using integral K-BKZ constitutive equation, crystallization kinetics using Nakamura model and thermal boundary conditions to simulate extrudate swell under non-isothermal conditions is presented in Chapter 5. Moreover, the comparison between the non-isothermal extrudate swell simulation results and experimental measurements are also presented in Chapter 5 (Objective 4). The conclusions summarizing the thesis and the recommendation for future research are presented in Chapter 6. 11 Chapter 2: Material and Experimental Methods This chapter describes the apparatuses and the methodology associated with the experimental techniques used in this work. The apparatuses/instruments used in this study include a rotational rheometer (with parallel plate, CPP, and SER fixtures), a pressure driven capillary rheometer, a Simultaneous Thermal Analyzer (STA) to perform Differential Scanning Calorimetry experiments. 2.1 Material A high molecular weight industrial grade HDPE (with weight average molecular weight, MW=206 kg/mol, Polydispersity index, PI = 10.8) with MFI=5.72 g/10min was used in the present study. 2.2 Rotational rheometer The polymer was rheologically characterized using a stress/strain controlled AntonPaar MCR501 rotational rheometer. The linear viscoelastic properties of the polymer used in this work were measured using small amplitude oscillatory shear (SAOS) or frequency sweep tests and the non-linear shear characteristics of the present resin were obtained by performing start-up of steady shear experiments. These experiments have been performed with the 25-mm parallel disk geometry and a gap of about 1 mm. The linear viscoelastic and non-linear shear properties can also be obtained using the cone and plate fixture. Further, the polymer was also characterized in uniaxial extension using the SER-2 universal testing platform fixture suitable for the AntonPaar MCR501 rheometer. However, it is not possible to obtain reliable rheological data of viscoelastic polymer melts for higher strains and strain rates using the conventional parallel plate and cone and plate fixtures due to the well-known edge fracture instability (Dealy and Wissbrun 1996; Larson 1992).Thus the non-linear shear measurements were conducted for a broader range of deformation/deformation rates using the newly available cone-partitioned plate (CPP) geometry mounted onto AntonPaar MCR702 with TwinDrive® technology. All the experiments have been carried under Nitrogen gas to avoid any thermal degradation. 12 2.2.1 Parallel plate (PP) geometry Parallel plate geometry is one of the most common rheological fixture to produce shear flows. As depicted in FIGURE 2.1, it has two parallel concentric disks where the upper disk rotates with respect to the lower, thus producing shear. This geometry is used primarily for the measurement of linear viscoelastic properties of polymer melts and is the preferred geometry for viscous melts for small strain material functions. Advantages of this rheometer is flow regularity and ease of sample preparation. On the other hand, non-homogeneous strain field and inability to reach high shear rates and strains due to edge fracture are its disadvantages (Macosko 1994). FIGURE 2.1: Schematic representation of parallel plate rheometer. A series of frequency sweeps (SAOS tests) were performed at different temperatures ranging from 160-220ºC with the 25-mm parallel disk geometry and a gap of about 1 mm. The time–temperature superposition (TTS) was used to generate the master curves at the reference temperature of 200ºC at which the extrudate swell measurements were performed and the results are presented in Chapter 3. 13 2.2.2 Cone and plate (CP) geometry Cone and plate geometry is one of the most popular rheometers for studying the viscoelastic properties of molten polymers due to its constant rate of shear and direct measurement of first normal stress difference, N1. This geometry is based on the use of a circular disk and a small angle cone as shown in FIGURE 2.2. The sample is inserted between these two elements, and one is rotated while the other is held stationary. Unlike parallel plate geometry, cone and plate generates homogenous deformation, but it is also limited by elastic edge failure, and loss of sample at edges at high shear rates and strains. A series of steady shear experiments have been performed using a cone and plate geometry of the rotational rheometer AntonPaar MCR 501 at a reference temperature of 200ºC at various shear rates ranging from 0.05 to 1 s-1. The experimental results are given in Chapter 3. FIGURE 2.2: Schematic representation of cone and plate rheometer. 14 2.2.3 Cone partitioned plate (CPP) geometry This new CPP fixture mounted onto AntonPaar MCR 702 is used to complete the rheological characterization by conducting step-strain stress relaxation experiments for shear strains ranging from 0.1 to 7.5 and the start-up of steady shear experiments for shear rates in the range of 0.05 to 20 s-1. The rheological data thus obtained are discussed in detail in Chapter 4. FIGURE 2.3 illustrates the schematic of the setup, which consists of a cone geometry with a 4° angle at the bottom and a partitioned plate at the top. The effects of edge distortions can be avoided using a cone-partitioned-plate (CPP) fixture (Meissner et al. 1989; Snijkers and Vlassopoulos 2011). In this geometry, another partition has been added, only to shield off edge fracture. As depicted in FIGURE 2.3, the top geometry consists of a standard plate (center plate) attached to the transducer, surrounded by a coaxial stationary plate (outer stationary ring). Since the transducer takes into account only torque contributions from the center plate, the effects of the edge fracture will not be sensed immediately. Therefore, reliable shear flow data can be obtained even at high rates and strains. Small amount of sample, no trimming and uniform shear are other advantages of this fixture. With the help of CPP fixture, the well-known edge fracture instability (Dealy and Wissbrun 1996; Larson 1992) is postponed to the higher strains/strain rates. Snijkers and Vlassopoulos (2011) have validated the rheological measurements for polystyrene and polyisoprene for a similar fixture. It is worth noting that the present setup is specifically made for rheometers with two motors like the AntonPaar MCR702. 