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Modeling and flow simulation of polytetrafluoroethylene (PTFE) paste extrusion Patil, Pramod Dhanaji 2007

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Modeling and Flow Simulation of  Polytetrafluoroethylene (PTFE) Paste Extrusion. by PRAMOD DHANAJI PATIL Bachelor of  Engineering (Chem. Eng.), Shivaji University, 1998 Master of  Applied Sciences (Chem. Eng.), Indian Institute of  Science, 2002 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Chemical and Biological Engineering) THE UNIVERSITY OF BRITISH COLUMBIA January 2007 © Pramod Dhanaji Patil, 2007 Abstract Constitutive rheological equations are proposed for  the paste extrusion of polytetrafluoroethylene  (PTFE) that take into account the continuous change of  the microstructure during flow,  essentially through fibril  formation.  The mechanism of fibrillation  is captured through a microscopic model for  a structural parameter, , that is the percentage of  fibrillated  domains in the paste. This model essentially represents a balance of  fibrillated  and unfibrillated  domains through a first  order kinetic differential equation. The rate of  fibril  formation  is assumed to be a function  of  the strain rate and a flow  type parameter, which describes the relative strength of  straining and rotation in mixed type flows.  The proposed constitutive equation consists of  a shear-thinning and a shear-thickening terms, the relative contribution of  the two being a function  of  £,. To improve the physics of  the constitutive equation and in order to develop a truly predictive flow  model, another constitutive equation is proposed which consist of  a viscous (shear-thinning) and an elastic (strain-hardening) term. A modified  Mooney-Rivlin model is used to model the elastic behavior of  the paste. The viscous and elastic parameters are determined by using shear and extensional rheometrical data on the paste. Finite element simulations using the proposed constitutive relations predict accurately the variation of  the extrusion pressure with the apparent shear rate and die geometrical parameters. An approximate analytical mathematical model for  polytetrafluoroethylene  (PTFE) paste extrusion through annular dies is also developed. This model takes into account the elastic-plastic and viscous nature of  the material in its non-melt state due to the formation of  fibrils  and presence of  lubricant. The radial flow  hypothesis (RFH) has been used to describe the flow  kinematics of  PTFE paste in the conical annular section of  the die. The validity of  this hypothesis is demonstrated by performing  numerical simulations using the developed shear thinning and shear thickening model. Model predictions are presented for  various cases and are found  to be consistent with experimental results of  macroscopic pressure drop measurements in rod and tube extrusion. Table of Contents Abstract ii Table of  Contents iii List of  Tables vi List of  Figures vii Acknowledgements xiii Dedication xiv 1 Properties and Processing of  Polytetrafluoroethylene  (PTFE) Paste 1 1.1. Introduction 1 1.2. Synthesis of  PTFE 2 1.3. Chemical and physical properties of  PTFE 2 1.4. PTFE fine  powder resin processing and application 6 1.5. Mathematical modeling of  PTFE paste flow  9 1.6. Bibliography 13 2 Paste Extrusion: General Review 15 2.1. Introduction 15 2.2. Paste flow  and extrusion 16 2.2.1. Paste formulation  16 2.2.2. Preforming  17 2.2.3. Phase migration and extrusion 18 2.2.4. Sintering 19 2.2.5. Mechanism of  PTFE paste flow  20 2.3. Experimental observations 21 2.3.1. PTFE paste extrusion 21 2.3.2. Effect  of  geometrical characteristics of  die on the extrusion pressure 22 2.4. Constitutive equations proposed to predict pressure drop in capillary die flow  .....25 2.5. Numerical simulation studies of  paste flow  30 2.6. Bibliography 32 3 Scope of  Work 35 3.1. Introduction 35 3.2. Thesis objectives 35 3.3. Thesis organization 36 3.4. Bibliography 38 4 Constitutive Modeling and Flow Simulation of  Polytetrafluoroethylene (PTFE) Paste Extrusion 39 4.1. Introduction 39 4.2. Theoretical modeling and numerical method 42 4.2.1. Governing equations 42 4.2.1.1 Constitutive equation 42 4.2.1.2 Flow type parameter^ 45 4.2.2. Boundary conditions 46 4.2.2.1 Slip boundary condition 47 4.2.3. Finite element method 49 4.3. Results and discussion 52 4.3.1. Effect  of  die entrance angle 54 4.3.2. Effect  of  apparent shear rate 57 4.3.3. Effect  of  die reduction ratio 58 4.3.4. Effect  of  die length-to-diameter ratio 59 4.3.5. Structural parameter 61 4.4. Conclusions 64 4.5. Bibliography 66 5 Viscoelastic Modeling and Flow Simulation of  Polytetrafluoroethylene (PTFE) Paste Extrusion 68 5.1. Introduction 68 5.2. Theoretical modeling and numerical method 69 5.2.1. Governing equations .....69 5.2.1.1 Constitutive equation 69 5.2.1.2 Parameter Estimation 73 5.2.1.3 Structural parameter 78 5.2.2. Boundary conditions 79 5.2.3. Finite element method 79 5.2.3.1 Particle tracking 81 5.2.3.2 Incorporating computed stresses into flow  solution 82 5.3. Results and discussion 84 5.3.1. Structural parameter 86 5.3.2. Effect  of  die entrance angle 89 5.3.3. Effect  of  apparent shear rate 91 5.3.4. Effect  of  die reduction ratio 93 5.3.5. Effect  of  die length-to-diameter ratio 94 5.4. Conclusions 96 5.5. Bibliography 98 6 An Analytical Flow Model for  Polytetrafluoroethylene  Paste Through Annular Dies 101 6.1. Introduction 101 6.2. Validation of  radial flow  103 6.2.1. Cylindrical dies 106 6.2.2. Annular die with varying diameter mandrel pin 108 6.2.3. Annular die with axisymmetric cylindrical mandrel pin 110 6.3. Mathematical Model 112 6.3.1. Annular die without die land (L/Da = 0) 112 6.3.2. Annular die with die land (L/Da?i 0) 117 6.4. Model predictions and comparison with experiments 118 6.5. Conclusions 123 6.6. Bibliography 124 7 Conclusions, Recommendations and Contribution to the Knowledge 126 7.1. Introduction 126 7.2. Conclusions 126 7.3. Contribution to knowledge 128 7.4. Recommendations for  future  work 129 7.5. Bibliography 130 List of Tables Table 4.1 Physical properties of  the Isopar® M lubricant used in the slip velocity measurements 48 Table 4.2 Parameters for  the shear-thinning and the shear-thickening terms of Eq.(4.3) ; 55 Table 5.1 Physical properties of  PTFE fine  powder resin studied in this work as provided by the supplier 73 Table 5.2 Parameters for  the shear-thinning terms of  Eq. (4.3) (F104 HMW) 75 Table 5.3 Material parameters for  PTFE samples subjected to different  Hencky strain rate 76 Table 6.1 Values of  material constants and coefficient  of  friction  needed in Eq. (6.22) to predict the extrusion pressure for  paste slow in cylindrical and annular dies 119 List of Figures Figure 1.1 Schematic diagram of  a chain segment of  PTFE molecule 3 Figure 1.2 Partial phase diagram of  PTFE (Sperati, 1989)..., 5 Figure 1.3 Schematic diagram of  (a) the preforming  unit and (b) Instron capillary rheometer used for  paste extrusion 7 Figure 1.4 Tube extrusion equipment of  PTFE fine  powder (Daikin technical bulletin) 8 Figure 1.5 A typical tube extrusion dies for  the PTFE fine  powder (Daikin technical bulletin) \ 9 Figure 1.6 SEM micrographs of  PTFE paste at various stages of  the paste extrusion process (a) before  processing (nearly no fibrillation),  (b) during processing (partially fibrillated  sample with small ^), and (c) after  processing (nearly fully  fibrillated  sample with larger t,) (Patil et al. 2006) 10 Figure 2.1 Typical start up of  pressure transient obtained in PTFE paste extrusion [Ochoa, 2006] 22 Figure 2.2 The effect  of  lubricant (ISOPAR® G) concentration on the steady-state extrusion pressure for  resin 3 (PTFE resin). Solid lines are model predictions [Ariawan et al., (2002)] 23 Figure 2.3 The effect  of  reduction ratio on the steady-state extrusion pressure for  different  PTFE resins. Solid lines are model predictions [Ariawan et al., (2002)] 24 Figure 2.4 The effect  of  die UD a ratio on the steady state extrusion pressure at different  reduction ratios for  resin 3 (PTFE resin). Solid lines are model predictions [Ariawan et al., (2002)] 24 Figure 2.5 The effect  of  die entrance angle on the steady-state extrusion pressure at different  extrusion rates for  resin 3 (PTFE resin). Solid lines are model predictions [Ariawan et al., (2002)]. Also dotted line shown is the prediction using Benbow-Bridgwater equation (1993) 25 Figure 4.1 A conical entry die used in the paste extrusion of  PTFE: The left  half illustrates the extrusion of  PTFE particles and the gradual structure formation  through particle fibrillation,  whereas the shaded area depicts the axisymmetric domain used in the simulations 40 Figure 4.2 Flow fields  corresponding to different  values of  flow  type parameters, v|/ 45 Figure 4.3 The apparent flow  curves of  PTFE paste extruded through three dies having the same L/D ratio and different  diameter 49 Figure 4.4 Mooney plot based on the experimental data of  Figure 4.4 prepared to calculate the slip velocity. The shear stress values that correspond to individual lines at various shear stress values are also shown 50 Figure 4.5 The slip velocity, V s , as a function  of  the wall shear stress, <rw for  a PTFE paste used in this work. A linear slip model seems adequate to capture the experimental results 50 Figure 4.6 Geometrical domain used for  simulations with enlarged section of the rounded corner shown on the right side 51 Figure 4.7 Radial velocity profiles  at various axial locations for  conical die with entrance angle of  90° 52 Figure 4.8 Simulated surface  plots of  flow  type parameter, vy inside the conical die with die entrance 2a = 60° 53 Figure 4.9 The effect  of  die entrance angle on the extrusion pressure: Comparison between experimental and simulation results 54 Figure 4.10 The effect  of  die entrance angle on the cross-sectional average structural parameter at the exit, £,exjt, (from  simulations) and the tensile strength of  dried extrudates (from  experiments) 56 Figure 4.11 The effect  of  apparent shear rate on the extrusion pressure of  PTFE paste extrusion: Comparison between experimental and simulation results 57 Figure 4.12 The effect  of  apparent shear rate on the cross-sectional average structural parameter at the exit, ^ e x i t , (from  simulations) and the tensile strength of  dried extrudates (from  experiments) 58 Figure 4.13 The effect  of  the die reduction ratio, RR, on the extrusion pressure: comparison between experimental and simulation results 59 Figure 4.14 The effect  of  die reduction ratio on the average structural parameter at the exit, £,exjt, (from  simulations) and the tensile strength of dried extrudates (from  experiments) 60 Figure 4.15 The effect  of  length-to-diameter ratio (L/D) on the extrusion pressure: Comparison between experimental and simulation results 60 Figure 4.16 The effect  of  die L / D ratio on average structural parameter at the exit, ^ e x i l , (from  simulations) and the tensile strength of  dried extrudates (from  experiments) 61 Figure 4.17 Axial profiles  of  structural parameter at various radial locations for a conical die having an entrance angle of  60° 62 Figure 4.18 Axial profiles  of  structural parameter along the centerline of  a conical die having an entrance angle of  60° for  various apparent shear rates indicated in the figure  63 Figure 4.19 Axial profiles  of  structural parameter along the centerline of conical dies having various entrance angles 64 Figure 5.1 Plot of  viscosity vs stress obtained from  controlled stress experiments for  two different  gap sizes 74 Figure 5.2 Corrected viscosity vs. shear rate obtained from  Eq. (5.13 and 5.14) and solid line shows the curve fitting  by using Carreau model 74 Figure 5.3 Uniaxial extension of  F104 HMW samples stretched at different Hencky strain rates (lines show the fits  of  Eq. (5.12)) 76 Figure 5.4 Uniaxial extension of  F104 LMW samples stretched at different Hencky strain rates, (lines show the fits  of  Eq. (5.12)) 77 Figure 5.5 Uniaxial extension of  F301 samples stretched at different  Hencky strain rates, (lines show the fits  of  Eq. (5.12)) 77 Figure 5.6 Uniaxial extension of  F303 samples stretched at different  Hencky strain rates, (lines show the fits  of  Eq. (5.12)) 78 Figure 5.7 Comparison between vortex intensities obtained in current work and those obtained by Olley et al. (1999), for  axisymmetric flow  of LDPE through a 4:1 abrupt contraction 84 Figure 5.8 Radial velocity profiles  at various axial locations for  conical die having an entrance angle of  90°. Thick and thin lines denote the velocity profiles  from  viscoelastic and STT model respectively 85 Figure 5.9 Radial velocity profiles  at various axial locations inside the die land for  a.conical die having an entrance angle of  90°. Thick and thin lines denote the velocity profiles  from  viscoelastic and STT model respectively 86 Figure 5.10 Axial profiles  of  structural parameter at various radial locations for a conical die having an entrance angle of  60°: thin and thick lines show the structural parameter values from  STT and viscoelastic models respectively 87 Figure 5.11 Axial profiles  of  structural parameter along the centerline of  a conical die having an entrance angle of  60° for  various apparent shear rates indicated in the figure:  thin and thick lines show the structural parameter values from  STT and viscoelastic models respectively.... ..: 88 Figure 5.12 Axial profiles  of  structural parameter along the centerline of conical dies having various entrance angles: thin and thick lines show the structural parameter values from  STT and viscoelastic models respectively 89 Figure 5.13 The effect  of  die entrance angle on the extrusion pressure: Comparison between experimental results, predictions from  STT and Viscoelastic model 90 Figure 5.14 The effect  of  die entrance angle on the cross-sectional average structural parameter at the exit, £,exit, (from  Viscoelastic and STT simulations) and the tensile strength of  dried extrudates (from experiments) 92 Figure 5.15 The effect  of  apparent shear rate on the extrusion pressure of  PTFE paste extrusion: Comparison between experimental and simulation results from  viscoelastic and STT model 93 Figure 5.16 The effect  of  apparent shear rate on the cross-sectional average structural parameter at the exit, 2,exjt, (from  viscoelastic and STT simulations) and the tensile strength of  dried extrudates (from experiments) 93 Figure 5.17 The effect  of  the die reduction ratio, RR, on the extrusion pressure: comparison between experimental and simulation results from viscoelastic and STT model 94 Figure 5.18 The effect  of  die reduction ratio on the average structural parameter at the exit, ^ e x i t , (from  STT and viscoelastic simulations) and the tensile strength of  dried extrudates (from  experiments) 95 Figure 5.19 The effect  of  length-to-diameter ratio (L/D) on the extrusion pressure: Comparison between experimental and simulation results from  viscoelastic and STT model (Patil et al., 2006) 95 Figure 5.20 The effect  of  die L / D ratio on average structural parameter at the exit, <|exit, (from  viscoelastic and STT simulations) and the tensile strength of  dried extrudates (from  experiments) 96 Figure 6.1 Schematic illustration of  the "radial flow"  hypothesis. The hypothesis assumes the existence of  a virtual surface  of  radius r as measured from  the die apex, on which all paste particles moving towards the apex have the same velocity: (a) cylindrical die for  rod extrusion and (b) annular die with inside cylinder of  varying diameter (mandrel pin) for  tube extrusion, and (c) annular die with inside cylinder of  constant diameter (mandrel pin) for  tube extrusion 104 Figure 6.2 Velocity profiles  along the spherical surfaces  at radius r = 5.8 x 10 -3 m (a), 1.16 x 10~2 m (b), and 1.54 x 10 -2 m (c) for  cylindrical die (0 = 0 corresponds to the centerline). In our model, there is only one radial velocity component, which is computed in a cylindrical coordinate system 105 Figure 6.3 Percentage variation of  velocity normalized by the centerline velocity, from  the centerline of  the die to the die wall plotted with die entrance angle. The three surfaces  are defined  by the cylindrical radius R= 1.5xl0~3 m (a), 3xl0~3 m (b) and 4x l0" 3 m (c) 107 Figure 6.4 Velocity profiles  along the spherical surfaces  at radius r = 5.8 x 10~3 m (a), 1.16 x 10~2 m (b), and 1.54 x 10~2 m (c) for  an annular die with inside cylinder of  varying diameter (0 = 15° corresponds to the outside wall, 0 = 0 ° does not exist due to the presence of  the internal mandrel pin) 109 Figure 6.5 Percentage variation of  velocity from  the wall of  the varying diameter mandrel pin to the die wall normalized by the velocity at the wall of  the varying diameter cylinder, plotted with die entrance angle at three different  spherical locations (a), (b) and (c) defined  in Figure 6.3 109 Figure 6.6 Velocity profiles  along the spherical surfaces  at radius r = 5.8 x 10 3 m (a), 1.16 x 10~2 m (b), and 1.54 x 10~2 m (c) for  an annular die having a mandrel pin of  constant diameter I l l Figure 6.7 Percentage variation of  velocity from  the wall of  the constant diameter mandrel pin to the die wall normalized by the velocity at the wall of  the constant diameter cylinder, plotted with die entrance angle at three spherical locations (a), (b) and (c) defined  in Figure 6.3 I l l Figure 6.8 (a) Annular die with varying diameter mandrel pin with volume element and (b) its dimensions in the annular conical zone of  a tapered die according to "radial flow"  hypothesis 113 Figure 6.9 Force balance on volume element in the die capillary zone 117 Figure 6.10 The effect  of  apparent shear rate on the extrusion pressure of  PTFE paste for  a cylindrical (rod extrusion) and an annular conical die (tube extrusion). The experimental data refer  to rod extrusion using an orifice  die (L/D=0), having RR=352, 2 a =60° 120 Figure 6.11 The effect  of  die reduction ratio on the extrusion pressure of  PTFE paste for  a cylindrical (rod extrusion) and an annular die (tube extrusion) 121 Figure 6.12 The effect  of  die entrance angle (2a) on the extrusion pressure of PTFE paste for  a cylindrical (rod extrusion) and an annular die (tube extrusion) 122 Figure 6.13 The effect  of  apparent shear rate on the extrusion pressure of  PTFE (resin B) paste for  an annular die (tube extrusion) 122 Acknowledgements A number of  people have helped me with this project over the last few  years, and 1 would like to take this opportunity to thank them. Firstly, I wish to express my sincere gratitude and appreciation to my supervisor Prof.  