Applied Science, Faculty of
Chemical and Biological Engineering, Department of
DSpace
UBCV
Ghahramani, Nikoo
Doctor of Philosophy - PhD
University of British Columbia
Thermoplastic vulcanizates (TPVs) are polymer blends that have good processability like their thermoplastic phase and good elasticity like their elastomeric phase. They consist of a high amount of dynamically cured rubber particles which make flow instability such as melt fracture intense and complicated. In this study a comprehensive rheological analysis is performed to gain a deep understanding of TPVs’ flow instabilities and identify the key parameters that control them. First, a thorough linear viscoelastic analysis is performed using several groups of TPVs which are systematically different in curing level, types of polymer components, and cured rubber content. All the TPVs show a non-terminal behavior reaching to an equilibrium modulus at low frequencies/high relaxation times. The equilibrium modulus, Gy, is an indication of the existence of yield stress and the linear modulus, G(t) can be modeled by a simple power-law model in the form of (𝐺(𝑡)−𝐺𝑦)∝𝑡−𝑝, where t is the time. A yield stress is also observed in the extensional tests and the relationship between extensional and shear yield stress is investigated. Due to the presence of oil and high amount of rubber particles, TPVs show wall slip. The multimode integral Kaye-Bernstein-Kearsley-Zapas (KBKZ) constitutive model considering wall slip is applied to study and model the TPVs’ non-linear behavior. To incorporate slip into the model, a new way is used by applying a fraction of imposed nominal strain to match the experimental data. Moreover, it is assumed that the material does not slip in the linear unyielding region i.e., for shear stresses less than the yield stress. Applying these assumptions, the KBKZ captures the experimental data well. Finally, to study processing parameters that affect the melt fracture phenomenon, capillary experiments are performed. It is observed that TPVs slip massively in capillary flow. Surface fracture of the TPVs gets better with shear rate indicating that the origin of melt fracture is different with that of TPV’s pure component. Yield stress controls flow instability of the TPVs and to overcome and/or mitigate melt fracture, high shear rates are needed to cause flow and eliminate unyielded regions in the complex geometries used in real processing.
Flow instabilities of PP filled with crosslinked EPDM rubber: using rheology to understand and control its occurrence by Nikoo Ghahramani B.Sc., Amirkabir University of Technology, Tehran, Iran, 2014 M.Sc., Amirkabir University of Technology, Tehran, Iran, 2017 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) September 2021 © Nikoo Ghahramani, 2021 ii The following individuals certify that they have read, and recommend to the Faculty of Graduate and Postdoctoral Studies for acceptance, the dissertation entitled: Flow instabilities of PP filled with crosslinked EPDM rubber: using rheology to understand and control its occurrence submitted by Nikoo Ghahramani in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemical and Biological Engineering Examining Committee: Savvas G. Hatzikiriakos, Chemical and Biological Engineering, UBC Supervisor Mark Martinez, Chemical and Biological Engineering, UBC Supervisory Committee Member Jeffrey Giacomin, Chemical Engineering, Queen’s University External Examiner Simcha Srebnik, Chemical and Biological Engineering, UBC University Examiner Reza Vaziri, Civil Engineering, UBC University Examiner Additional Supervisory Committee Members: Sheldon Green, Mechanical Engineering, UBC Supervisory Committee Member Antonios Doufas, ExxonMobil Chemical Company Supervisory Committee Member Krishnan Iyer, ExxonMobil Chemical Company Supervisory Committee Member iii Abstract Thermoplastic vulcanizates (TPVs) are polymer blends that have good processability like their thermoplastic phase and good elasticity like their elastomeric phase. They consist of a high amount of dynamically cured rubber particles which make flow instability such as melt fracture intense and complicated. In this study a comprehensive rheological analysis is performed to gain a deep understanding of TPVs’ flow instabilities and identify the key parameters that control them. First, a thorough linear viscoelastic analysis is performed using several groups of TPVs which are systematically different in curing level, types of polymer components, and cured rubber content. All the TPVs show a non-terminal behavior reaching to an equilibrium modulus at low frequencies/high relaxation times. The equilibrium modulus, Gy, is an indication of the existence of yield stress and the linear modulus, G(t) can be modeled by a simple power-law model in the form of (𝐺(𝑡) − 𝐺𝑦) ∝ 𝑡−𝑝, where t is the time. A yield stress is also observed in the extensional tests and the relationship between extensional and shear yield stress is investigated. Due to the presence of oil and high amount of rubber particles, TPVs show wall slip. The multimode integral Kaye-Bernstein-Kearsley-Zapas (KBKZ) constitutive model considering wall slip is applied to study and model the TPVs’ non-linear behavior. To incorporate slip into the model, a new way is used by applying a fraction of imposed nominal strain to match the experimental data. Moreover, it is assumed that the material does not slip in the linear unyielding region i.e., for shear stresses less than the yield stress. Applying these assumptions, the KBKZ captures the experimental data well. Finally, to study processing parameters that affect the melt fracture phenomenon, capillary experiments are performed. It is observed that TPVs slip massively in capillary flow. Surface fracture of the TPVs gets better with shear rate indicating that the origin of melt fracture is different with that of TPV’s pure component. Yield stress controls flow instability of the TPVs and to overcome and/or mitigate melt fracture, high shear rates are needed to cause flow and eliminate unyielded regions in the complex geometries used in real processing. iv Lay Summary Thermoplastic vulcanizates are polymer blends which contain a high amount of rubber particles that are dispersed in a thermoplastic matrix. Due to the presence of a high amount of rubber, they exhibit yield stress i.e. the applied forces should be higher than a certain value to cause flow. TPVs also slip massively and exhibit surface distortions (melt fracture) when they flow out of dies in processing. In this thesis, the rheological properties of (TPVs) are studied in detail and used to understand the origin of flow instability such as melt fracture. v Preface This thesis entitled “Flow Instabilities of PP Filled with Crosslinked EPDM Rubber: Using Rheology to Understand and Control its Occurrence” presents the research the author performed during her PhD study under the supervision of Professor Savvas G. Hatzikiriakos. Rheology and flow behavior of different groups of TPVs have been extensively studied to fully understand their flow instabilities. The following journal papers have been published or submitted for publication from the research work presented in this dissertation: Chapter 5 has been prepared based on a published manuscript. Ghahramani, N., Iyer, K., Doufas, A., Hatzikiriakos, S.G., (2020). Rheology of thermoplastic vulcanizates (TPVs). Journal of Rheology, 64(6), pp.1325-1341. The materials covered in chapter 6 is based on a published manuscript. Ghahramani, N., Iyer, K., Doufas, A., Hatzikiriakos, S.G., (2021). Rheological modelling of Thermoplastic Vulcanizates (TPVs) using the Kaye–Bernstein, Kearsley, Zapas (K–BKZ) constitutive law. Physics of Fluids, 33(8), p.083107. Chapter 7 has been prepared based on a published manuscript. Ghahramani, N., Zhang, S., A Iyer, K., Doufas, A.K. and Hatzikiriakos, S.G., 2021. Melt fracture and wall slip of thermoplastic vulcanizates. Polymer Engineering & Science, 61(4), pp.942-958. Capillary experiments were done by Shiling Zhang. All manuscripts were written by the author of this thesis and revised by the supervisor (Prof. S.G. Hatzikiriakos) and the other co-authors. vi Table of Contents Abstract ........................................................................................................................................................ iii Lay Summary ............................................................................................................................................... iv Preface .......................................................................................................................................................... v Table of Contents ......................................................................................................................................... vi List of Tables ............................................................................................................................................... ix List of Figures ............................................................................................................................................... x List of Symbols ........................................................................................................................................... xv List of Abbreviations ................................................................................................................................ xviii Acknowledgements ..................................................................................................................................... xix Dedication .................................................................................................................................................... xx 1 Introduction ........................................................................................................................................... 1 1.1 Production of TPVs ....................................................................................................................... 2 1.2 Formation Mechanism .................................................................................................................. 2 2 Literature Review .................................................................................................................................. 5 2.1 Structural Parameters Affecting Rheology of TPVs ..................................................................... 6 2.1.1 Effect of polymer type .......................................................................................................... 6 2.1.2 Effect of the composition ratio (R/P) .................................................................................... 6 2.1.3 Effect of the content of crosslinking agent ........................................................................... 7 2.2 Yield Stress ................................................................................................................................... 7 2.2.1 Ways of obtaining yield stress .............................................................................................. 7 2.3 Slip Velocity ................................................................................................................................. 9 2.4 Melt Fracture ............................................................................................................................... 12 2.5 Knowledge Gap .......................................................................................................................... 14 3 Objectives and Thesis Organization ................................................................................................... 16 3.1 Research Objectives .................................................................................................................... 16 3.2 Thesis Organization .................................................................................................................... 17 4 Materials and Experimental Methods ................................................................................................. 19 4.1 Materials ..................................................................................................................................... 19 4.2 TPVs Sample Preparation ........................................................................................................... 20 4.3 Rheological Measurements ......................................................................................................... 22 4.3.1 Rotational rheometer ........................................................................................................... 22 vii 4.3.2 Capillary rheometer............................................................................................................. 22 5 Rheology of Thermoplastic Vulcanizates (TPVs) .............................................................................. 24 5.1 Theoretical Background .............................................................................................................. 24 5.2 Results and Discussion ............................................................................................................... 25 5.2.1 Rheology of pure polymeric constituents of TPVs ............................................................. 25 5.2.2 Linear viscoelasticity of TPVs ............................................................................................ 32 5.2.3 Viscoelastic spectrum of TPVs ........................................................................................... 34 5.2.4 Uniaxial extension of TPVs ................................................................................................ 41 5.3 Conclusions ................................................................................................................................. 45 6 Rheological Modelling of Thermoplastic Vulcanizates (TPVs) using the K-BKZ Constitutive Law 46 6.1 Theoretical Background .............................................................................................................. 46 6.1.1 Constitutive modelling for TPVs ........................................................................................ 46 6.1.2 Implementation of slip in simple shear ............................................................................... 48 6.2 Results and Discussion ............................................................................................................... 48 6.2.1 Linear viscoelastic behavior (LVE) .................................................................................... 48 6.2.2 Linear relaxation modulus .................................................................................................. 50 6.2.3 Non-Linear viscoelastic behavior ....................................................................................... 52 6.2.4 Startup of steady shear and K-BKZ predictions ................................................................. 53 6.2.5 Uniaxial extension and K-BKZ predictions ........................................................................ 55 6.3 Conclusions ................................................................................................................................. 56 7 Melt Fracture and Wall Slip of Thermoplastic Vulcanizates (TPVs) ................................................. 57 7.1 Linear Viscoelasticity (LVE) of TPVs ........................................................................................ 57 7.2 Capillary Flow ............................................................................................................................ 60 7.2.1 Capillary flow of PPs and EPDMs ...................................................................................... 60 7.2.2 Capillary flow of TPVs ....................................................................................................... 62 7.3 Slip of TPVs and Their Basic Constituent Polymers (PP and EPDM) ....................................... 65 7.4 Melt Fracture ............................................................................................................................... 69 7.4.1 Melt fracture of PPs and EPDMs ........................................................................................ 69 7.4.2 Melt fracture of TPVs ......................................................................................................... 73 7.5 Extensional Viscosity .................................................................................................................. 77 7.6 Conclusions ................................................................................................................................. 78 8 Conclusions and Recommendations ................................................................................................... 80 8.1 Conclusions ................................................................................................................................. 80 viii 8.2 Future Recommendations ........................................................................................................... 81 References ................................................................................................................................................... 83 ix List of Tables Table 4-1. Polymers that are the main constituents of the TPVs studied in the present work and their molecular characteristics. ..................................................................................................... 19 Table 4-2. Characteristics of TPVs used in this study (commercial polymers). .......................... 20 Table 4-3. Characteristics of TPVs whose composition is varied systematically. ....................... 20 Table 5-1. The Maxwell relaxation spectrum and nonlinear rheological parameters for the K-BKZ/Wagner model (Eq.4) of all polymers listed in Table 4-1 at Tref = 190°C. ......................... 31 Table 5-2. Yield strains and related critical yield stresses of TPVs listed in Table 4-2 at T=190 ℃. .................................................................................................................................................. 34 Table 5-3. Parameters of the storage modulus function (Eq 5-3) of all TPVs at T=190 ℃. ........ 37 Table 5-4. Yield stresses in shear and extension of TPVs at 190oC and their ratio .................... 44 Table 6-1. Relaxation parameters at T=190 ℃. .......................................................................... 51 Table 6-2. Equilibrium Modulus, Yield stress and strain for TPVs S1-S3. ................................ 53 Table 7-1. Parameters of the relaxation modulus (Equations 7-1 and 7-3) of all TPVs. ............ 