15 FIGURE 2.3: Schematic of cone and partitioned plate (CPP) fixture. The flow is generated by the lower part cone while the resulting torque is measured at the upper partitioned plate. 2.2.4 SER (Sentmanat Extension Rheometer) universal testing platform Sentmanat Extensional Rheometer (SER) is a universal testing platform consisting of dual wind-up drums (referred to as master and slave drums) which can be used to generate uniaxial extensional data (Sentmanat 2004). Master drum rotates by the rotation of drive shaft, and the slave drum counter-rotates via intermeshing gears, that connect the two drums (see FIGURE 2.4). The sample is fixed between two drums by means of securing clips, and undergoes a uniform uniaxial stretching over an unsupported length, L0. Using small amount of sample resulting in better controlling the temperature is one of the most important advantages of this fixture. For a constant drive shaft rotation rate, Ω, the Hencky strain rate applied to the sample is defined as (Sentmanat, 2004):𝜀?̇? = 2𝛺𝑅𝑊/𝐿0, where Rw is the radius of the wind-up drums, and L0 is the fixed, unsupported length of the stretched sample which is equal to the centerline distance between two drums. 16 A series of uniaxial extensional rheology measurements were carried out for the present resin using the SER-2 Universal Testing Platform fixture suitable for the AntonPaar MCR501 for different Hencky strains in the range of 005 to 5 s-1. FIGURE 2.4: Schematic of SER (Sentmanat Extension Rheometer) testing platform. 2.3 Capillary rheometer Capillary rheometer is the most commonly used instrument to measure the viscosity of the polymers. Compared to shear flow driven rheometers, the main advantage of a capillary rheometer includes its operation at high shear rates, closer to industrial processing conditions. The schematic representation of capillary rheometer is shown in FIGURE 2.5. The pressure drop and the velocity of the piston are the raw data obtained from this device and the following relations can be used to determine the apparent shear rate, ?̇?𝐴 and wall shear stress, τw: 332DQA  (2.1) Sample Master drum Slave drum Clips Intermeshing gears Lo RW Ω Ω 17 4WD pL (2.2) where Q, Δp, D and L are volumetric flow rate, applied pressure, die diameter and die length, respectively. Similarly, for a rectangular channel (slit) of height H and width W with WH, so that end effects can be neglected, the following expressions can be derived: 26WHQA  (2.3) 2WH pL (2.4) In this project, a pressure-driven Instron (model #4465 equipped with a 9.52mm diameter reservoir) capillary rheometer of constant piston speed has been used to study the processing/slip behavior of the present polymer melt at 200ºC. It consists of four individual heating zones, which ensures precise temperature control along its length. Capillary extrusion experiments using dies of various length-to-diameter ratios (L/D = 5, 16, and 33, with D=0.79 mm) and diameters (such as D=0.43, 0.79, and 2.11 mm with L/D=16) were carried out to apply all the necessary corrections such as Bagley and slip (Mooney analysis) to the raw data as presented in Appendix A. The polymer is extruded at a reference temperature of 200ºC for most of the swell measurements. 18 FIGURE 2.5: Schematic representation of capillary rheometer. 2.4 Extrudate swell measuring system The experimental set-up for the extrudate swell measurements in capillary/slit dies is shown in FIGURE 2.6. The experimental set-up consists of a pair of Keyence® LS-7030M optical micrometers. These micrometers are located near the die exit and can be moved both radially and axially to determine the complete extrudate swell profile. This set-up is useful for measuring the swell in both capillary and slit dies. In addition, the set-up includes a pair of heaters placed below the die exit on both sides of the extrudate. These heaters help to keep the ambient air at a desired temperature. Thus both isothermal and non-isothermal extrudate swell measurements (see FIGURE 2.6) can be performed. More specifics on the extrudate swell measuring system used in this study are given in Appendix B. 19 The extrudate swell measurements were performed using the following procedure: when the extrusion pressure reached steady-state, the extrudate was cut at the exit of the die and allowed to flow from the die exit in the downstream direction. The extrudate swell was recorded at the tip of the extrudate, when it reached micrometers’ location. This measurement is taken as the gravity-free extrudate swell value. The extrudate swell evolves with distance down the exit due to the continuous relaxation of the normal stresses (that represent memory of the polymer melt) and cannot be characterized by a single value (Huang and White 1980; Luo and Mitsoulis 1989; Behzadfar et al. 2015). This is true for both isothermal and non-isothermal measurements and is valid even for long extrudates. However, it is tedious to measure the complete extrudate swell profile for all the cases and most of our swell measurements were done only at a distance 20mm below the die exit, at which the extrudate swell reaches at least 90% of its final value. The measurements were repeated multiple times and only the average swell values are plotted in the figures reported in Chapters 3, 4, and 5. The reproducibility of swell readings is within ±5% for most of the readings. More details on the experimental set-up describing how to measure gravity-free and isothermal/non-isothermal swell measurements are presented elsewhere (Behzadfar et al. 2015). FIGURE 2.6: The Instron capillary rheometer along with the novel extrudate swell measurement set-up placed under the die exit (on the left) and a schematic (on the right) representing the ability of the present set-up to measure both isothermal and non-isothermal measurements. 