Savvas G. Hatzikiriakos for  his unflagging  assistance and encouragement throughout my research. His insight and ideas have greatly contributed to the modeling and experimental aspects of  this work. I would also like to thank my co-supervisor, Dr. James J. Feng for  his encouragement and inputs in modeling and simulation aspects of  this work. I learned a lot from  the discussion I had with him and enjoyed his enthusiastic participation during the weekly research meetings. Thanks to Daikin Industries Ltd for  financial  support and the supply of  the polymer samples. My colleagues from  RHEOLAB at UBC helped in various ways. Especially, I wish to thank Isaias Ochoa for  providing me with experimental data at times and for having insightful  discussion about the modeling and experimental analysis. On a more personal level, I would like to thank my mother and father  for  their continued support, and my sisters for  their continuous encouragement during last three years. Most of  all, I thank my wife  Gayatri (Shobha) who has been an unending source of strength and motivation for  success, especially during the preparation of  this dissertation. Finally, I would like to thank my dear friends,  Rochish and Kiran for  their mentorship during this work. I would also like to thank the Indian community at UBC, "UTSAV" for  arranging nice cultural programs along with delicious Indian food. To my parents CHAPTER 1 Properties and Processing of Polytetrafluoroethylene (PTFE) Paste 1.1 Introduction Polytetrafluoroethylene  (PTFE) is a material of  great commercial value. Since its discovery by Dr. Roy Plunkett in 1938 (Plunkett, 1941), it has "revolutionized the plastic industry and led to various application not otherwise possible" (Plunkett, 1987). Polytetrafluoroethylene  (PTFE) was synthesized the first  time by an accident, but once its physical and chemical properties were disclosed the wide gamut of  applications was envisioned. These outstanding properties include a high melting point, exceedingly high molecular weight and melt viscosity, and high chemical resistance due to its limited solubility. The development of  innovative fabrication  processes resembling those used with metal powders was a crucial step in the emergence of  PTFE products. Processes currently in use include coating from  aqueous dispersions, compression molding and ram extrusion of  granular powders and paste extrusion of  lubricated fine  powder (Mazur, 1995). It is the last process of  paste extrusion that is being studied in this work. During the process of  extrusion, PTFE paste starts as a two phase system (lubricant and solid PTFE fine  particles) and ends essentially as a solid. Structural changes are taking place during flow  and thus rheological changes are significant.  Therefore,  in order to understand the process of  paste extrusion, the rheological changes that occur during extrusion should be first  understood. In particular, the following  questions should be considered; the relationship between the microstructure of  the PTFE paste and the rheological changes that take place during extrusion; the relationship between the microstructure formation  (fibrillation)  and the final  mechanical properties of  the extrudate; the influence  of  the lubricant concentration on the rheological properties; the influence  of  the physical properties of  the lubricant on various stages of  the process; the effects  of  the geometric characteristics of  the extrusion die used on the overall process as well as on the mechanical properties of  the final  extrudates. These questions consist also the scope of  the present work and they are discussed in more detail in chapter 3. This chapter presents a general overview of  the main object of  this work, namely polytetrafluoroethylene  (PTFE). The chemistry of  PTFE and its various physical properties are discussed. An overview of  the various applications of  PTFE is also presented. Emphasis is placed on the mathematical modeling of  industrial processes of PTFE and in particular on the paste extrusion process. 1.2 Synthesis of  PTFE Two different  methods of  polymerization are common for  production of  different types of  PTFE. Both of  them are carried out in an aqueous medium involving an initiator, a surfactant,  other additives and agitation brought to high temperature and pressure. The main differences  are the amount of  surfactant  added to the polymerization reactor and the shear rate applied during the reaction. The first  procedure is known as suspension polymerization.  This is the route to produce granular  resins which are processed as molding powder. The second technique of  polymerization is called emulsion or dispersion  polymerization. By this way dispersion  and fine  powder  PTFE products are manufactured.  The details about these polymerization techniques can be found  elsewhere (Ebnesajjad, 2000; Gangal, 1994). Although the two procedures result in the same high molecular weight PTFE polymer, the products are distinctly different.  The granular product can be molded in various forms.  However, the resin obtained from  aqueous dispersion polymerization cannot be molded, but has to be fabricated  by dispersion coating, in the case of  the concentrated dispersion, or by paste extrusion in the case of fine  powder resin (Blanchet, 1997; Ebnesajjad, 2000). 1.3 Chemical and physical properties of  PTFE Fluoropolymer or perfluoropolymer  are the names given to designate those polymers whose molecules are mainly consisting of  carbon (C) and fluorine  (F) atoms. Those names let us distinguish them from  other polymers that are just partially fluorinated.  An example of  a linear fluoropolymer  is polytetrafluoroethylene  (PTFE). PTFE, with chemical formula  [(-CF2-CF2-)N], can be compared with polyethylene [(-CH2-CH2-)n] where all the hydrogen atoms have been substituted by fluorine  atoms. Of  course, polyethylene and PTFE are prepared in totally different  ways. The basic properties of  fluoropolymers  arise from  the atomic structure of  fluorine and carbon and their covalent bonding in specific  chemical structures. Figure 1.1 depicts the straight chain molecular configuration  of  PTFE. The fluorine  atoms, in cyan color, are placed helically around the carbon backbone (in grey color) providing a protective shield from  virtually any chemical attack thus imparting chemical inertness and stability to the molecule (Gangal, 1994; Ebnesajjad, 2000). The helical conformation  of  the fluorine atoms assures that the hysteric repulsion is minimized. The two types of  covalent bonds present in the PTFE molecule, C-F and C-C, are extremely strong (Cottrell, 1958; Sheppard and Sharts, 1969) causing PTFE to have excellent mechanical strength and resistance to heat. The fluorine  shield is also responsible for  the low surface  energy (18 dynes/cm) causing PTFE to have a low coefficient  of  friction  on steel (0.05-0.8 static) and non-stick properties (Gangal, 1994). PTFE, with its thermal and chemical stability, makes an excellent electrical insulator. Figure 1.1: Schematic diagram of  a chain segment of  PTFE molecule. The slippery PTFE can not be dissolved in any solvent, acid, or base and upon melting forms  a stiff  clear gel without flow.  Consequently, its molecular weight cannot be determined by conventional techniques. In practice, the number average molecular weight (Mn) is usually estimated from  the standard specific  gravity (SSG) of  the polymer. Higher SSG implies greater crystallinity and hence, lower molecular weight (Gangal, 1994; DuPont, 2001; Suwa, 1973). Due to the linearity of  PTFE molecules, the crystallinity of  a virgin PTFE resin may be as high as 92-98% (Gangal, 1989). As a result, the SSG of  PTFE is high for  a polymer, typically ranging from  2.1 to 2.3. Following the standard procedure for  measuring SSG (ASTM D4895), the number average molecular weight can be determined from Equation 1.1 is applied to 100% homopolymer resins with SSG >2.18 (DuPont technical information,  2001). The calculated molecular weights for  PTFE with SSG <2.18 are quite large (probably unrealistic), due to the asymptotic behavior of  Equation 1.1 in this range. The number average molecular weight of  a 100% homopolymer has also been correlated to the second heat of  recrystallization (AH C). The second heat of recrystallization is obtained by melting and crystallizing a sample of  PTFE twice by Differential  Scanning Calorimetry (DSC) (DuPont technical information,  2001). It was found  that M n = 2.1 x 10'°(A//C)~516 [1.2] where AH C is in cal/g. The applicable cooling rate is 4-32°C/min, over which the heat of second crystallization remained constant for  a given polymer. Typically, Mn is in the 106 to 107 range (Gangal, 1994). Comparison of  PTFE molecular weight, regardless of whether or not the resins contain other comonomers, can be made by considering the resin melt creep viscosity instead. The melt creep viscosity, as detailed in US Patent 3,819,594 (Holmes et al., 1974), is higher for  a higher molecular weight PTFE resin (DuPont, 2001). The melting point of  virgin PTFE (first  melting temperature) is 342°C (Sperati, 1989), which is high for  a thermoplastic polymer. The second melting temperature is 327°C (Ebnesajjad, 2000), which is the value often  reported in the literature. It means that a previous melted PTFE does not recover the original crystallinity back making the resin less crystalline (Gangal, 1994). During melting, a volume increase of  30% is typical (Sperati, 1989). The melt is stable, since even at 380°C, the melt viscosity is relatively high at approximately 10 GPa.s (Gangal, 1994). 8 7 6 -Q • 4 L_ 3 if)  _ w 3 E a. 2 1 0 0 20 40 60 80 100 120 Temperature (°C) - Figure 1.2: Partial phase diagram of  PTFE (Sperati, 1989). Besides the melting point, PTFE has other transition temperatures, two of  which are particularly important due to their proximity to the ambient temperature. These are shown in the partial phase diagram of  PTFE in Figure 1.2 (Sperati, 1989). Under ambient pressure conditions, the first  transition occurs at 19°C. At this temperature, the PTFE molecule chain segments change from  a perfect  three-dimensional order to a less ordered one undergoing a slight untwisting. Above 30°C, the second transition temperature, the extent of  disorder of  the rotational orientation of  molecules about their long axis is increased. In other words, below 19°C PTFE resin is strong enough to endure premature mechanical damage. Above 19°C, molecules are packed more loosely and shearing will cause the unwinding of  crystallites, creating fibrils  (Mazur, 1995; Ebnesajjad, 2000). At temperatures greater than 30°C, a higher degree of  fibrillation  can be achieved. This property has made it possible to process PTFE with paste extrusion near ambient temperature, producing a mechanically strong extrudate as these fibrils  are formed  and oriented in the flow  direction (Ebnesajjad, 2000). 1.4 PTFE fine  powder resin processing and applications The process of  PTFE paste extrusion is generally carried out in four  steps: paste preparation, preforming,  paste extrusion and sintering (Daikin technical bulletin). Paste is essentially a suspension of  solid particles in a liquid phase. In PTFE paste preparation, fine  powder resin of  individual particle diameters of  approximately 0.2 |um is first  mixed with a lubricating liquid (lube) in a desired mass proportion at a temperature lower than 19°C to form  a paste. A typical lube concentration varies from  16 to 25 wt. % (DuPont, 1994). Lubricant includes those from  the ISOPAR® series. Mixing is carried out below the PTFE transition temperature to ensure that the resin is not damaged prior to extrusion. The mixing container is then placed in a horizontal roll mixer that rotates at 15 rpm for approximately 1-2 hours. The resulting mixture (paste) is aged at room temperature for 24 hours before  extrusion experiments in order to allow uniform  wetting of  the resin particles by the lubricant. Preforming  is normally done before  extrusion using a capillary rheometer with a blank die as shown in Figure 1.3(a). The purpose of  preforming  is to remove air from  the material and compact the resin to achieve the maximum amount of  material for  extrusion. The paste inside the preforming  unit is compressed by means of  a piston at a pressure of 2 MPa over a period of  30 s to produce a cylindrical preformed  billet that is free  of  air voids (Ariawan et al., 2001; Ochoa, 2006). It has been shown that the preforming pressure and duration significantly  affect  the quality of  preform  as it influences  liquid migration and density. The effect  of  physical properties of  lubricant such as viscosity and surface  tension on processing behavior of  the PTFE paste has been studied by Ochoa (2006). The improvement in preforming  quality was found  with an increase in lubricant viscosity and with improvement in the wettability characteristics of  the lubricant with PTFE (Ochoa and Hatzikiriakos, 2004; Ochoa, 2006). It was also found  that a lubricant with higher viscosity produces a more uniform  preform  as liquid migration is minimal. In addition lubricant with increasing wettability (low surface  tension) with PTFE produces better mixture/pastes. Sintered PTFE Capillary Rheometer Load Cell Preforming  Unit (Aluminum Pipe) Capillary Rheometer Test Frame Steel Plug Electrical Heaters Die entrance (contraction) zone Die capillary zone Tapered Capillary Die (a) (b) Figure 1.3: Schematic diagram of  (a) the preforming  unit and (b) Instron capillary rheometer used for  paste extrusion. The next step involves the extrusion of  the preform  using a ram extruder at a temperature slightly higher than 30°C (Ebnesajjad, 2000) as shown in Figure 1.3(b). During this step PTFE paste is extruded to obtain its final  shape. The physical properties of  lubricant were found  to play a significant  role in the extrusion of  PTFE pastes (Ochoa and Hatzikiriakos, 2005; Ochoa, 2006). Increasing the wettability of  lubricant with PTFE and decreasing the lubricant viscosity causes a reduction in the extrusion pressure and an increase in the tensile strength of  the extrudates. Raman spectroscopy has been used on the extrudate to describe quantitatively the degree of  fibril  orientation in the extrudates (Ariawan, 2002; Ariawan et al., 2002b). The effects  of  extrusion conditions such as extrusion temperature and speed, on the steady-state extrusion pressure have been reported by Ariawan et al., (2002a). The effect  of  die design have been studied through the dependence of  extrusion pressure on die reduction ratio, die entrance angle and die L/Da ratio (Ariawan, 2002; Ariawan et al., 2002a; Ochoa, 2006). The last step is the evaporation of  the lube by passing the extrudate through an oven. This is followed  by sintering at temperature above the melting point of  PTFE, for processes such as wire coating and tube fabrication.  To analyze the mechanical properties of  the extrudate, dried extrudates are tested for  their tensile strength according to ASTM D1710-96. The thermal properties of  the extrudate are determined using a DSC according to ASTM D3418-82. Typical extrusion processes include tube extrusion, wire coating and calendaring. For tube extrusion, the basic equipment is illustrated in Figure 1.4. The extruder consists of  a cylinder, a ram, a driving mechanism (hydraulic or screw type), a die and a mandrel. The cylinders generally used in extruders range from  50-200 mm in diameter, and from 500-1800 mm in length. A typical schematic of  a tube extrudate die appears in Figure 1.5. Figure 1.4: Tube extrusion equipment of  PTFE fine  powder (Daikin technical bulletin). Tube extrusion die "Die: T«np. (50-60*0) (122*-140sF) eater (50-60*0 (122-140»F) R,R.= Dc- Cylindar Inside diameter Dm: MarwJral outside ciiatneter cto: Di® orifice  inside dbamotof dp: Cora pin outs ids dumeter do - dp' Figure 1.5: A typical tube extrusion die for  the PTFE fine  powder (Daikin technical bulletin). 1.5 Mathematical modeling of  PTFE paste flow The paste is a mixture of  solid particles and liquid lubricant and the deformation of  a paste can be accompanied by motion of  the liquid component of  the paste relative to the solid (Sherwood, 2002). In addition, creation of  fibrils  between PTFE particles during PTFE paste flow  makes the flow  mechanism different  from  that of  the pastes of  other materials. During the extrusion process, compacted resin particles entering the die conical zone are highly compressed due to reduction in the flow  cross-sectional area. As the particles are squeezed against each other under the application of  pressure, PTFE crystallites across the interface  in neighbouring particles begin to mechanically interlock. This results into the interconnection of  adjacent particles through the formation  of  fibrils. Scanning Electron Microscope (SEM) analysis has shown that fibrils  are created in the conical zone of  the die where flow  is extensional (Mazur, 1995; Ariawan, 2002; Ariawan et al., 2002b). Simple shearing action of  loosely compacted paste at a relatively low pressure does not result in a practically useful  extent of  fibrillation.  Figure 1.6 shows typical SEM micrographs of  the paste before,  during and after  the extrusion. Figure 1.6a shows the absence of  fibrils  in the unprocessed paste, where PTFE particles essentially retain their spherical identity. Figure 1.6b shows the existence of  fibrils  as the paste flows downstream in the conical die zone. At the exit of  the die and depending on its geometrical characteristics, the paste might become nearly fully  fibrillated  (Figure 1,6c). It is these submicron-diameter fibrils  between polymer particles that essentially give the dimensional stability and strength to the final  extruded product. (c) ^ , < ^ < 1 Figure 1.6: SEM micrographs of  PTFE paste at various stages of  the paste extrusion process (a) before  processing (nearly no fibrillation),  (b) during processing (partially fibrillated  sample with small E, ), and (c) after  processing (nearly fully  fibrillated  sample with larger £,) (Patil et al. 2006a). Mathematically, the flow  of  PTFE paste is treated by using the "radial flow hypothesis" (Snelling and Lontz, 1960; Ariawan, 2002) which states that paste particles at the same radial distance from  the apex of  the die conical zone move towards the die apex at the same velocity. However, this approach is only empirical arising from experimental evidence. The 1-D model proposed by Ariawan et al., (2002b), assuming paste as an elasto-viscoplastic material through contribution from  strain hardening and viscous resistance, was able to predict the steady state extrusion pressure reasonably well. However, this model assumes a velocity profile  which makes it only empirical in nature. In addition, fibril  formation  is not taken into account i.e. structure formation (Ariawan, 2002; Ariawan et al., 2002b; Benbow and Bridgwater, 1993; Horrobin and Nedderman, 1998). Fibril formation  during PTFE paste flow  has to be considered as an important parameter in modeling the paste flow  dynamics, and it is the main focus  of  the present study. Furthermore, experiments have shown that the degree of  fibrillation depends on the operating parameters as well as design characteristics of  the die (Ariawan, 2002; Ariawan et al., 2002a; Ariawan et al., 2002b; Ochoa and Hatzikiriakos, 2004). Numerous works have been done in modeling paste flow  in general using various approaches. For example, Kolenda et al., (2003) have solved the solid and liquid conservation equation separately for  flow  of  ceramic paste using a Lagrangian frame  of reference.  Burbidge and Bridgwater (1995) have also modeled the flow  of  ceramic paste based on the "radial flow  hypothesis" in paste flow  dynamics but their model always overpredicted the measured stress magnitude. Therefore,  a more complete model is required to predict the extrusion of  PTFE paste. A rational approach seems to be; first  to formulate  an approximate rheological constitutive equation that takes into account the structure formation  in paste flow  through fibrillation;  then to use this equation to simulate the paste extrusion process; and finally  compare the calculated with the experimental extrusion pressure as a function  of  operating and die geometrical characteristics. In this work, new constitutive equations are formulated  for  PTFE paste flow.  The rate-induced microstructural changes during PTFE paste processing, essentially consist of change of  paste from  a liquidlike to a solidlike state. This is incorporated in the constitutive model by introducing the concept of  structural parameter, , that represents the mass fraction  of  the paste which is fibrillated.  A kinetic model for  the structural parameter, , is proposed to describe the evolution of  with flow.  