60 Table 7-2. Variation of model coefficients with the temperature. .............................................. 60 Table 7-3. Critical shear rates and stresses for the onset of Melt Fracture (MF) phenomena of PPs and EPDMs at 190C and 205C. .......................................................................................... 72 Table 7-4. The critical shear rates and stresses at 190oC and 205oC for the end of surface instabilities of TPVs listed in Table 4-2. ...................................................................................... 77 Table 7-5. Yield stresses in shear and extension of TPVs at 190oC ........................................... 77 x List of Figures Figure 1-1. Schematic figure of preparation methods of TPVs adopted from [2] ......................... 3 Figure 1-2. Morphological development mechanism proposed for the phase inversion of EPDM/PP blends during dynamic vulcanization adopted from [2,22]. .......................................... 4 Figure 1-3. Schematic diagram of the proposed mechanism for the development of PP/EPDM blends during dynamic vulcanization adopted from [2,23]. ........................................................... 4 Figure 2-1. Typical flow curve of a yield stress fluid showing the extrapolation to shear rate equal to zero. ................................................................................................................................... 8 Figure 2-2. Using 90% and crossover methods to find yield stress in strain amplitude sweep test a) viscoelastic moduli vs. shear strain amplitude, and b) Stress vs. shear strain amplitude obtained from the same test. (Typically, these curves are plotted in a log-log scale) .................... 8 Figure 2-3. Depiction of Mooney method a) flow curve at different gaps, and b) nominal shear rate vs. 1ℎ at a specific wall stress. ............................................................................................... 10 Figure 2-4. Velocity profiles in simple shear flow under no-slip and slip boundary conditions. 10 Figure 2-5. Velocity profiles in capillary flow under no-slip and slip boundary conditions. ..... 11 Figure 2-6. Flow curve deviation from LVE which can be used to calculate slip velocity. ....... 11 Figure 2-7. Various types of melt fracture. Adopted from [49]. ................................................. 13 Figure 2-8. Typical extrudate instability occurrence on a flow curve ........................................ 13 Figure 4-1. Representative AFM micrographs showing the morphology in two TPV samples: a) TPV1 and b) TPV4 ....................................................................................................................... 21 Figure 4-2. Schematics of the a) rotational rheometer equipped with the cone partitioned-plate and b) uniaxial extensional fixture that is used in conjunction with a rotational rheometer. ....... 23 Figure 4-3. Schematic representation of the capillary rheometer. ............................................... 23 Figure 5-1. (a) The linear viscoelastic moduli G, G and complex viscosity 𝜂 ∗, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂 +and (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸 + for L-PP (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. .......................................................... 27 Figure 5-2. (a) The linear viscoelastic moduli G, G and complex viscosity 𝜂 ∗, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂 +and xi (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸 + for B-PP (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. .......................................................... 28 Figure 5-3. (a) The linear viscoelastic moduli G, G and complex viscosity 𝜂 ∗, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂 +and (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸 + for L-EPDM (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. .................................... 29 Figure 5-4. (a) The linear viscoelastic moduli G, G and complex viscosity 𝜂 ∗, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂 +and (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸 + for B-EPDM (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. .................................... 30 Figure 5-5. Comparison of the viscoelastic moduli of the two PPs and the two EPDMs listed in Table 4-1 at Tref=190oC. Note the much higher material functions of PP1(linear) compared to B-PP (branched) and the similar material functions of the two EPDMs. It is also noted that EPDMs have in general material functions about 1 order of magnitude greater than those of PPs. .......... 31 Figure 5-6. Strain sweep experiments for TPV8 at various temperatures (190-220oC) to determine the limiting values of strain for linear viscoelastic response and the stress and strain for shear yielding (flow). .............................................................................................................. 33 Figure 5-7. The linear viscoelastic moduli of a) TPV2 and b) TPV3 compared with the corresponding moduli of their constituents at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus. .................................................................... 36 Figure 5-8. Fitting the experimental data with the model of Eqs 5-3 and 5-6 to determine the parameters listed in Table 5-3 a) storage and loss moduli, and b) shear stress and relaxation modulus. ........................................................................................................................................ 37 Figure 5-9. The yield stress,𝜎𝑦, and yield strain, 𝛾𝑦, as functions of the equilibrium modulus 𝐺𝑦′ ................................................................................................................................................. 38 Figure 5-10. The linear viscoelastic moduli of a) TPV1 and 2, and b) TPV5 and 6 compared to show the effect of different thermoplastic used in the TPV formulation at Tref=190 oC. Solid symbols represent storage modulus and open symbols represent loss modulus. .......................... 38 xii Figure 5-11. The linear viscoelastic moduli of a) TPV3 and TPV4, and b) TPV7 and TPV8 compared to show the effect of different thermoplastic used in the TPV formulation at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus........................................................................................................................................................ 39 Figure 5-12. The effect of curing level on the linear viscoelastic moduli of TPV2 (high curing) and TPV3 (normal curing) with other parameters the same (left) and TPV6 (high curing) and TPV7 (normal curing) with other parameters the same (right) at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus. ........................................ 40 Figure 5-13. The effect of rubber concentration (high, normal) on the linear viscoelastic moduli of TPVs with other parameters the same (constituent polymers and level of curing) at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus. ................ 41 Figure 5-14. Comparison of the extensional behavior of TPV4 and TPV5 with those of their constituent components. ................................................................................................................ 42 Figure 5-15. Uniaxial stress growth coefficient of TPV4 at 190oC. a) Uniaxial stress growth vs. time, and b) Uniaxial stress vs. strain. Arrows show the yielding point at different Hencky strain rates. .............................................................................................................................................. 44 Figure 6-1. Storage and loss moduli vs. angular frequency of all the TPVs at T=190 ℃ ........... 49 Figure 6-2. LVE of TPVs at different curing levels at T=190 ℃. a) low cured, b) medium cured, and c) high cured samples. ............................................................................................................ 50 Figure 6-3. Master curve for LVE behavior of the TPVs at T=190 ℃. ....................................... 51 Figure 6-4. a) Stress relaxation of TPV S1, b) Shifted curves to find damping function. All experiments were done T = 190 ℃. .............................................................................................. 52 Figure 6-5. Damping function for S1-S3 at T=190 ℃. ................................................................ 52 Figure 6-6. Start-up experimental results compared to model predictions, Left: no slip condition. Right: applying slip condition to the model .................................................................................. 54 Figure 6-7. a) Strain fractions at different shear rates, b) The slip velocity of TPVs as a function of wall shear stress ........................................................................................................................ 55 Figure 6-8. KBKZ extensional predictions using power-law relaxation function. ...................... 55 Figure 7-1. The linear viscoelastic moduli of all TPVs listed in Table 4-2 at (a) T=190oC, (b) 205°C, and (c) 220 °C. .................................................................................................................. 58 xiii Figure 7-2. Fitting the universal viscoelastic moduli of TPVs with the model of Eq 7-1 to determine the parameters listed in Table 7-1 and Table 7-2. ...................................................... 59 Figure 7-3. (a) The apparent flow curves of L-PP using three capillary dies having the same D and different L/D at 205oC, (b) the Bagley plot to determine the ends pressure, (c) the ends pressure versus the apparent shear rate (d) the flow curves corrected for the ends pressure effects........................................................................................................................................................ 62 Figure 7-4. The flow curves of PPs and EPDMs at 190 ℃. While slip effects in PP are minimal, EPDMs do slip significantly at high shear rates (significant differences between LVE and capillary flow curves) ................................................................................................................... 63 Figure 7-5. The flow curves of PPs and EPDMs at 205 ℃. While slip effects in PP are minimal, EPDMs do slip significantly at high shear rates (significant differences between LVE and capillary flow curves) ................................................................................................................... 64 Figure 7-6. (a) The apparent flow curves of TPV1 using three capillary dies having the same D and different L/D at 190oC, (b) the Bagley plot to determine the ends pressure, (c) the ends pressure versus the apparent shear rate and (d) the flow curves corrected for the ends pressure effects. ........................................................................................................................................... 65 Figure 7-7. (a) The flow curves of TPV1 using two capillary dies having the same L/D and different D at 190oC to demonstrate slip effects in TPVs. ............................................................ 66 Figure 7-8. Start-up of steady shear tests of TPV1 at various shear rates at 190C. ................... 67 Figure 7-9. The slip velocity of TPV1 from capillary and parallel-plate rheometers as a function of wall shear stress at 190oC. ........................................................................................................ 68 Figure 7-10. The slip velocity of EPDMs at 190 and 205C, compared to that of TPV1 at 190C........................................................................................................................................................ 68 Figure 7-11. The flow curves of all TPVs listed in Table II at 190°C and the corresponding slip velocities at 190C. ....................................................................................................................... 69 Figure 7-12. Extrudate images of L-PP (first row) and B-PP produced (second row) by capillary extrusion at 190C. ....................................................................................................................... 71 Figure 7-13. Extrudate images of L-EPDM (first row) and B-EPDM (second row) produced by capillary extrusion at 190C. ........................................................................................................ 72 Figure 7-14. Typical extrudate images and their corresponding surface profiles that exhibit minor and severe surface distortions. ............................................................................................ 73 xiv Figure 7-15. The effect of curing level (TPV2 and TPV3 at 190℃ ) and that of temperature (TPV3 at 190 and 205℃) on the surface extrudate distortion of TPVs. ....................................... 75 Figure 7-16. Comparison of extrudate surfaces with different amount of crosslinked rubber (TPV1 and TPV5 at 190 ℃) ......................................................................................................... 76 Figure 7-17. Comparison of extensional behavior of TPV4 and TPV5 with those of their constituent components. The data for TPV5 and TPV4 have been multiplied by 6 times for the sake of clarity. ............................................................................................................................... 78 Figure 7-18. Extensional behavior of a) TPV4, and b) TPV 1 at different temperatures. ........... 78 xv List of Symbols 𝐶𝑡 Cauchy-Green tensor 𝐶𝑡−1 Finger tensor oC The degree Celsius (centigrade) D Die diameter (m) Ea Energy of activation (J/mol) f Fraction G Relaxation modulus (Pa) G* Complex modulus (Pa) G' Storage modulus (Pa) G" Loss modulus (Pa) 𝐺𝑐" Loss modulus coefficient (constant) (Pa) 𝐺𝑐 Relaxation modulus constant (proportionality constant) Gy Equilibrium relaxation modulus (Pa) 𝐺𝑦′ Equilibrium storage modulus (Pa) 𝐺𝑐′ Storage modulus coefficient (constant) (Pa) ℎ(𝛾) Damping function 𝐼𝐶𝑡 First invariant of the Cauchy-Green tensor L Die length (m) m (t-t’) Memory function Mw Weight average molecular weight (g/mol) n Material constant used in the damping function N1 First normal stress difference (Pa) N2 Second normal stress difference (Pa) p Power-law exponent 𝛥𝑃 Pressure drop (Pa) xvi 𝛥𝑃𝑒𝑛𝑑 Ends pressure (Pa) Q Volumetric flow rate (m3/s) 𝑅 Die radius (m) T Temperature (K) t Time (s) 𝑉𝑠 Slip velocity (m/s) Greek symbols β Material constant used in the damping function 𝛾 Strain ?̇? Shear rate (s-1) ?̇?𝐴 Apparent shear rate (s-1) ?̇?𝐴,𝑆 Apparent shear rate corrected for the effect of slip (s-1) 𝛾𝑦 Yield strain 𝛾𝑛 Nominal strain ?̇?𝑛𝑅 Nominal shear rate (s-1) ?̇?𝑅 True shear rate (s-1) 𝛿 Damping factor 𝜀̇ Hencky strain rate 𝜂 Viscosity (Pa.s) 𝜂∗ Complex viscosity (Pa.s) 𝜂𝐸+ Uniaxial tensile stress growth coefficient (Pa.s) 𝜂+ Shear stress growth coefficient (Pa.s) 𝜃 Material constant defining the ratio of the normal stress differences λ Relaxation time (s) 𝜎 Stress (Pa) 𝜎𝐴 Apparent shear stress (Pa) xvii 𝜎𝑦 Shear yield stress (Pa) 𝜎𝑊 Wall shear stress (Pa) 𝜎𝑦,𝐶 Crossover yield stress (Pa) 𝜏𝑅 Characteristic time (s) 𝜔 Frequency (s-1) xviii List of Abbreviations CD Crosslinked density ENR Epoxidized natural rubber EPDM Ethylene propylene diene monomer EOC Ethylene-octene copolymer EVM Ethylene-vinyl acetate rubber GMF Gross melt fracture HDPE High-density poly(ethylene) K-BKZ Kaye-Bernstein-Kearsley-Zapas LCB Long chain branching LVE Linear viscoelasticity NBR Nitrile butadiene rubber NR Natural rubber PDI Polydispersity index PLA Poly(lactic acid) PP Poly (propylene) PVDF Poly (vinylidene fluoride) SiR Silicon rubber TPE Thermoplastic elastomer TPV Thermoplastic vulcanizate tTS Time-temperature superposition xix Acknowledgements First and foremost, I would like to thank my supervisor Prof. Savvas G. Hatzikiriakos for his sincere support, advice and encouragement throughout my PhD research. Without his guidance, this work would never have been done. I would also like to acknowledge my supervisory committee members Professor Mark Martinez, Dr. Sheldon Green, Dr. Antonios Doufas and Dr. Krishnan Iyer for their time and helpful comments. Sincere gratitude to my wonderful colleagues and friends Shohre, Shiling, Tanja, Benjamin, Marziyeh, Tina, Ziyue, Marina, Muaad, Zhe, Shahryar, Yu, Zahra, and Ali for their heartwarming friendships and for their great support along my journey over the past three years. I would also like to thank my roommate Parya for all her support, encouragement, and effort to create a peaceful environment to live. My acknowledgement also goes to ExxonMobil Chemical Co. for their research grant that provided the necessary financial support and the materials for this research. The most profound and loving thanks go to my family; my wonderful sisters and brothers, Soybeh, Pourya, Sandia, and Arya for their endless love, encouragement and caring, and my cousin Pardis who was always there through my ups and downs. My deepest gratitude goes to my parents, Hedayat and Najimeh whom supported me and loved me each and every day. xx Dedication To my lovely parents مزیزع ردام و ردپ هب میدقت1 1 Introduction Generally, polymer blends can be categorized into two broad classes: miscible and immiscible. In the first category polymers exist in a homogeneous, single phase state with properties typically different from their primary components. In the second category, the blend forms two distinct phases. There is a third category of polymers blends which is technologically compatible blends or alloys. These blends exist in two or more phases in micro-scale but show macroscopic properties similar to a single phase material [1]. Thermoplastic elastomers (TPEs) are immiscible polymer blends that contain a thermoplastic and a rubbery phase (category 2). Therefore, while they show high elastic properties similar to traditional vulcanized rubbers, they have a good processability analogous to thermoplastics [2]. TPEs are widely used in footwear, wire insulation, medical devices and sporting goods and adhesives [3,4]. TPEs can be classified into two following categories. The first category is thermoplastic block copolymers and the second thermoplastic blends including thermoplastic-rubber blends undergoing a dynamic vulcanization process. These blends are referred to as thermoplastic vulcanizates (TPVs). In TPVs, a high content of rubber (> 50 𝑤𝑡.