20 2.5 Differential scanning calorimetry (DSC) A PerkinElmer Simultaneous Thermal Analyzer (STA) 6000, which was calibrated with indium was used to study the thermal behavior of the present HDPE resin. All the measurements were carried out under N2 at a flow rate of 20ml/min to avoid thermal degradation of the test samples. The melting and crystallization peaks of the HDPE resin were obtained by using multiple heating and cooling protocols applied on to 20 to 25mg of sample polymer in a ceramic pan. The sample was heated from a temperature of 50 to 200°C at the rate 20°C /min and was equilibrated for about 15 min to eliminate any thermal or flow histories. It was then cooled down to 50ºC/min at the cooling rate of 20°C/min and reheated to 200°C at the same heating rate (i.e., 20°C/min) to determine the crystallization, Tc and melting, Tm peak temperatures, respectively. The values of crystallization, Tc and melting, Tm points determined are the averages of three different runs. The isothermal crystallization was studied at different temperatures between the determined crystallization, Tc, and melting, Tm peak temperatures. In each test, the specimen was first melted at 200ºC for 15 min to eliminate thermal history and then cooled to the desired crystallization temperature at the cooling rate of 10°C/min and the corresponding thermal behavior was recorded. The isothermal analysis is performed at four different temperatures 116, 118, 119, and 120ºC. The relative crystallinity (α=X(t)/Xf) as a function of time t was defined to be the fractional area confined between the rate-time curve and the baseline (Kase and Matsuo 1965) and is calculated using Equation 2.5 (refer to schematic FIGURE 2.7): 00)(tttf dtdtdHdtdtdHXtX (2.5) where Xf is the total crystallinity at the end of primary crystallization process, H is the enthalpy, t0 is time at the start of crystallization process. The non-isothermal crystallization was studied at different cooling rates of 5, 10, 20, 30, and 40°C/min. The standard protocol of history elimination was used for all the non-isothermal 21 DSC experiments that is the sample was kept at 200°C for 15 min in each test. Similarly, the relative crystallinity developed during the cooling process is determined to be the fractional area confined between the rate-time curve and the baseline. At a given cooling rate, the relative crystallinity (α=X(T)/Xf) which is a function of temperature is calculated (Equation 2.6) by integrating the area under the crystallization peak of the DSC thermogram from the equilibrium melting temperature, 𝑇𝑚0 up to the desired temperature and can be given as follows (Patel and Spruiell 1991): fmmTTTTfTdTHTdTHXTX00)()()( (2.6) where Tf is the temperature corresponding to the intercept of the exotherm with the baseline in the glass transition temperature region. FIGURE 2.7: Schematic representation of relative crystallinity fraction, α computation. 22 Chapter 3: Extrudate Swell of HDPE: Modeling using Integral and Differential Constitutive Equations The HDPE polymer used is rheologically characterized using the multimode integral (Kaye-Bernstein-Kearsley-Zapas referred to as the K-BKZ), the differential Phan-Thien-Tanner (referred to as the PTT), the Giesekus and the Double Convected POM-POM (referred to as the DCPP) models. The simulation results computed using ANSYS POLYFLOW® are compared with experimental swell measurements studying the effects of apparent shear rate (?̇?𝐴), temperature, Length-to-Diameter ratio (L/D), and the presence of reservoir (contraction). A detailed analysis on the applicability and validity of the aforementioned rheological (integral and differential/molecular) models in describing extrudate swell of the studied HDPE melt in capillary dies at high shear rates (in the range of 5 to 100s-1) is presented here. Such detailed analysis would help one to identify a suitable rheological/constitutive model that can accurately represent the extrudate swell phenomenon especially for a highly viscoelastic polymer melt such as the one used in the present study. 3.1 Mathematical modeling 3.1.1 Governing equations The governing equations for the flow of incompressible fluids such as polymer melts under isothermal, creeping, gravity-free and steady flow conditions are given as: 0 u (3.1) 0p  τ (3.2) where u is the velocity vector, p is the pressure, τ is the extra stress tensor of the polymer melt. For viscoelastic fluids, such as polymer melts, constitutive equations that relate the stress tensor in terms of velocity gradients are required to complete the system of equations. 23 3.1.2 Constitutive equations In this chapter, mainly the integral K-BKZ (with Wagner and Papanastasiou-Scriven-Macosko (PSM) damping functions) (Papanastasiou et al. 1983; Wagner 1976) and the differential PhanThien Tanner (PTT) (Phan-Thien and Tanner 1977) models were used for the extrudate swell simulations. In addition, the differential Giesekus (Giesekus 1982) and DCPP (Clemeur et al. 2003) models were also used for few computations to evaluate their ability in predicting extrudate swell in comparison with the K-BKZ and PTT constitutive equations. A detailed review comparing various constitutive equations for representing the rheological data of polymer melts has been reported by Larson (1987) which provides useful insights on selecting appropriate viscoelastic rheological models. A version of K-BKZ model proposed by Papanastasiou et al. (1983) and later modified by Luo and Tanner (1988) used in the simulations, is given by:   /11( , ) [ ( ) ( )]1it Nt tin ige h t t dt       -1 tt-1C t tCτ I I C C (3.3) where λi and gi are the relaxation