Steady shear and extensional rheological experiments are performed  on PTFE paste in order to determine the parameters of  the rheological model. Finite element flow  simulations are performed and the results are compared with experimental results in order to check the validity and usefulness  of  the proposed rheological constitutive equation. In addition, flow simulations are used to predict the extrusion pressure as a function  of  the operating and die geometrical characterictics and to explore the relationship between the tensile strength of  the extrudate and the degree of  fibrillation.  To model the process of  tube extrusion, an approximate analytical model is also proposed and validated through comparison with experiemental results. 1.6 Bibliography Ariawan, A. B., S. Ebnesajjad and S.G. Hatzikiriakos, Preforming  Behavior of  PTFE Pastes, Powder Technology 121, 249-258 (2001). Ariawan, A. B., Ebnesajjad, S. and Hatzikiriakos, S. G. Paste extrusion of polytetrafluoroethylene  (PTFE)  fine  powder  resins. Can. Chem. Eng. J., 80, 1153-1165 (2002a). Ariawan, A.B., Ebnesajjad, S. and Hatzikiriakos, S. G. Properties  of polytetrafluoroethylen  (PTFE)  paste extrudate.  Polym. Eng. Sci., 42, 1247-1253 (2002b). Ariawan, A. B. Paste Extrusion  of  Polytetrafluoroethylene  Fine  Powder  resins, The University of  British Columbia. Dept. of  Chemical and Biological Engineering. Thesis Ph. D., 2002. Benbow, J. J., and J. Bridgwater, Paste Flow  and  Extrusion,  Oxford  University Press, Oxford,  1993. Blanchet, T.A., Polytetrafluoroethylene,  Handbook  of  Thermoplastics,  Marcel Dekker, NY, 1997. Burbidge, A. S., J. Bridgwater, and Z. Saracevic, Liquid  Migration  in Paste Extrusion, Chem. Eng. Res. Design, 73, 810-816 (1995). Cottrell, T. L., The  strength  of  chemical bonds,  2 n d ed., Butterworths, Washington, D. C., 1958. Daikin Industries Ltd., Daikin PTFE  fine  powder,  Technical Bulletin, 2003. DuPont Fluoroproducts, Teflon®  PTFE  Fluoropolymer  Resin - Processing Guide for  Fine Powder Resins, Technical Bulletin, Wilmington, Delaware, 1994. DuPont Fluoroproducts, Molecular weight of  PTFE, Technical Bulletin, Wilmington, Delaware, 2001. Ebnesajjad Sina, Fluoroplastics,  Vol  1 Non-Melt  Processible  Fluoroplastics,  Plastic Design Library. William Andrew Corp, NY, 2000. Gangal, S. V., Polytetrafluoroethylene,  Homopolymers  of  Tetrafluoroethylene,  in Encyclopedia of  Polymer Science and Engineering, 2 n d ed., John Wiley & Sons, New York, 1989, 577-600. Gangal, S. V., Polytetrafluoroethylene,  in Encyclopedia  of  Chemical  Technology,  4 t h ed., John Wiley & Sons, New York, 621-644,1994. Holmes, D. A., Fasig, E. W., Plunkett, R. J., US Patent 3,819,594, assigned to DuPont de Nemours and Company, June 1974. Horrobin, D. J., and R. M. Nedderman, Die Entry  Pressure Drops in Paste Extrusion, Chem. Eng. Sci., 53, 3215-3225 (1998). Kolenda, F., Retana, P., Racineux, G. and Poitou, A., Identification  of  rheological parameters  by the squeezing test,  Powder Technology, 130, 56-62 (2003). Mazur, S., Paste Extrusion  of  Poly(tetrafluoroehtylene)  Fine  Powders  in Polymer Powder technology,441-481, Narks, M., Rosenzweig, N., Ed. John Wiley & Sons, 1995. Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, Constitutive  modeling  and  flow simulation of  polytetrafluoroethylene  (PTFE)  paste extrusion, J. Non-Newt. Fluid Mech. 139, 44-53 (2006a). Ochoa, I., Hatzikiriakos, S. G. Polytetrafluoroethylene  {PTFE)  paste performing: Viscosity  and  surface  tension effects.  Powder Technology, 146(1-2), 73-83 (2004). Ochoa, I., Hatzikiriakos, S. G. Paste extrusion of  polytetrafluoroethylene  (PTFE): Surface  tension and  viscosity effects,  Powder Technology (2005), 153(2), 108-118. Ochoa, I., Paste Extrusion  of  Polytetrafluoroethylene  Fine  Powder  resins: The  effect  of the processing aid  physical properties,  Ph.D., The University of  British Columbia. Dept. of  Chemical and Biological Engineering. Thesis Ph. D., 2006. Sheppard, W.A., Sharts, C. M., Organic Fluorine  Chemistry,  W. A. Benjamin, Inc., New York, 1969. Sherwood, J. D., Liquid-solid  relative  motion during  squeeze flow  of  pastes, J. Non-Newt. Fluid Mech., 104, 1-32 (2002). Snelling, G. R., and J. F. Lontz, Mechanism  of  Lubricant-Extrusion  of  Teflon®  TFE-Tetrafluoroethylene  Resins, J. Appl. Polym. Sci. Ill, 257-265 (1960). Sperati, C. A., Physical Constants  of  Fluoropolymers,  Polymer Handbook, John Wiley and Sons, NY, 1989. Suwa, T., M. Takehisa and S. Machi, Melting  and  Crystallization  Behavior of Poly(tetrafluoroethylene):  New  Method  for  Molecular  Weight  Measurement  of Poly(tetrafluoroethylene)  using a Differential  Scanning  Calorimetry,  J of  Appl. Polymer Sci., 17, 3253-3257 (1973). CHAPTER 2 Paste Extrusion: General Review 2.1 Introduction Paste extrusion is a widely used process in different  industries such as chemical, food  and pharmaceutical. Less common products but not less important includes ceramic components, catalyst supports, bricks, and many others. The increasing demand of  these products has attracted the attention of  researchers around the world and increased the interest to study the paste extrusion process. Many complicated structures, such as thin-walled honeycomb catalytic supports, rely on the uniformity  of  the extrudate to provide certain "high performance"  properties. Ram extrusion has made possible the characterization of  the rheological properties of  pastes when other techniques cannot be used. Among the most important factors  to be considered in ram extrusion of  pastes are paste formulations,  paste densification,  extrusion rate and die geometry. All these factors  together will allow a complete understanding of  paste extrusion for  the designing of  the optimum processing conditions for  a given extruded product. So far,  most of  the work on paste extrusion has been done with alumina pastes due to its importance in the catalyst and electronics industry. However, PTFE prepared by emulsion polymerization finds  paste extrusion as a good alternative for  product manufacturing.  Thanks to paste extrusion it is possible to process thin hoses, thick tubes (liners), wire insulation and unsintered tapes of  PTFE. PTFE paste extrusion is still under study and several topics need further  development. In this chapter, literature related to the subject of  PTFE rheology and its relation to the process of  paste extrusion is reviewed. Discussion is subdivided into paste flow and extrusion and into modelling of  paste flow.  In addition, some definitions  used in other chapters are introduced. The principles of  operation of  the equipment used to study the rheology of  PTFE paste extrudates are also included. 2.2 Paste flow  and extrusion 2.2.1 Paste formulation In simple terms, paste is a suspension of  solid particles in liquid phase, the relative amounts being such that the resulting material can be moulded readily (Benbow and Bridgwater, 1993). However, this definition  is not definitive  and other definitions  are made depending on the perceived mechanical response (Khan, 2001). For example, sometimes is indicated that the composition of  the paste is that to render a material soft and plastic, but the object so formed  should be able to retain its shape to allow any further processing. PTFE fine  powder resins are extremely sensitive to pressure and shear, so much so that they are shipped in specially constructed drums, which are designed to minimize compaction and shearing. Shearing would cause a phenomenon called fibrillation  that may lead to the formation  of  lumps, which cannot be broken up easily in order to produce a uniform  paste [DuPont, Processing Guide  for  Fine  Powder  Resins (1994); Daikin Industries Fluoroplastic, Product  Information  Guide  for  TFE  Fine  Powder  resins (1997)]. As already discussed fine  powder PTFE is processed by paste extrusion. PTFE resin is combined with a minimal quantity of  lubricant (an inert liquid hydrocarbon) and then extruded at a modest temperature (typically 30-35°C) into preforms  of  various shapes and dimensions with substantial mechanical integrity (Mazur, 1995). The liquid phase serves the purpose of  lubrication between the solid particles and also prevents solid particles from  mechanical damage. The lubricant also plays the role of  filling  the voids between particles. This way the paste becomes resistant to compressive load, without increasing inter-particle contact area. Also, the adhesive forces  between particles are reduced since the interfacial  tension of  polymer-lubricant is much less than that of polymer-air. This allows particles to rearrange more easily in response to mechanical force  without deformation.  Due to this, the particles will also remain isotropic in nature after  compression, which is not the case for  dry powder. The amount of  the lubricant and its properties critically affect  the extrusion process and, hence, the quality of  the final  product. The concentration of  the processing aid in the mixture depends on the type of  the product, equipment design and the desired extrusion pressure. Its content should be as low as possible but not so low that the extrusion pressure would be excessively high. The range of  lubricant content was found to be between 15 and 25% of  the total weight of  the compound and corresponds typically to volume fraction  between 0.34 and 0.45 (Mazur, 1995; Ebnesajjad 2000). As the amount of  liquid added to the powder increases above a critical value, the pressure required to extrude the mixture falls  dramatically (Benbow et al., 1998; Ariawan, 2002). For a typical commercial fine  powder, a 2% increase in lubricant causes a 40% decrease in extrusion pressure (Daikin technical bulletin). As more liquid is added, the material soon becomes too soft  to retain its shape. On the other hand, if  an inadequate amount of lubricant is used, the extrudate tends to be rough and irregular (Mazur, 1995). Regarding the properties of  the processing aid, any difference  in density and/or viscosity implies different  rheological properties. The viscosity of  the lubricant has a great effect  on the quality of  the paste. For example, the use of  a more viscous liquid as a lubricant results in a less uniform  mixture (Ochoa and Hatzikiriakos, 2004). Consequently the paste would not extrude into a continuous body, but many microcracks would be developed during the drying process after  extrusion (Ebnesajjad, 2000). In addition, the extrusion pressure will exhibit higher values when a processing aid with high viscosity is used (Benbow, 1998; Ochoa and Hatzikiriakos, 2005). Ideally, the lubricant should have a lower surface  tension than the critical surface  tension of  PTFE. That increases the wettability of  the lubricant with the resin particles (Ebnesajjad, 2000; Ochoa and Hatzikiriakos, 2005). The extrusion aid must be easily removable from  the extrudate without leaving a residue, which could alter the colour of  the final  product. Other requirements of  lubricants include high purity, low odour, low polar components, high auto-ignition temperature, and low skin irritation. 2.2.2 Preforming Another aspect related to paste extrusion is preforming.  During this step, the paste is placed in a cylindrical billet and by means of  a piston the pressure is gradually increased to remove the air from  the voids that will render a final  product mechanically weak. In this way a cylindrical rod that is fed  into the extruder's barrel is formed.  In PTFE paste processing, the preforming  stage is carried out at room temperature although it is not temperature sensitive (Mazur, 1995). However, the application of  stress introduces another problem since it may cause the liquid component of  a paste to move through the solid matrix in the radial and axial directions causing a liquid maldistribution throughout the paste (Yu, 1999). The extent of  the preforming  pressure and its duration significantly  affect  the quality of  the preform.  In fact,  to produce a preform  of  uniform density the magnitude of  the pressure depends on the molecular weight (standard specific gravity) of  the resin (Ariawan, 2002). Lack of  adequate pressure will result in a preform of  non-uniform  density which will extrude unsteadily, resulting in an unacceptable final product. During preforming,  the applied pressure compacts the particles making those ones adjacent to the wall of  the preforming  unit undergo plastic deformation  that results in a smooth film  of  deformed  powder surrounding the preform.  Because of  this layer, the rest of  the resin particles remain spherical even after  high pressure preforming  (Mazur, 1995). 2.2.3 Phase migration and extrusion Phase migration is a phenomenon that occurs not only during preforming  but also during extrusion (Yu, 1999). It is determined by relative motion of  the liquid through the voids between the solid-phase particles. This migration eventually results in non-uniform distribution of  lubricant in the mixture. This effect  is enhanced with time, especially in the presence of  high extrusion pressure. As the paste becomes drier, the extrusion pressure rises and the liquid loss increases consequentially. Eventually high frictional forces  may occur due to direct contact between particles and particles, and between particles and containing walls (Benbow et al., 1998; Blackburn, 1993). The packing characteristics of  the paste depend on the particle size, shape and size distribution and there is a direct correlation between the permeability of  a consolidated paste and its porosity (Rough, 2002). Thus, since the solid and liquid phases, move at significantly  different  rates under application of  a pressure gradient, some of  the liquid escapes from  the paste. If  the permeability through the packed particles is high and the liquid viscosity is low, conditions for  the liquid to move forward  faster  than the solid will be promoted. The result will be that the paste becomes effectively  drier, the extrusion pressure rises, and the process can be halted in extreme cases (Benbow and Bridgwater, 1993). After  extrusion, the extrudates can exhibit surface  fracture  depending on the processing conditions. Benbow and Bridgwater (1993) reported the effect  of  die shape, operating conditions, and paste formulation  on surface  defects  of  final  products. Domanti and Bridgwater (2000) studied extensively the effect  of  die land length, extrusion rate, die entry angle, extrusion ratio and water content on the surface  fracture  in the extrusion a-alumina paste mixed with Bentonite clay and carbohydrates. To reduce the severity of the extrudate distorsion, several options are available such as decreasing the extrusion rate, increasing the lubricant concentration in the paste mixture, using extrusion dies with long length to diameter ratio and small entrance angle (Benbow et al., 1987; Benbow and Bridgwater, 1993), altering the viscosity and yield properties of  the liquid phase, and by blending fine  and coarse powders in order to decrease the average pore size (Blackburn and Bohm, 1993). These alternatives have advantages and disadvantages or even they may not work at all for  all kind of  pastes. As far  as PTFE paste processing is concerned, there is an optimum value for  the entrance angle and length to diameter ratio of  the die as well as lubricant and viscosity concentration (Ariawan, 2002; Ochoa and Hatzikiriakos, 2005; Ochoa 2006). A very interesting phenomenon that occurs during PTFE paste extrusion is fibrillation.  It is the formation  of  fibrils  that interconnect the various particles together and these essentially give dimensional stability to the final  product (Figure 1.7 b,c). Lewis and Winchester (1953) first  reported that fibrillation  occurs during paste flow through the contraction area of  the die. Later, Ariawan (2002) found  the same through SEM analysis of  paste in the die entry region. Mazur (1995) explained this phenomenon by making reference  to particles that reorganize themselves to pass through the die during the initial stage of  extrusion. After  passing that region, particles are deformed  due the shear/extensional stresses, resulting in the formation  of  fibrils  which contribute to the mechanical strength of  the extrudates. 2.2.4 Sintering Sintering is the process during which a granular material, such as polymer powder of  PTFE, is heated to a temperature near its melting point (Hooper, 2000). In this process the particles of  the loose powder or pressed compact material weld together to form  an interconnected solid (Mackenzie and Shuttleworth, 1949). As a result, the density of  the compact changes. The coalescence of  contacting polymer particles is important to provide the final  product with suitable and improved mechanical properties. Previous studies of  sintering revealed that the surface  tension was the driving force  for  this phenomenon to occur. However, more recent reports have shown that the degree of sintering is governed by the particle size, viscosity, interfacial  tension, molecular architecture and molecular weight distribution (Hooper, 2000). In PTFE paste processing, the elastic phenomena dominate the sintering process (Mazur, 1995). During PTFE sintering, the net volume of  the material changes but the changes in linear dimensions are highly anisotropic. During the heating cycle, the sample contracts in the axial direction and expands in the radial direction making a net shrinkage of  about 4% (Mazur, 1995). In fact,  axial contraction is the resultant of  a contraction and expansion occurring simultaneously. Apparently, the axial contraction is driven by molecular orientation while the expansion is the response to release the stress accumulating during the former (Mazur, 1995). The resultant sintered extrudate exhibit higher tensile strength than the unsintered sample as an example of  the improvement in the mechanical properties (Ochoa and Hatzikiriakos, 2005). 2.2.5 Mechanism of  PTFE paste flow The flow  mechanism associated with PTFE paste extrusion differs  significantly from  polymer melt flow.  It is because, microscopically, solid state PTFE molecules are confined  in their crystallite and spherulite configurations,  while in polymer melt, molecules are randomly positioned, not conformed  to a specific  shape and are significantly  more mobile. In a number of  ways, the process of  PTFE paste extrusion is similar to ceramic paste processing (Benbow and Bridgwater, 1993) and the pastes used in pharmaceutical applications: see, for  example, Rough et al., (2000), Burbidge et al. (1995), and Yu et al. (1999). However, since fibrillation  is involved in the mechanism of PTFE paste flow,  the resulting extrudate is relatively stronger. To determine the morphological changes that take place during the course of PTFE paste extrusion, SEM analysis has been performed  on the PTFE paste before  and after  preforming,  as well as after  extrusion as shown in Figure 1.6. After  extrusion the rheology of  the material is quite different  as the PTFE particles are interconnected with fibrils  mostly oriented in the direction of  flow. 2.3 Experimental Observations 2.3.1 PTFE paste extrusion As compared to the various applications of  PTFE paste, little work has been done to understand the theoretical aspects of  PTFE paste flow.  