%) is simultaneously crosslinked and dispersed in a thermoplastic matrix [2]. Different TPV blends such as ethylene propylene diene monomer (EPDM)/polypropylene (PP) [2,5–11], ethylene-octene copolymer/polypropylene (PP) [12], epoxidized natural rubber/PP [13], natural rubber /PP [14], Natural rubber/high density polyethylene (HDPE) [15], nitrile butadiene rubber/PP TPVs [16], oil resistant ethylene-vinyl acetate rubber /poly(vinylidene fluoride) (PVDF) TPVs [17], high temperature resistant silicon rubber /PVDF TPVs [18], bio-based TPVs, such as poly (lactic acid) /Natural Rubber TPVs [19] have been studied in the literature. The subject of the present thesis is rheology and processing of TPVs. Most of TPV properties such as mechanical, electrical, and rheological properties are highly influenced by their microstructure [10,20,21]. In order to reach the matrix disperse morphology at the end, a co-continuous morphology is required at the beginning of the melt mixing process [1] and the inversion from a co-continuous phase to a matrix disperse phase is the key factor in the 2 fabrication of TPVs [2]. It should be mentioned that a fine dispersion of rubber particles is required to obtain good mechanical properties [2]. Due to TPVs’ complex microstructure, a substantial knowledge of the parameters affecting their processability is required. Rheological properties determine the processing behavior of TPVs. Therefore, understanding thoroughly their rheological behavior can be extremely useful to optimize the processability of TPVs and reduce major processing problems i.e., flow instability (slip and melt fracture). Although the rheological properties of TPVs have been investigated to some extent, there is no systematic and comprehensive study on rheological properties of TPVs in terms of their structure and composition and their correlation with the processability. Thus, this thesis aims to perform a thorough experimental study on the structural and rheological properties of TPVs attempting to obtain a better understanding of their flow behavior and their relation with processability, essentially targeting at improving their processing. 1.1 Production of TPVs TPVs are usually produced by melt blending. They can be produced by either a continuous process using compounders (mixing extruders) or co-rotating twin screw extruders or batch process by internal mixers [1]. There are three main feeding procedure for producing TPVs, illustrated in Figure 1-1. In methods 1 and 2 rubber and plastic are mixed together at high temperature, and high shear rates. On the other hand, in method 3 rubber and crosslinking agent undergo a cold mixing to form a pre-mix. Afterwards, this premix is blended with the plastic and vulcanized simultaneously. The difference between method 1 and 2 is that in method 1, the rubber-plastic pre-mix goes through a cold mixing with the crosslinking agent., Then the dynamic vulcanization is applied to the mixture under high temperature and high shear rate. However, in method 2, the rubber-plastic pre-mix is mixed to crosslinking agent and vulcanized at the same time [2]. 1.2 Formation Mechanism Two factors are of high importance during the preparation of TPVs, namely the phase inversion and the fine dispersion of the rubber phase into the plastic matrix. Three different mechanisms have been proposed for phase inversion. Machado et al. [22] proposed a mechanism for the phase inversion in the manufacturing of PP/EPDM TPVs depicted in Figure 1-2. They hypothesized that before vulcanization of the rubber phase, the plastic droplets are dispersed in the rubber matrix 3 due to the high phase ratio of the rubber to the plastic phase (Figure 1-2-a). Then at the early stage of the dynamic vulcanization, the viscosity of the rubber phase increases which results in a co-continuous structure (Figure 1-2-b). Subsequently, the rubber phase forms an unstable fibrillar structure (Figure 1-2-c). Finally, with more increment in the crosslinking density, the viscosity of the rubber phase increases dramatically which results into an increase of the stress on the rubber phase and as a result the rubber phase breaks up into particles (Figure 1-2-d). Figure 1-1. Schematic figure of preparation methods of TPVs adopted from [2] Goharpey et al. [23] proposed another mechanism, depicted in Figure 1-3. The proposed model consists of four stages. First, the melt-melt dispersion before the onset of curing where the rubber droplet breakup and coalescence occur simultaneously. In the second stage, the rubber micro-droplets formed by the breakup of the rubber phase, cannot coalescence, instead, they can be transformed into micro-particles forming agglomerations. As the vulcanization proceeds, the interfacial adhesion between the two phases increases. In the last stage, due to the shrinkage of the 4 rubber particle (lower the surface energy), the rubber agglomerations break down to rubber micro-particles which results into a fine dispersion and thus phase inversion. Figure 1-2. Morphological development mechanism proposed for the phase inversion of EPDM/PP blends during dynamic vulcanization adopted from [2,22]. Figure 1-3. Schematic diagram of the proposed mechanism for the development of PP/EPDM blends during dynamic vulcanization adopted from [2,23]. 5 2 Literature Review One of the main advantages of TPVs over traditional cured rubbers is their processability, which is determined by their rheological behavior. Therefore, rheological studies of TPVs have received significant attention in the literature. The rheological properties can be studied by using parallel-plate modular compact rheometers as well as capillary rheometers. It should be noted that since TPVs contain a high amount of cured rubber particles, their rheological properties are significantly different than uncured blends due to the formation of a strong cured rubber network [2]. In order to do rheometry with modular compact rheometers, polymeric samples are placed between two circular plates (parallel-plate geometry). The upper plate is gradually lowered to touch the sample and compresses it until building up a high normal force. Due to buildup of the high normal force, the gap cannot be controlled at an exact specified value, and as such cone-and-plate geometry is not a good choice for TPVs. Instead, the parallel-plate geometry having a normal force controlled moving profile is a better choice. It should also be noted that during sample preparation for the rheometry, if the measuring plate approaches the sample at high velocity, a high normal force builds up abruptly which cannot be subsequently relaxed. Therefore, a low velocity moving profile near the sample surface coupled with stopping controlled by normal force is suggested as a good moving profile before starting the rheological measurements. Another important point is that to reduce slip, the TPV samples should be placed in the convectional oven at the desired temperature for around 30 minutes before starting the measurement. Due to the existence of the strong rubber network, TPVs are expected to show yield stress [24] as well they are expected to slip violating the no-slip boundary condition of fluid mechanics [25,26]. Thus, in addition to structural parameters such as rubber/thermoplastic ratio, pure polymers’ structure, and curing content, two major phenomena that dictate the rheological behavior of TPVs are yield stress and wall slip. Therefore, in this thesis, emphasis is placed on the study and explanation of these two phenomena. Finally, the melt fracture of the TPVs as a criterion for their processability has been emphasized in this thesis as well. Melt fracture is manifested as surface distortion on the surface of polymers when the flow exits a die and typically occurs if the flow rate exceeds a critical value. 6 2.1 Structural Parameters Affecting Rheology of TPVs TPVs microstructure highly influences their properties such as mechanical and rheological [10,20,21]. As it was mentioned above, TPVs’ rheological properties are of high importance because they affect their processability and can be used to understand and optimize it [27,28]. Since there is a strong rubber network in TPVs due to the presence of the cured rubber particles, their complex viscosity at low shear rates are much higher than that of the thermoplastic matrix [29–31]. There are other significant parameters such as the ratio of rubber phase to plastic phase, the size of the rubber domains, the curing level of the rubber phase, and the addition of plasticizers and fillers which determine their rheological properties [13,27–29,32–36]. It has been widely reported that by increasing the crosslinking agent concentration, the crosslinking density increases and consequently the complex viscosity is enhanced accordingly [1,18,23]. Generally, while a small amount of filler has only a minor effect, a large amount increases the viscosity of TPVs, [1]. It has also been reported that incorporating filler disrupts the rubber network and decreases the complex modulus, G*, of TPVs [37]. 2.1.1 Effect of polymer type Goharpey et. al. [23] studied several sets of PP/EPDM blends with different PP viscosity. They reported that by increasing the PP melt viscosity, the EPDM/PP viscosity ratio decreases, which results in a better dispersion of rubber aggregates and consequently more stabilized morphology and better mechanical properties of the TPE samples. Prut et. al. [38] also compared PP/oil-free EPDM TPVs to PP/Oil-extended EPDM ones and reported that the processability of oil-extended TPVs are significantly better than the oil free ones. 2.1.2 Effect of the composition ratio (R/P) With increasing the rubber concentration in TPVs, the number and density of the rubber particles increase and form a strong rubber network. In addition, the rubber phase has much higher shear viscosity than that of the thermoplastic phase. Therefore, the viscosity increases with the concentration of rubber particles. Meanwhile, because the thickness of the plastic ligaments decreases with increasing the rubber content, it also causes an increase in the viscosity of TPVs and deterioration of processability [2]. 7 2.1.3 Effect of the content of crosslinking agent It has also been widely reported in various TPVs systems that with increase of the crosslinking agent both the crosslinked density (CD) and the viscosity of TPVs increase, which can have a negative impact on their processability [2,17,27]. Although the rigidity of the rubber particles increases with CD, it decreases the deformability of the system, resulting in strengthening of the rubber network in TPVs [29,39]. Moreover, an increase in the curing agent concentration could result in the formation of smaller rubber particles in TPVs, and consequently an increase in viscosity of TPVs [34]. Unbound/uncontrolled increase of viscosity typically have negative effects on the processability. 2.2 Yield Stress TPVs contain a high amount of rubber particles dispersed in a polypropylene matrix. As such they are expected to possess a yield stress. At large deformations and shear rates, the continuous rubbery network will eventually yield and flow and possibly destruction and reformation of the network will be occurring [18,27]. Determining the yield stress is of high importance in studying the flow behavior of these materials as it can be correlated with the curing degree, percentage of curing agent and level of particle dispersion (quality control). Li et al. [24] reported that the higher rubber content results in higher modulus, viscosity and yield stress in PP/EPDM TPVs. 2.2.1 Ways of obtaining yield stress There are different ways of measuring and calculating yield stress such as fitting models to their flow curve (shear stress versus shear rate), creep test experiments, oscillatory amplitude sweep, as well as strain recovery experiments [40]. Among them, fitting of the flow curve and oscillatory amplitude sweep are the most common. Examples on yield determination are given below. To determine the yield stress using flow curve, the experimental data are fitted by an equation and yield stress would be the extrapolation of shear stress at diminishingly values of the shear rate (Figure 2-1). On the other hand, calculating yield stress based on oscillatory amplitude sweep can be done via two methods. First, the stress that the material starts deviating from linear viscoelastic behavior that represents the point where initial plastic deformations are forming. Second, stress related to the crossover of storage and loss moduli can also be used particularly when the loss 8 modulus exhibits a maximum which indicated the occurrence of flow. The crossover could be related to yield stress since it points out to a strain above which loss modulus gets a higher value than storage modulus; therefore, dissipation or fluid like behavior surpasses solid like behavior. It is noted that this point should be considered as yield stress with caution because the definition of 𝐺′ and 𝐺′′ is only valid in linear viscoelastic region. However the crossover is definitely in the non-linear region [40]. Both methods are presented in Figure 2-2. One way to define the onset of non-linearity initiation is to find a point where the storage modulus drops to 90% of its limiting (plateau) value [41]. Using the strain related to this point and stress-strain curve, the yield stress can be calculated. Figure 2-1. Typical flow curve of a yield stress fluid showing the extrapolation to shear rate equal to zero. Figure 2-2. Using 90% and crossover methods to find yield stress in strain amplitude sweep test a) viscoelastic moduli vs. shear strain amplitude, and b) Stress vs. shear strain amplitude obtained from the same test. (Typically, these curves are plotted in a log-log scale) 9 The flow curves can be obtained using both capillary and modular compact rheometers. However, since TPVs show wall slip due to the presence of strong rubber network, determining the yield stress from flow curves from capillary rheometry might not be of high accuracy. Yield stress in simple shear flow field has been well characterized [40,42,43]. However, polymeric materials experience complex flow fields including elongational components in many real processes such as extrusion, squeezing, blade coating, etc. where the material undergoes a shrinkage in one direction while being extended in the perpendicular direction [44]. Therefore, studying the extensional yield stress would be extremely helpful to understand the behavior of yield stress fluids in these flow fields. 2.3 Slip Velocity Due to the presence of oil and rubbery phase these systems are expected to exhibit wall slip phenomena. It has been reported that molten linear polymers such as polyethylene, and elastomers slip no matter how small the applied shear stresses are. In addition slip velocity increases with increase of temperature, decrease of molecular weight and polydispersity [45,46]. A review of wall slip phenomena in polymers can be found in [45]. Ebrahimi [47] has explained different methods for analyzing experimental data to determine the wall slip of polymers in detail. A common method to determine slip velocity is based on the gap dependence measurements of flow curves in simple geometries (parallel plate and capillary), known as Mooney method [48]. In this method, the flow curves at different gaps are determined experimentally and if there is a gap dependence Figure 2-3-a, the experimental data from simple shear flow can be analyzed based on Equation 2-1, which can be easily derived by using the velocity profiles under slip and no-slip depicted in Figure 2-4. 10 Figure 2-3. Depiction of Mooney method a) flow curve at different gaps, and b) nominal shear rate vs. 1ℎ at a specific wall stress. ?̇?𝑛𝑅 = ?̇?𝑅 +2𝑉𝑠ℎ 2-1) where ?̇?𝑛𝑅 and ?̇?𝑅 are nominal and true shear rates at the edge of the disk, respectively, 𝑉𝑠 is slip velocity and ℎ is the gap between the plates. At a fixed shear stress value, the ?̇?𝑛𝑅 is plotted versus 1/h, with the slope of the line to be equal to 2Vs (Figure 2-3-b). The procedure is repeated for several wall shear stress values to result the functional dependence of slip velocity on wall shear stress. Figure 2-4. Velocity profiles in simple shear flow under no-slip and slip boundary conditions. The slip velocity in pressure driven capillary flows (Figure 2-5) can be calculated from Eq. 2-2 ?̇?𝐴 = ?̇?𝐴,𝑆 +4𝑉𝑠𝑅= ?̇?𝐴,𝑆 +8𝑉𝑠𝐷 2-2) 11 where ?̇?𝐴, and ?̇?𝐴,𝑆 are the apparent shear rate and apparent shear rate corrected for the effect of slip, respectively, 𝑅 is the radius and 𝐷 is the diameter of the capillary die [47]. Similar to simple shear, at a fixed shear stress value the apparent shear rate ?̇?𝐴 is plotted versus 1/D, with the slope of the line to be equal to 8Vs. The procedure is repeated for several wall shear stress values to result the functional dependence of slip velocity on wall shear stress. Figure 2-5. Velocity profiles in capillary flow under no-slip and slip boundary conditions. Another method to determine slip velocity is based on the deviation of flow curve from LVE [45]. A similar approach to the gap dependency is used here, as well (Figure 2-6), which means that at a specific shear stress, the slip velocity is obtained from Eq. 2-3 8𝑉𝑠 = (?̇?𝐴 − ?̇?𝐴,𝑆). 𝐷 2-3) Figure 2-6. Flow curve deviation from LVE which can be used to calculate slip velocity. 12 2.4 Melt Fracture During the extrusion of most polymers when the wall shear stress exceeds a critical value, melt fracture phenomena, such as sharkskin (small amplitude periodic distortions occurring on the surface of the melt), stick-slip or oscillating melt fracture and gross melt fracture (GMF) (large amplitude/ chaotic distortions affecting the whole volume of the melt), have been known to occur [49,50]. Figure 2-7 shows images for the various types of melt fracture. Typically, by increasing shear rate, extrudate shows smooth surface, sharkskin, stick-slip, and gross melt fracture respectively (Figure 2-7). Sharkskin is of great importance as it is the first instability that occurs and thus the first problem to be addressed. This periodic distortion that is only limited to the surface of the extrudate, is a function of die geometry, temperature, and molecular structure [49]. Stick-slip instability is characterized by oscillation in the pressure or shear rate in controlled throughput capillary extrusion. It can also be observed on the extrudate surface as alternating rough and smooth regions resulting from the transition from weak to strong slip condition. It can be also distinguished from a plateau or discontinuity (a transition zone) in the flow curve (Figure 2-8). Typically, stick-slip instability occurs at shear rates above sharkskin and below gross melt fracture. However, there are cases in which gross melt fracture happens at shear rates below stick-slip melt fracture. Stick-slip is dependent on capillary length, capillary length-to-diameter ratio, die temperature, molecular weight, and molecular weight distribution. GMF is a profound chaotic distortion that involves the whole cross section of the extrudate. Therefore, it is frequently referred to as “volume melt fracture” as opposed to sharkskin which is a “surface melt fracture”. GMF is a result of severe instability of the melt at the entrance of a converging die while sharkskin forms at the die exit. Although helical melt fracture is a type of GMF, sharkskin can also cause a helical distortion that is limited to the surface. Since sharkskin occurs at shear rates lower that GMF, it is an important issue as it restricts the production rate and needs to be first addressed [49]. 