times and relaxation moduli, respectively, N is the number of relaxation modes, IC and IC-1 are the first invariants of the Cauchy–Green tensor Ct, and its inverse Finger strain tensor Ct-1,  is a material constant given by N2/N1 = /(1-), N1 (≡τ11-τ22) and N2 (≡τ22-τ33) are first and second normal stress differences, respectively, with the value of  between 0.2 to 0.1 observed from experimental measurements. The Cauchy–Green tensor Ct is calculated by using the deformation gradient tensor Ft as follows (Luo and Tanner 1986a): tt tTC F F (3.4) The deformation gradient tensor Ft is obtained from the following relation: 24 )()()(ssDssDttFLF (3.5) IF 0)(sts (3.6) where Lt(s) is the velocity gradient tensor, s=t-t' is a dummy variable used to integrate along the path of the particle and I is the unit tensor. Further, the inverse Finger strain tensor Ct-1 is simply determined by inverting Ct. More particulars on the determination of Cauchy-Green tensor Ct and Finger strain tensor Ct-1 are available in the literature (Luo and Tanner 1986a, 1988). The function h is a strain-dependent memory (or damping) function. We have considered two damping functions, namely the Wagner and the PSM functions. The Wagner damping function is given by (Wagner 1976): 0.5( , ) exp( ( (1 ) 3) )h n      -1 -1t tt tC CC CI I I I (3.7) where n and β are nonlinear model constants to be determined from shear and elongational flow data, respectively. The PSM (Papanastasiou et al. 1983) damping function is given by: ( , )( 3) (1 )pph     -1tt-1ttCCCCI II I (3.8) where αp and β are model constants to be determined from shear and elongational flow data, respectively. The PTT model has also extensively been used in the simulation of complex viscoelastic flows and was found to be one of the simplest differential models to represent both the shear and elongational properties of polymer solutions in entry flows (Quinzani et al. 1995). For a spectrum of N relaxation modes (six are used in the present simulation), the viscoelastic extra-stress tensor τ is obtained as the sum of N individual contributions τi : 1,Ni iτ τ (3.9) 25 Each individual contribution τi is described by the PTT constitutive equation (Phan-Thien and Tanner 1977; Phan-Thien 1978): exp ( ) 1 22 2i i i ii iitr                   i i i iτ τ τ τ D (3.10) The rate of deformation tensor D is defined as: ( ) / 2   TD v v , where ▽v is the velocity gradient tensor. The symbols ▽ and Δ over the stress tensor () represent the upper and lower convective time derivatives, while ξi and εi are non-dimensional shear and elongational material parameters, respectively. The Giesekus model is also one of the most realistic differential viscoelastic models and is best suited for shear flows, which can predict shear thinning viscosity and also normal stresses in all directions (Giesekus 1982). Like the PTT model, a multi-mode Giesekus model is used in the present work and the viscoelastic extra-stress tensor τ is obtained as the sum of N individual contributions (Equation 3.9). The individual contribution of the extra stress tensor for the Giesekus model is defined by: 2i ii ii       i iI τ τ D (3.11) αi is a material constant for each mode, which indicates the anisotropic mobility stimulated by flow characteristics. The differential POM-POM model (based on a molecular theory) best suited for branched polymers, is also used in the present study for comparison. The original differential form of the POM-POM model was developed by McLeish and Larson (1998). The proposed model has undergone several modifications and changes to make it appropriate for numerical schemes/formulations and to represent viscoelastic flow under complex deformations. The formulation developed by Clemeur et al. (2003) referred to as Double Convected Pom-Pom (DCPP) model is used in the present paper. For a multi-mode DCPP model, the total extra stress tensor τ is defined as the sum of individual components and the extra stress tensor contribution by 26 each mode is given in terms of an orientation tensor S, and a stretching scalar Λ (state variables) as shown by the Equation below: 2(3 )1iiG  iτ S I (3.12) where, Gi and ζi are shear modulus and nonlinear material constants of each mode, respectively. The state variables, S (orientation tensor) and Λ (stretch variable) are obtained from the following differential equations: 031]:2)[1(2212 ISSSDSS iiii (3.13)     01:12inqisiseDtDSv (3.14) where λi and λsi are the relaxation times related to the orientation and stretching mechanisms, respectively and qn represents the number of dangling arms at the ends of the pom-pom molecule. 3.1.3 Boundary conditions The conservation equations (Equations 3.1 and 3.2) combined with the constitutive equation (either the integral or differential models) need to be solved along with suitable boundary conditions. The present problem is solved by applying well-known boundary conditions such as described in the literature (Béraudo et al. 1998; Barakos and Mitsoulis 1996; Mitsoulis 2010; Ansari et al. 2013). The boundary conditions for axisymmetric flows are discussed below. Similar boundary conditions will be applied for planar flows studied in the present work. Please refer to FIGURE 3.1 representing all the necessary boundary conditions to solve the extrudate swell problem of capillary extrusion. (a) Along the domain entry, a fully developed velocity profile is imposed, which is in turn related to the volumetric flow rate. 27 (b) Along the center line (for example in the case of axisymmetric flows), and because of symmetry, the radial velocity component is set to zero, as well as the shear stresses. (c) Along the solid walls (i.e., capillary and reservoir walls), usually the no-slip velocity boundary condition is imposed. If the polymer melt slips at the wall, which is the case for the HDPE studied here, a slip law that relates slip velocity (Vs) to the shear stress (τw) of the form Vs = βsl τwb (with βsl = 1565.7 mm/s/MPab and b=3.4 corresponding to extrusion temperature 200ºC) is used. The slip law corresponding to extrusion temperatures other than the reference temperature of 200ºC are obtained by correcting the values of slip coefficient βsl using the time-temperature shifting factor aT. The values of slip coefficient βsl corresponding to extrusion temperatures 160ºC, 170ºC, and 180ºC are 793.6 mm/s/MPab, 925.0 mm/s/MPab, and 1131.5 mm/s/MPab, respectively. The slip law exponent b remains same for all the extrusion temperatures. The nonlinear slip law in extrudate swell simulation is a challenging problem. An auxiliary sweep method is used to address this issue. At first, an evolution scheme on flow rate (i.e., flow rate is increased gradually to the desired value) with full slip boundary condition. Then the solution is used as initial condition to solve the problem at desired slip with evolution on slip coefficient βsl keeping the slip law exponent b constant. This method allowed solving the extrudate swell problem with nonlinear slip condition without any major convergence issues. (d) Along the domain exit, usually zero surface tractions and zero transverse velocity (i.e., considering fully developed profile at the exit). (e) At the free surface, zero surface tractions along with a kinematic boundary condition of no flow normal to the surface is applied, i.e., ?̂?.u=0, where ?̂? is the unit outward normal vector to the free surface. Similarly, a number of relevant boundary conditions need to be defined in the case of non-isothermal extrudate swell simulations, as will be discussed in Chapter 5. 28 FIGURE 3.1: Schematic representation of flow domain for a typical capillary extrudate swell simulation problem along with all the necessary boundary conditions. 29 3.1.4 Method of solution The finite element method (FEM) for the flow simulations uses the u-v-p-h formulation (primitive variable approach) for the integral models, such as the K-BKZ Wagner/PSM as in viscous flows, and EVSS (mixed variable approach) (Rajagopalan et al. 1992) or DEVSS formulation (enhanced mixed variable approach) (Guénette and Fortin 1995) for the differential models, such as the multi-mode PTT, Giesekus and DCPP models, combined with special streamline-upwind/Petrov-Galerkin (SUPG) formulation (Barakos and Mitsoulis 1995; Béraudo et al. 1998). The weak formulation of the problem is obtained using biquadratic approximation for the velocities, temperature, and free surface location and bilinear approximation for the pressures. The free surface location is determined in a coupled manner along with the solution of other primary variables. Despite the fact that the SUPG FEM formulation is found to be appropriate for solving stresses for the given velocity field, the method needs some improvements to solve highly convective and/or highly viscoelastic flows. To implement SUPG FEM formulation for such highly convective and/or viscoelastic flows, a special discretization scheme called non-consistent streamline upwind (SU) method is used. More particulars on this method can be found elsewhere (Marchal and Crochet 1987). The viscoelastic extra stress tensor is added as a body force (decoupled method) in the discretized momentum conservation equations. Further, the integral models include an appropriate numerical integration method (such as Gauss-Laguerre quadrature method) to determine the stress components along the streamlines (Luo and Tanner 1986). The above numerical schemes have been used in the numerical package ANSYS POLYFLOW®, which has been licensed for use in the present work. Independent numerical results using the integral method outlined by Luo and Tanner (1986a, 1986b) and Luo and Mitsoulis (1989) have confirmed the numerical results obtained from ANSYS POLYFLOW®. A typical FEM grid for a capillary die with length-to-diameter ratio of L/D = 16 is presented in FIGURE 3.2. The flow domain shows a 12:1 abrupt contraction with the diameter of the reservoir (barrel) (Db) being 9.52 mm and the diameter of the die being 0.79 mm (D). The length-to-diameter ratio is 16, long enough to ensure fully developed flow conditions. The extrudate length (Lext) considered in the simulations is 20 mm (Lext/D=25), necessary to capture the memory effects associated with the viscoelastic flow simulations. The flow domain is meshed using a 30 mapped meshing technique and the arrangement of meshes/grids is such that the grids become denser towards the entry singularity, i.e. near the entrance of the die, and coarser grids are used away from the entry singularity. Similarly, a finer grid or more mesh elements are used towards the die exit to better capture the exit singularity. The mesh is chosen based on the experience from earlier studies (both viscous and viscoelastic) of our group (Ansari et al. 2013; Ansari and Mitsoulis 2013). The grid consists of 1900 elements and a mesh four times denser is also used by dividing each element into four sub-elements to check the mesh independency of the numerical results. The mesh independency is checked such that the differences between the overall pressures and swell ratios from both the meshes are <1%. FIGURE 3.2: A typical finite element grid for the simulation of the 12:1 abrupt circular contraction with L /D = 16, Lext /D = 25, D=0.79 mm and 2a = 180o. 31 The simulations were performed at various apparent shear rates, ?̇?A, ranging from 5 to 100 s-1 (experimental data are available over this range). The corresponding Weissenberg numbers (defined as mean relaxation time times the shear rate, ?̅??