Therefore,  relevant literature should be drawn from  work on paste extrusion of  other materials such as: ceramics, alumina-based materials, food,  and other polymers although the operating and design parameters are different  from  those applied to PTFE paste extrusion. Figure 2.1 represents a typical start up pressure transient response obtained during a PTFE paste extrusion by means of  a capillary rheometer (Ariawan, 2002; Ochoa, 2006). Three operating zones can be seen. The maximum in the extrusion pressure obtained in zone I, is essentially due to the finite  compressibility and the yield stress that causes jamming of  the paste in the barrel. Until this point is reached, the paste flows  in the die at very low speed. The paste is being compressed in the barrel and as a result the pressure increases gradually. During this compression period, the paste is in a state of  jamming, which is defined  as the conversion of  a liquid  system into a solid  by imposed  stress (Haw, 2004). This essentially means that there is a number of  immobile clusters of  particles in the upstream to the die entrance direction that are responsible of  the jamming (Breedveld, 2003; Manoharan and Elsesser, 2003). Collapse of  these immobile clusters of  particles initiate the flow  and this happens once the yield pressure is reached (Haw, 2004). Zone II is taken to be the steady state part of  the extrusion process. The recorded average pressure in this zone is reported as the extrusion pressure. Finally in zone III, the pressure gradually increases due to the fact  that the final  part of  the preform  becomes drier due to liquid migration. The network of  PTFE particles plays the role of  an apparently immobile screen. The net result of  this is that the lubricant is moving slightly faster  than the assembly of  the particles and therefore  causes the last part of  the preform to become drier (lower lubricant concentration) and therefore  to be extruded at a higher pressure (Ochoa and Hatzikiriakos, 2004). 3 (/> in 0) JL. QL c o 35 •5 HI Distance in the barrel Figure 2.1: Typical start up of  pressure transient obtained in PTFE paste extrusion [Ochoa, 2006]. 2.3.2 Effect  of  geometrical characteristics of  die on the extrusion pressure Lewis and Winchester (1953) have studied the effect  of  extrusion pressure, temperature, and die design on the process of  PTFE paste extrusion. They reported that fibrillation  occurs during the paste flow  through a conical die. Later, Ariawan (2002) found  the same through SEM analysis of  paste in the die entry region. Mazur (1995) explained this phenomenon by making reference  to particles that reorganize themselves to pass through the die during the initial stage of  extrusion. As paste advances further  in the conical die, particles are deformed  due to shear and extensional stresses, resulting in the formation  of  fibrils  which contribute to the final  mechanical strength of  the headings. Ariawan et al., (2002a) have performed  detailed experiments to show the effect  of volumetric flow  rate, die entrance angle and the die reduction ratio on the extrusion pressure of  PTFE (Figures 2.2-2.5). o. u c o </> 100 90 80 70 F-60 50 40 30 20 10 0 20 Resin 3 + ISOPAR G, 35°C R = 352:1, a = 45°, L/D, = 0 30 40 16 wt .% ISOPAR G i 18 wt .% ISOPAR G 22 wt .% ISOPAR G -J_ 50 60 70 80 Volumetric Flow Rate (mm /s) Figure 2.2: The effect  of  lubricant (ISOPAR® G) concentration on the steady-state extrusion pressure for  resin 3 (PTFE resin). Solid lines are model predictions [Ariawan et al., (2002)]. Figure 2.2 shows the effect  of  volumetric flow  rate on the extrusion pressure for three different  lubricant concentrations for  a PTFE resin. The authors have concluded that a lower lubricant concentration result into a higher extrusion pressure that may cause fibril  breakage. On the other hand, a higher lubricant concentration results into a wet and weak extrudate. The steady-state extrusion pressure generally increases with increase of the extrusion rate (volumetric flow  rate). Figures 2.3 and 2.4 show that the extrusion pressure increases with increase in the reduction ratio and L/D ratios of  the die respectively. This is due to increased levels of strain hardening and frictional  losses, respectively. Figure 2.5 represents the dependence of  the extrusion pressure on die entrance angle. The extrusion pressure initially decreases and subsequently increases with increase of  the die entrance angle. Figure 2.3: The effect  of  reduction ratio on the steady-state extrusion pressure for different  PTFE resins. Solid lines are model predictions [Ariawan et al., (2002)]. Die L/D„ Ratio Figure 2.4: The effect  of  die L/D a ratio on the steady state extrusion pressure at different reduction ratios for  resin 3 (PTFE resin). Solid lines are model predictions [Ariawan et al., (2002)]. 65 60 55 I 50 s 45 p D 40 1 35 b 30 s 2b 20 15 10 Resin 3 + 18wt.% ISOPAR G, 35°C R = 352:1, L/D =0 75.4 m m 3 / s Benbow-Bridgwater (1 9 9 3 ) : 10 20 30 Entrance Angle, a 40 50 Figure 2.5: The effect  of  die entrance angle on the steady-state extrusion pressure at different  extrusion rates for  resin 3 (PTFE resin). Solid lines are model predictions [Ariawan et al., (2002)]. Also dotted line shown is the prediction using the Benbow-Bridgwater equation (1993). 2.4 Constitutive equations proposed to predict pressure drop in capillary die flow Snelling  and  Lontz Mode l. Snelling and Lontz (1960) have assumed steady state flow  through an orifice  die (L/D  = 0) of  conical entry angle 2a, and used the following rheological model to predict the pressure drop: • Cy"  +rj dy dt [2.1] where r is the shear stress, y is the strain and C,rj,  m and n are constants. By also utilizing the "radial flow  hypothesis", they derived the following  relationship for  the total pressure drop AP • 4 C 3(h + 1) 3 In £o_ D + Arj 3 m 12gsin3 a n{\  - cos a)D3 [2.2] where C, n, r/,  and m are constants to be evaluated experimentally, (DJD>) 2 is the reduction ratio and Q is the volumetric flow  rate. The authors have modeled the problem by considering a constitutive equation that includes a strain hardening term (elastic term) and a shear thinning term (viscous resistance). Eq. 2.2 addresses the dependence of extrusion pressure on the flow  rate, die dimensions, and lubricant concentrations. Additionally, by following  the redistribution of  pigmented paste within the die during extrusion, the authors have managed to experimentally determine the velocity field  in the die entry region. As discussed above the authors have used the "radial flow"  hypothesis which assumes that the paste particles at the same radial distance from  the virtual apex of the conical zone of  the die move towards the die apex at the same velocity. Thus, the velocity of  a point on a spherical surface  at distance r from  the apex is [Snelling et. al Doraiswamy Mode l. Doraiswamy et al. (1991) have proposed a non-linear rheological model for  concentrated paste. This model considers the elastic, viscous, and yielding behavior of  the material by introducing a recoverable strain term, y . In addition, this model has the advantage of  using data easily accessible by means of  a parallel plate rheometer. Thus, they have suggested the following  constitutive equation for  a material that exhibits yield stress: (I960)]: dr dt 2tt{\  - cos a)r [2.3] T  = Gy Y  \< v. [2.4] Y  1= Y c [2.5] Y  l< Yc [2.6] dt Y  1= Yc  < Y [2.7] where G is the elastic modulus, yc is the critical strain value at yielding, y is the recoverable strain tensor, and K  and n are power law constants. Note that the viscosity, evaluated as the term in brackets in Eq. 2.5, approaches a Newtonian viscosity at low shear rates and a power law viscosity at high shear rates. Benbow and  Bridgwater  Model  (1993). Simple equations for  paste flow  through dies of various entry angles, cross sectional shapes and L/D ratios have been proposed analytically by the authors. Particularly, for  a steady state flow  through a capillary die of entry angle 2a, and performing  separated force  balances on the entry region of  the die and on the actual die land, the following  relationship for  the total pressure drop was derived: AP = 2(g0 + £ ,V m + xq cot cc)ln £ o D P , V n ' D v D o ; cot a 4L D (xo + P j V n ) [2.8] where <r0, £ m, r0, fit  and  n are parameters to be determined experimentally and V  and D0 are the mean paste velocity and barrel diameter, respectively. The first  term accounts for  the change in cross sectional area in the conical entry (extensional  and shear term), while the second term accounts for  pressure drop in the die land {shear  term). The rheological model used to obtain equation (2.8) was: a = cr, + a • V [2.9] Zhensa  et al. (2003)  model . The mathematical model proposed in this work was for  the extrusion of  water-oxide pastes in the production of  a-Fe203 catalysts. This model had enabled the authors to determine the thickness of  the adsorption-solvation shell on solid-phase particles and also the concentration of  the components of  the dispersion (continuous) phase as a function  of  time. The authors have proposed the following expression for  the extrusion pressure as a function  of  the average extrudate flow  velocity: [2.10] where a0, cp, k , n, and a are unknown parameters. a0 is the yield stress at the molding channel inlet, q> is velocity development factor,  and k,  n are power law constants. The parameter ' a ' is a function  of  the diameter of  a molding channel and decreases with increasing channel diameter. The authors have fitted  these five  parameters by using the random search method. Coussot  et al. (2003 ) have proposed a simple thixotropic model to predict the steady and transient behaviour of  the paste. The empirical model proposed by Coussot et al. (2003) was used to capture the thixotropic behaviour of  the paste by Roussel et al. (2004). The model parameters were estimated by fitting  the experimental data obtained from controlled stress rheometer. The authors have concluded that the model could qualitatively predict the steady state and a transient velocity profile  measured by NMR for  a bentonite suspension is a coaxial cylinder geometry. The authors have proposed the following  relation for  the stress: where r | 0 , n , 0 and a are four  material parameters and X is the structure parameter. The weaknesses of  the thixotropic model proposed by these authors were that this model is incapable of  predicting the characteristics of  the start-up flow  correctly; moreover a larger number of  empirical parameters would be required to achieve more accurate predictions. The  Ariawan Model  (2002 ). Using the "radial flow"  hypothesis proposed by Snelling and Lontz (1960), Ariawan et al. (2002a) have proposed a one-dimensional mathematical model to describe the effect  of  operating parameters and die design parameters on the extrusion pressure of  PTFE paste. This model considers the paste as an elasto-viscoplastic material that exhibits both strain hardening and viscous resistance effects  during flow. Ariawan et al. (2002a) have used the following  constitutive equation: Tl = T l 0 ( l + r ) [2.11] [2.12] <Te-°r=CY'L*+VY\ n max m [2.13] where errand ar are the principal stresses in 0 and r directions respectively and ^ and y are the maximum values of  the strain and strain rate, respectively. The extrusion pressure in the conical zone was found  to be: i], n, m and/are material constants that have to be determined experimentally. This model was found  to describe adequately most experimental observations. For example, the continuous lines in Figures 2.2-2.5 represent fits  to experimental results. Overall it was found  that this model can describe quantitatively and qualitatively the effects  of  die entrance angle, reduction ratio, length-to-diameter ratio of  the die, and PTFE properties on the extrusion pressure. However, it assumes the velocity distribution by using the "radial flow"  hypothesis. From the above discussion it is evident that a model capable of  predicting the velocity profile  for  paste flow  needs to be formulated.  To this respect, the following difficulties  have to be overcome: 1. The flow  profile  of  the PTFE paste inside the conical die is 2-D and it should be fully calculated by a flow  model. 2. The assumption of  the "radial flow"  hypothesis is empirical in nature and therefore,  its validity should be checked. 3. Experimental analysis has shown that the degree of  fibrillation  depends on the operating parameters as well as geometrical characteristics of  the die. Thus, a complete model should also consider the mechanism of  fibrillation. 4. The rheological parameters of  the chosen constitutive equation should be determined by fitting  the data obtained from  rheological testing. P J  =  CT extrusion rb [2.14] where crra is the stress at the die exit, R is the reduction ratio, defined  as (Db/Da)2, and C, 2.5 Numerical simulation studies of  paste flow While there are numerous reports available on the experimental and theoretical studies of paste flow,  there is none that refers  to the case of  PTFE paste. Below some of  the numerical simulations studies most relevant to paste flow  are discussed. Adams,  M.  J.  et al. (1997 ) have reported a finite  element analysis of  the sqeeze flow  of elasto-viscoplastic paste materials placed in between two circular horizontal plates. The model was based upon the assumptions that linear elastic deformations  occurs prior to yielding and that the yield surface  is strain rate hardening as defined  by an associated viscoplastic flow  rule. The authors have assumed that the elastic and viscoplastic strain rates are additive i.e.: e ^ + e ? [2.15] where s^1 and iv? are the elastic and viscoplastic components of  the strain rate tensor respectively. The authors have formulated  and used the following  constitutive relation: where y v p is the viscoplastic shear strain rate and t o is the shear yield stress. The viscoplastic material parameters were determined by using capillary measurements (shear flow).  The finite  element analysis was carried out using the code ABAQUS (Hibbit, Karlsson and Sorenson, USA, version 5.4). The prediction of  the displacement field  and the normal force  with gap was reported to be in reasonable agreement with the experimental measurements. Horrobin  et al. (1998 ) have described paste flow  by using an elastic-plastic finite  element method to calculate load in the paste extrusion. The numerical results were compared with the first  term in the Benbow-Bridgwater (1993) equation. They have used the ABAQUS finite  element package, version 5.5 for  their numerical simulations. The elastic response of  the material is modeled using Hooke's law and Von Mise's yield criterion. The authors have concluded that viscoplastic materials, displaying rate dependence can be modeled by using lubricated Bingham or Herschel-Buckley fluids. Ozkan et al. (1999 ) performed  a rheological analysis of  ceramic pastes (alumina paste). The flow  behavior of  the paste is approximated by an elasto-viscoplastic constitutive T  =  T  + O [2.16] model and implemented by using an established finite  element code, ABAQUS (Hibbitt, Karlson and Sorensen, Inc., version 5.4). The Herschel-Buckley model was used in shear and uniaxial deformation  form  to describe the material as a combination of  elastic, plastic and viscous properties. A coulombic friction  boundary condition was implemented at the contact between paste and platen. The authors have reported reasonably good agreement between the experimental measurement and the finite  element simulations. The authors have also studied the flow  profile  of  the paste for  lubricated and unlubricated boundary conditions. Domanti et al. (2002 ) have devised some theoretical criteria for  predicting the onset of surface  fracture  in ram extrusion, using the elastic-plastic finite  element method. The availability of  the literature on the numerical study of  polymer paste flow  by using finite element simulation is very limited. This could be because of  the complexity in simulating the polymer paste flow. In the present study, finite  element simulations are performed  by using proposed constitutive models. The gradual change of  PTFE paste from  a liquid-like (before processing) phase to a solid-like phase is modeled through shear thinning and strain-hardening term premultiplied by functions  of  the structural parameter, , that represents the percentage of  the domains which are fibrillated.  The mechanism of  fibrillation  which transforms  paste from  shear-thinning fluid  to strain-hardening fluid  is modeled through a kinetic model for  the structural parameter, t, . The parameters used in the model are obtained from  rheological experiments. Unlike the previous models reported in this chapter, this model is a rather complete model, which considers the effect  of  fibrillation on rheology and processing of  PTFE paste. The details of  the proposed consitutive models and simulations performed  in the present work are discussed in later chapters of this thesis. 2.6 Bibliography Adams, M. J., I. Aydin, B. J. Briscoe, and S. K. Sinha, A Finite  Element  Analysis of  the Squeeze Flow  of  an Elasto-Viscoplastic  Paste Material,  J. Non-Newt. Fluid Mech., 71, 41-57 (1997). Ariawan, A. B., Ebnesajjad, S. and Hatzikiriakos, S. G. Paste extrusion of polytetrafluoroethylene  (PTFE)  fine  powder  resins. Can. Chem. Eng. J., 80, 1153-1165 (2002a). Ariawan, A. B. Paste Extrusion  of  Polytetrafluoroethylene  Fine  Powder  resins, The University of  British Columbia. Dept. of  Chemical and Biological Engineering. Thesis Ph. D., 2002. Benbow, J. J., Oxley, E. W., and Bridgwater, J. The  extrusion mechanics of  pastes-The influence  of  paste formulation  on extrusion parameters.  Chem. Eng. Sci., 42 (9), 2151-2162, (1987). Benbow, J. J., and J. Bridgwater, Paste Flow  and  Extrusion,  Oxford  University Press, Oxford,  1993. Benbow, J. J., Blackburn, S., and Mills, H., The  effects  of  liquid-phase  rheology  on the extrusion behaviour of  paste. Journal of  Material Science, 33, 5827-5833 (1998). Blackburn, S., Bohm, H., The  influence  of  powder  packing  on paste extrusion behaviour. TransIChemE, Vol 71, Part A, 250-256 (1993). Breedveld V., and D. J. Pine. Microrheology  as a tool for  high-throughput  screening. Journal of  Materials Science, 38, 4461-4470 (2003). Burbidge, A. S., J. Bridgwater, and Z. Saracevic, Liquid  Migration  in Paste Extrusion, Chem. Eng. Res. Design, 73, 810-816 (1995). Coussot, P., Nguyen, Q. D., Huyuh, H. T., Bonn, D., Viscosity  bifurcation  in thixotropic, yielding  fluids,  J. Rheol., 46, 573-589 (2002). Domanti, A. T. J., Bridgwater J., Surface  fracture  in axisymmetric paste extrusion. Trans IChemE, Vol 78, Part A, 68-78 (2000). Domanti, A. T. J., Horrobin, D. J., and Bridgwater, J., An investigation  of  fracture criteria  for  predicting  surface  fracture  in paste extrusion, Int. J. Mech. Sci., 44, 1381-1410(2002). Doraiswamy, D., A. N. Munumdar, I. Tsao, A. N. Beris, S. C. Danforth,  and A. B. Metzner, The  Cox-Merz  Rule Extended:  A Rheological  Model  for  Concentrated Suspensions and  Other Materials  with a Yield  Stress,  J. Rheol., 35, 647-685 (1991). DuPont Fluoroproducts, Teflon®  PTFE  Fluoropolymer  Resin - Processing Guide for  Fine Powder Resins, Technical Bulletin, Wilmington, Delaware, 1994. Ebnesajjad Sina, Fluoroplastics,  Vol  1 Non-Melt  Processible  Fluoroplastics,  Plastic Desgin Library. William Andrew Corp, NY, 2000. Haw M. D., Jamming,  Two-fluid  behaviour, and  self-filtration  in concentrated particulate  suspension. Phys Rev Letters, 92 (18) 18506 (2004). Hooper, R., Macosko, C. W., Derby, J. J. Assessing a flow-based  finite  element model  for the sintering  of  viscoelasticparticles.  Chem. Eng. Sci., 55, 733-746, (2000). Horrobin, D. J., and R. M. Nedderman, Die Entry  Pressure Drops in Paste Extrusion, Chem. Eng. Sci., 53, 3215-3225 (1998). Khan, A. U., Briscoe, B. J., Luckham, P. F., Evaluation  of  slip in capillary  extrusion of ceramic pastes. Journal of  the European Ceramic Society, 21 (2001), 483-491. Lewis, E. E., and C. M. Winchester, Rheology of  Lubricated  Polytetrafluoroethylene Compositions  - Equipment and  Operating  Variables,  Ind. Eng. Chem., 45, 1123-1 127, (1953). Mackenzie, J. K., and Shuttleworth, R., A phenomenological  theory of  sintering,  Proc. Phys. Soc. B 62, 833-852 (1949). Manoharan, V. N., M.T. Elsesser, and D J Pine, Dense Packing  and  Symmetry  in Small Clusters  of  Microspheres.  Science, 301, 483-487 (2003). Mazur, S., Paste Extrusion  of  Poly(tetrafluoroehtylene)  Fine  Powders  in Polymer Powder technology,441-481, Narks, M., Rosenzweig, N., Ed. John Wiley & Sons, 1995. Ochoa, I., Hatzikiriakos, S. G. Polytetrafluoroethylene  {PTFE)  paste performing: Viscosity  and  surface  tension effects.  