13 Figure 2-7. Various types of melt fracture. Adopted from [49]. Figure 2-8. Typical extrudate instability occurrence on a flow curve 14 These phenomena can be problematic in the processing of polymers by setting an upper limit to the throughput applied during extrusion. PPs and EPDMs (basic constituents of TPVs) exhibit these phenomena. Many reports exist in the literature for PPs [35,36] and EPDMs [51,52]. However, very few studies exist on the TPV melt fracture in the literature. Most of these studies are not systematic as they do not address explicitly the role of the molecular characteristics of constituent polymers of TPVs, their relative composition, curing degree and temperature. The single study by Wu et.al. [53] on TPV melt fracture reported that by increasing the time of TPV extrusion via performing the extrusion twice, also with increasing the extrusion temperature and time, smoother extrudate surfaces are achieved. 2.5 Knowledge Gap PP/EPDM TPVs have been investigated in both industry and academia to a certain extend. There are several studies specifically on linear [5,6,11,23,54] and nonlinear [9,55] rheological properties, studying mainly the effects of various parameters such as the amount of processing oil, curing level and composition, and molecular architecture of constituent polymers on TPVs’ microstructure and properties. To the best of my knowledge, there is no study on the influence of microstructure and molecular architecture on both linear and non-linear rheological properties of TPVs in both shear and extension. For instance, how yield stress values determined in shear manifest themselves in extension. While the rheology of TPVs in drag shear flow has been studied to a certain extent, their flow behavior in pressure driven flows needs further attention [24,56]. The presence of oil and rubbery phase causes TPV systems to exhibit wall slip phenomena [45,46]. Such studies on the wall slip phenomena of TPVs during their flow in contact with solid walls do not exist. It is extremely useful to derive slip velocity relationships for these systems in modeling TPV extrusion for process optimization. Moreover, these highly elastic systems exhibit melt fracture phenomena that manifest themselves as distortions on the surface of extruded TPV samples [53]. While several melt fracture studies exist for polymer melts such as polyethylenes [57–59], polypropylenes [60,61], polystyrenes [62], polydimethylosiloxanes [63], polylactides and polycaprolactones [64–66], fluoropolymers [67], 15 polybutylenes [68], ionomers [69] as well as polymer blends and composites [70–73], specific studies focused on TPVs are rare. To wrap it up, as rheological properties of TPVs determine their processability, there is an evident need to have a deep understanding of rheological behavior of TPVs to reduce their significant processing flaw in the industry which is melt fracture. Therefore, the three main elements of the present thesis, is (i) a systematic rheological study of a series of TPVs having different compositions including determination of yield stress (ii) study of the slip behavior of TPVs in simple shear and pressure driven drag flows (iii) a comprehensive study of the instabilities of TPVs in capillary flow. The objectives of this PhD thesis are discussed in detail in the next chapter. 16 3 Objectives and Thesis Organization 3.1 Research Objectives This work is essentially aiming to use rheology as a powerful tool to completely understand the flow behavior of TPVs and subsequently reduce or eliminate TPVs’ processing troubles originated from flow instabilities. The main objective of this work is to perform a thorough systematic study on the linear and non-linear rheological behavior of TPVs, determine the main parameters affecting their flow instabilities and correlate rheological parameters to their processability as this can be reflected from the occurrence of flow instabilities such as melt fracture. The main question to answer is how to optimize the formulation of TPVs to avoid and/or postpone flow instabilities such as melt fracture (distortions on the surface of final products) to higher rates of production. In more detail, the following objectives of this PhD study are: 1 To perform a thorough rheological characterization of several TPVs and the critical parameters that affect both their linear and nonlinear (shear and extensional) rheology. The rheology of TPVs will be compared with that of their constituent components, namely the uncured EPDM and PP. 2 To determine and draw relationships between the yield stress values in shear and extension and see how the yield stress values determined in shear (easily observed) manifest themselves in extension (never studied and observed before). 3 To study the effect of the degree of curing and plastic/rubber (P/R) ratio on the rheological properties of TPVs in relation to the rheology of their main constituents, PP and EPDM 4 To study the wall slip of TPVs in simple shear and capillary flow relative to the slip behavior of their constituent individual components (PP and EPDM). 5 To model the rheological behavior of PPs, EPDMs, and TPVs using the integral constitutive relationship of K-BKZ 6 To study the melt fracture behavior of several TPVs as a function of shear rate, temperature, molecular characteristics of constituent polymers, degree of curing, and concentration of 17 filler will be studied relative to the melt fracture of their individual components (PP and EPDM). 7 To develop interrelationships between the rheological properties (yield stress) and processability. 3.2 Thesis Organization The organization of this dissertation is as follows. A brief introduction of thermoplastic vulcanizates (TPVs), their properties and fabrication mechanism is presented in Chapter 1. In Chapter 2 a literature review on the structural and processing parameters affecting TPVs’ rheological and processing properties is discussed. The research objectives and thesis organization are presented in Chapter 3. In Chapter 4, the materials used in this study are presented, their composition and their method of fabrication. Details of the apparatuses used in this study, including a rotational rheometer (AntonPaar MCR 502 or AntonPaar MCR 702 equipped with cone-and-plate or parallel-plate geometry and partitioned plate fixture), a capillary rheometer (Instron), an Olympus MIC-D digital microscope (to take images of the polymer samples extruded) are also presented in Chapter 4. The experimental methodology associated with these experimental techniques is also described. Chapter 5 is focused on the linear rheological properties of TPVs and their constituent polymers used in their fabrication. Use of the K-BKZ model to model the rheological properties is also presented. The yield stresses of these materials were determined in both shear and extensional flows (objectives 1, 2, and 3). This chapter is based on a published paper (N. Ghahramani, K. Iyer, A.K. Doufas and S.G. Hatzikiriakos, “Rheology of Thermoplastic Vulcanizates (TPVs),” J. Rheology, 64, 1325-1341 (2020) Chapter 6 presents the non-linear rheological behavior of TPVs behavior in shear and extensional flow fields. The K-BKZ constitute model is considered to model their rheological behavior including the inclusion of yield stress and wall slip (objectives 4, 5). This chapter is based on a paper recently submitted to Physics of Fluids. Finally, in Chapter 7 the wall slip and melt fracture of TPVs are investigated in order to gain a better understanding of TPVs’ processing. Relationships between yield stress and extensional 18 rheological properties with the processability of these polymers are also presented and discussed. The flow instabilities (melt fracture) are also studied in capillary extrusion and detailed images are presented to identify and illustrate these instabilities. (Objectives 6 and 7). This thesis is concluded in Chapter 8 where recommendations for future work are also presented. 19 4 Materials and Experimental Methods 4.1 Materials Two different polypropylenes (PPs, one linear labelled as L-PP and one branched labelled as B-PP) and two ethylene propylene diene monomer rubbers (EPDMs, one linear labelled as L-EPDM and one branched labelled as B-EPDM) were obtained from ExxonMobil Chem., Co. (Baytown, TX, USA) listed in Table 4-1, which are the main constituents of the fabricated TPVs listed in Table 4-2. Table 4-1 lists the molecular characteristics of the basic constituent resins, including the weight molecular weight Mw, Polydispersity Index (PDI), and the degree of long chain branching (LCB-g’). Both EPDMs contain oil (L-EPDM: 42.9 wt. %, B-EPDM: 50 wt.%). Table 4-1. Polymers that are the main constituents of the TPVs studied in the present work and their molecular characteristics. Sample* Mw (kg/mol) PDI LCB-g’ Activation energy, Ea (kJ/mol) Vertical shift factor (bT) L-PP (linear) 562 7.5 1.00 18.5 1.00 to 1.08 B-PP (branched) 540 19 0.86 26.5 1.00 to 1.25 L-EPDM (linear) 610 2.5 1.00 13.9 1.00 to 1.05 B-EPDM (branched) 509 4 0.87 15.5 0.97 to 1.00 Table 4-2 list the various TPVs and their compositions examined in this work. These materials were also obtained from ExxonMobil for this study. Table 4-2 specifies two groups of TPVs. The main difference is that TPVs in Group A (PP: 27.5 wt. %, EPDM: 72.5 wt. %) contain a higher % wt. of crosslinked rubber and higher % wt. of PP compared to those of Group B (PP: 35 wt. %, EPDM : 65 wt. %). As such, TPVs of Group B have higher hardness (70-75 Shore A) relative to the TPVs of Group A (60-65 Shore A). As also seen the level of curing or curing degree varies as an important factor to be studied in the present work. The amount of EPDM used in all TPVs was 20 more than 50wt% in PP/EPDM blends, which will be evident from the linear viscoelastic properties of these polymers [1,27]. This set of TPVs are mostly commercial. A new series of TPVs has been prepared by ExxonMobil where the composition has been varied in a more systematic way (Table 4-3). The rheological study of these series can shed light into the relative effects of various parameters related to their composition. Table 4-2. Characteristics of TPVs used in this study (commercial polymers). Sample PP-Type EPDM-Type Cure level (Group A) TPV1 B-PP L-EPDM High TPV2 L-PP L-EPDM High TPV3 L-PP L-EPDM Normal TPV4 L-PP B-EPDM Normal (Group B) TPV5 B-PP L-EPDM High TPV6 L-PP L-EPDM High TPV7 L-PP L-EPDM Normal TPV8 L-PP B-EPDM Normal Table 4-3. Characteristics of TPVs whose composition is varied systematically. Formulation (phr) B-EPDM L-EPDM L-PP B-PP filler oil curative accelerative total S1 175.00 - 45.30 - 44.00 66.75 5.00 1.11 337.16 S2 175.00 - 45.00 - 44.00 65.00 7.50 1.67 338.17 S3 175.00 - 44.54 - 44.00 62.38 11.25 2.51 339.68 S4 175.00 - - 45.30 44.00 66.75 5.00 1.11 337.16 S5 175.00 - - 45.00 44.00 65.00 7.50 1.67 338.17 S6 175.00 - - 44.54 44.00 62.38 11.25 2.51 339.68 S7 - 200.00 45.30 - 44.00 41.75 5.00 1.11 337.16 S8 - 200.00 45.00 - 44.00 40.00 7.50 1.67 338.17 S9 - 200.00 44.54 - 44.00 37.38 11.25 2.51 339.68 S10 - 200.00 - 45.30 44.00 41.75 5.00 1.11 337.16 S11 - 200.00 - 45.00 44.00 40.00 7.50 1.67 338.17 S12 - 200.00 - 44.54 44.00 37.38 11.25 2.51 339.68 *All samples are cured using a phenolic resin with the amounts listed in the table **Acid chloride salts are used as accelerator 4.2 TPVs Sample Preparation ExxonMobil Co. prepared all TPV samples used in this work. The procedure of sample preparation is as follows: 21 EPDM was fed into the feed throat of an extruder. The thermoplastic resin (polypropylene) was also fed into the feed throat along with other reaction rate control agents such as zinc oxide and stannous chloride. Fillers, such as clay and carbon black, were also added into the extruder feed throat. Process oil was injected into the extruder at two different locations along the extruder. The curative was injected into the extruder after the rubber, thermoplastics and fillers commenced blending. The second process oil (post-cure oil) was injected into the extruder after the curative injection. Rubber crosslinking reactions were initiated and controlled by balancing a combination of viscous heat generation due to application of shear, barrel temperature set point, use of catalysts, and residence time. The extrusion mixing was carried out at 325 revolutions per minute (RPM). A barrel metal temperature profile in °C, ranging from barrel section near feed throat down towards the die was maintained at 160 to180 °C (wherein the last value is for the die) was used. The TPV samples were cut to fit into an atomic force microscope (AFM) sample holder (Leica) and then cryo-microtomed in the desired plane at -120 °C using a Leica Ultramicrotome. All samples were dried in a desiccator before AFM imaging. The AFM images were collected using a Bruker Dimension Icon AFM. The scans were acquired with a cantilever of spring constant of 42 N/m, and resonance frequency of ~320 kHz. The data channels monitored were height and phase. Figure 4-1 below presents typical TPV morphologies of two of the TPVs examined in this work indicating bi-continuous morphology with typical domain sizes of the order ∼ 1𝜇𝑚. Figure 4-1. Representative AFM micrographs showing the morphology in two TPV samples: a) TPV1 and b) TPV4 22 4.3 Rheological Measurements 4.3.1 Rotational rheometer A rotational rheometer (Anton Paar, MCR-501) with parallel-plate geometry is used to study the linear viscoelastic (LVE) properties of the polymers listed in Table 4-1 to 4-3. The following tests are performed to study their rheological response. First frequency sweep tests are carried out at different temperature ranges depending on the polymer (190-220oC for PP, 130-210oC for EPDMs, 190-220oC for TPVs). Maximum temperature for EPDMs is slightly lower to avoid ant degradation or crosslinking at high temperatures. Then master curves of the linear viscoelastic moduli are constructed using the time-Temperature Superposition (tTS) principle and the results are presented at the reference temperature of Tref=190oC. The Anton Paar MCR 702 rheometer equipped with a cone partitioned-plate geometry (Figure 4-2-a) is used for experiments in the nonlinear viscoelastic region, such as stress relaxation and start-up of steady shear of pure componentsand TPVs. Using the partitioned plate geometry, significantly reduces the edge fracture which occurs during non-linear measurements using regular geometries [74]. Uniaxial extensional experiments are conducted at various temperatures same with those used in shear) using the second generation Sentmanat Extensional Rheometer (SER2) fixture, Figure 4-2-b. Specimen of these polymers are prepared as films with a thickness of about 0.6 mm (PPs and EPDMs) and 0.3 mm (TPVs), respectively. Various Hencky strain rates are used in the range of 0.1 to 5 s-1 for pure components and 0.001-0.5 s-1 for TPVs. 4.3.2 Capillary rheometer An Instron pressure driven constant-speed capillary rheometer, Figure 4-3, is used to study the effect the wall slip of TPVs in capillary flow and compare it with that obtained from rotational rheometry (Anton Paar MCR-702). This comparison will address the consistence of slip measurements. Two capillary dies of different diameter, D (0.43 mm and 0.89mm) and fixed length-to-diameter ratio, L/D=33 to keep the pressure approximately constant at a given shear stress value, are used to study the slip behavior of basic constituent polymers and TPVs. Capillary dies of fixed diameter (D = 0.89 mm) and various L/D (4, 14 and 33) are used to identify and apply 23 the end pressure correction to the experimental data in order to determine the true flow curves of PPs, EPDMs and TPVs. An Olympus MIC-D digital microscope is used for taking various images of the collected extrudate samples. Figure 4-2. Schematics of the a) rotational rheometer equipped with the cone partitioned-plate and b) uniaxial extensional fixture that is used in conjunction with a rotational rheometer. Figure 4-3. Schematic representation of the capillary rheometer. 24 5 Rheology of Thermoplastic Vulcanizates (TPVs) In this chapter, the rheology of several TPVs is studied with focus on their linear viscoelasticity. The rheological behavior of their base components, namely, PP and EPDM is also performed in detail. The effects of long chain branching of the primary constituents (prior to crosslinking), the level of curing, and the amount of crosslinked rubber on their rheology are studied in detail and found to affect the rheology to various degrees. The rheology of TPVs in relation to their base polymers is also discussed in detail. The rheological data are modeled by using an integral constitutive equation, known as the K-BKZ model which is presented next. 5.1 Theoretical Background The K-BKZ constitutive model proposed by Wagner [75] and modified by Luo and Tanner [76] was used to model the rheology of individual TPV constituents (PP and EPDM): ?̄̄? =11 − 𝜃∫ ∑𝐺𝑘𝜆𝑘𝑒𝑥𝑝 (−𝑡 − 𝑡′𝜆𝑘) 𝑒𝑥𝑝 [−𝑛 (𝛽𝐼𝐶𝑡−1 +(1 − 𝛽)𝐼𝐶𝑡 − 3)12] [𝐶𝑡−1(𝑡′) + 𝜃𝐶𝑡(𝑡′)]𝑑𝑡′𝑁𝑘=1𝑡−∞ 5-1) where λk and Gk are the relaxation times and relaxation modulus coefficients, N is the number of relaxation modes, n and β are material constants, and 𝐼𝐶𝑡, 𝐼𝐶𝑡−1 , are the first invariants of the Cauchy-Green tensor 𝐶𝑡 and its inverse 𝐶𝑡−1, the Finger strain tensor. The parameter n can be determined from step-strain experiments in simple shear by the determination of the single exponential Wagner damping function, ℎ(𝛾) = 𝑒𝑥𝑝( − 𝑛𝛾), and the parameter 𝛽 can be calculated from fitting uniaxial extension experiments. The material constant 𝜃 is given by 𝑁2𝑁1=𝜃1 − 𝜃 5-2) where N1 and N2 are the first and second normal stress differences, respectively. It is noted that 𝜃 is not zero for polymer melts, which possess a non-zero second normal stress difference [38]. Its usual range is between -0.1 and -0.2 in accordance with experimental findings and numerical simulations [38,39]. Its effect on the predictions are minor [77,78]. The relaxation modulus (Maxwell model) 𝐺(𝑡) = 𝐺𝑘 𝑒𝑥𝑝(−(𝑡 − 𝑡′)/𝜆𝑘) is suitable for the modeling of the individual constituent polymers (PPs and EPDMs) as presented and discussed 25 below. However, a much simple power–law form is suitable for the relaxation modulus of TPVs, nearly universal for all TPVs (discussed below in detail and used for modelling the TPV rheological behavior in Chapter 6). 