̇?A) of the resin used in this study are in the range of 107.3 to 2146 with the mean relaxation time of 21.46 s (obtained using Equation 3.15). It should be noted that the simulations were performed considering the upstream contraction, and no convergence related problems were found within the range of Weissenberg numbers studied. FIGURE 3.3 shows the evolution of the global residual with the number of iterations at different flow rates for a typical 2D axisymmetric simulation of capillary swell L/D=16 (D=0.79mm) using the integral K-BKZ Wagner model, indicating stable computations that reach convergence within the range of apparent shear rates/Weissenberg numbers studied. Number of iterations1 10 100Tolerance 10-410-310-210-1100 = 5 s-1 = 11 s-1 = 26 s-1 = 64 s-1 = 100 s-1..... FIGURE 3.3. Computational variable residual with the number of iterations at different apparent shear rates ranging from 5 to 100 s-1. 32 3.2 Results and discussion 3.2.1 Rheological characterization The measurement of linear viscoelastic properties using a small-amplitude oscillatory shear (SAOS) test is well-established. The SAOS or frequency sweeps tests were carried using parallel plate geometry (refer to section 2.2.1 for more details). The master curves of the dynamic moduli G'(ω) and G"(ω) of the HDPE resin (with a Melt Flow Index, MFI=5.72g/10min) at the reference temperature of 200C along with the Maxwell model predictions (using a six-mode relaxation spectrum) are shown in FIGURE 3.4. Constrained non-linear optimization technique is used to determine the discrete Maxwell relaxation spectrum parameters (Owens and Philips 2002). The fitted Maxwell parameters, relaxation times, λi, and the relaxation moduli, gi, are listed in TABLE 3.1. Furthermore, the average relaxation time, , and the zero shear viscosity, η0, can be calculated from: 211Nk kkNk kkgg, (3.15) 01Nk kkg  . (3.16) The values of these parameters are  =21.46 s, 0=215,196 Pas, indicating a highly viscoelastic polymer melt. The nonlinear model parameters n and β of the Wagner damping function, and the material parameters αp and β of the PSM damping function are obtained by fitting the shear and elongational rheological data such as the start-up of steady shear stress growth coefficient (ηs+) and uniaxial stress growth coefficient (ηE+). The corresponding model fittings along with the experimental rheological data are shown in FIGURE 3.5 and 3.6. As previously mentioned, the start-up of shear experiments were carried using cone and plate geometry (refer section 2.2.2 and 2.2.3 for more details) and the data could be collected only for shear rates up to 1s-1 due to the instabilities such 33 as edge fracture and sample ejection observed at higher strains/shear rates. However, it is found that obtaining such data at higher strains/shear rates is crucial for the accurate determination of damping function and thus extrudate swell, which would be discussed in greater detail in the next chapter. A value of θ=-0.15 is used study the effect of non-zero second normal stress difference on extrudate swell predictions and is found to decrease the swell only about 2 to 7% (depending on shear rate) and thus θ is not considered in the present simulations. This observation is in agreement with previous studies (Ansari and Mitsoulis 2013; Goublomme and Crochet 1993; Barakos and Mitsoulis 1996). The values of the relaxation spectrum and the material constants of the HDPE polymer melt at 200C for both Wagner and PSM damping functions of the K-BKZ equations are listed in TABLE 3.1. FIGURE 3.4: The master curves of storage (G') and loss (G") moduli and the corresponding fitting using a six-mode Maxwell spectrum at the reference temperature of Tref = 200oC (see TABLES 3.1, 3.2, 3.3 and 3.4). Frequency,  (rad/s)10-2 10-1 100 101 102 103Dynamic modul i, G', G'' (Pa)103104105106G'G''K-BKZ PTTGiesekusDCPPHDPE MFI 5.72g/10minT=200°C34 TABLE 3.1: Relaxation spectrum at temperatures 160oC, 180 oC, and 200oC for HDPE melt with K-BKZ model parameters (Wagner and PSM damping functions, =0). Mode # Relaxation spectrum K-BKZ Parameters 160oC 180oC 200oC gi (Pa) λi (s) gi (Pa) λi (s) gi (Pa) λi (s) Wagner 1 387808 0.0017 387808 0.0012 387808 0.00086 n β 2 185307 0.015 185307 0.011 185307 0.0075 0.21 0.9 3 93338 0.108 93338 0.074 93338 0.0548 4 37766 0.796 37766 0.547 37766 0.403 PSM 5 12934 5.92 12934 4.07 12934 2.99 αp β 6 5025 60.76 5025 41.75 5025 30.78 7.6 0.6 TABLE 3.2: Relaxation spectrum at temperatures 160oC, 180 oC, and 200oC for HDPE melt with PTT model parameters. Mode # Relaxation spectrum PTT Parameters 160oC 180oC 200oC ηi (Pa.s) λi (s) ηi (Pa.s) λi (s) ηi (Pa.s) λi (s) εi ξi 1 658.26 0.0017 452.32 0.0012 336.66 0.0009 8.29 0 2 2736.25 0.015 1880.2 0.011 1395.33 0.0075 1.88 0 3 10101.7 0.108 6941.31 0.074 5473.37 0.0548 9.42 0 4 30052.3 0.796 20650.3 0.547 13400.3 0.403 0.21 0 5 76606.2 5.92 52639.6 4.07 36232.6 2.99 6.57 0 6 305387 60.76 209845 41.75 155795 30.78 0.23 0 35 Time (s)10-1 100 101 102Shear stress growth coefficient, S (Pa.s)103104105K-BKZ (Wagner)K-BKZ (PSM)PTT GiesekusDCPPHDPE MFI 5.72g/10min T=200°C 0.05 s-10.5 s-11 s-1 FIGURE 3.5: Experimental data (symbols) and model fits (lines) of the shear stress growth coefficient S, using both integral (K-BKZ Wagner and K-BKZ PSM) and differential (PTT, Giesekus and DCPP) constitutive models. Time (s)10-2 10-1 100 101 102Uniaxial stress growth coefficient,  (Pa.s)104105106K-BKZ (Wagner)K-BKZ (PSM)PTTGiesekusDCPPHDPE MFI 5.72 g/10min T=200°C0.05 s-10.5 s-15 s-1 FIGURE 3.6: Experimental data (symbols) and model fits (lines) of uniaxial stress growth coefficient E, using both integral (K-BKZ Wagner and K-BKZ PSM) and differential (PTT, Giesekus and DCPP) constitutive models. 