Powder Technology, 146( 1 -2), 73-83 (2004). Ochoa, I., Hatzikiriakos, S. G. Paste extrusion of  polytetrafluoroethylene  (PTFE): Surface  tension and  viscosity effects..  Powder Technology (2005), 153(2), 108-118. Ochoa, I., Paste Extrusion  of  Polytetrafluoroethylene  Fine  Powder  resins: The  effect  of the processing aid  physical properties,  Ph.D., The University of  British Columbia. Dept. of  Chemical and Biological Engineering. Thesis Ph. D., 2006. Ozkan, N., Oysu, C., Briscoe, B. J., and Aydin Ismail., Rheological  Analysis of  Ceramic Pastes, Journal of  the European Ceramic Society, 19, 2883-2891 (1999). Rough, S.L., Bridgwater, J. & Wilson, D.I., Effects  of  liquid  phase migration  on extrusion of  microcrystalline  cellulose  pastes, Intl. J. Pharm., 204, 117-126 (2000). Roussel, N., Roy, R. L., Coussot, P., Thixotropic  modeling  at local  and  macroscopic scales. J. Non-Newt. Fluid Mech., I l l , 85-95 (2004). Snelling, G. R., and J. F. Lontz, Mechanism  of  Lubricant-Extrusion  of  Teflon®  TFE-Tetrafluoroethylene  Resins, J. Appl. Polym. Sci. Ill, 257-265 (1960). Yu, A. B., Bridgwater, J., Burbidge, A. S., Saracevic, Z. Liquid  maldistribution  in particle  paste extrusion. Powder Technology, 103 (1999), 103-109. Zhensa, A. V., Kol'tsova, E. M., Petropavlovskii, I. A., Kostyuchenko, V. V. and Fillippin, V. A., Mathematical  Modeling  of  Extrusion  of  Water-Oxide  Pastes in the Production  of  a -Fe203  Catalysts,  Theoretical Foundation of  Chemical Engineering, 37(2), 197-203 (2003). CHAPTER 3 Scope of Work 3.1 Introduction Due to its outstanding properties, PTFE has become a technologically important material and as such has found  a wide variety of  applications ranging from  wire insulation to body part replacement. It is undeniable that the processing techniques by means of  which PTFE is manufactured  have been improved since their introduction, although there are still several issues to be understood in order to further  optimize them; the microstructural formation  during the PTFE paste processing is one important issue. Being a thermoplastic, the first  idea is to attempt to melt process PTFE as a commodity polymer. However, due to its high melting point and high viscosity such techniques are almost impossible. The second possibility is to treat PTFE as paste. Techniques for  pastes are known for  other materials such as ceramic pastes and food stuffs  (Chevalier, 1997; Steffe,  1996; Rough, 2000). Since the PTFE manufacturing process facilitates  the production of  powders, it is not surprising that PTFE is processed using techniques such as sintering, pressing and paste extrusion. While paste extrusion has been studied for  other materials to great extent, it is only recently that PTFE paste extrusion has been a subject of  scientific  studies (Ariawan, 2002; Huang, 2005; Ochoa, 2006). Mathematically, previous work on the development of  constitutive equations to model the flow  of  polymer melts is of  limited relevance to the PTFE paste system. This is because both solid and liquid phases are present in PTFE pastes, with the solid particles being fibrillated  during extrusion. The present work intends to contribute to our understanding of  several modeling aspects of  PTFE paste rheology and its role in extrusion. The various objectives of  this work are discussed in detail in this chapter. 3.2 Thesis Objectives This project is mostly devoted to the modeling and simulation aspects of  PTFE paste flow.  An effort  has been made to understand the relation between microstructure formation  and its effect  on the rheology and processing of  PTFE paste. The objectives of  this work can be summarized as follows: 1. To measure the rheological properties of  PTFE paste using parallel plate rheometer. 2. To perform  the capillary extrusion experiments in order to analyse the slip behavior of  the PTFE paste. 3. To develop a constitutive model to predict the rheological behavior of  the paste. The constitutive model should be capable of  predicting the gradual structural changes that occur during the PTFE paste flow. 4. To develop a model which can describe the mechanism of  fibrillation  with consideration of  creation and breakage of  fibrils  during flow. 5. To perform  numerical simulations using finite  element method in order to predict the flow  behavior of  the PTFE paste during extrusion. 6. To compare the dependence of  the extrusion pressure on die design parameters and processing conditions with experimental observations. 7. To propose an approximate 1-D analytical model which can predict the processing behavior of  PTFE paste in tube extrusion similar to that developed by Ariawan et al., (2002). 3.3 Thesis Organization Chapter 1 of  the thesis discusses basic information  related to tetrafluoroethylene (TFE) polymerization techniques as well as the basic physical and chemical properties of PTFE. The industrial processes relevant to PTFE are also discussed with particular emphasis to fine  powder PTFE processes. This chapter is also devoted to addressing the microstructure formation  and its effect  on the rheology of  PTFE paste. Complexity in mathematical modeling of  PTFE paste flow  is also demonstrated. Chapter 2 presents the literature related to paste extrusion. The study about the processing behaviour of  pastes using experimental observations and modeling is discussed. Chapter 3 includes the objectives of  the present work, as well as describes the organization of  the thesis. In chapter 4 the viscous constitutive model as a function  of  fibrillation  is proposed to predict the processing behavior of  PTFE paste. A kinetic model for  structural parameter is proposed which models the mechanism of  fibrillation.  Capillary extrusion experiments used to determine the slip behavior of  PTFE paste is discussed. The predictions from  the finite  element simulations are compared with the experimental observations. This chapter is based on a journal paper that has already been published (Patil, P. D., Feng, J. J., and Hatzikiriakos, S. G., "Constitutive modeling and flow simulations of  PTFE paste extrusion," J. of  Non-Newtonian Fluid Mech., 139, 44-53, 2006) Chapter 5 presents a constituve model where the PTFE paste is treated as a viscoelastic material. The same model for  the evolution of  the structural parameter is used as reported in chapter 4. The details of  the numerical scheme used for  the viscoelastic simulations using the finite  element method are reported. The results from two different  models are compared with each other and also with experimental observations. The hypothesis of  a higher degree of  fibrillation  leads to a higher tensile strength of  the extrudate is explored in this section. This chapter is based on a journal paper which has been submitted (Patil, P. D., Ochoa, I., Feng, J. J., and Hatzikiriakos, S. G., "Viscoelastic modeling and flow  simulations of  PTFE paste extrusion," J. Rheol. 2006) Chapter 6 focuses  on the derivation of  an analytical model for  PTFE paste extrusion through annular dies. Model predictions are presented for  various cases and are compared with experimental results of  macroscopic pressure drop measurements in rod and tube extrusion. This chapter is based on a journal paper that has been accepted for publication (Patil, P. D., Feng, J. J., and Hatzikiriakos, S. G., "An approximate flow model for  polytetrafluoethylene  paste through annular dies," accepted in AIChE J., 2006). Finally, the conclusions and contributions to knowledge are discussed in Chapter 7. A general summary of  the most significant  modelling aspects resulted from  this work and some recommendations for  future  work are presented here. 3.4 Bibliography Ariawan, A. B., Ebnesajjad, S. and Hatzikiriakos, S. G. Paste extrusion of polytetrafluoroethylene  (PTFE)  fine  powder  resins. Can. Chem. Eng. J., 80, 1153-1165 (2002). Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, Constitutive  modeling  and  flow simulation of  polytetrafluoroethylene  (PTFE)  paste extrusion, J. Non-Newt. Fluid Mech., 139, 44-53 (2006a). Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, An analytical  flow  model  for polytetrafluoroethylene  paste through  annular dies,  accepted in AIChE J. (2006b). Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, Viscoelastic  modeling  and  flow simulation of  polytetrafluoroethylene  (PTFE)  paste extrusion, manuscript under preparation (2006c). CHAPTER 4 Constitutive Modeling and Flow Simulation of Polytetrafluoroethylene (PTFE) Paste Extrusion1 4.1 Introduction Owing to its high melting point and melt viscosity, it becomes almost impossible to melt-process polytetrafluoroethylene  (PTFE) (Blanchet, 1997; Ebnesajjad, 2000; Sperati, 1989; and Ochoa and Hatzikiriakos, 2004). Instead, techniques involving cold processing, paste extrusion and sintering have to be employed. In PTFE paste extrusion that is of interest to the present work, fine  powder of  the PTFE resin (primary particle diameter of approximately 0.20 ^m) is first  mixed thoroughly with a lubricating liquid to form  a paste. The paste is consequently extruded through a conical die at a low temperature, typically 35 °C. A schematic diagram of  the die used to process the PTFE paste is shown in Figure 4.1. It consists of  two parts: (i) the die entry, the region where mixed shear and extensional deformation  occurs; and (ii) the die land that is a capillary of  constant cross section attached at the bottom of  a conical section of  the die. The die is attached at the bottom of  a capillary rheometer. Due to the presence of  the lubricant, slippage of  the paste at the solid boundaries of  the die is possible and the flow  is rather complicated. The conical die entry shown in Figure 4.1 is defined  by the entry angle (2  a), an important parameter in paste extrusion; the reduction ratio, RR, defined  as the ratio of  the initial to the final  cross-sectional area of  the conical entry, D ^ / D 2 ; and the length-to-diameter ratio, L / D . During the PTFE paste extrusion through such a die, complex structural changes occur in the polymer paste that significantly  influence  its rheology. These structural changes have been determined by scanning electron microscopy (SEM) analysis (Mazur, 1995; Ebnesajjad, 2000), as already shown in Figure 1.6 which depicts typical SEM micrographs of  the paste before,  during and after  the extrusion. The most remarkable feature  is the formation  of  fibrils  between neighboring polymer particles. ' A version of  this chapter has been published. Patil, P. D., Feng, J. J., and Hatzikiriakos, S. G. (2006) Constitutive Modeling and Flow Simulation of  Polytetrafluoroethylene  (PTFE) Paste Extrusion, J. of  Non-Newtonian Fluid Mech. 139: 44-53. Db Figure 4.1: A conical entry die used in the paste extrusion of  PTFE: The left  half illustrates the extrusion of  PTFE particles and the gradual structure formation  through particle fibrillation,  whereas the shaded area depicts the axisymmetric domain used in the simulations. This is due to the mechanical interlocking of  crystallites of  neighboring particles which unwind as the particles enter the converging conical die (extensional flow)  (Ebnesajjad, 2000; Mazur, 1995; Ariawan et al., 2001; Ariawan, 2002; Ariawan et al., 2002a). Figure 1.6a shows the absence of  fibrils  in the unprocessed paste, where PTFE particles essentially retain their spherical identity. Figure 1.6b shows the existence of  fibrils  as the paste flows  downstream in the conical die zone. At the exit of  the die and depending on the geometrical characteristics of  the die, the paste might become nearly fully  fibrillated (Figure 1.6c). It is these submicron-diameter fibrils  between polymer particles that essentially give the dimensional stability and strength to the final  extruded product. The experimental dependence of  extrusion pressure on reduction ratio (RR), entrance angle (2 a ) and die land L / D ratio of  conical dies has been studied for  several types of  PTFE and lubricants (Ochoa and Hatzikiriakos, 2004; Ariawan, 2002; Ariawan et al., 2002b). A somewhat unexpected observation is that the extrusion pressure varies non-monotonically with the entrance angle of  the conical die; 2a , and it achieves a minimum for  an intermediate entrance angle, 2a « 30°. It has also been recognized that the rheological behavior of  PTFE pastes is strongly dependent on the number of  fibrils formed  between PTFE particles during the extrusion. This implies that the rheology of the paste continuously changes and this complicates the flow  modeling. The formation  of fibrils  has not been given any consideration in previous models (Ariawan, 2002; Ariawan et al., 2002b; Benbow and Bridgwater, 1993; Horrobin and Nedderman, 1998), and it is the main focus  of  the present study. Furthermore, experiments have shown that the degree of  fibrillation  depends on the operating parameters as well as design characteristics of  the die (Ochoa and Hatzikiriakos, 2004; Ariawan et al., 2002a; Ariawan et al., 2002b). Therefore,  a rational approach seems to be to first  formulate  an approximate rheological constitutive equation that takes into account the structure formation  through fibrillation;  and then to use this equation to simulate the paste extrusion process and compare the calculated with the experimental extrusion pressure, as a function  of  operating and die geometrical characteristics. In this paper, first  a constitutive equation is proposed based on the concept of  a structural parameter, 2,. This parameter represents the mass fraction  of  the PTFE paste that is fibrillated  and takes values of  0 and 1 for  the unfibrillated  and fully  fibrillated cases respectively. The evolution of  the structural parameter is described by a first-order kinetic differential  equation, which is developed based on concepts borrowed from network theory of  polymeric liquids (Bird et al., 1987; Jeyaseelan and Giacomin, 1995; Liu et al., 1984). Similar concepts have been adopted in studies of  the rheology of  filled polymers and dilute polymer solutions (Hatzikiriakos and Vlassopoulos, 1996; Leonov, 1990). The constitutive equation is subsequently used to simulate the flow  of  paste through the conical die depicted in Figure 4.1. The finite-element  results are compared with the experimental results as a check on the validity and usefulness  of  the rheological constitutive equation. In addition, the flow  simulation is used to predict the extrusion pressure as a function  of  the operating and die geometrical characteristics and to explore the relationship between the tensile strength of  the extrudate and the degree of fibrillation.  In this last task, the average structural parameter at the exit, ^ e x i t , is related with the experimental tensile strength for  a range of  the operating and die geometrical parameters. 4.2 Theoretical Modeling and Numerical Method 4.2.1 Governing Equations The steady state mass and momentum conservation equations coupled with the rheological constitutive model are solved to simulate the flow  of  the PTFE paste. The axisymmetric r-z domain (Figure 4.1) has been used to perform  the simulations. The velocity field  v is subject to the incompressibility constraint, (small volume changes due to fibril  formation  that involve phase change are assumed to be small): V • V = 0 . [4.1] Due to the high viscosity of  the paste, the inertial terms in the momentum equation are neglected: V p - V • T = 0 , [4.2] where p is the pressure and T is the stress tensor, which depends on the structural parameter, , through a constitutive equation developed below. 4.2.1.1 Constitutive  Equation The rheology of  the PTFE paste depends on the formation  and evolution of  a network of fibrils  connecting PTFE polymer particles during the extrusion. To model this complex flow  behavior, a rheological constitutive equation is proposed which explicitly accounts for  the evolution of  fibrils.  Prior work has modeled flow  induced structure formation  in concentrated suspensions, polymer solutions and filled  polymers as a combination of shear-thinning behavior at low shear rates and shear-thickening behavior at high shear rates (Jeyaseelan and Giacomin, 1995; Dunlap and Leal, 1987). A similar concept has been adopted in the present study to model the PTFE paste flow  behavior. The rheology of  the paste continuously changes as it flows  through the conical sections. It starts as a two phase fluid-like  system, an oversaturated suspension (Figure 1.6a) and it ends as a highly fibrillated  solid-like system. While the paste initially behaves as a shear-thinning fluid,  after  the appearance of  fibrils  in its structure, it behaves more and more as a shear-thickening fluid.  Thus, it is assumed that the stress tensor consists of  two contributions coming from  the unfibrillated  and fibrillated  domains of  the paste, represented respectively by a shear-thinning and a shear-thickening viscous stress. The relative significance  of  the two contributions should depend on the structural parameter, . Recall that the structural parameter,^ is the mass fraction  of  the paste that is fibrillated, and takes values between 0 and 1. Thus, the total viscous stress can be written in the following  form: -c = ( i - O ^ i i t + tn 2 y , [4.3] where y is the rate of  strain tensor, and rj, and r|2 are the shear-thinning and shear-thickening viscosities that are expressed by a Carreau model (Bird et al., 1987): "Hi ^ ^ - ^ M M I , ) 2 ) ^ 2 . where i = 1 refers  to shear-thinning (n, < 1) and i = 2 refers  to shear-thickening (n 2 >1). The values of  parameters r | M , r | o i , ri; and ?^are discussed and reported in the following section. The creation of  fibrils  has been attributed to the unwinding of  mechanically locked crystallites due to the extensional nature of  the flow  in the conical region of  the die (Ebnesajjad, 2000; Mazur, 1995). The extensional flow  also causes elongation of newly formed  fibrils  which might also break depending on the total local Hencky strain. Therefore,  both creation and breakage are possible. A kinetic model is proposed for  the structural parameter, which is a balance of  the fibrillated  and unfibrillated  domains of  the paste and whose dynamics are controlled by the rates of  creation and breakage: v • V£, = f  - g , [4.4] where f  and g denote the rate of  creation and elimination of  fibrillated  domains in the paste. These functions  are given by: •N f(y,vj/)  = a y V y , ^ = J where a and p are dimensionless rate constants for  fibril  creation and breakage, both assumed to be 1 in our simulations; \j/ is the flow  type parameter; and y is the magnitude of  the strain rate tensor. The flow  type parameter, y , indicates the relative strength of straining and rotation in a mixed flow  (Fuller et al., 1980; Fuller and Leal, 1980; Dealy and Wissbrun, 1990). Its magnitude ranges from  -1 to 1 depending on the flow  type as shown in Figure 4.2. Since fibrils  are mainly created due to elongation and never due to rotation, \\i is taken to vary only between 0 and 1. Negative values of  are reset to 0. While the function  f  involves the formation  of  fibrils  as unwinding of  crystallites of  neighboring particles, the function  g represents their breakage. This is not to be taken as disappearance of  the circular fibrils  (see Figures 1.6b, 1.6c) and reformation  of  perfect spherical particles. Broken fibrils  do not contribute significantly  to the overall pressure drop and to structural strength of  the extrudate. A final  remark for  the parameter is as follows:  It represents the percentage of the domains of  the system that are fibrillated,  and as such should take values between 0 and 1. This can be fixed  by limiting the ratio of  a/p < l(note that both parameters have been assigned the value of  1). For example at steady-state conditions, Eq. (4.4) results = a-Jy , which essentially limits ^ to less than 1. Analytical solutions for  one-dimensional axisymmetric flows  where the velocity profile  can be taken approximately known (fully  developed) can be derived for  Eq. (4) and these show that is always less than 1. It should be noted that £, can take the value of  1 in pure elongational flow,  where \|/ becomes 1. Pure Elongation v|/ = 1.0 Shear and elongation = 0.5 Pure Shear \\i = 0.0 Pure Rotation \\i = -1.0 Figure 4.2: Flow fields  corresponding to different  values of  flow  type parameters, \\i. 4.2.1.2 Flow  type parameter,  \|/ The concept of  the flow  type parameter, v|/, has been used to account for  the dependence of  the structural parameter on the relative amount of  straining and rotation in the flow field.  