5.2 Results and Discussion 5.2.1 Rheology of pure polymeric constituents of TPVs As explained above various rheological tests were performed to determine the parameters of the K-BKZ model for each polymer listed in Table 4-1. Figure 5-1- Figure 5-4 present the rheological results for all polymers in the same order as listed in Table 4-1. The linear viscoelastic moduli (G and G) have been used to determine the relaxation moduli and relaxation times {𝐺𝑘, 𝜆𝑘} of each polymer, the stress relaxation after the imposition of sudden strain data to determine the parameter n (in the damping function of Equation 5-1) and the uniaxial extensional data to determine the parameter (in the damping function of Equation 5-1). The normal stress ratio has been assumed constant and equal to -0.15 for all polymers in the absence of experimental data, although typical data for polymers melts report values between -0.2 and -0.1 [79]. As shown by Tomkovic et al. [80] the results are nearly insensitive to the value of -N2/N1 when this ratio is kept within the range of {0.1, 0.2}. The continuous lines in Figure 5-1 - Figure 5-4 represent the fits of the K-BKZ model. The fitted parameters in the K-BKZ model for all polymers simulated are listed in Table 5-1. The main conclusions drawn from Figure 5-1- Figure 5-4 are as follows: First, the linear viscoelasticity (LVE) of EPDMs are more complex compared to that of their PP counterparts as more Maxwell relaxation times are needed (typically 6 relaxation modes for PPs versus 10 relaxation modes for EPDMs). Due to their elastomeric nature, no terminal relaxation has been reached for both EPDMs; in fact, a tendency for a plateau in G at low frequencies is observed, which is typical for elastomers. The viscosity material functions and viscoelastic moduli of EPDMs are nearly one order of magnitude higher compared to those of PPs (Figure 5-5). Although B-PP is branched, its complex viscosity is lower than that of L-PP throughout the angular frequency range, which is due to its lower molecular weight. The slightly higher complex viscosity of B-EPDM compared to that of L-EPDM is also due to the branched structure of B-EPDM. 26 However, from the practical point of view both EPDMs have similar linear viscoelastic moduli (Figure 5-5). No strain hardening is observed for L-PP and L-EPDM (Figure 5-1-d and Figure 5-3-d) due to their linear molecular architecture. Unlike L-PP and L-EPDM, B-PP and B-EPDM exhibit strong strain hardening as can be observed from Figure 5-2-d and Figure 5-4-d, which is a sign of the presence of long chain branching in their structure. A stronger strain hardening is shown by B-PP compared to that of B-EPDM due to its higher degree of strain hardening (LCB-g values listed in Table 4-1). Similar conclusions can be drawn by inspecting the values of the energy of activation, Ea, listed in Table 4-1. First the Ea for both PPs are lower than the reported values in the literature 27-40 kJ/mol [81–83]. This might be attributed to the use of Nitrogen that avoided degradation. The activation energy, Ea, of B-PP is clearly higher than the one for L-PP due to its branched microstructure [59,84] . The Ea for L-EPDM (linear) is as low as Ea=13.9 kJ/mol. The EPDM has unsaturated covalent bonds that can react at temperatures above 145oC, and therefore the activation energy is low. Typical reported values for the Ea for EPDMs vary significantly 𝐸𝑎 = 10 − 50 kJ/mol [54,85,86] depending on the molecular architecture. For example, the Ea value for B-EPDM (branched) Ea=15.5 kJ/mol is higher compared to that of L-EPDM (linear) Ea=13.9 kJ/mol indicative of its long chain branching structure. 27 Figure 5-1. (a) The linear viscoelastic moduli G, G and complex viscosity |𝜂 ∗|, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂+and (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸+ for L-PP (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. 28 Figure 5-2. (a) The linear viscoelastic moduli G, G and complex viscosity |𝜂 ∗|, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂+and (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸+ for B-PP (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. 29 Figure 5-3. (a) The linear viscoelastic moduli G, G and complex viscosity |𝜂 ∗|, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂+and (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸+ for L-EPDM (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. 30 Figure 5-4. (a) The linear viscoelastic moduli G, G and complex viscosity |𝜂 ∗|, (b) the damping function ℎ(𝛾) in shear, (c) the shear stress growth coefficient in start-up of steady shear, 𝜂+and (d) the uniaxial tensile stress growth coefficient in start-up of uniaxial extension, 𝜂𝐸+ for B-EPDM (Table 4-1) at 190oC. Continuous lines are fits of the K-BKZ/Wagner model to the experimental data with the values of the parameters listed in Table 5-1. 31 Table 5-1. The Maxwell relaxation spectrum and nonlinear rheological parameters for the K-BKZ/Wagner model (Eq.4) of all polymers listed in Table 4-1 at Tref = 190°C. Relaxation time λk(s) Relaxation modulus Gk (Pa) Relaxation time λk(s) Relaxation modulus Gk (Pa) Sample L-PP Sample B-PP 0.009 95201 0.009 49263 0.070 40786 0.087 14008 0.373 19552 0.656 5294 1.971 5990 5.058 1565 10.370 1144 39.18 402 58.840 139 350.3 71 n=0.23, 𝛽 =1, 𝜃 = -0.15 n=0.28, 𝛽 =0.02, 𝜃 = -0.15 Sample L-EPDM Sample B-EPDM 0.177 34512 0.818 5196 0.008 60070 0.016 70989 0.820 18848 85.13 6422 9.007 4986 8.536 5863 2.357 11990 0.327 17403 0.379 22222 2.281 8772 0.025 53037 0.056 50479 0.0705 48082 0.818 8117 0.001 69134 0.004 123397 91.240 3486 0.157 26516 n=0.36, 𝛽 =1, 𝜃 =-0.15 n=0.41, 𝛽 =0, 𝜃 =-0.15 Figure 5-5. Comparison of the viscoelastic moduli of the two PPs and the two EPDMs listed in Table 4-1 at Tref=190oC. Note the much higher material functions of L-PP(linear) compared to B-PP (branched) and the similar material functions of the two EPDMs. It is also noted that EPDMs have in general material functions about 1 order of magnitude greater than those of PPs. 32 5.2.2 Linear viscoelasticity of TPVs As discussed above the linear viscoelastic behavior of all TPVs listed in Table 4-2 was studied in detail and compared to that of their basic constituents. The results are presented in this section. Strain sweep: Figure 5-6 presents strain sweep results for TPV8 at three different temperatures using a frequency of 0.1 Hz. As seen, the temperature does not have a significant effect on the elastic response, G (same observation for all TPVs examined). While G decreases with increase of temperature as expected, a small increase of G has been observed with increase of temperature (counterintuitive observation). The elastic modulus G of TPV is possibly dominated by the presence of rubber (entropic effect) giving rise to this small increase which has been universally observed for all TPVs. The effect of temperature is more significant on the viscous response of the material, G. In fact, the G exhibits a maximum, which is related to the onset of flow/shear melting (increase of viscous response), although flow should have been initiated at the point where G starts increasing [87]. The existence of this maximum is an indication of the existence of yield stress. In fact a sequence of processes start at the point where G starts increasing (increase of dissipation implying flow initiation) that is also the point where the stress (|𝐺 ∗|)-strain 𝛾 relationship deviates from linearity [43,87,88]. This is where destruction (yielding) of the rubber network starts and reconstruction due to the movement of matrix (PP) and thus the maximum in G appears [43,87–89]. Two methods to estimate the yield stress were used. The first one is the transition from linear to non-linear regime [42]. To consistently determine the limiting value of strain for linear viscoelastic response and the stress for shear yielding, 𝜎𝑦, we consider the strain at which G drops to 90% of its linear viscoelastic value at which the strain value is defined as 𝛾𝑦which we refer to as the yield strain [41] [39]. Table 5-2 lists the yield strain,𝛾𝑦, and the related yield stress, 𝜎𝑦, of different TPVs at 190oC. Typical values of 𝜎𝑦are in the range from 3.6 to 7.8 kPa. TPVs 1-4 (Group A) overall possess higher yield stresses due to the relatively higher amounts of crosslinked rubber particles. In addition, the effect of curing degree renders the materials stiffer and thus they fail at a smaller stress and strain. For example, compare TPV2 with TPV3 and TPV6 with TPV7 which only differ in the level of curing. TPV3 and TPV7 fail at a higher strain and shear stress. The more crosslinked system is more likely to form stronger rubber network (more discussion below) and hence yields at a lower strain. On the other hand, the less crosslinked rubber 33 might form smaller domains and can result in a weaker network that is more amenable to larger strains. Another method to determine the yield stress is using the crossover of G and G, labelled as 𝜎𝑦,𝐶 with values listed in Table 5-2 (3-4 times higher). It should be mentioned that although this point is traditionally associated with yield, it must be made with caution as the definition of G and 𝐺" are only valid in the linear regime and the crossover point is certainly in the non-linear flow regime [40]. However, although yield stress is defined as a single point, it should be realized that the process of yield occurs over a wide range of shear stress values (distribution of values rather than a single point), where parts of the fluid yield locally at different values particularly for inhomogeneous fluids. Therefore, the two yield stress values listed in Table 5-2 provide the range over which the yield process is initiated and completed (transition from solid-like to fluid-like behavior). Figure 5-6. Strain sweep experiments for TPV8 at various temperatures (190-220oC) to determine the limiting values of strain for linear viscoelastic response and the stress and strain for shear yielding (flow). 34 Table 5-2. Yield strains and related critical yield stresses of TPVs listed in Table 4-2 at T=190 ℃. Sample Yield strain, 𝜸𝒚 (%) Yield stress, 𝝈𝒚 (kPa) Yield stress at crossover, 𝜎𝑦,𝐶 (kPa) TPV 1 5.3 4.6 18.5 TPV 2 7.6 6.4 15.4 TPV 3 11.3 7.8 22.5 TPV 4 8.4 4.8 18.9 TPV 5 6.0 3.6 12.3 TPV 6 7.0 5.1 22.7 TPV 7 10.4 6.0 22.7 TPV 8 8.8 4.6 25.7 5.2.3 Viscoelastic spectrum of TPVs Several factors that influence the LVE of TPVs is examined in this section. Figure 5-7 depicts the viscoelastic moduli of TPV2 (left) and TPV3 (right) compared with those of their constituent polymers at Tref=190oC to see how the LVE response of the individual constituents of TPVs affects their LVE behavior. At higher frequencies, the TPV response is controlled by the segmental dynamics of its constituent polymers (Rouse time response) and as such the LVE moduli of TPVs are between those of their constituent polymers (closer to the response of the matrix which is the PP). However, at lower frequencies the effect of the cured EPDM interface with the matrix (PP) plays a significant role as there is no sign of terminal relaxation or tendency for relaxation. Both G and G show solid-like behavior with a small dependence on frequency (G much higher than G). This is typically observed for all TPVs independent of the type of constituent polymeric components, curing degree (high or normal), and level of crosslinked rubber concentration (high in Group A or low in Group B). Considering the particle network existence in TPVs, which results in a non-terminal behavior in storage modulus (storage modulus does not take value zero when frequency is going to zero), the “simple” mode of relaxation known as power-law relaxation typically observed in the response of foods such as dough and yogurt [58-59] can be applied: 𝐺′ = 𝐺𝑐′(𝜔𝜏𝑅)𝑝 + 𝐺𝑦′ 𝑎𝑛𝑑 𝐺′′ = 𝐺𝑐′′(𝜔𝜏𝑅)𝑝 5-3) Where 𝐺𝑐′ and 𝐺𝑐" are the storage and loss moduli coefficients (constants), 𝜏𝑅 is a characteristic time in the transition zone from unyielding to yielding zone and p is the power-law coefficient indicating the dependence of 𝐺′ 𝑎𝑛𝑑 𝐺′′on frequency, 𝜔. This characteristic time 𝜏𝑅 can be 35 calculated as the inverse frequency at which the power-law expression 𝐺′ = 𝐺𝑐,𝑖′ 𝜔𝑝 = 𝐺𝑦′ holds (𝐺𝑐,𝑖′ = 𝐺𝑐′𝜔𝑝). Based on this definition of 𝜏𝑅, 𝐺𝑐′ = 𝐺𝑦′ , and Eq. 5-3 is reduced to: 𝐺′ 𝐺𝑦′⁄ = 1 + (𝜔𝜏𝑅)𝑝 𝑎𝑛𝑑 𝐺′′ 𝐺𝑦′⁄ = (𝐺𝑐′′ 𝐺𝑦′⁄ )(𝜔𝜏𝑅)𝑝 5-4) The term 𝐺𝑦′ (equilibrium modulus) indicates the existence of yield stress and it is related to the rubber network strength. It also correlates with both the yield stress, and yield strain (see discussion below). Based on Equations 5-3, the following model can be written, which provides a direct link between the microstructure of the material and its linear rheological properties with a minimum number of parameters (mainly the exponent p, the constants 𝐺𝑐′ and 𝐺𝑦′ ) within the context of linear viscoelasticity [41,90,91]. 𝜏 = ∫ 𝐺(𝑡 − 𝑡′)?̇?(𝑡′)𝑑𝑡′𝑡−∞ 5-5) Where G(t) is the relaxation modulus and ?̇? is the shear rate. The form of the relaxation modulus function can be written in view of Eq 5-3, as: 𝐺(𝑡) = 𝐺𝑐𝑡−𝑝 + 𝐺𝑦 5-6) Obviously, 𝐺𝑦 = 𝐺𝑦′ in view of Eq. 5-3 (𝐺′′(0) = 0). Since the damping factor, 𝑡𝑎𝑛𝛿(𝜔)=𝐺"(𝜔) 𝐺′(𝜔)⁄ is small in case of the TPVs, Eq. 5-7 can be used to calculate the relaxation modulus [92]. 𝐺(𝑡) ≅ 𝐺′(𝜔) − 0.560𝐺"(𝜔 2⁄ ) + 0.200𝐺"(𝜔) 5-7) Stress relaxation test was also performed to compare the accuracy of Eq. 5-7 with the experimental data. Figure 5-8 depicts fits of typical experimental data by means of the proposed relaxation model (Eqs 5-3 and 5-6). Table 5-3 summarizes the parameters of the storage modulus of all TPVs studied in the present work at 190oC. The equilibrium modulus, 𝐺𝑦′ and 𝐺𝑐′ for the samples with higher rubber amount (TPV1-4 of Group A) and higher curing degrees (TPV2 vs TPV3 and TPV6 vs TPV7 see Table 4-2 for details) are higher, indicating that the level of curing and amount of crosslinked rubber form stronger networks. Thus, the 𝐺𝑦′ value can be used as a measure of the network strength. However, a stronger network (stiffer material) may fail and yield at smaller shear 36 stress values and smaller yield strains. This can be seen from Figure 5-9, where the yield stress𝜎𝑦, and yield strain, 𝛾𝑦, are plotted as functions of the equilibrium modulus 𝐺𝑦′ (note the relationship between these quantities, 𝜎𝑦 = 𝐺𝑦′𝛾𝑦). The trends are similar in both groups of TPVs (A and B) for both the yield stresses and strains indicating consistency of the results. A maximum appears in both yield strain and yield stress at an optimum network strength (𝐺𝑦′ ). This (strongest network) is formed by combining the two linear polymers (PP and EPDM) in the network at a medium curing level. A high curing degree increases the network strength (high 𝐺𝑦′ ), and as stiffer fails at lower yield strains, thus resulting smaller yield stress values. In addition, it appears that the use of the B-EPDM weakens the network and results in the smallest yield stress values in both Groups of TPVs (TPV4 and TPV8). These parameters (𝜎𝑦, 𝐺𝑦′ , 𝛾𝑦) are expected to have a significant impact on the processing of these materials. Figure 5-7. The linear viscoelastic moduli of a) TPV2 and b) TPV3 compared with the corresponding moduli of their constituents at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus. 37 Figure 5-8. Fitting the experimental data with the model of Eqs 5-3 and 5-6 to determine the parameters listed in Table 5-3 a) storage and loss moduli, and b) shear stress and relaxation modulus. Table 5-3. Parameters of the storage modulus function (Eq 5-3) of all TPVs at T=190 ℃. Sample 𝐺𝑦′ (kPa) 𝜏𝑅 (s) p (Group A) TPV1 78.3 0.003 0.258 TPV2 69.7 0.027 0.231 TPV3 45.1 0.213 0.222 TPV4 33.3 2.864 0.200 (Group B) TPV5 61.8 0.009 0.274 TPV6 55.6 0.049 0.270 TPV7 39.8 0.191 0.257 TPV8 28.7 0.928 0.234 Universal constants of Eq 5-3 for all TPVs, p = 0.215 and 𝐺𝑐"𝐺𝑦′⁄ = 0.34 38 Figure 5-9. The yield stress,𝜎𝑦, and yield strain, 𝛾𝑦, as functions of the equilibrium modulus 𝐺𝑦′ 5.2.3.1 Effect of thermoplastic type (L-PP vs B-PP) Figure 5-10 compares the viscoelastic moduli of TPV2 (B-PP+L-EPDM) and TPV3 (L-PP+L-EPDM) and those of TPV5 (B-PP+L-EPDM) and TPV6 (L-PP+L-EPDM) at low and high rubber concentrations. Although L-PP (linear) has viscoelastic moduli much higher than those of B-PP (branched), their effect is minimal on G in both cases since relaxation is dominantly controlled by the EPDM phase and the cured interface. There is a relatively higher influence on G (viscous component) over the whole frequency range. However, because G>>G, the overall effect of the loss modulus G can be ignored. Figure 5-10. The linear viscoelastic moduli of a) TPV1 and 2, and b) TPV5 and 6 compared to show the effect of different thermoplastic used in the TPV formulation at Tref=190 oC. Solid symbols represent storage modulus and open symbols represent loss modulus. a b 39 5.2.3.2 Effect of rubber type (L-EPDM vs B-EPDM) Figure 5-11 compares the viscoelastic moduli of TPV3 (L-PP+L-EPDM) and TPV4 (L-PP+B-EPDM) and those of TPV7 (L-PP+L-EPDM) and TPV8 (L-PP+B-EPDM) at high (Group A) and low (Group B) rubber concentrations, respectively. The functions G and G are slightly higher at lower frequencies when the L-EPDM (linear) is used over B-EPDM (branched) although the individual viscoelastic moduli of B-EPDM are higher than those of L-EPDM. This can be attributed to the higher amount of oil in the L-EPDM (lower amount of rubber) compared to that in the B-EPDM, which causes a higher crosslink density at the same curing level during dynamic vulcanization. In addition, L-EPDM has a higher Mw, which makes it more elastomeric. Therefore, crosslinked L-EPDM can form a stronger rubber network due to a higher crosslink density and Mw. This was also discussed in view of the results plotted in Figure 5-9, where the B-EPDM appeared to produce the weakest network (smallest 𝐺𝑦′ values). Figure 5-11. The linear viscoelastic moduli of a) TPV3 and TPV4, and b) TPV7 and TPV8 compared to show the effect of different thermoplastic used in the TPV formulation at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus. 5.2.3.3 Effect of curing level Figure 5-12 compares the linear viscoelastic properties of TPV2 (high curing) and TPV3 (normal curing) that are made up of the same of L-PP and L-EPDM with the same filler concentration (left side) and the linear viscoelastic properties of TPV6 (high curing) and TPV7 (normal curing) that are made up of the same L-PP and L-EPDM with the same filler concentration (right side). Higher curing increases the G (elastic) significantly due to the stronger cured interface formed between a b 40 the two constituent polymers. In fact, as the values of 𝐺𝑦′ listed in Table 5-3 for these systems also show, a stronger rubber network is formed at a higher curing level. Figure 5-12. The effect of curing level on the linear viscoelastic moduli of TPV2 (high curing) and TPV3 (normal curing) with other parameters the same (left) and TPV6 (high curing) and TPV7 (normal curing) with other parameters the same (right) at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus. 5.2.3.4 Effect of crosslinked rubber concentration Figure 5-13-a to d compares the linear viscoelastic properties of several TPVs in four different cases in each group. Group A has higher amount of EPDM. In all cases, when the amount of EPDM increases, the linear viscoelastic moduli (both G and G) increase accordingly because either EPDM phase has higher elasticity or higher amount of rubber (proportionally lower amount of PP matrix) increases the linear voscoelastic moduli (both G and G) accordingly. b a 41 Figure 5-13. The effect of rubber concentration (high, normal) on the linear viscoelastic moduli of TPVs with other parameters the same (constituent polymers and level of curing) at Tref=190oC. Solid symbols represent storage modulus and open symbols represent loss modulus. 