36 The relaxation spectrum and the nonlinear material constants for the differential models (six-mode PTT, Giesekus and DCPP models) were obtained by fitting the corresponding constitutive equations simultaneously to the dynamic moduli, start-up of steady shear, and uniaxial elongation data. The detailed procedure for the fitting of PTT (Langouche and Debbaut 1990), Giesekus (Hulsen and Zanden 1991) and DCPP (Clemeur et al. 2003) model parameters are available in the literature and the calculated model parameters at 200ºC are listed in TABLES 3.2, 3.3 and 3.4, respectively. The nonlinear materials parameters ξi of the PTT model are set to zero as the effect of N2 is not significant on extrudate swell and also to reduce the number of material parameters and thus the complexity of calculations associated with the model. The corresponding model fittings of differential models PTT, Giesekus and DCPP are also included in FIGURES 3.4, 3.5, and 3.6. It can be seen from FIGURES 3.4, 3.5, and 3.6, the rheological data (linear and nonlinear) are represented well by all constitutive equations. TABLES 3.1 and 3.2 further include the relaxation spectrums at temperatures 160C and 180C that are necessary to study the effect of extrusion temperature on extrudate swell by using the integral K-BKZ and differential PTT models. TABLE 3.3: The relaxation spectrum along with Giesekus model parameters at 200oC. Mode # ηi (Pa.s) λi (s) αi 1 348.90 0.0009 0.01 2 1371.21 0.0075 0.39 3 5484.34 0.055 0.48 4 15264.5 0.403 0.45 5 37988.7 2.99 0.35 6 154673 30.78 0.40 TABLE 3.4: The relaxation spectrum along with DCPP model parameters at 200oC. Mode # Gi (Pa) λi (s) λsi (Pa) qni ζi 1 372942 0.0009 0.0006 2 0.62 2 184712 0.0075 0.0055 2 0.02 3 93454 0.055 0.0372 7 0.04 4 37761 0.403 0.01 8 0.08 5 12862 2.98 0.113 11 0.02 6 5007 30.77 24.81 14 0.82 37 3.2.2 Flow pressure drop comparison The pressure drop predictions obtained from the FEM flow simulations using all the constitutive equations, i.e., both integral and differential models, are compared with the experimental data of capillary extrusion in FIGURE 3.7. All the constitutive equations predict the total pressure drop of capillary extrusion reasonably well (within  10% error for most cases). Apparent shear rate, A (s-1)10 100Total pressure dr op, P(MPa) 110K-BKZ (Wagner)K-BKZ (PSM)PTTGiesekusDCPPExperimental.HDPE MFI 5.72T=200°C, L/D=164 20050 FIGURE 3.7: Experimental data (solid symbols) and model predictions of the total pressure drop for flow in a capillary having L/D=16 and D=0.79 mm, using the K-BKZ Wagner, K-BKZ PSM, PTT, Giesekus, DCPP models for the HDPE polymer melt at 200oC. 3.2.3 Extrudate swell simulations Extrudate swell simulations were carried out to study its dependence on the capillary die geometric parameters, such as length-to-diameter ratio, L/D, presence of reservoir, and on the operating conditions, such as the apparent shear rate, ?̇?𝐴 and temperature. 38 3.2.3.1 Integral versus differential constitutive models FIGURE 3.8 shows the variation of extrudate swell ratio, BD with apparent shear rate, simulated using the aforementioned constitutive equations, with reservoir, for the die with L/D=16 and D=0.79 mm. The values of the swell ratio, BD, measured and the corresponding predicted values by the various models are those at a distance 20 mm (Lext/D=25) from the die exit. At this distance the extrudate swell has been well developed and takes values close to about 90% of its ultimate value as can be seen below when the complete extrudate swell profiles are presented. The swell ratio predicted using both integral (K-BKZ Wagner and K-BKZ PSM) and differential (PTT, Giesekus and DCPP) models increases with an increase of the apparent shear rate conforming to the general trend of experimental data. However, the integral K-BKZ model overpredicts the extrudate swell (≈250% at high shear rates), while the differential models underpredict the results. In the case of the integral K-BKZ model, due to the memory effects, the melt attempts to attain its original shape in the reservoir as it exits the die (Barakos and Mitsoulis 1995, 1996; Ansari and Mitsoulis 2013). This is intriguing in a sense that while both integral and differential constitutive models represent the rheological data equally well, their predictions of extrudate swell are so distinctly different. This can be ascribed to the high elastic energy/memory effect associated with the integral K-BKZ model compared to that of differential models. On the other hand, all differential models used resulted into similar predictions. In an attempt to understand this problem, a more detailed analysis of the predictions of material functions by the two constitutive equations is performed (see Appendix C). 39 Apparent shear rate, A (s-1)10 100Extrudate Swell, BD123456K-BKZ (Wagner)K-BKZ (PSM)PTT GiesekusDCPPExperimental.HDPE MFI 5.72g/10min, T=200°C D=0.79 mm, L/D=164 200 Figure 3.8: Experimental data (solid symbols) and model predictions of extrudate swell ratios predicted by using integral K-BKZ models with Wagner and PSM damping functions and differential PTT, Giesekus and DCPP models with reservoir. Apparent shear rate, A (s-1)10 100Extrudate Swell, BD123456K-BKZ (Wagner)K-BKZ (PSM)K-BKZ (Wagner) without reservoirK-BKZ (PSM) without reservoirPTT PTT without reservoirExpt.HDPE MFI 5.72g/10min, T=200°C D=0.79 mm, L/D=164 200 FIGURE 3.9: Experimental data (solid symbols) and model predictions of extrudate swell ratios predicted by using K-BKZ models with Wagner and PSM damping functions and the PTT model with and without reservoir. 40 Extrudate swell simulations were carried without reservoir using the integral K-BKZ (Wagner and PSM) and differential PTT models and the results are presented in FIGURE 3.9. When the reservoir is excluded from the simulations, the predictions of the PTT, Giesekus and DCPP models only slightly change compared to those with the reservoir. On the other hand, for the K-BKZ case, excluding the reservoir causes significant decrease to the predictions. Therefore, memory effects are significant in flow simulations when integral models are used. 3.2.3.2 Swell profiles comparison As discussed above, a single value of BD cannot characterize completely the extrudate behavior of the polymer as its profile is continuously been developed in the downstream direction once it is exiting the capillary die. The experimental extrudate swell profiles are compared with the simulated profiles using the K-BKZ PSM and PTT models at low and high