As an example, a linear planar flow  has a velocity field  V = T  • X , where T is the velocity gradient tensor (Fuller et al., 1980; Fuller and Leal, 1980; Dealy and Wissbrun, 1990): r o r w o r = y [4.6] The flow  of  primary interest in our study is the strong flow  for  which 0 < v|/ < 1. Thus, for a planar flow,  the largest eigenvalue of  the velocity gradient tensor has the form  of  y-Jy (Fuller et al., 1980; Fuller and Leal, 1980; Dealy and Wissbrun, 1990). On the other hand the flow  parameter vj/ can also be written as: |D | - |W I [4.7] |D| + |W| where D and W denote the deformation  and vorticity tensor respectively. In the case of a axisymmetric flow,  the velocity gradient tensor has a different  form  than Eq. (4.6), and its the largest eigenvalue is no longer equal to y^Jy . Nevertheless, we can still define  \\> according to Eq. (4.7) and therefore,  it retains the significance  of  a flow  type parameter. It appears reasonable to use such a i|/ in our kinetic Eq. (4.5). The magnitude of  D and W in cylindrical coordinate can be written as follows: 9v j 1 f  du 3v dz)  2\dz  dr 1 1 V 2 V  3z dr  J The flow  type parameter controls the magnitude of  structural parameter inside the flow domain with the maximum amount of  fibrillation  to occur at the center and much less at the die wall. This also ensures that the fibril  creation and evolution mainly takes place inside the conical section; very little changes in fibril  structure occur inside the die land where the flow  becomes pure shear. This picture is supported by experimental evidence. This was confirmed  by performing  experiments with L/D ratios 0 and 20 (Ochoa and Hatzikiriakos, 2004; Ariawan et al., 2002a; Ariawan et al., 2002b). Therefore phenomenologically, the modeling concepts incorporated into our flow  model agree well with the experimental observations. The agreement between the measured and calculated extrusion pressure, and the relationship between the structural parameter and the dimensional strength of  extrudate remain to be seen. 4.2.2 Boundary Conditions The boundary conditions used in the simulations are listed below. (i) Inlet boundary conditions (z = 0): The numerical fully  developed velocity profile  for a shear-thinning Carreau fluid  model has been imposed at the inlet with V r = 0 . The no-fibrillation  boundary condition is also assumed: E, = 0 (ii) Outlet boundary conditions: The normal stress boundary condition and zero radial velocity are imposed: n • ( - pi + x)n = - p 0 ; vr =0 . (iii) Slip boundary condition at the die wall: The Navier slip condition has been used at the die wall, which relates slip velocity with the wall shear stress, C7W : V s = C c t w . where C = 1.92 m/M Pa s. This value of  C has been calculated from  experimental data which will be discussed in the next subsection. (iv) The axisymmetric boundary condition is used at r = 0: v r = 0 , dv z/dr  = 0. In capillary flow,  the shear stress at the wall, a w , can be determined by correcting for  the pressure losses associated with the end effects  by using the Bagley correction (Macosko, 1994; Snelling and Lontz, 1960): 4.2.2.1 Slip Boundary Condition No experimental data on the slip velocity of  polymer pastes was available, and thus it was decided to perform  a series of  capillary extrusion experiments to determine the relation between slip velocity and wall stress using the Mooney analysis technique (Mooney, 1931). The paste was prepared using a high molecular weight PTFE resin (F-104) mixed with 18 wt. % ISOPAR® M as the lubricant; these are the same as those used by Ochoa and Hatzikiriakos (Ochoa and Hatzikiriakos, 2004). The physical properties of isoparaffinic  lubricant (ISOPAR® M) are listed in Table 4.1. In capillary flow,  the shear stress at the wall, <rw , can be determined by correcting for  the pressure losses associated with the end effects  by using the Bagley correction (Dealy and Wissbrun, 1990; Macosko, 1994): = 4 ( L / A D + e ) - 1 4 8 1 where Ap is the total pressure drop over the capillary, L is the length and D is the diameter of  the capillary, and "e" is the Bagley end correction in terms of  an equivalent length associated with the end correction. Table 4.1: Physical properties of  the Isopar M lubricant used in the slip velocity measurements. Property Isopar® M Density, g/cm3 25°C 0.79 Surface  Tension, dynes/cm 25°C 26.6 Vapour Pressure, mmHg, 38°C 3.1 Viscosity, mPa-s 25°C 2.70 a (g2 s"1 cm"5) 608.8 The equivalent length can be determined either by extrapolating the Ap versus the L /D ratio curves to L / D = 0 or by performing  experiments with a die having L / D = 0 (Mooney, 1931). The latter approach has been followed  here to find  the corrected pressure drop and therefore  the wall shear stress. The apparent shear rate, y A , is determined from  8V / D , where V is the average fluid  velocity of  the fluid,  which is equal to the velocity of  the piston in capillary extrusion. To determine the slip velocity, capillary dies having the same length to diameter ( L / D ) ratios are used. Figure 4.3 depicts the apparent flow  curves of  the paste for  the same L/D ratio and different  die land diameters. The analysis proposed by Mooney for fully  developed, incompressible, isothermal, and laminar flow  in circular tubes with a slip velocity of  Vs at the wall yields: • / x 8 v s YA = Y A , S O w ) + - p - S M where yA s is the apparent shear rate corrected for  the slip, solely a function  of  ctw . Thus, a plot of  the apparent shear rate, y A , versus 1/D at constant a w values should result straight lines with slopes equal to 8v s provided that the Mooney analysis is correct. 1 aT Q-0.1 b ui ID Q) W 0.01 ro CD .C CO :> 0.001 100 1000 10000 1 Apparent Shear Rate, yA (s ) Figure 4.3: The apparent flow  curves of  PTFE paste extruded through three dies having the same L/D ratio and different  diameter. This is done in Figure 4.4. The individual data points do not fall  on perfect  straight lines, which might be due to experimental error. In spite of  this, and since an approximate expression for  the slip velocity is needed, the slopes of  best fit  straight lines passing through those points in Figure 4.4 were calculated in order to determine the slip velocity as a function  of  g w . The plot of  slip velocity, v s , versus a w results a straight line with slope equal to 1.92 m/M Pa s as shown in Figure 4.5. This relation has been used in the simulations reported in the present study as a slip boundary condition at the die wall. PTFE resin (F104 HMW) 2a = 90° L/D = 20 o - o - • — RR=56(DC = 0.127 cm) O - RR = 156 (Dc = 0.076 cm) • RR = 352 (Dc = 0.051 cm) 1/D (m"1) Figure 4.4: Mooney plot based on the experimental data of  Figure 4.3 prepared to calculate the slip velocity. The shear stress values that correspond to individual lines at various shear stress values are also shown. 0.06 0.08 0.10 0.12 0.14 0.16 0.1 J Wall Shear Stress, ctw (M Pa) 0.20 Figure 4.5: The slip velocity, V s , as a function  of  the wall shear stress, crw for  a PTFE paste used in this work. A linear slip model seems adequate to capture the experimental results. 4.2.3 Finite Element Method The equations of  motion coupled with the proposed constitutive and structural parameter models were solved using the Galerkin finite  element method. All simulations reported in this paper were performed  by using the commercial finite  element code FEMLAB 3.1. As the problem considered here is axisymmetric, two-dimensional meshes are used on the computational domain. These unstructured meshes comprise triangular elements of  widely varying sizes, small and large elements being employed in regions where the rates of  strain were large and small, respectively. The smallest elements are required near the die corners, especially the re-entrant corner. The total numbers of elements used are in the range of  3,000 to 10,000. The corners are also rounded slightly to avoid geometrical singularity, and the local element size is chosen to be smaller than the fillet  radius at the corner. The fillet  radius is a small proportion of  the capillary radius, and so the solution obtained from  the analysis is expected to be close to the solution for  a die with perfectly  sharp corners. Using meshes of  the form  shown in Figure 4.6, the solutions were found  to be insensitive to the number of  mesh elements. outlet Figure 4.6: Geometrical domain used for  simulations with enlarged section of  the rounded corner shown on the right side. Inlet Axisymmetri B.C. \ The simulations are carried out for  various die design parameters: the die reduction ratio, (RR = D j ; /D 2 ) , the die land length to diameter ratio (L/D) and die entrance angle ( 2 a ) . Simulations are also carried out for  different  inlet flow  rates that correspond to various apparent shear rate values, y A . The run time of  the simulations is in the range of  200 - 1000 seconds on Intel Pentium IV 2.8 GHz with 1 GB RAM machines. 4.3 Results and discussion In this section, the simulation results are reported and compared with the experimental findings  of  Ochoa and Hatzikiriakos (2004) for  pastes prepared by mixing a high molecular weight PTFE (F-104) with 18 wt. % Isopar®. To gain a better understanding of the structure of  PTFE paste flow,  typical velocity profiles  at various axial location inside the conical die (2a=90 and L/D=20) are plotted in Figure 4.7. U) £ o o <u > 0.225 0.200 0.175, 0.050. 0.025 0.000 - 2a = 9 0 ° L /D = 2 0 z = 0 R = 4.76E-3 m z = 2E-3 R = 2.74E-3 m z = 4E-3 R = 7.54E-4 m z = 5E-3 R = 2.54E-4 m z = 7E-3 R = 2.54E-4 m z = 8E-3 R = 2.54E-4 m z = 9E-3 R = 2.54E-4 m 0.0 0.2 0.4 0.6 0.8 1.0 D i m e n s i o n l e s s R a d i a l D i s t a n c e (m) R (mm) = 4.76 .74 0.75 z (mm) = o 1.2 Figure 4.7: Radial velocity profiles  at various axial locations for  conical die with entrance angle of  90°. The x-axis in Figure 4.7 is a dimensionless radial distance that is the radial distance normalized with the capillary radius at the corresponding axial location. The flow  inside the conical section is mostly elongational (note the significant  slip at the wall) and this essentially causes the formation  of  significant  amount of  fibrils  continuously increases). As 4 increases, it causes an increase of  the breakage term in Eq. (4.4). This in turn might cause a decrease in \ mostly before  the entrance to the die land. Flow in the die land is simple shear with significant  slip and there the velocity profile  soon attains almost a fully  developed shape. The true shear rate is very small and this essentially causes very small changes in the velocity profiles  which are quite insignificant  (see Figure 4.7 where the three profiles  in the die land coincide and they differ  very little with the profile  at the entrance to the die land). Figure 4.8 shows the typical surface  plots of  flow  type parameter / inside the conical die obtained from  the present simulations for  no slip and with slip boundary condition at the die wall. The different  colors represents different  magnitude of  the v|/ as shown in the figure. w«t: t.rn No Slip slip Figure 4.8: Simulated surface  plots of  flow  type parameter, inside the conical die with die entrance 2a = 60°. The effect  of  the entrance angle on the evolution of  fibrils  has also been explored. As will be seen later, the non-monotonic variation of  the extrusion pressure with entrance angle has been captured with the present model. All the experiments reported in this section are for  a high molecular weight PTFE (F104) resin mixed with 18 wt. % of ISOPAR® M as lubricant (Ochoa and Hatzikiriakos, 2004). 4.3.1 Effect  of  Die Entrance Angle Simulations were performed  for  conical dies with RR = 352 , L/D = 20 and various entrance angles in the range of  8° < 2a < 90°. The simulated dependence of  the extrusion pressure on die entrance angle is shown in Figure 4.9. The agreement between the predicted extrusion pressure and that obtained from  the experimental analysis is very good. Entrance Angle (2a) Figure 4.9: The effect  of  die entrance angle on the extrusion pressure: Comparison between experimental and simulation results. The parameters of  Eq. (4.3) namely, r)„, r | o i , r|; and A.,., are determined by a trial and error method until the best fit  to the experimental data results. At first  simulations were performed  using parameter values chosen arbitrarily. Then depending on the difference between the calculated and experimental values new values were chosen. This is repeated till the best agreement between simulations and experimental values is obtained. The values of  the parameters are reported in Table 4.2. Table 4.2: Parameters for  the shear-thinning and the shear-thickening terms of  Eq. (4.3). parameters Shear-thinning Shear-thickening r| „ (Pa s) 0 0 r| 0 (Pa s) 4000 1600 X  (s"1) 0.3 1 n 0.5 1.3 It can be seen that the initial decrease in the extrusion pressure with entrance angle is similar to the trend seen in capillary extrusion of  polymer melts and other viscous liquids. This trend can be predicted by using the lubrication approximation assumption (Horrobin and Nedderman, 1998). However, lubrication approximation is only valid for  small entrance angles and use of  this for  larger entrance angles continues to predict decrease of  the extrusion pressure monotonically. In fact,  the extrusion pressure of  PTFE increases significantly  with increase of  entrance angle beyond a certain critical value 2a « 30° . Such a behaviour is commonly observed in the extrusion of  elastic solids (for  example, see Horrobin and Nedderman (Horrobin and Nedderman, 1998) and the references  therein). At very small entrance angle PTFE paste behaves mostly as a shear-thinning fluid  with little fibrillation  (small value of  ) and this is captured by the present model. The flow  type parameter,;)/, for  small entrance angle is also close to zero and that ensures that the dominant contribution to the stress tensor comes from  the shear-thinning part. As the entrance angle increases, the flow  becomes more extensional and this has an impact on v|/ and subsequently on E, , with both dramatically increasing. The paste now becomes more solidlike and this can be modelled by the shear-thickening term included in the constitutive rheological model of  Eq. (4.3). The dominant contribution at high entrance angles comes from  the shear-thickening term which causes the significant  increase in the extrusion pressure. It should be mentioned that our primary focus  in modelling the PTFE paste extrusion process was to predict correctly this trend, that is the non-monotonic variation of  extrusion pressure with entrance angle. Such an observation was initially countertuitive and mainly comes from  the gradual change of  the nature of  the material from  liquidlike to solidlike one. Figure 4.10 depicts the variation of  the average structural parameter at the exit, 4exit' the entrance angle. It can be seen that the model predicts an increase in the degree of  fibrillation  with an increase of  the entrance angle. This increase in the degree of fibrillation  can now be related to the tensile strength of  the extrudates, as more fibrils  are expected to increase the dimensional stability of  the extrudates (Ochoa and Hatzikiriakos, 2004). 20 40 60 Entrance angle (2a) 80 100 CL-OT c a> -*—' V) 0) '</) c aj Figure 4.10: The effect  of  die entrance angle on the cross-sectional average structural parameter at the exit, , (from  simulations) and the tensile strength of  dried extrudates (from  experiments). Figure 4.10 also plots the effect  of  the entrance angle on the tensile strengths of  dried extrudates reported by Ochoa and Hatzikiriakos (2004). It can be seen that the tensile strength goes through a minimum with increase of  the entrance angle. The initial decrease of  tensile strength with entrance angle is certainly countertuitive and not predicted by the model in terms of  4e x i t • It might be related to the initial decrease of  the extrusion pressure with entrance angle. A higher pressure can mechanically lock crystallites of  particle more tightly and this facilitates  fibrillation.  In our model the mechanism of  fibrillation  does not depend on local pressure and therefore  such effects  can not be predicted. 4.3.2 Effect  of  apparent shear rate Simulations were performed  for  various apparent shear rate values for  a conical die having an entrance angle 2a - 30° and L /D = 20. The dependence of  the extrusion pressure on apparent shear rate, y A (s32Q/7iD3) is shown in Figure 4.11, where the agreement between the experimental and simulation results is excellent. Figure 4.11: The effect  of  apparent shear rate on the extrusion pressure of  PTFE paste extrusion: Comparison between experimental and simulation results. It is noted again that the same parameters have been used for  all comparisons with experimental results. The evolution of  the structural parameter with apparent shear rate contributes to the monotonic increase of  the extrusion pressure. Figure 4.12 plots the average structural parameter at the exit ^ e x j t , as a function  of  the apparent shear rate, y A . The simulation predicts a very small increase in fibrillation  with increase in apparent shear rate, and this effect  saturates quickly with increase in the apparent shear rate. An increase in y A causes an increase in the structural parameter , and this in turn increases the rate of  fibril  breakage which results in slower increase of  £, at higher apparent shear rate. The experimental data of  tensile strength show essentially no effect  of  the apparent shear rate on the tensile strength of  the extruded paste (Figure 4.12). o.ie RR j= 352:1, L/D = 20, 2a = 30° T = 35°C 0.16 0.14 0 .12 • Tensile Strength (experimetal) Structural parameter (simulation) 10 TO CL •tt 0.10 X <D |u_P 0.08 0.06 0.04 0.02 O) 1= 0) tn 0) '</) c (U 0.00 1000 2000 3000 4000 5000 6000 7000 8000 9000 •1 Apparent Shear Rate, yA (s ) Figure 4.12: The effect  of  apparent shear rate on the cross-sectional average structural parameter at the exit, ^ e x i t , (from  simulations) and the tensile strength of  dried extrudates (from  experiments). 4.3.3 Effect  of  Die Reduction Ratio Simulations were performed  for  dies having L /D = 20, 2a = 60° and various reduction ratios in the range of  56 < RR < 352 . Figure 4.13 depicts the effect  of  die reduction ratio on the extrusion pressure of  the pastes. The agreement between the simulated and experimental dependence of  extrusion pressure on die reduction ratio is again good. The extrusion pressure increases with the increase in the reduction ratio in a nonlinear fashion,  which is captured by the simulated results. Figure 4.14 shows that the structural parameter at the die exit ^ e x i t increases with the reduction ratio. Experimental data show that the tensile strength also increases with the reduction ratio initially, but reaches a maximum at RR = 156. Reduction Ratio Figure 4.13: The effect  of  the die reduction ratio, RR, on the extrusion pressure: comparison between experimental and simulation results. 