5.2.4 Uniaxial extension of TPVs As discussed above, uniaxial extension has been performed for TPVs at several Hencky strain rates at 190oC. Figure 5-14 depicts the uniaxial stress growth coefficient, 𝜂𝐸+, of TPV4 and TPV5 at 190oC together with those of their constituent polymers. Note that the uniaxial stress growth coefficients for TPVs have been multiplied by 6x for the sake of clarity, essentially to separate and show the data more clearly. It is clear from the plotted data that TPVs exhibit an “instability” at the early times of their extension due to the yielding of the material. In fact, at these early times the coefficient, 𝜂𝐸+ go below the LVE envelop (continuous lines representing 3𝜂+) change concavity and at longer times grow and follow practically the LVE curve again. This is definitely not due to sagging effects that are impossible to happen due to the strong network these materials form. This can also be seen from the corresponding coefficients, 𝜂𝐸+, of their constituent polymers 42 (PP and EPDM) which, although much softer at the same temperature, show no signs of sagging or other instabilities. The observed effect is inherent to the yield stress in extension of these materials, which manifests itself as a change in the concavity where at some time the slope of the coefficient, 𝜂𝐸+, approaches zero. This effect is shown for the first time in the case of polymer melt-like materials. Similar effects have been observed and reported previously for a thixotropic suspension [93]. These experimental results were modelled by Varchanis et al. [94] who developed a constitutive equation for an elastoviscoplastic material and has shown computationally the yielding obtained in uniaxial extension deformations. The computed elastic behavior during startup of uniaxial extension is initially observed, while at times beyond the elastic limit the slope of the stress curve decreases to zero showing its yielding behavior. Another important observation to be noted is that the strong strain hardening of B-PP (Figure 5-14) is not manifesting itself when it is part of TPV5. Similarly, the strong strain hardening of B-EPDM is suppressed when it is part of TPV4 (Figure 5-14b). Figure 5-14. Comparison of the extensional behavior of TPV4 and TPV5 with those of their constituent components. Figure 5-15-a depict typical uniaxial extension data for TPV4. Figure 5-15-b shows the same data in terms of uniaxial stress at various Hencky strain rates, multiplied by various factors to show the yield tensile strength at various Hencky strain rates more clearly. It can be concluded that the tensile yield stress is independent of the Hencky strain rate up to strains of 0.1s-1 (also see Table 5-4), which we define as the tensile yield stress, 𝜎𝑦,𝐸. Table 5-4 lists the tensile yield stress, 𝜎𝑦,𝐸 of TPVs as an average at different Hencky strain rates (0.001-0.1 s-1). Similar to our discussion of 43 yield in shear, yield in extension should also occur over a range of tensile stress values. The process starts at the point where the slope of the tensile stress curve decreases to zero or having the tendency to decreasing to zero [94] and it continues till flow initiation where the tensile curve reaches back again the LVE (shown with an arrow in Figure 5-15 for different Hencky strain rates). Therefore, the tensile stress of fully flow initiation point is considered as the tensile yield stress𝜎𝑦,𝐸. It has also been observed that the extensional yield stress slightly increases with the temperature (perhaps within experimental error). The temperature effect in extension is similar to that observed from the strain sweep results (Figure 5-6), where it was concluded that the temperature has no significant influence on the shear and thus on the extensional yield stress. In solid mechanics, for sufficiently slow flows [44], the von Mises criterion relating these two yield stresses, defines this ratio, 𝜎𝑦,𝐸/𝜎𝑦, as √3 ≈ 1.73. However, this criterion assumes homogeneous extensional flow. For many complex materials this ratio has been reported to be much higher [44]. It should also be mentioned that normal forces in the solid phase prior to yielding also affect the von Mises criterion [95]. In particular this ratio has been calculated for different materials to be in the range between √3 and 3√3 for various suspensions and pastes [44,96–98]. Table 5-4 compares the ratio of tensile and shear yield stresses for TPVs studied in this work. As seen from the corresponding values listed in Table 5-4, in the case of TPVs this ratio of 𝜎𝑦,𝐸/𝜎𝑦takes values in the range of 1.4-3.25. The ratio of 𝜎𝑦,𝐸/(𝐺𝑦′×𝛾𝑦) was also calculated using the critical yield strain determined from linear viscoelastic measurements, 𝛾𝑦 (listed in Table 5-2) and the equilibrium storage modulus, 𝐺𝑦′ , obtained from fitting Eq 5-3 to linear viscoelastic moduli (listed in Table 5-3). This procedure results in slightly higher yield stress ratio values (2-4.5) but still in the range of √3 and 3√3 . The deviation of this ratio from √3 could also be related to (i) inhomogeneity in the elongational flow and (ii) the normal stresses that develop in the solid state prior to yielding [95]. Table 5-4 also lists the yield stress ratio by using as yield stress in shear obtained from the crossover of G and G (Table 5-2) which falls below √3 for all samples (practically less than 1). Since 𝜎𝑦 and 𝐺𝑦′×𝛾𝑦 in Table 5-4 represent the lower limit of shear yield stress and 𝜎𝑦,𝐶 the upper limit, a single value for this ratio could be somewhere in between those limits, still consistent with the von Mises criterion. Finally, considering the values listed in Table 5-4, one can conclude that the extensional yield stress increases with rubber particle concentration (Group A TPVs have higher values compared to those in Group B). Finally, the extensional yield 44 stress follows the same pattern as 𝐺𝑦′ (apart from TPV8), therefore it can be concluded that it correlates with rubber strength. As a concluding remark, to the best of our knowledge this is the first time that yield evidence in extension is reported for polymer melts and definitely further investigation is needed. Figure 5-15. Uniaxial stress growth coefficient of TPV4 at 190oC. a) Uniaxial stress growth vs. time, and b) Uniaxial stress vs. strain. Arrows show the yielding point at different Hencky strain rates. Table 5-4. Yield stresses in shear and extension of TPVs at 190oC and their ratio Sample 𝜎𝑦,𝐸 (kPa) at different Hencky strain rates Average𝜎𝑦,𝐸 (kPa) Various measure of yield in shear (kPa) Yield stress ratio in extension over shear 0.001 (𝑠−1) 0.01 (𝑠−1) 0.1 (𝑠−1) 0.5 (𝑠−1) Mean (0.001-0.1) 𝜎𝑦 𝐺𝑦′× 𝛾𝑦 𝜎𝑦,𝐶 𝜎𝑦,𝐸𝐺𝑦′ × 𝛾𝑦 𝜎𝑦,𝐸𝜎𝑦 𝜎𝑦,𝐸𝜎𝑦,𝐶 TPV1 14.1 15.8 14.5 23.1 14.8±0.9 4.6 4.15 18.5 3.56 3.21 0.8 TPV2 15.5 15 13.6 26 14.7±1 6.4 5.29 15.4 2.77 2.29 0.95 TPV3 13.3 13.5 13.7 20 13.5±0.2 7.8 5.09 22.5 2.65 1.90 0.6 TPV4 10.9 11.9 15 22.4 12.6±2.1 4.8 2.8 18.9 4.5 2.60 0.67 TPV5 11.6 11.4 12 15.6 11.7±0.1 3.6 3.71 12.3 3.15 3.25 0.95 TPV6 9.45 10.5 10.1 17.5 10±0.5 5.1 3.89 22.7 2.57 1.96 0.44 TPV7 7.52 8.3 9.5 18.9 8.4±1 6 4.2 22.7 2 1.40 0.37 TPV8 7.9 8.69 13.6 17 10.1±3 4.6 2.53 25.7 4 2.19 0.39 45 5.3 Conclusions In this chapter, the effects of several parameters including the thermoplastic matrix and rubber types, amount, and curing levels on the rheological behavior of TPVs were investigated. Linear viscoelastic results showed that the storage modulus of TPVs follows the behavior of the thermoplastic matrix at high frequencies, while at the lower frequencies the strong interfacial associations between the two components dominate their relaxation. The storage modulus of TPVs is nearly independent of the frequency and always higher than the loss modulus by one order of magnitude at least. The effects of long chain branching of the TPV constituents, the level of curing, and amount of crosslinked rubber particles on their rheology were found to affect the TPV rheology to various degrees. For example, increase of curing or increase of level of crosslinked rubber increases the rheological properties of TPVs (G and G), rendering the material stiffer and thus they yield and flow at larger levels of imposed stresses. The relaxation modulus of TPVs follows a power law relation, 𝐺(𝑡) = 𝐺𝑐𝑡−𝑝 + 𝐺𝑦′ , where 𝐺𝑦′ is the equilibrium modulus indicating the elastomeric nature of TPVs which was suggested to be related to their microstructure. Only three parameters are needed to describe the relaxation modulus. It remains to be seen if this modulus coupled with the damping function can predict the nonlinear rheological behavior of TPVs. These observations indicate that TPVs behave as nearly elastic solids at small deformations (linear viscoelasticity). When large deformations are imposed on TPVs, yield phenomena occur in both shear and uniaxial extension. For these transient tests it was possible to obtain the yield stresses which were found to be similar in these two different types of deformation with their ratio (yield stress in extension to that in shear) to be in the range of √3 and 3√3, which agrees with the von Mises criterion in solid mechanics for yield stress fluids. 46 6 Rheological Modelling of Thermoplastic Vulcanizates (TPVs) using the K-BKZ Constitutive Law In this chapter, several model TPVs of known composition are studied rheologically in both shear and elongation to understand and model their response using the K-BKZ rheological law. The power-law relaxation model obtained in the previous chapter is applied on the model TPVs, as well, to prove its applicability to them. Afterwards, yield stress and wall slip are applied on K-BKZ model. Overall, it is shown that the use of the K-BKZ with yield stress and slip can capture well their rheological response. 6.1 Theoretical Background 6.1.1 Constitutive modelling for TPVs In the previous chapter a simple expression for the relaxation modulus for TPVs was developed, which can be written as follows. 𝐺(𝑡) = 𝐺𝑐𝑡−𝑝 + 𝐺𝑦 6-1) where 𝐺𝑐 is a proportionality constant, p is the power-law exponent (universal for all TPVs) and 𝐺𝑦is the equilibrium modulus indicating the elastomeric nature of TPVs and absence of terminal relaxation. The memory function is defined by 𝑚(𝑡 − 𝑡′) ≡ −𝑑𝐺(𝑡−𝑡′)𝑑𝑡′ and therefore can be written as: 𝑚(𝑡 − 𝑡′) = −𝑝𝐺𝑐(𝑡 − 𝑡′)−𝑝−1 6-2) The K-BKZ constitutive model proposed by Wagner [75] and modified by Luo and Tanner [76] was used to model the rheology of TPVs. 𝜏(𝑡) =11 − 𝜃∫ 𝑚(𝑡 − 𝑡′) ℎ(𝐼𝐶 , 𝐼𝐶−1) [𝐶−1(𝑡′) + 𝜃𝐶(𝑡′)]𝑑𝑡′𝑡−∞ 6-3) Where ℎ is damping function and 𝑰𝑪, 𝑰𝑪−𝟏 are the first invariants of the Cauchy-Green tensor 𝐶𝑡 and its inverse𝐶𝑡−1, the Finger strain tensor. The material constant 𝜃 is given by: 47 𝑁2𝑁1=𝜽1 − 𝜽 6-4) where N1 and N2 are the first and second normal stress differences, respectively. It is noted that 𝜃 is not zero for polymer melts, which possess a non-zero second normal stress difference [38]. Its usual range is between -0.1 and -0.2 in accordance with experimental findings and numerical simulations [38,39]. Its effect on the predictions is minor [77,79,80]. The damping function used in the present work and shown below that it can capture the non-linear stress relaxation of TPVs can be written as [99]. ℎ(𝐼𝐶 , 𝐼𝐶−1) =𝛼𝛼+(𝛽𝐼𝐶+(1−𝛽)𝐼𝐶−1−3)𝑏2⁄ 6-5) Where 𝛼 and 𝛽 are material constants. The parameter 𝛼 can be determined from step-strain experiments in simple shear and the parameter 𝛽 can be calculated from fitting uniaxial extension experiments. In simple shear, Equation 6-5 can be written (simplified) as: ℎ(𝐼𝐶 , 𝐼𝐶−1) =𝛼𝛼+𝛾𝑏 6-6) Yield stress materials, before yielding, behave nearly elastically and typically exhibit none or very little dissipation. This can be considered by the following formulation: 𝜏(𝑡) ={ ∫ (𝐺𝑐(𝑡 − 𝑡′)−𝑝𝑡−∞+ 𝐺𝑦)?̇? 𝑑𝑡 𝑜𝑟 𝐺𝑦𝛾 𝜏 ≤ 𝜏𝑦𝜏𝑦 +11 − 𝜃∫ −𝑝𝐺𝑐(𝑡 − 𝑡′)−𝑝−1 ℎ(𝐼𝐶 , 𝐼𝐶−1) [𝐶−1(𝑡′) + 𝜃𝐶(𝑡′)]𝑑𝑡′𝑡𝑡𝑦 𝜏 > 𝜏𝑦 6-7) Where 𝜏𝑦 is yield stress, which is essentially 𝜏𝑦 = 𝐺𝑦𝛾𝑐, where 𝛾𝑐 is the critical strain beyond which the material yields – marking the transition from elastic to visco-elasto-plastic behavior. The expression for the elastic part of the deformation has been written in two different forms. For the TPVs of high curing level, the response is purely elastic (no dissipation) and the stress can be calculated simply by 𝐺𝑦𝛾. However, for TPVs of low curing level the response deviates slightly from purely elastic and exhibits viscoelastic response and the form of Boltzmann’s linear viscoelastic response is more appropriate resulting excellent predictions (presented below). For example, for Gy high enough 𝐺𝑦 ≫ 𝐺𝑐(𝑡 − 𝑡′)−𝑝 and the two expressions become identical. 48 6.1.2 Implementation of slip in simple shear Due to the presence of oil and rubber particles, TPVs slip significantly. Steady-shear experiments are analyzed in this work by means of K-BKZ Equation 6-7 to determine slip. Since Equation 6-7 is expressed in terms of the finite deformation tensor (strain), the steady shear simulations are performed by imposing only a portion of the nominal strain that best matches the experimental results. It is assumed that there is no slip in the linear response region (before yielding), therefore; slip only happens after yielding and flow initiation. Using the no-slip boundary condition, the K-BKZ over predicts the shear stress response of the material. However, using only a certain fraction f (to be determined) the calculations match the experimental data well. Therefore, we impose that the true strain,  is: 𝛾 = 𝑓𝛾𝑛 6-8) Where is the nominal strain defined by the linear velocity of the plate divided by the distance between the plates, and 𝑓 defines the fraction of the strain that the material follows. 𝑓 is applied on strain tensor in non-linear shear flow. Based on these calculations to match the experimental data, the slip velocity can be calculated by: 𝑉𝑠 = (1 − 𝑓)?̇?𝑛ℎ/2 6-9) and plotted as a function of the wall shear stress. The results are presented below. 6.2 Results and Discussion 6.2.1 Linear viscoelastic behavior (LVE) 6.2.1.1 Effect of curing level on LVE Figure 6-1 shows the frequency sweep of all the TPVs in this study. As can be seen TPVs with high curing level have higher storage modulus than the medium and low curing level ones. This is related to the stronger rubber network that is created at high curing level. The medium and low curing levels almost have similar linear viscoelastic behavior. 6.2.1.2 Effect of polymer type on LVE Figure 6-2 shows effect of components of TPV on their LVE behavior at different curing levels. At low and medium curing level, storage modulus has a higher value when L-EPDM is the rubbery 49 component. That is due to higher molecular weight of L-EPDM compared to B-EPDM. The thermoplastic phase (PP) seems not to have a significant influence on the LVE behavior. On the other hand, for TPVs with high curing level, all storage moduli almost overlap over the whole frequency range. Thus, at high curing level, due to the strong elastic network through curing of the interface, the effect of components is negligible. These effects have been discussed in detail previously (previous chapter). Figure 6-1. Storage and loss moduli vs. angular frequency of all the TPVs at T=190 ℃ 50 Figure 6-2. LVE of TPVs at different curing levels at T=190 ℃. a) low cured, b) medium cured, and c) high cured samples. 6.2.2 Linear relaxation modulus In the previous chapter it was shown that the viscoelastic moduli of TPVs, G' and G" can be modeled by (in agreement with Equation 6-1): 𝐺′𝐺𝑦′= 1+(𝜔𝜏𝑅)𝑝 6-10) 𝐺"𝐺𝑦′= 𝐺𝑐"𝐺𝑦′ (𝜔𝜏𝑅)𝑝 6-11) Where 𝐺𝑐" is the loss modulus coefficient (constant), 𝜏𝑅 is a characteristic time in the transition from the unyielding to the yielding zone and p is the power‐law coefficient indicating the dependence of 𝐺′ and 𝐺" on frequency, 𝜔. Since the damping factor, 𝑡𝑎𝑛𝛿(𝜔)=𝐺"(𝜔)/𝐺′(𝜔) is small in case of TPVs, Eq. 6-12 can be used as an approximation to calculate the relaxation modulus from the viscoelastic moduli [92]. This has been shown to indeed approximate the relaxation modulus of TPVs extremely well based on G' and G" data, 𝐺(𝑡) ≅ 𝐺′(𝜔) − 0.560𝐺"(𝜔 2⁄ ) + 0.200𝐺"(𝜔) 6-12) Table 6-1 presents the relaxation fitting parameters of all the TPVs in this study. Based on Table 6-1, by increasing curing level 𝐺𝑦′ increases, while 𝜏𝑅 decreases. This means that at high curing level, the transition from liquid like to solid like behavior happens at shorter frequencies (higher relaxation times) and the relaxation modulus gets close to its finite low-frequency limit quickly. Therefore, the TPV mostly behaves similar to a solid like material. It can also be observed that the exponent p is similar for all TPVs pointing to a universal relaxation behavior. 51 Table 6-1. Relaxation parameters at T=190 ℃. sample Curing level GC 𝐺𝑦′ (kPa) P 𝜏𝑅 (s) S1 Low 2.47×104 5.8 0.252 466 S2 Medium 2.00×104 10.8 0.242 13 S3 High 2.69×104 36.7 0.232 0.408 S4 Low 2.44×104 8.7 0.233 135 S5 Medium 1.73×104 14.3 0.224 3.694 S6 High 2.17×104 39.2 0.236 0.129 S7 Low 3.94×104 11.8 0.228 303 S8 Medium 2.63×104 25.8 0.219 1.75 S9 High 2.69×104 38.3 0.222 0.322 S10 Low 3.53×104 13.4 0.206 174 S11 Medium 2.16×104 23.8 0.212 1.006 S12 High 1.60×104 55 0.230 0.007 The universal behavior of normalized storage and loss moduli vs. 𝜔𝜏𝑅 for all the TPVs are displayed in Figure 6-3. As can be seen all the TPVs follow a master curve with parameters 𝑝 =0.232 and 𝐺𝑐"𝐺𝑦′ = 0.36. ( )'0'lim y ytG G G→→= = Figure 6-3. Master curve for LVE behavior of the TPVs at T=190 ℃. 52 6.2.3 Non-Linear viscoelastic behavior To use K-BKZ model, damping function of the material should be determined. Therefore, the stress relaxation behavior of TPV S1 at 190oC is obtained using various step strains ranging from 0.05 to 2, (Figure 6-4-a). The line labelled as LVE in this figure represents the linear viscoelastic relaxation of the material and it has been calculated based on Equation 6-12. Figure 6-4-b shows the shifted curves to superpose on LVE which is used to calculate the damping function. Apart from the data at the early time scales<<1s, the data superposes well. Figure 6-4. a) Stress relaxation of TPV S1, b) Shifted curves to find damping function. All experiments were done T = 190 ℃. Using shifting values acquired from stress relaxation test, damping functions of S1-S3 are plotted in Figure 6-5. The damping functions are fitted with Equation 6-6 which seems to represent the results well. Figure 6-5. Damping function for S1-S3 at T=190 ℃. 