apparent shear rates (5 s-1 and 100 s-1), in FIGURES 3.10 (a) and (b), respectively. The predicted extrudate swell profiles using integral K-BKZ model keep developing along the length of the extrudate in a similar fashion with the corresponding experimental ones. It is also seen that the use of the integral K-BKZ PSM model without the presence of the reservoir results into good prediction of the whole extrudate profile. The swell predicted using the PTT model reaches its ultimate value very close to the die exit itself, indicating that the memory effects fade away fast (Guillet et al. 1996) Distance from die exit or Extrudate length (mm)0 5 10 15 20 25Ext rudate swe ll, BD1.01.21.41.61.82.02.22.42.6K-BKZ (PSM) K-BKZ (PSM) without reservoirPTTExperimental(a) Distance from die exit or Extrudate length (mm)0 5 10 15 20 25Extrudate swell, BD12345K-BKZ (PSM) K-BKZ (PSM) without reservoirPTTExperimental(b) FIGURES 3.10: Comparison of experimental extrudate profiles with simulations for the capillary die having L/D=16 and D=0.79 mm at (a) a low shear rate, 5 s−1 and (b) a high shear rate, 100 s-1. 41 3.2.3.3 Pressure and stress profiles comparison The model predictions of pressure p, shear stress τ12, and first normal stress difference N1,w along the wall and free-surface and first normal stress difference N1,sym, along the axis of symmetry using K-BKZ (PSM) and PTT models, at an apparent shear rate of 26 s-1, are depicted in FIGURES 3.11 (a) to (d), respectively. These profiles are in agreement with the general trends reported in earlier studies (Saramito and Piau 1994; Ahmed 1995; Béraudo et al. 1998; Ganvir et al. 2009). Distance from flow domain inlet (mm)0 10 20 30 40 50Pr essure (MPa)02468101214K-BKZ (PSM) PTT(a)Die EntryDie Exit Distance from flow domain inlet (mm)0 10 20 30 40 50Shear st ress,  12 (MPa)-0.2-0.10.00.10.20.30.4(b)Die EntryDie Exit 42 Distance from flow domain inlet (mm)0 10 20 30 40 501st normal stress difference, N1,W (MPa)-1.0-0.50.00.51.01.5(c)Die EntryDie ExitDistance from flow domain inlet (mm)0 10 20 30 40 501st normal stress difference, N1,sym (MPa)-0.4-0.20.00.20.40.60.8(d)Die EntryDie Exit FIGURE 3.11. Comparison of (a) Pressure p (b) Shear stress τ12 (c) First normal stress differences N1,W along the wall and free-surface and (d) First normal stress differences N1,sym, along the axis of symmetry, using integral K-BKZ PSM and differential PTT models, at an apparent shear rate of ?̇?A = 26 s-1. As can been from FIGURES 3.11 (a) to (d), all these material function profiles along the flow domain predicted by integral K-BKZ PSM and differential PTT model are fairly similar. Some small differences exist near the die entry and exit singularities (especially for first normal stress difference along the symmetry axis) which cannot explain the significant differences in the extrudate swell predictions of the two models. Also, the first normal stress difference N1,sym along the axis of symmetry, predicted using the differential PTT model relaxes slightly faster than that predicted by integral K-BKZ model (FIGURE 3.11 (d)). These differences could be attributed to the variations in the numerical schemes/formulations associated with these two types of models. 3.2.3.4 The effect of die length The effect of length-to-diameter ratio (L/D) on extrudate swell simulations using the K-BKZ PSM model is studied using three capillary dies of different L/D ratios (L/D= 5, 16, and 33 all with diameter D=0.79 mm). The results are compared with experimental measurements in FIGURE 3.12 (a). The simulation results show that the swell ratio decreases with increasing the L/D ratio, in qualitative agreement with the experimental observations. Flow in a longer die makes the memory effects fade away in the reservoir and causes a decrease in the extrudate swell. 43 Excluding the upstream reservoir eliminates the effect of die length on extrudate swell as can be seen in FIGURE 3.12 (b). A similar trend was observed for extrudate swell simulations using the PTT model as well. The results are plotted in FIGURE 3.13. However, the effect of die length on swell is insignificant in comparison with integral model predictions and also with the actual measurements. It is worthy of mentioning that the simulation results using PTT model with and without reservoir are very similar. The comparison is consistent qualitatively, although there are significant quantitative differences. The decrease of the extrudate swell ratio with the length of the die (see also FIGURE 3.12(a) related to the fading memory of the viscoelastic polymers melt (Barakos and Mitsoulis 1995). Apparent shear rate, A (s-1)10 100Ext rudate swell, BD123456L/D = 5 ExptL/D =16 ExptL/D = 33 ExptL/D = 5 K-BKZ (PSM)L/D =16 K-BKZ (PSM)L/D =33 K-BKZ (PSM).D = 0.79mm4 200(a) Apparent shear rate, A (s-1)10 100Extrudate swell, BD1.01.52.02.53.0L/D = 5 ExptL/D =16 Expt L/D =33 Expt L/D = 5 K-BKZ (PSM)L/D =16 K-BKZ (PSM)L/D =33 K-BKZ (PSM).D = 0.79mm4 200(b) FIGURE 3.12: Extrudate swell simulations using the K-BKZ PSM model for three different dies with L/D=5, 16, and 33 and their comparison with experimental results (a) including the reservoir in the flow simulation (b) excluding the reservoir in the flow simulation. 44 Apparent shear rate,  (s-1)10 100Extrudate swell, BD1.01.21.41.61.82.02.22.4L/D = 5 ExptL/D = 16 ExptL/D = 33 Expt L/D = 5 PTTL/D = 16 PTTL/D = 33 PTT.D = 0.79mm FIGURE 3.13: Extrudate swell simulations using the 6-mode PTT model for three different dies with L/D=5, 16, and 33 and their comparison with experimental results. Similar simulation results with or without the reservoir. It would be very interesting to calculate the die length necessary for complete relaxation of memory effects associated with the integral K-BKZ model, when the reservoir is included in the flow simulations. For this purpose, the extrudate swell is calculated in capillary dies having various length-to-diameter ratios in the range of 5