4.3.4 Effect  of  die length-to-diameter ratio Simulations were performed  for  dies with RR = 352, 2a = 90° and several L/D ratios in the range of  0 to 40 at the apparent shear rate of  5869 s 1 . Figure 4.15 plots the effect  of the length-to-diameter ratio of  the die, L/D, on the extrusion pressure of  paste. The agreement between the simulated and experimental dependence of  extrusion pressure on the L/D ratio is excellent. The extrusion pressure increases linearly with increase of  the L/D ratio. Note that most of  the resistance to flow  is due to the conical zone. 0.5 0.4 0.1 0.0 2a = 60° L/D = 20 r A = 5869 s"1 T = 35°C , S 0.3 0 . 2 -Structural parameter (simulations)) • Tensile strength (experimental). 50 100 150 200 250 300 Die Reduction Ratio (RR) 350 400 ro Q. 05 C d) W 0) w c CD Figure 4.14: The effect  of  die reduction ratio on the average structural parameter at the exit, £,exjt, (from  simulations) and the tensile strength of  dried extrudates (from experiments). 140 TO CL C o 1— •R LLI RR = 352:1, 7a = 5869 s"1, 2a = 90° 120 -100 80 60 40 FEM simulations • Experimental 10 20 L/D 30 40 50 Figure 4.15: The effect  of  length-to-diameter ratio (L/D) on the extrusion pressure: Comparison between experimental and simulation results. The pressure needed to extrude the polymer through the conical zone is about 60 MPa compared to an extra of  about 25 MPa needed to extrude it through the straight section of L/D = 40 of  the capillary die. The simulation results also predict an initial increase in the magnitude of  S,exit with increase in L/D ratio as shown in Figure 4.16. This effect saturates for  L / D > 10 . A similar trend is shown by the variation of  the tensile strength with L/D ratio (Ochoa and Hatzikiriakos, 2004). 2.0 Structural Parameter (simulations) Tensile strength (experimental) 1.5 S 1.0 |WJ> 2a = 90°, RR = 352:1, r A = 5869 s"1 T = 35°C 0.5 3 « - 2 O) c a) i CO C 1 ,<» 0.0 10 20 L/D 30 40 50 Figure 4.16: The effect  of  die L /D ratio on average structural parameter at the exit, e^xit > ( f r o m simulations) and the tensile strength of  dried extrudates (from  experiments). 4.3.5 Structural Parameter The structural parameter is a quantitative measure of  the degree of  fibrillation  in the sample during extrusion. Local axial variations of  the structural parameter at various radial locations for  a die having an entrance angle of  60° are shown in Figure 4.17. Similar trends have been seen for  other conical dies having other entrance angles. The structural parameter, , increases from  0 at the entrance of  the conical die and reaches a maximum before  entering the die land. It can also be seen that once t, relaxes to a constant level, it remains constant throughout the die land section. Axial Distance (m) Figure 4.17: Axial profiles  of  structural parameter at various radial locations for  a conical die having an entrance angle of  60°. Initially £, increases as more and more fibrils  are created due to the extensional flow  in the conical die. As the end of  the conical section is approached, the structural parameter, E, increases further  and this makes the rate of  breakage higher than that of  the creation. Overall this decreases the degree of  fibrillation  and as a result a maximum in the axial profile  of  appears. The flow  in the land region is pure shear and therefore  no additional fibrils  are created. Figure 4.18 shows the effect  of  the apparent shear rate on the rate of  creation and breakage of  fibrils.  For small values of  the apparent shear rate, increases till the exit of the conical section and remain constant thereafter  instead of  reaching maximum. Above a certain apparent shear rate value, y A , of  about 55 s"1, the rate of  breakage overcomes the rate of  creation near the exit of  the conical section, resulting the maximum in the axial profile  of  ^ • 0) "55 E ro k_ CL 15 o i— C/5 0.000 0.002 0.004 0.006 0.008 Axial Distance (m) 0.010 0.012 0.014 center l ine Figure 4.18: Axial profiles  of  structural parameter along the centerline of  a conical die having an entrance angle of  60° for  various apparent shear rates indicated in the figure. Figure 4.19 shows the variation of  structural parameter with axial distance along the centerline of  dies having various entrance angles. The increase in the maximum value of  structural parameter with entrance angle shows the increase in the degree of fibrillation.  The increase in the structural parameter, , with entrance angle is because of the increase in the elongational rate. The simulation results show that the creation of fibrils  takes place inside the conical region with very little change in fibrillation  in the die land region. This is quite similar to the experimental observation by Ariawan et al. (Ariawan, 2002; Ariawan et al., 2002b), and Ochoa and Hatzikiriakos (2004). 1.2 1.0 JLT 0 8 a5 E ro 0.6 nj CL 2 0.4 o 2 0.2 C/) 0.0 0.00 0.02 0.04 0.06 0.08 Axial distance (m) Figure 4.19: Axial profiles  of  structural parameter along the centerline of  conical dies having various entrance angles. 4.4 Conclusions In this work, finite  element simulations of  PTFE paste extrusion are presented in order to predict the dependence of  extrusion pressure on apparent shear rate, die reduction ratio, die L/D ratio and die entrance angle. The rheological constitutive equation proposed, with the total stress comprising a shear-thinning term and a shear-thickening term, is capable of  capturing the main features  of  the process as previously documented by experiments (Ochoa and Hatzikiriakos, 2004; Ariawan, 2002; Ariawan et-al., 2002b). The PTFE paste has been treated as a shear-thinning fluid  before  the occurrence of  fibrillation.  The fibrils  gradually turn the paste to exhibit more shear-thickening behaviour. Change in the nature of  the paste from  a fluidlike  behaviour to a solidlike one, is implemented by the introduction of  a microscopic structural parameter, E, . An evolution equation has been developed for  based on the kinetic network theories (Jeyaseelan and Giacomin, 1995; Liu etal., 2984; Hatzikiriakos and Vlassopoulos, 1996). RR = 352:1 along the center line YA = 5869 s"1 2a = 8° 2a = 15° 2a = 30° 2a = 60° 2a = 90° I  i \ II  ' li The phenomena of  fibril  formation,  evolution and breakage have been captured through this kinetic model. Simulation results were found  to be in agreement with the experimental findings reported by Ochoa and Hatzikiriakos (Ochoa and Hatzikiriakos, 2004). Based on this agreement it can be concluded that the proposed constitutive equation is suitable for modeling the behaviour of  the paste. In addition, the structural parameter, E,, was related to the tensile strength of  the pastes. The predicted effects  of  the die geometrical parameter and operating condition on the E, are generally in agreement with the observed ones on the tensile strength. 4.5 Bibliography Ariawan, A. B., S. Ebnesajjad and S.G. Hatzikiriakos, Preforming  Behavior of  PTFE Pastes, Powder Technology 121, 249-258 (2001). Ariawan, A. B., Ebnesajjad, S. and Hatzikiriakos, S. G. Paste extrusion of polytetrafluoroethylene  (PTFE)  fine  powder  resins. Can. Chem. Eng. J., 80, 1153-1165 (2002a). Ariawan, A.B., Ebnesajjad, S. and Hatzikiriakos, S. G. Properties  of polytetrafluoroethylen  (PTFE)  paste extrudate.  Polym. Eng. Sci., 42, 1247-1253 (2002b). Ariawan, A. B. Paste Extrusion  of  Polytetrafluoroethylene  Fine  Powder  resins, The University of  British Columbia. Dept. of  Chemical and Biological Engineering. Thesis Ph. D., 2002. Benbow, J. J., and J. Bridgwater, Paste Flow  and  Extrusion,  Oxford  University Press, Oxford,  1993. Bird, R. B., O. Hassager, and R. C. Armstrong, Dynamics of  Polymer Liquids.  Vol.  1, 2 n d ed., John Wiley & Sons, Inc., New York, 1987. Bird, R. B., O. Hassager, R. C. Armstrong, and C. F. Curtiss, Dynamics of  Polymeric Liquids  Vol.  2, Kinetic  theory, John Wiley & Sons Inc., New York, 1987. Blanchet, T.A., Polytetrafluoroethylene,  Handbook  of  Thermoplastics,  Marcel Dekker, NY, 1997. Dealy, J. M., and K. F. Wissbrun, Melt  Rheology and  its Role in Plastics  Processing -Theory  and  Applications,  Van Nostrand Reinhold, New York, 1990. Dunlap P. N., and L. G. Leal, Dilute polystyrene  solutions in extensional flows birefringence  and  flow  modification,  J. Non-Newt. Fluid Mech. 23, 5-48 (1987). Ebnesajjad Sina, Fluoroplastics,  Vol  1 Non-Melt  Processible  Fluoroplastics,  Plastic Desgin Library. William Andrew Corp, NY, 2000. Fuller, G. G., Rallison, J. M., Schmidt, R. L. and Leal, L. G., The  measurements of velocity gradients  in laminar flow  by homodyne  light-scattering  spectroscopy, J. Fluid. Mech. 100(3), 555-575 (1980). Fuller, G. G., and Leal, L. G., Flow  birefringence  of  dilute  polymer solutions in two-dimensional  flows,  Rheol. Acta 19, 580-600 (1980). Hatzikiriakos S. G., and D. Vlassopoulos, Brownian dynamics  simulations of  shear-thickening  in dilute  polymer solutions,  Rheo. Acta 35, 274 - 287 (1996). ; Horrobin, D. J., and R. M. Nedderman, Die Entry  Pressure Drops in Paste Extrusion, Chem. Eng. Sci. 53, 3215-3225 (1998). Jeyaseelan R. J., and A. J. Giacomin, Structural  network  theory for  a filled  polymer melt in large  amplitude  oscillatory  shear, Polymer Gels and Networks 3, 1 17-133 (1995). Laun, H. M., R. Bung, S. Hess, W. Loose, O. Hess, and P. Linder, Rheological  and  small angle neutron scattering  investigation  of  shear induced  particle  structures  of concentrated  polymer dispersions  submitted  to plane Poiseulle  and  Couette  flow,  J. Rheol. 36(4) (1992) 743-787. Leonov, A. I., On the rheology  of  filled  polymers,  J. Rheol. 34(7), 1039-1068 (1990). Liu, T. Y., D. S. Soong, and M. C. Williams, Transient  and  steady  rheology  of polydisperse  entangled  melts:  Predictions  of  a kinetic  network  model  and  data comparisons, J. Polym. Sci.: PI. Phys. Ed. 22, 1561-1587 (1984). Macosko, C. S., Rheology principles, measurements, and  applications,  VCH publishers, Inc., New York, 244-247 (1994). Mazur, S., Paste Extrusion  of  Poly(tetrafluoroehtylene)  Fine  Powders  in Polymer Powder technology,441-481, Narks, M., Rosenzweig, N., Ed. John Wiley & Sons, 1995. Mooney, M., Explicit  Formulas  for  Slip  and  Fluidity,  J. Rheol. 2, 210-222 (1931). Ochoa, I., Hatzikiriakos, S. G. Polytetrafluoroethylene  (PTFE)  paste performing: Viscosity  and  surface  tension effects.  Powder Technology 146(1-2), 73-83 (2004). Ochoa, I., Hatzikiriakos, S. G. Paste extrusion of  polytetrafluoroethylene  (PTFE): Surface  tension and  viscosity effects..  Powder Technology 153(2), 108-118 (2005). Ochoa, I., Paste Extrusion  of  Polytetrafluoroethylene  Fine  Powder  resins: The  effect  of the processing aid  physical properties,  Ph.D., The University of  British Columbia. Dept. of  Chemical and Biological Engineering: Thesis Ph. D., 2006. Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, Constitutive  modeling  and  flow simulation of  polytetrafluoroethylene  (PTFE)  paste extrusion, J. Non-Newt. Fluid Mech. 139, 44-53 (2006a). Sperati, C. A., Physical Constants  of  Fluoropolymers,  Polymer Handbook, John Wiley and Sons, NY, 1989. CHAPTER 5 Viscoelastic Modeling and Flow Simulation of Polytetrafluoroethylene (PTFE) Paste Extrusion 2 5.1 Introduction In the previous chapter an ad hoc constitutive equation was developed that accounts for  the increasing fibrillation  through a shear-thickening viscosity. This approach has two weaknesses: (i) the rheological consequence of  fibrillation  should be elastic strain-hardening, not shear-thickening, and (ii) the parameters are treated as adjustable to a large extent, and cannot be directly related to measured properties of  the paste. The present study aims to remove both shortcomings. First, a viscoelastic constitutive equation based on the concept of  a structural parameter, E, is proposed. This parameter represents the mass fraction  of  the PTFE material that is fibrillated  and takes values of  0 and 1 for  the unfibrillated  and fully fibrillated  cases, respectively. The evolution of  the structural parameter is described by a first-order  kinetic differential  equation, which is developed based on concepts borrowed from  network theory of  polymeric liquids (Bird et al., 1987; Jeyaseelan and Giacomin, 1995; Liu et al., 1984). The total stress tensor consists of  a viscous part modeled by using the shear-thinning Carreau model, and an elastic part by a modified  Moony-Rivlin model. The latter is borrowed from  hyperelastic modeling of  rubbers (Mooney, 1940; Rivlin, 1948a; Rivlin, 1948b; Amin et al., 2006; Selvadurai and Yu, 2006). The model parameters in the constitutive equation are determined from rheological measurements on the pastes with parallel-plate and extension rheometers. The viscoelastic constitutive equation is subsequently used to simulate the flow  of  paste through the conical die depicted in Figure 4.1. The finite-element  results are compared with the experimental results. In addition, the flow  simulation is used to predict the extrusion pressure as a function  of  the operating and die geometrical characteristics and 2 A version of  this chapter will be submitted for  publication. Patil, P. D., Feng, J. J., and Hatzikiriakos, S. G. (2006) Viscoelastic Modeling and Flow Simulations of  Polytetrafluoroethylene  (PTFE) Paste Extrusion. to explore the relationship between the tensile strength of  the extrudate and the degree of fibrillation.  In this last task, the average structural parameter at the exit, ^ e x i t , is correlated with the experimental tensile strength for  a range of  the operating and die geometrical parameters. 5.2 Theoretical model ing and numerical method 5.2.1 Governing Equations The steady state mass and momentum conservation equations coupled with the rheological constitutive model are solved to simulate the flow  of  the PTFE paste. The axisymmetric r-z domain (Figure 4.1) has been used to perform  the simulations. The velocity field  v is subject to the incompressibility constraint (volume changes due to fibril  formation  are assumed to be small): V • V = 0. [5.1] Due to the high viscosity of  the paste, the inertial terms in the momentum equation are neglected: V p - V - T = 0 , [5.2] where p is the pressure and T is the stress tensor, which depends on the structural parameter, , through a viscoelastic constitutive equation discussed below. 5.2.1.1 Constitutive  Equation The rheology of  the PTFE paste depends on the formation  and evolution of  a network of fibrils  connecting PTFE polymer particles during the extrusion. To model this complex flow  behavior, a constitutive equation is proposed which explicitly accounts for  the evolution of  fibrils.  Prior work has modeled flow-induced  structure formation  in concentrated suspensions, polymer solutions and filled  polymers as a combination of shear-thinning behavior at low shear rates and strain-hardening behavior at high shear rates (Jeyaseelan and Giacomin, 1995; Dunlap and Leal, 1987). A similar concept has been adopted in the present chapter. While the paste initially behaves as a shear-thinning fluid,  after  the appearance of  fibrils  in its structure, it behaves more and more as a strain-hardening fluid.  Thus, it is assumed that the stress tensor consists of  two contributions coming from  the unfibrillated  and fibrillated  domains of  the paste, represented respectively by a shear-thinning viscous stress and a strain-hardening elastic stress. The relative significance  of  the two contributions depends on the structural parameter, (Patil et al., 2006), the mass fraction  of  the paste of  the system that is fibrillated: T = 0 - ^ ) r l iY + [5.3] where y is the rate of  strain tensor, and r|, is a shear-thinning viscosity that is expressed by a Carreau model (Jeyaseelan and Giacomin, 1995): 7, - Vo\L 1 + \ a \ Y )  J • [5.4] The term Te denotes the elastic stress due to the fibrillated  domains. The strain-hardening behavior of  fibrillated  PTFE paste is well captured through the elastic stress term in the present model. This is superior to the model proposed by Patil et al. (2006), where fibrillation  was modeled by a shear-thickening model. To model the elastic stress xE , a hyperelastic model is used (Mooney, 1940; Rivlin, 1948a; Rivlin, 1948b; Amin et al., 2006; Selvadurai and Yu, 2006). When a solid body is subjected to a large deformation,  the relationship of  positions in deformed  and undeformed  configurations  is described by a deformation  gradient tensor F whose eigen values A,, X2 , and X3 are the stretching ratios in the principal directions. The Mooney-Rivlin equation relates the Cauchy stress tensor t e to the Cauchy-Green strain tensor B = F F t by: ai, si2 where the strain energy density function,  W , is a function  of  the strain invariants (Rivlin, 1948a; Rivlin, 1948b). W = W ( I , , I 2 , I 3 ) [5.7] /,  = tr  B I 2=\{trBf-tr{BB) / 3 = detB Since PTFE paste can be assumed to be incompressible, the value of  / 3 is taken to be unity. Thus, in a strain-invariant based incompressible hyperelasticity model, the description of  W  as a function  of  /, and / 2 forms  the basis of  the approach. Alternatively, on the basis of  the Valanis-Landel hypothesis (Valanis and Landel, 1967), W  can also be expressed directly as a function  of  the three principal stretches, namely, A.,, X 2, and A,3. To capture the rate dependent response of  hyperplastic materials, W should also depend on the strain rate y (Selvadurai and Yu, 2006): W  = W(y,I ],I 2), [5.7] In the present study, we use the following  form  of  the strain energy function: W(I„I 2) = C,(I i-3)+C2 l n f r / y j / , - 3 ) "  +C3 ln(y/y Jl,  - 3)m + C,(l 2-3). [5.8] where C,, C2, C3, C4, n, and m are material parameters, and y c is a threshold strain-rate below which rate-dependence is absent (C 2 = C 3 = 0 for  y < y c ) . Eq. (5.8) embodies ideas from  both Selvadurai and Yu (2006) and Amin et al. (2006). The power-law terms come from  Amin et al. (2006). The C2 term, with C2>0, predicts the hardening feature observed at higher srain levels (Yamashita and Kawabata, 1992; Amin et al., 2006). Similarly, the C3 term, with C3<0, accounts for  the initial stiffness  encountered in elastic materials (Amin et al., 2002). The log-factors  that incorporate rate-dependence are borrowed from  Selvadurai and Yu (2006). Let us consider the predictions of  Eq. (5.8) in steady uniaxial elongation, with principal stretch ratio in the loading direction,' and compression in the other two directions, with X 2 - = X~ V 2 owing to isotropy and incompressibility. Now the deformation  gradient tensor, F, and Cauchy-Green deformation  tensor, B, can be written as: F = X, 0 0 0 0 0 B ^ 0 0 0 ± 0 A, 0 0 '1  J I [5.9] i J and the strain invariants as: / 2= — + 2X l A, [5.10] where A., = L / L 0 , L 0 is the original length of  the sample and L is the length of  the deformed  sample. Using Eqs (5.5) and Eq. (5.8), the expression for  elongational stress component xE M can be written as (Kawabata and Kawai, 1977; Amin et al., 2006): x = 2 X. vdW  1 dW^ "i y Kdl,  \,dl 2j [5.11] Using Eq. (5.8), Eq. (5.11) can be written as: C,+C 2 ln(y/yc) x = 2 f  1 ^  V 2 12 + - 3 + + C3 ln(y/yc) • + A - 3 + -£± K [5.12] By measuring the elongational stress at different  elongational strain and strain rates, the parameters C,, C 2 , C 3 , C4, m, n and yc can be determined by fitting  the data to Eq. (5.12). 