53 6.2.4 Startup of steady shear and K-BKZ predictions Figure 6-6 shows experimental data obtained from start-up of steady shear experiments in terms of the shear stress growth coefficient as a function of time for several values of the shear rate ranging from 0.01 to 20 s-1. To find the yield stress that is used in Equation 6-7, it is assumed that the rubber particles form a viscoelastic network which governs the response of the materials at strains bellow critical yield strain. The yield strain is obtained from strain sweep test which was done at a frequency equal to 0.1 Hz. The yield stress corresponding to the yield strain in the strain sweep test was referred to as experimental yield stress. The experimental yield stresses obtained from strain sweep test, the yield stress from pure elastic assumption (𝜏𝑦 = 𝐺𝑦𝛾𝐶), and the critical strain for the onset of yielding and the corresponding value of the linear modulus are listed in Table 6-2. Critical yield strains obtained from strain sweep test assuming yielding happens at the region where the storage modulus is between 90% storage modulus of the plateau. The predictions of the K-BKZ model as formulated by Equation 6-7 without slip (left) and with slip (right) are also shown in Figure 6-6. Predictions under no slip, overpredict the experimental data. On the other hand, predictions under slip match the experimental data well in all cases. As discussed above the level of slip in each case was calculated as a portion of the nominal shear strain to match the experimental data. Both the experimental data and predictions remarkably show the onset of yielding by a sudden deviation from the viscoelastic response (unyielded). The resemblance of the transients curves (experimental and calculated) is remarkable. As discussed above the slip was imposed in terms of strain as a fraction of the nominal shear strain. Based on these values the slip velocity can be calculate by Equation 6-9. The slip velocity is plotted as a function of wall shear stress in Figure 6-7-b. The slip velocity is expressed as a power law (mS wV A= ) with a slope of 3.58 and A=1.92× 105𝑀𝑃𝑎−𝑚𝑚𝑚/𝑠), which is represented in Figure 6-7 by the continuous line. The fitting was done using bi-square robust regression. Table 6-2. Equilibrium Modulus, Yield stress and strain for TPVs S1-S3. Sample Eqm storage Modulus 𝐺𝑦′ (kPa) Critical yield strain 𝛾𝐶 (%) Experimental yield stress (kPa) (𝜔 = 0.1 𝐻𝑧) Elastic yield Stress, 𝜏𝑦 = 𝐺𝑦𝛾𝐶 (kPa) S1 5.8 17.8 4.5 1.03 S2 10.8 16.6 3.5 1.79 S3 36.7 7.5 3.9 2.75 54 Figure 6-6. Start-up experimental results compared to model predictions, Left: no slip condition. Right: applying slip condition to the model 55 Figure 6-7. a) Strain fractions at different shear rates, b) The slip velocity of TPVs as a function of wall shear stress 6.2.5 Uniaxial extension and K-BKZ predictions Figure 6-8 shows a comparison of uniaxial extensional results with the K-BKZ model (Equation 7). Similar to shear, the extensional yield stress was implemented into the model for uniaxial extension. To produce these results the yield stress in extension, 𝜏𝑦,𝐸, is set by using according to von Misses criterion [44]. Model captures the correct curvature of experimental data (Figure 6-8). Initially the response is elastic (before yield), consequently the material yields exhibiting a strain hardening with concavity downwards (decreasing slow with time). However, at stress lvels well above the yield point, the concanity points upwards with a strong strain hardening behaviour due to the presence of the cured interface. The experimental data exhibit a similar curvature. Figure 6-8. KBKZ extensional predictions using power-law relaxation function. 56 6.3 Conclusions In this chapter, the K-BKZ constitutive model was used to predict the rheology of TPVs by using the power-law relaxation function. The effect of yield stress and slip velocity were also added into the model. The von Misses criterion was used to include the yield stress in extension. A convenient method was developed to include the effect of slip into the K-BKZ modelling in startup of steady shear. Essentially a fraction of the imposed nominal shear strain was used in the deformation tensor and this was calculated to match the experimental data. Based on these calculated portions, the slip velocity of TPVs was determined as a function of the wall shear stress and it was found to be similar for all TPVs. In general, it was concluded that the nonlinear behavior of TPVs can be successfully predicted by the K-BKZ model considering all the aforementioned phenomena. The model successfully predicts shear behavior of TPVs. It deviates from extensional data at high curing levels. However, it predicts the curvature well. 57 7 Melt Fracture and Wall Slip of Thermoplastic Vulcanizates (TPVs) In this chapter, a number of different TPVs with various characteristics, including the degree of curing and amount of cross-linked rubber are examined in capillary extrusion at two different temperatures (190°C and 205°C) relevant to real processing. First, the effect of the temperature on the yield stress is investigated using rheological measurements. Consequently, the flow behavior of the TPVs in capillary flow is studied concluding that TPVs slip massively (nearly plug flow) due to the presence of lubricant and the vulcanized rubber phase. Although there is little slip observed in PP samples, EPDMs themselves exhibit severe slip and melt fracture. Therefore, the TPV samples essentially follow the slip behavior of EPDMs. Finally, the melt fracture analysis of several TPVs is shown that with increase of temperature and amount of cross-linked rubber, the severity of TPVs' surface defects increases accordingly. 7.1 Linear Viscoelasticity (LVE) of TPVs The linear rheological properties of the basic polymeric constituents of TPVs listed in Table 4-1 were studied in Chapter 5 in detail as well as those of most TPVs listed in Table 4-2. Here we simply present the complex modulus, *( )G  of all TPVs studied in Figure 7-1 at 190C, 205C, and 220 C. The effect of T is not significant, and, in some cases, the experimental data is within experimental error. All data plotted in Figure 7-1, exhibit the tendency to approach a limiting value (equilibrium modulus, *yG ) at smaller frequencies due to their elastomeric nature. This implies the absence of terminal relaxation as expected for elastomeric materials. 58 Figure 7-1. The linear viscoelastic moduli of all TPVs listed in Table 4-2 at (a) T=190oC, (b) 205°C, and (c) 220 °C. As concluded in Chapter 5, at higher frequencies the TPV response is controlled by the segmental dynamics of its constituent polymer (Rouse time response), however, at lower frequencies the effect of the interface of cured EPDM (dispersed) with PP (matrix) plays a significant role as there is no sign of terminal relaxation (see also Figure 7-1). Both G and G show solid-like behavior with a small dependence on frequency (G much less than G) presented in Chapter 5. This was typically observed for all TPVs independent of the type of constituent polymeric components, level of curing (high or normal) and level of rubber concentration (high or low). This “simple” relaxation known as power-law relaxation has been observed also in the linear viscoelastic response of foods such as dough and yogurt [90,91,103] and it can be modelled by: ( ) ( )( )' ' ' ''/ 1 ''/ /p p'y R y c y RG G and G G G G = + = 7-1) Where 𝐺𝑐" is the loss moduli coefficient (constant), R is a characteristic time in the transition from the unyielding to the yielding zone and p is the power-law coefficient indicating the dependence of G and G on frequency, and 'yG indicates the existence of yield stress. The relaxation modulus of TPVs can be written as: 𝐺(𝑡) = 𝐺𝑐𝑡−𝑝 + 𝐺𝑦 7-2) Where G(t) is the relaxation modulus and 'y yG G= . 59 Finally, Equations 7-1 and 7-2 can be combined to derive an equation for the complex modulus: |𝐺∗(𝜔)| 𝐺𝑦∗⁄ =1 + (𝜔𝜏𝑅)𝑝 7-3) It is noted that *yG should be very close to'yG . The universal viscoelastic moduli, G' and G'', of all TPVs and their fitting using Equation 7-1, are plotted in Figure 7-2 at the three temperatures of 190C, 205C, and 220C. Figure 7-2. Fitting the universal viscoelastic moduli of TPVs with the model of Eq 7-1 to determine the parameters listed in Table 7-1 and Table 7-2. Table 7-1 lists the parameter values of 'yG , R , p, for all TPVs at 190C, 205C, and 220C. The parameter 'yG and *yG can be viewed as fingerprints of the morphological structure of these complex multiphase systems. Table 7-2 lists the coefficients of Equation 7-1 at different temperatures. From Table 7-1 it can be concluded that the equilibrium (storage) modulus 'yG increases with temperature from 190C to 205C, which implies that the rubber particle network strengthens with temperature. From 205C to 220C in some cases, 'yG slightly decreases (within experimental error) or remains constant. The increase of 'yG with temperature might be either due to further crosslinking or increase of entropy in rubber upon deformation which consequently increases the modulus. In addition, 'yG increases with increase in the level of curing (TPV1 and TPV2 have higher cure level compared to TPV3 and TPV4, also TPV5 and TPV6 have higher cure level compared to TPV7 and TPV8), and with increase in the amount of rubber (TPVs of Group 60 A have more EPDM than their counterparts in Group B). These parameters will be used below to relate them with the onset of flow instabilities. Finally, the exponent, p, in the relaxation modulus also increases with temperature from 0.215 to 0.239 reflecting the influence of T on the relaxation modulus, essentially implying that the relaxation of the network occurs faster at a higher temperature. Table 7-1. Parameters of the relaxation modulus (Equations 7-1 and 7-3) of all TPVs. Sample 𝐺𝑦′ (kPa) p R (s) Temperature (℃) 190 205 220 190 205 220 190 205 220 (Group A) TPV1 78.3 91.4 88.4 0.258 0.245 0.273 0.003 0.002 0.001 TPV2 69.7 75.7 69.1 0.231 0.251 0.236 0.027 0.011 0.012 TPV3 45.1 48.8 43.7 0.222 0.232 0.232 0.213 0.100 0.074 TPV4 33.3 34.3 36.2 0.200 0.202 0.214 2.864 1.535 0.631 (Group B) TPV5 61.8 62.9 57.7 0.274 0.311 0.316 0.009 0.005 0.008 TPV6 55.6 50.7 45.0 0.270 0.274 0.277 0.049 0.041 0.035 TPV7 40.5 46.0 50.0 0.242 0.266 0.282 0.191 0.084 0.055 TPV8 28.7 33.2 31.5 0.234 0.236 0.250 0.928 0.547 0.315 Table 7-2. Variation of model coefficients with the temperature. 𝟏𝟗𝟎 ℃ 𝟐𝟎𝟓 ℃ 𝟐𝟐𝟎 ℃ 𝑮′ 𝑮𝒚′⁄ = 𝟏 + (𝝎𝝉𝑹)𝟎.𝟐𝟏𝟓 𝑮" 𝑮𝒚′⁄ = 𝟎. 𝟑𝟒(𝝎𝝉𝑹)𝟎.𝟐𝟏𝟓 |𝑮∗| 𝑮𝒚∗⁄ = 𝟏 + (𝝎𝝉𝑹)𝟎.𝟐𝟎𝟗 𝑮′ 𝑮𝒚′⁄ = 𝟏 + (𝝎𝝉𝑹)𝟎.𝟐𝟐𝟐 𝑮" 𝑮𝒚′⁄ = 𝟎. 𝟑𝟓(𝝎𝝉𝑹)𝟎.𝟐𝟐𝟐 |𝑮∗| 𝑮𝒚∗⁄ = 𝟏 + (𝝎𝝉𝑹)𝟎.𝟐𝟏𝟕 𝑮′ 𝑮𝒚′⁄ = 𝟏 + (𝝎𝝉𝑹)𝟎.𝟐𝟑𝟗 𝑮" 𝑮𝒚′⁄ = 𝟎. 𝟑𝟕(𝝎𝝉𝑹)𝟎.𝟐𝟑𝟗 |𝑮∗| 𝑮𝒚∗⁄ = 𝟏 + (𝝎𝝉𝑹)𝟎.𝟐𝟑𝟓 7.2 Capillary Flow 7.2.1 Capillary flow of PPs and EPDMs The basic polymeric constituents of TPVs were first tested in capillary flow at 190C and 205C to determine possible pressure effects on their viscosity (flow curve) and the existence of wall slip. Figure 7-3 presents an example for the L-PP. First Figure 7-3-a depicts the apparent flow curves of L-PP using three capillary dies having the same diameter and different lengths. These are plotted 61 in terms of the apparent shear stress ( / 4( / )A P L D   as a function of the apparent shear rate ( )332 /A Q D  , where P is the pressure drop along the capillary die including the entrance and exit, Q is the volumetric flow rate and D is the capillary diameter. These apparent flow curves lie above the Linear Viscoelastic data (LVE) obtained from parallel-plate rheometry (reported previously in Chapter 5) due to the effect of ends pressure, endP . The latter should be determined in order to calculate the true wall shear stress. Figure 7-3-b shows the Bagley plot (pressure as a function of apparent shear rate) typically used to determine the ends pressure, endP . This is plotted in Figure 7-3-c as a function of the apparent shear rate. Using the ends pressure, the true shear stress can be calculated from ( )( ) / 4w endP P L D =  − and the true shear rate from [(3 1) / 4 ]w An n  + where n is defined by (log ) / (log )w An d d  , that is the local slope of the apparent flow curve [104]. The flow curve is compared with the LVE data in Figure 7-3-d and the agreement implies the consistency of the experimental data from the two different flow geometries. For this high molecular weight PP of relatively narrow molecular weight distribution, slip effects are minimal and due to minor effects of slip there is a slight decrease of capillary data compared to LVE [105,106]. The procedure was repeated for the B-PP and the two EPDMs at 190C and 205C. The ends pressure corrected flow curves are plotted in Figure 7-4 and Figure 7-5 at 190oC and 205oC, respectively. First, slip effects are minimal for the case of PPs as discussed above [45,46]. This is due to the high molecular weight and relatively narrow molecular weight distribution of L-PP and the branched structure of B-PP [105,106]. However, both EPDMs do slip significantly particularly at higher shear rates. There appear significant differences between LVE and capillary flow curves consistent with the assumption of slip. In particular, the flow curve for both EPDMs deviate significantly from the LVE (no slip) at high shear rates, which imply strong slip effects and possible appearance of volume defects/melt fracture. Based on these data, the slip velocity for both EPDMs has been calculated and the results are presented below together with those of TPVs. Similar results with those plotted in Figure 7-4-c and Figure 7-4-d have also been reported for polyisobutylene elastomers [68]. 62 Figure 7-3. (a) The apparent flow curves of L-PP using three capillary dies having the same D and different L/D at 205oC, (b) the Bagley plot to determine the ends pressure, (c) the ends pressure versus the apparent shear rate (d) the flow curves corrected for the ends pressure effects. 7.2.2 Capillary flow of TPVs Capillary experiments were also performed for all TPVs quite similar to those presented for PPs and EPDMs. Figure 7-6 presents typical results for TPV1 at 190C. First, Figure 7-6-a shows the comparison between LVE and the apparent flow curves of TPV1 at T=190oC without the application of the ends pressure. Figure 7-6-b is the Bagley plot to determine the ends pressure, endP . This is plotted in Figure 7-6-c. Finally, Figure 7-6-d shows the comparison between LVE and the capillary flow curves corrected for the ends pressure effects. The capillary data at different L/D ratios now collapse into a single curve (compare Figure 7-6-a with that of Figure 7-6-d). 63 Moreover, the capillary data are significantly lower than that of LVE. This is mainly due to two reasons (i) existence of the yield stress which causes failure of the Cox-Merz rule (in yield stress materials the Cox-Merz rule does not apply) [102] and (ii) due to significant slip effects. The slip effects in TPVs are studied next. Figure 7-4. The flow curves of PPs and EPDMs at 190 ℃. While slip effects in PP are minimal, EPDMs do slip significantly at high shear rates (significant differences between LVE and capillary flow curves) 64 Figure 7-5. The flow curves of PPs and EPDMs at 205 ℃. While slip effects in PP are minimal, EPDMs do slip significantly at high shear rates (significant differences between LVE and capillary flow curves) 65 Figure 7-6. (a) The apparent flow curves of TPV1 using three capillary dies having the same D and different L/D at 190oC, (b) the Bagley plot to determine the ends pressure, (c) the ends pressure versus the apparent shear rate and (d) the flow curves corrected for the ends pressure effects. 7.3 Slip of TPVs and Their Basic Constituent Polymers (PP and EPDM) To demonstrate slip, capillary experiments were performed for one TPV using two dies having a different diameter, D and same L/D ratio (similar pressure gradients). The results in terms of the end pressure-corrected flow curves are plotted in Figure 7-7. Clearly the flow curve depends on the die diameter and the experimental data are consistent with the assumption of slip. Using the Mooney technique, slip was calculated and it was found essentially to correspond to a nearly plug flow [48]. Mooney analysis has resulted shear rates corrected for the effects of slip around zero and in some cases slightly negative. This implies that for all practical purposes, TPVs can be 66 assumed to exhibit a nearly plug flow velocity profile. Thus at a given apparent shear rate, the slip velocity, Vs can be calculated by / 8S AV D= which results from the definition of the apparent shear rate, 332 /A Q D  by setting2 / 4SQ D V= . The results are presented below after examining the slip of TPVs in parallel plate geometry. Figure 7-7. (a) The flow curves of TPV1 using two capillary dies having the same L/D and different D at 190oC to demonstrate slip effects in TPVs. Figure 7-8 plots the shear stress growth coefficient of TPV1 in simple shear at various shear rates at 190°C. The LVE curve is also plotted as reference. Since these materials possess a yield stress and slip, the agreement of steady-shear experiments and LVE is lost at low strain levels below the yield stress value. As seen from Figure 7-8, LVE lies well above the shear stress response, similar to what is depicted and discussed in Figure 7-7 for yield stress materials (failure of the Cox-Merz rule). We have used these experimental data and the assumption of plug flow to calculate slip in simple shear. Essentially slip can be calculated by, / 2S nV h= , where n is the nominal shear rate (values are specified in Figure 7-8). 67 Figure 7-8. Start-up of steady shear tests of TPV1 at various shear rates at 190C. Figure 7-9 shows the slip velocity of TPV1 as a function of wall shear stress obtained from capillary and parallel-plate at 190C. The two sets of data from parallel-plate and capillary are consistent (one set is extension of the other) and thus the assumption of nearly plug flow is validated, although the two sets of capillary data do not agree (although they exhibit the same trend). The slip velocity expressed as a power law (mS wV A= ) results a slope of 3.29 and A=8.5× 104𝑀𝑃𝑎−𝑚𝑚𝑚/𝑠), which is represented by the continuous line in Figure 7-9. The significant slip of TPVs is due to the presence of EPDM, which do slip easily due to their elastomeric nature (high elasticity which at high shear rates renders the materials behaving solid-like). Figure 7-10 plots the slip velocity of the two EPDMs at different temperatures and compares this with that of TPV1 at 190°C. First, the slip velocity of EPDM increases with increase of temperature consistent the time-Temperature superposition (tTS) principle. Second, the linear EPDM slips more than the branched EPDM again consistent with previous findings for polyethylene (linear PEs slip significantly more than their branched counterparts) [29,30]. Finally, TPV1 shows a similar slip trend with that of EPDMs indicating the consistency of the experimental results that 68 the EPDM phase controls the slip of TPVs (similar behavior). Similar results have been reported for polyisobutylenes which are also elastomeric in nature similar to EPDMs [49,50]. Figure 7-9. The slip velocity of TPV1 from capillary and parallel-plate rheometers as a function of wall shear stress at 190oC. Figure 7-10. The slip velocity of EPDMs at 190 and 205C, compared to that of TPV1 at 190C. 69 Using the assumption of plug flow (validated above for TPV1), the flow curves of all TPVs determined from capillary rheometry (Figure 7-11-a) were used to calculate their slip velocity at 190C. These are plotted in Figure 7-11-b. The general trend is that the lower the flow curve in terms of shear stress, the higher the slip velocity calculated (expected). The slip velocities of TPVs in Group B are higher than those in Group A, although no firm conclusion can be drawn as these slip calculations are based on the assumption of plug flow. Figure 7-11. The flow curves of all TPVs listed in Table II at 190°C and the corresponding slip velocities at 190C. 7.4 Melt Fracture 7.4.1 Melt fracture of PPs and EPDMs Melt fracture phenomena, such as sharkskin (small amplitude periodic distortions on the extrudate surface), stick-slip or oscillating (alternating smooth and distorted surfaces) and gross melt fracture (GMF) (large amplitude/ chaotic distortions affecting the whole volume of the melt) either as helical or chaotic distortions, have been known to occur during the extrusion of most polymers (more dominant in linear polymers) when the wall shear stress exceeds a critical value [49,50]. These phenomena cause problems in the processing of polymers by setting an upper limit to the throughput applied during extrusion. PPs and EPDMs also exhibit these phenomena and many reports exist in the literature for PPs [35,36] and EPDMs [51,52] and other elastomeric materials such as PDMS [63], PIB [68,107], and SBR [26]. The stick-slip instability with large pressure oscillation is normally observed for linear polymers after the sharkskin defect, which is a result of 70 a cyclic transition between weak and strong slip. The size of pressure oscillations depends on the material compressibility and the geometrical characteristics of the die used for the extrusion. On the other hand, for PS [108], or SBR[26] a volume defect (helical or chaotic) develops above a critical stress, corresponding to a change in the slope of the flow curve [26]. A similar behavior has been reported for EPDMs where there is a change in the slope that occurs simultaneous with volume defect development [51,52]. Figure 7-12 and Figure 7-13 present extrudate images obtained in the capillary extrusion of the two PPs and EPDMs at various shear rates at 190oC. The geometrical characteristics of the die used were L/D=33 and D=0.89 mm. As seen from Figure 7-12, PPs at certain shear rates possess the shape of regular helices (type of GMF) with no sharkskin (surface) melt fracture. At higher rates the helical distortions take the form of chaotic ones (severe form of melt fracture). The critical shear rates and shear stresses for the onset of these phenomena are listed in Table 7-3. While PP exhibit relatively mild melt fracture phenomena at relatively high shear rates, EPDMs exhibit well-developed sharkskin and chaotic GMF even at very low shear rate (as low as 5 s-1) due to their elastomeric nature. In fact, a more or less helical volume instability is superimposed on sharkskin and it develops severely with increasing shear rate leading to a chaotic GMF. Similar phenomena have been reported for other elastomeric polymers such as PDMS [63], PIB [107] and SBR [26]. 71 L-PP, T=190 ℃, D=0.889mm, L/D=33 ?̇? = 5 𝑠−1, Smooth ?̇? = 170 𝑠−1, Gross (helical) ?̇? = 260 𝑠−1, Gross (helical) ?̇? = 1000 𝑠−1, Gross B-PP, T=190 ℃, L/D=33 ?̇? = 5 𝑠−1, Smooth ?̇? = 260 𝑠−1, Gross ?̇? = 640 𝑠−1, Gross ?̇? = 1000 𝑠−1, Gross Figure 7-12. Extrudate images of L-PP (first row) and B-PP produced (second row) by capillary extrusion at 190C. 72 L-EPDM, D=0.889mm, L/D=33 ?̇? = 5 𝑠−1, Sharkskin (Major) ?̇? = 100 𝑠−1, Helical ?̇? = 400 𝑠−1, Gross ?̇? = 1000 𝑠−1, Gross B-EPDM, T=190 ℃, L/D=33 ?̇? = 5 𝑠−1, Sharkskin ?̇? = 100 𝑠−1, Helical ?̇? = 400 𝑠−1, Gross ?̇? = 1000 𝑠−1, Gross Figure 7-13. Extrudate images of L-EPDM (first row) and B-EPDM (second row) produced by capillary extrusion at 190C. Table 7-3. Critical shear rates and stresses for the onset of Melt Fracture (MF) phenomena of PPs and EPDMs at 190C and 205C. Resin Critical shear rate (s-1)/shear stress (MPa) Sharkskin Helical instability Gross melt fracture L-PP T= 190 ℃ - 170/0.15 820/0.311 T= 205℃ - 170/0.1437 820/0.225 B-PP T= 190 ℃ - - 450/0.133 T= 205℃ - 170/0.074 - L-EPDM T= 190℃ 5/0.102 17/0.148 285/0.178 T= 205℃ 5/0.087 17/0.137 70/0.145 B-EPDM T= 190 ℃ 5/0.100 70/0.201 330/0.240 T= 205℃ 5/0.860 70/0.141 260/0.232 73 7.4.2 Melt fracture of TPVs As discussed above the melt fracture behavior of TPVs was studied in capillary rheometry at 190oC and 205oC. As a general remark the distortions observed in the capillary extrusion of TPVs are much milder compared to those seen in PPs and EPDMs at similar shear rates and temperatures. Two types of instabilities have been observed, which are classified in the context of this work as minor and severe surface distortions. Essentially no gross melt fracture appears in the case of TPV extrusion. Figure 7-14 illustrates these two types of extrudate distortion (minor and severe surface melt fracture) using the real extrudate image and the processed image with photoshop to observe more clearly the edge profile. It has to be mentioned that although these instabilities/distortions are limited to the surface, they are not periodic such as those seen in EPDM and therefore they cannot be referred to as sharkskin (term reserved for periodic surface distortions). Below the effects of temperature, shear rate, curing degree and EPDM concentration on melt fracture is examined. Minor surface distortions Severe surface distortions Figure 7-14. Typical extrudate images and their corresponding surface profiles that exhibit minor and severe surface distortions. Effect of curing level and temperature: The effect of curing level on the melt fracture of the TPVs is shown in the Figure 7-15 (TPV2 and TPV3 at 190℃). As it can be observed from all images, the defects are limited to the surface and they become less severe or disappear with increase of the shear rate. It appears that the first surface distortions appear at wall shear stresses 74 in the range of 45-78 kPa (listed in Table 7-4) of the same order of magnitude with the yield stress values reported by Ghahramani et al [2] (shear yield stresses in the range of 20-25 kPa and yield stresses in extension in the range of 9-15 kPa, all listed in Table 7-5). The polymer extruded at such shear rates include yielded and un-yielded regions that relax differently upon extrusion resulting these minor instabilities, which are definitely non-periodic. At higher shear rates/stresses, well above the yield stress, most of polymer volume is in a yielded state (except the core region or dead flow regions) resulting more uniform and smoother extrudates. The surface defects in TPV3 are more severe than those in TPV2 at the same shear rate. This is due to higher yield stress of TPV3 [2]. Therefore, a higher flow rate is needed in the case of TPV3 to eliminate these surface instabilities. Temperature also has an unexpected effect in the processing of TPVs. Figure 7-15 presents images of TPV3 at 190 and 205℃. It can be observed that the severity of surface defects increases with temperature. As discussed above the equilibrium storage and shear moduli, 'yG and *yG (fingerprints of the morphological structure of these complex multiphase systems related to the yield stress) increase with temperature. In other words, increase of temperature increases the strength of the rubber network strengthening, causing this surprising result. 75 TPV2, T=190 ℃, L/D=33 ?̇? = 5 𝑠−1, (Minor) ?̇? = 11 𝑠−1, (Minor) ?̇? = 170 𝑠−1, (Minor) TPV3, T=190 ℃, L/D=33 ?̇? = 5 𝑠−1, (Minor) ?̇? = 11 𝑠−1, (Minor) ?̇? = 170 𝑠−1, (Minor) TPV3, T=205 ℃, L/D=33 ?̇? = 5 𝑠−1, (Major) ?̇? = 11 𝑠−1, (Major) ?̇? = 170 𝑠−1, (Minor) Figure 7-15. The effect of curing level (TPV2 and TPV3 at 190℃ ) and that of temperature (TPV3 at 190 and 205℃) on the surface extrudate distortion of TPVs. The effect of crosslinked rubber concentration: Figure 7-16 shows surface fractures of TPV1 (Group A) and TPV5 (Group B) at 190 ℃. While TPV1 shows some minor sharkskin, TPV5 is almost completely smooth at all shear rates. Therefore, increase of the amount of crosslinked rubber particles, increases the severity of surface defects. This again can be explained on the basis 76 of a higher yield stress (Table 7-4), and higher equilibrium storage and shear moduli, 'yG and *yG (fingerprints of the morphological structure) of TPV1 compared to those of TPV5. Finally, it was also concluded that the thermoplastic type (L-PP vs B-PP) or rubber type (L-EPDM vs B-EPDM) does not have a clear influence on the TPV’s surface fracture. TPV1, T=190 ℃, 𝐿/𝐷 =33 ?̇? = 5 𝑠−1, (Minor) ?̇? = 11 𝑠−1, (Minor) ?̇? = 170 𝑠−1, (Minor) TPV5, T=190 ℃, L/D=33 ?̇? = 5 𝑠−1, Smooth ?̇? = 11 𝑠−1, Smooth ?̇? = 170 𝑠−1, Smooth Figure 7-16. Comparison of extrudate surfaces with different amount of crosslinked rubber (TPV1 and TPV5 at 190 ℃) 77 Table 7-4. The critical shear rates and stresses at 190oC and 205oC for the end of surface instabilities of TPVs listed in Table 4-2. Sample Critical shear rate s-1/ shear stress (MPa) for end of instability Major Sharkskin Minor Sharkskin 190 ℃ 205 ℃ 190 ℃ 205 ℃ TPV1 <5/0.063 <5/0.065 >1000/0.173 >1000/0.171 TPV2 <5/0.077 <5/0.066 >1000/0.190 >1000/0.162 TPV3 <5/0.073 18/0.083 640/0.164 820/0.156 TPV4 <5/0.057 18/0.068 >1000/0.168 >1000/0.163 TPV5 <5/0.078 <5/0.074 <5/0.078 640/0.218 TPV6 8/0.078 <5/0.069 64/0.116 820/0.175 TPV7 <5/0.066 <5/0.068 >1000/0.190 820/0.168 TPV8 <5/0.045 <5/0.037 170/0.090 820/0.106 Table 7-5. Yield stresses in shear and extension of TPVs at 190oC Sample Yield stress in Shear y (kPa) Yield stress in Extension ,y E (kPa) TPV1 18.5 14.8 TPV2 15.4 14.7 TPV3 22.5 13.5 TPV4 18.9 12.6 TPV5 12.3 11.7 TPV6 22.7 10 TPV7 22.7 8.4 TPV8 25.7 10.1 7.5 Extensional Viscosity Figure 7-17 shows the comparison of extensional behavior of TPV 4 and TPV 5 with their components. The change in concavity of TPVs’ curves represents yield stress [102]. As can be seen, the strong strain hardening of components is not observed in the TPVs. However, Figure 7-18 shows that the strain hardening becomes more obvious at elevated temperatures. The yield stress is also remains the same or slightly increases with the temperature. This is probably due to network strengthening with the temperature similar to 𝐺𝑦′ . 78 Figure 7-17. Comparison of extensional behavior of TPV4 and TPV5 with those of their constituent components. The data for TPV5 and TPV4 have been multiplied by 6 times for the sake of clarity. Figure 7-18. Extensional behavior of a) TPV4, and b) TPV 1 at different temperatures. 7.6 Conclusions In this chapter, the effects of several parameters including the temperature, thermoplastic matrix and rubber particles types, filler and curing levels on the melt fracture of TPVs were investigated. Depending on the molecular characteristics of the constituent polymers, the level of curing, and amount of crosslinked rubber, their processing (extrusion) exhibits various difficulties such as melt fracture (extrudate distortions). 79 Based on the results of this work, raising the temperature or the amount of rubber particles increases the rubber network strength and consequently creates more surface defects. It also causes more strain hardening in the extensional flow field for all combinations of PP and EPDM independent of linear or branched structure. Moreover, the slip velocity of EPDMs increases with the temperature and linearity of the polymer. It was also concluded that the slip effect for PPs is negligible due to the high molecular weight and relatively narrow molecular weight distribution of L-PP and the branched structure of B-PP. The results showed that TPV’s slip behavior follows its constituent EPDM at high shear rates and based on Mooney analysis [48] it was concluded that the TPV flow in the capillary is almost a plug flow. Melt fracture analysis showed that TPV surface fracture is increased with decreasing the curing level (especially at high shear rates) which is probably due to yield stress increment in the TPVs. In summary, yield stresses (in shear and extension) and the equilibrium storage and shear moduli, 'yG and *yG (fingerprints of the morphological structure) play a critical role in the melt fracture behavior of TPVs. It appears that these parameters correlate inversely proportional with the critical shear stress for the onset of surface instabilities. 80 8 Conclusions and Recommendations In this PhD thesis, the linear and non-linear viscoelastic and flow behavior of several groups of TPVs was fully investigated using rheology to understand how to reduce the flow instabilities such as slip and melt fracture. The critical parameters influencing flow instabilities of TPVs were determined and the relationship between these parameters in shear, extensional, and capillary (pressure driven flow field) was studied, as well. This chapter summarizes the important results obtained from the experiments and modeling reported in the thesis. Recommendations for further work are also provided. 8.1 Conclusions • All TPVs showed a universal non-terminal relaxation behavior i.e., they do not relax the stress at long times or low frequencies. The behavior can be modeled by a simple power-law model 𝐺(𝑡) = 𝐺𝑐𝑡−𝑝 + 𝐺𝑦′ , where 𝐺𝑦′ is the equilibrium modulus indicating the elastomeric nature of TPVs which was suggested to be related to their microstructure. • The yield stress increases by increasing the rubber content. It decreases by the curing level as the network becomes stiffer and the yield strain decreases which consequently decrease the yield stress. The TPVs behave more similar to their thermoplastic phase at high frequencies (rose timescale) and follow their rubber behavior at low frequencies. • For the first time the yield stress for polymer melts in the extensional flow field was observed. The yield stress manifests itself as a change of the curvature of the stress-strain curve. The ratio of extensional yield stress and shear yield stress was found to deviate from von Misses criterion and be in the range of √3 and 3√3 consistent with reported finding for other similar materials. • It was concluded that TPVs slip massively. The combination of the presence of slip and yield stress originated from the strong rubber network cause a significant reduction in the stress from capillary experiment compared to linear viscoelastic behavior. The slip velocity in capillary flow was obtained from Mooney technique. • The non-linear behavior of the TPVs can be modeled using K-BKZ model. To incorporate slip in the model, a fraction of the imposed strain should be applied on the model. It is supposed that the TPVs do not slip in the linear viscoelastic region and they show a pure 81 elastic behavior before yielding (at high curing levels). The slip measurements from parallel-plate and capillary flow were found to be consistent • TPVs mostly show surface fracture as opposed to melt fracture. The surface fracture is not sharkskin as it is not periodic, and it decreases by increasing shear rate. Therefore, the origin of surface fracture in TPVs differ from other polymers. It was concluded that a minimum stress is required to eliminate or reduce the surface fracture which is the yield stress. By increasing rubber content, the yield stress increases. Thus, more surface fracture is seen in TPVs with high rubber content. The fracture increases with the temperature, as well. • It was concluded that the yield stress is a key parameter determining the processability of the TPVs. Any parameter that increases yield stress, leads to higher flow instabilities, for example by increasing rubber content, yield stress and consequently flow instability increase. • As a general picture of the parameters affecting melt fracture, it can be concluded that higher curing level, higher rubber content, and higher molecular wight of the rubber phase contribute significantly to the melt fracture of the TPVs. The effect of the curing level is more profound than the effect of the molecular weight the rubber phase. Moreover, polymer components’ molecular structure and molecular weight of the PP phase have a minor influence on the melt fracture of the TPVs. In addition, lower temperature, and higher shear rate assist with the reduction of the melt fracture. 8.2 Future Recommendations • It is recommended to study the melt fracture of TPVs using different die geometries such as slit to address the edge and stress concentration effects on their melt fracture. In addition, other more complex geometries can be used that create dead zones for unyielded regions to study how unyielded regions affect melt fracture phenomena • It would be helpful to study the slip behavior at higher temperatures to investigate if the TPVs show the same behavior at higher temperatures. On the same time the K-BKZ model can be used to check if it is still valid. The slip behavior in capillary flow can be studied also by using capillary dies of different diameter to study the effect of pressure on slip. It 82 is typically expected that increasing pressure (normal force), the slip should decrease (analog of friction of elastomers on solid substrates). • The presence of oil has a significant influence on the slip behavior of TPVs. It would be also useful to study the effect of oil content on the rheological properties of TPVs and how much if affects the TPVs wall slip behavior.83 References [1] K. 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Thesis/Dissertation
eng
Chemical and Biological Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Hatzikiriakos, Savvas Georgios, 1961-
Flow instabilities of PP filled with crosslinked EPDM rubber : using rheology to understand and control its occurrence
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