5.2.1.2 Parameter Estimation To obtain the values of  the viscous parameters r |o l , n, and a , , steady shear experiments were performed  using a stress-controlled rheometer equipped with parallel plates (C-VOR Bohlin). The samples were prepared using various PTFE resins (listed in Table 5.1) and lubricant (Isopar® M). The physical properties of  various PTFE resins and isoparaffinic  lubricant (ISOPAR® M) are listed in Table 5.1 and Table 4.1 respectively. The paste was first  compacted and shaped into discs before  loaded onto the rheometer. Sand paper was glued onto the plates to suppress slippage. The experiments were done at two different  gap sizes of  1.0 mm and 2.1 mm at room temperature with the stress value ranging from  100 to 4000 Pa. Table 5.1: Physical properties of  PTFE fine  powder resin studied in this work as provided by the supplier. Resin Type Particle Diameter (fim) Specific Gravity F104 HMW Homopolymer 400-650 2.17-2.20 F104 LMW Homopolymer 400-650 2.16-2.18 F301 Modified 400-650 2.15-2.18 F303 Modified 400-700 2.14-2.16 Figure 5.1 plots the viscosity versus stress for  two different  gap sizes. By using sand paper, no slip occurs at low shear stress values. However, at higher stress values significant  slip was observed. Assuming that the slip velocity depends only on the shear stress ct„, , Yoshimura and Prud'Homme (1998) suggested the following  relation for correcting the strain rate and viscosity: t > J = g | U f f - ) ' " l U < 0 . [5.13] H\~ H2 and n f r V t 5 - 1 4 ] where yR is the corrected shear rate, and y ttX and y t t 2 are the apparent shear rates corresponding to gap sizes of  H l and H 2 respectively. Figure 5.2 plots the corrected viscosity as a function  of  the corrected shear rate. These data are used to determine the viscous parameters in Eq. (5.4). The best-fit  parameters are listed in Table 5.2, and the corresponding viscosity function  is also plotted in Figure 5.2. 107 106 105 104 103 F104 HMW Gap size H, = 2.1 E-3 m 101 102 103 104 Stress, a (Pa) Figure 5.1: Plot of  apparent viscosity vs stress obtained from  controlled stress experiments for  two different  gap sizes. 107 106 0 105 10" F104 HMW • • 10-6 10"5 10-4 10'3 10-2 10-1 10° 101 102 103 104 Shear Rate, n (s"1) re 5.2: Corrected viscosity vs. shear rate obtained from  Eq. (5) and solid line shows the curve fitting  by using Carreau model. Table 5.2: Parameters for  the shear-thinning terms of  Eq. (4.3) (F104 HMW). parameters Shear-thinning ti 0 (Pa s) 2.73 x 106 a , (s"1) 38.13 n 0.52 To determine the parameters of  the proposed hypereiastic model, Eq. (5.12) can be fitted  to measured elongational properties of  extruded PTFE paste (Ochoa, 2006). The uniaxial extension experiments are performed  on four  different  pastes with F104 HMW-Isopar® M, F104 LMW-Isopar® M, F301-Isopar® M and F303-Isopar® M at 18 wt. % of lubricant. The samples were prepared by extruding the pastes through a slit die (for  more details see Ochoa, 2006). The extruded samples are assumed to be fully  fibrillated Thus, subsequent measurements concern only the elastic term t e in Eq. (5.3). The prepared rectangular samples were loaded onto the Sentmanat Extensional Rheometer (SER) attached to a strain controlled rheometer (Sentmanat, 2003). The samples were thus stretched at constant Hencky strain rates in the range of  0.00113 s"1 to 1.13 s"1. The fitted  value of  yc is found  to be roughly the same for  all four  PTFE resins, that is 1.36 x 10 -5 s"1. Table 5.3 lists the parameter values determined for  the four  pastes. The experimental observations and predictions are plotted in Figures 5.3-5.6. Figures 5.3-5.6 show the fitting  of  the transient tensile stress by Eq. (5.12) for  two samples stretched at different  strain rates. For F104HMW, Eq. (5.12) fits  the data reasonably well. Resin F303 (co-polymer) displays a lower extensibility compared to the homopolymer F104HMW and the model is not able to predict the extensional behavior well (Figure 5.6). Overall, it can be said that the model works reasonably well and is useful.  These parameters are used in the numerical simulations of  PTFE paste flow  by using finite element method as discussed later in this chapter. Table 5.3: Material parameters for  PTFE samples subjected to different  Hencky strain rate. Resin c, c2 N c3 c4 M F104 HMW 7.66 2.56 8.1x10"' -9.98 4.4 1.35x10"' F104 LMW 1.23x10' 4.8 1.1 -1.17x10' 7.22 8.7 xlO"2 F301 8.1 1.88x10' 6.9x10"' ' - 7 . 3 3.66 5.2 xlO"2 F303 1.4 xlO"2 2.5x10' 2.6x10"' -1.41x10' 3.8x10"' 2.3x10"' Figure 5.3: Uniaxial extension of  F104 HMW samples stretched at different  Hencky strain rates (lines show the fits  of  Eq. (5.12)). 25 20 F104 LMW 'Henky strain rate, t H (s"1) TO Q. b 15 10 (/> <u CO _cl> in c 0) 10 • 0.00113 o 0.0113 • 0.113 A 1.13 2 3 4 Extensional Ratio, = L/L0 Figure 5.4: Uniaxial extension of  F104 LMW samples stretched at different  Hencky strain rates (lines show the fits  of  Eq. (5.12)). 20 Q-| b <n (/) (U •4—' CO 0) W3 C (U F301 Henky strain rate, e„ (s"1) 15 - • 0.00113 o 0.0113 T 0.113 A 1.13 2 3 4 Extensional Ratio, = L/L0 Figure 5.5: Uniaxial extension of  F301 samples stretched at different  Hencky strain rates (lines show the fits  of  Eq. (5.12)). 10 CTJ CL (/) Ul (1) CO o> 03 C 0) F303 Henky strain rate, eH (s ) • 0.00113 o 0.0113 • 0.113 A 1.13 0.8 1.0 1.2 1.4 1.6 1.8 2.0 Extensional Ratio, - L/L0 Figure 5.6: Uniaxial extension of  F303 samples stretched at different  Hencky strain rates (lines show the fits  of  Eq. (5.12)). 5.2.1.3 Structural Parameter The kinetic model for  the structural parameter, E, , previously proposed by Patil et al. (2006) is also used here. The variation of  E,, the mass fraction  of  fibrillated  paste, along a streamlines as determined by the rates of  fibril  creation and breakage can be written as: where a and P are dimensionless rate constants, v|/ is the flow  type parameter and y is the magnitude of  the strain rate tensor. The first  and second term on the right hand side of Eq. (4.4) denote the rate of  creation and breakage of  fibrils  in the paste, respectively. With E, = 0 at the inlet of  the die, E, is subsequently bounded between 0 and 1 if  a/p < 1. In this study, both rate constants a and P have been assigned the value of  1. The creation and evolution of  fibrils  is due to elongational flow  (Ariawan et al., 2002a; Patil et al., 2006), and this is reflected  by making the rate of  creation to depend on the flow type parameter, vj/. This indicates the relative strength of  straining and rotation in a mixed flow  (Dunlap and Leal, 1987; Fuller et al., 1980; Fuller and Leal, 1980). More details about this model can be found  in Patil etal.  (2006). 5.2.2 Boundary Conditions For the various boundary segments in the geometry shown in Figure 4.1 the following boundary conditions are used: (i) Inlet boundary conditions (z = 0): The fully  developed velocity profile  for  a shear-thinning Carreau fluid  model has been imposed at the inlet with radial velocity V r = 0 . The paste is completely unfibrillated  at this point: £, = 0. Assume the paste to be in an underformed  state. In addition, the initial strain is set to zero: F = 1 0 0 0 1 0 0 0 1 The strain of  the upstream shear is neglected relative to the much stronger deformation downstream in the die. (ii) Outlet boundary conditions: The normal stress boundary condition and zero radial velocity are imposed: n • ( - pi + x)n = - p 0 ; v;. = 0 . (iii) Slip boundary condition at the die wall: The Navier slip condition has been used at the die wall, which relates slip velocity with the wall shear stress, <yH,: v s = C a w , where C = 1.92 m/M Pa s as determined from  experimental data by Patil et al. (2006). (iv) The axisymmetric boundary condition is used at r = 0: v r = 0 , dvjdr  = 0 . 5.2.3 Finite Element Method The equation of  motion coupled with the proposed viscoelastic constitutive and structural parameter models were solved using the commercial finite  element (FE) code FEMLAB 3.2 with user-defined  MATLAB routines for  calculating the strain history of  the material and the viscoelastic stress tensor. As the problem considered here is axisymmetric, two-dimensional meshes are used on the computational domain (Patil et al., 2006). These unstructured meshes comprise triangular elements of  widely varying sizes, small and large elements being employed in regions where the rates of  strain were large and small, respectively. The smallest elements are required near the die corners, especially the re-entrant corner. The total number of  elements used are in the range of  3000 to 10000. The corners are also rounded slightly to avoid geometrical singularity, and the local element size is chosen to be smaller than the fillet  radius at the corner. The fillet  radius is a small portion of  the capillary radius, and thus the solution is expected to be slightly affected  by the corner rounding. Finite element simulations for  viscoelastic flows  have been performed  by many groups (Crochet et al., 1984; Owens and Phillips, 2002), using both differential  and integral types, of  constitutive equations. In particular, Olley and Coates (1997) and Olley et al. (1999) described FE methods with 'streamline upwinding' elements for  the K-BKZ model with strain and time damping components introduced by Papanastasiou et al. (1993). Since our constitutive model is based on the nonlinear strain tensor, time-integration along the streamlines is essential, and an approach similar to that of  Olley et al. (1999) has been adopted. Since the problem is time independent, the solution involves iterations among the velocity field,  the strain field,  and the stress field.  First, a Newtonian flow  field  is generated, subject to the proper boundary conditions. This trial flow  field  is used in two ways. On one hand, the kinetic equation for  the structural parameter t, is solved to obtain a distribution of  throughout the simulation domain. This will be required in the viscoelastic constitutive equations [Eq. (5.3)] for  stress computation. On the other hand, pathlines are computed based on this trial flow  field  to track the time history of  fluid particles along them. Elastic stress at a point is computed by tracing back the particle path to calculate the deformation  gradient tensor from  which the Cauchy-Green strain tensor B(t,t')  can be calculated. These quantities with parameters listed in Table 5.3 for resin F104 HMW are used to evaluate the elastic stress at that point as per Eq. (5.5) and (5.8). Once these elastic stresses are computed, the total stress components are obtained by using Eq. (5.3) and (5.4) and incorporated into the FEMLAB flow  solution as a body force  [Olley et al. (1999)]; from  this a new flow  solution is obtained using the FEMLAB finite  element solver. This new flow  solution is then used as the new trial flow  field,  and the above procedure is repeated. Convergence is reached when the fractional  velocity change between successive iterations is below a certain threshold (10~5) over all nodes. 5.2.3.1 Particle Tracking The stresses according to Eq. (5.3), (5.6) and (5.11) require the positional history of  a particle. For a steady flow  this can be achieved by an incremental procedure (Luo and Mitsoulis, 1990; Bernstein et al., 1994; Olley and Coates, 1997; and Olley et al., 1999): dt i2 f 2 dt dv r dv r r(t-t')  = r(t)-v rdt' + ^ - + , [5.15] \ or oz J >2 r a.. a.. , ( , - , ' ) = + + , [5.16] 2 \ or oz J where dt'  is a step in time, and v r and v z are velocity components in the r and z direction, respectively. After  each time step, a new position is reached. The current element is ascertained by finding  an element for  which certain conditions are true. If  no element is found,  such as when the particle is tracked back to the inlet, the particle is placed on the boundary of  its last known element at the nearest point. Once found,  the interpolation functions  for  that element are used to determine the tensor elements of  the velocity gradient tensor, L(/'). The pathlines are calculated using second order time integration as described by Olley et al. (1999). Then the deformation  gradient tensor, F(/,/'), is computed by integrating the following  equation along the pathlines: ^ = [5-17] dt using a fourth-order  Runge-Kutta method. L(t')  is the velocity gradient tensor experienced by a fluid  particle as it traverses the pathline. Given F at a relative time, / - / ' , t h e strain tensor B(t,t')  is calculated using the Eq. (5.5). L(,') = ^ 0 dr 0 ^ dr r 0 0 av, dz [5.18] 5.2.3.2 Incorporating computed stresses into flow  solution The stress distribution is stored in a file  in a format  suitable for  the FEMLAB software  to read. The values of  all the variables at integrations points are obtained by interpolating variables at nodal points. The subroutine also has a code to determine which triangular element an integration point is in, and then interpolate the nodal values to the integration point. The direct method is used to achieve velocity field  adjustment. In this method the updated velocities and pressures are calculated within the finite  element solver (FEMLAB) (Olley et al., 1999). In this method the body forces  that are required can be determined by comparing the equations that the CFD package solves, and the equations that are required to be solved. The steady state solution must obey V • a = 0 (neglecting body forces  and momentum). In case of  axisymmetric Newtonian flow  problem the FEMLAB software  solve the equations: l_d_ r dr 2 rji, a ^ dr •P +• d ( n av dz dz dz l_d_ r dr V v av r dv. — + \\ dz  dr + F  =0 J  J He av r av, — l + — -dz  dr \\ + F=0 [5.19] [5.20] J) where p,0 =cor|0 l , where co is the relaxation parameter and r |0 l is the zero shear rate viscosity, is the constant viscosity value provided to the code. The values of  body-force that the package must be provided with for  axisymmetric flow  comes from  equations: r dr dz <K dr 100 r f + -dz •Mi + -dr rxrz -rfi, v av ^ dv r av7 —'-  + —-dz  dr \ \ J) dr  dz J  J [5.21] [5.22] The value of  relaxation parameter was chosen to be 3, the reason behind selecting this value is that the convergence becomes independent of  co. The aim behind using higher viscosity value is to increase the Newtonian stresses to be of  the same order of  magnitude as the obtained viscoelastic stresses (Viriyayuthakorn and Caswell, 1980). Nodal values for  all terms in the Eqs. (5.19)-(5.22) can be deduced by differentiating  the nodal values for  viscoelastic stresses, structural parameter, velocity, and velocity gradient. Along the centerline, many of  the terms in the equations for  body forces  become indeterminate. The limiting value assessed at a small distance away from the centerline is used. A map of  these variables is stored in a file,  so that it will be available to the package. This allows the package to incorporate these values as body forces  in the next stake's flow  solution. The resulting solution is a solution of  Eq. (5.19) and (5.22) for  the current state of  stresses. This solution is then used to re-compute the stresses according to Eq. (5.3), and an iterative procedure continues until convergence. To validate our code, the same contraction flow  of  LDPE as reported by Olley et al. (1999) has been simulated. The K-BKZ model with strain damping function  given by Papanastasiou et al. (1983) is solved for  simulating the flow  of  LDPE melt through an abrupt 4:1 contraction. The two results are in excellent agreement. In particular, the growth of  the corner vortex size with increasing apparent shear rate was successfully captured. This is illustrated in Figure 5.7 by plotting the vortex intensity against stress ratio. The vortex intensity v|/ in given by: where v|/M, is the stream function  value at the wall and v|J c/ is the stream function  value at the centerline of  the channel. v|/ is the stream function  value at the center of  the vortex. The stress ratio S r is the ratio of  normal stress to shear stress on the donwstream tube wall of  the 4:1 contraction. The difference  between the two solutions is below 6.8 %. The convergence of  the simulation results with respect to number of  elements was also confirmed.  When doubling the number of  mesh elements from  3000 to 6000, the difference  in the velocity is below 3 % throughout the simulation domain. The run time of  the simulations was in the range of  1500 - 2500 seconds on Intel Pentium IV machines (2.8 GHz) with 1 GB RAM. nO £ c 0) -4—' c o > <1> > <D DC Stress Ratio, SR Figure 5.7: Comparison between vortex intensities obtained in current work, and those obtained by Olley et al. (1999), for  axisymmetric flow  of  LDPE through a 4:1 abrupt contraction. 5.3 Results and discussion In this section, the simulations are carried out for  different  flow  rates and for various die design parameters, namely the die reduction ratio, RR = Dj; / D 2 , the die land length to diameter ratio, L / D , and die entrance angle, 2 a . The simulation results are compared with the experimental findings  reported by Ochoa and Hatzikiriakos (2004) for pastes prepared by mixing a high molecular weight PTFE (F104HMW) with 18 wt. % Isopar® M. Comparisons are also done with the predictions of  the shear-thinning and shear-thickening (STT) model proposed by Patil et al. (2006). To gain a better understanding of  the structure of  PTFE paste flow,  typical velocity profiles  at various axial locations inside the conical die (2a=90 and L/D=20) are plotted in Figure 5.8 and 5.9. The x-axis in Figures 5.8 and 5.9 is the dimensionless radial distance normalized by the die radius at the corresponding axial location. The flow  inside the conical section is mostly elongational (note the significant  slip at the wall) and this causes significant  fibrillation  (discussed below). The two velocity profiles  for  the two models in the die land coincide with each other (Figure 5.9). Flow in the die land is simple shear with significant  slip and there the velocity profiles  soon attain fully developed shape. The behavior of  structural parameter, E, , with operating and geometrical parameters is discussed in the section below, followed  by the comparison of the simulations results with experimental observations. 0.04 0.03 0.02 E / o o <v > 0.00 2a = 90° L/D = 20 z = 0 R = 4.76E-3 m z = 2E-3 R = 2.74E-3 z = 4E-3 R = 7.54E-4 m 0.0 0.2 0.4 0.6 0.8 Dimensionless radial distance R (mm) = 4.76 z (mm) = o 2 1.0 1.2 Figure 5.8: Radial velocity profiles  at various axial locations for  a conical die having an entrance angle of  90°. Thick and thin lines denote the velocity profiles  from viscoelastic and STT models respectively (y A = 5869 s"'). w E 0.194 0.192 0.190 0.188 o 0.186 > 0.184 0.182 0.180 0.0 0.2 0.4 0.6 0.8 1.0 Dimensionless radial distance R (mm) = 4.76 1.2 0.25 0.25 z (mm) = o Figure 5.9: Radial velocity profiles  at various axial locations inside the die land for  a conical die having an entrance angle of  90°. Thick and thin lines denote the velocity profiles  from  viscoelastic and STT model respectively (y A = 5869 s"1). 5.3.1 Structural Parameter The structural parameter is a quantitative measure of  the degree of  fibrillation  in the sample during extrusion. Local axial variations of  the structural parameter at various radial locations for  a die having an entrance angle of  60° are shown in Figure 5.10. Similar trends have been seen for  other conical dies having other entrance angles. The structural parameter, 4 , increases from  0 at the entrance of  the conical die and reaches a maximum before  entering the die land. The E, relaxes rapidly upon entering the die land and remains constant further  downstream. The viscoelastic model predicts a markedly lower structural parameter than the STT model. This is probably a result of  the viscoelastic stresses suppressing the elongational flow  in the contraction. Axial Distance (m) Figure. 5.10: Axial profiles  of  structural parameter at various radial locations for  a conical die having an entrance angle of  60°: thin and thick lines show the structural parameter values from  STT and viscoelastic models respectively. The axial profiles  of  the structural parameter, depicted in Figure 5.10 can be explained as follows:  initially increases as more and more fibrils  are created due to the extensional flow  in the conical region of  the die. As the end of  the conical section is approached the structural parameter, E, , makes the rate of  breakage higher than that of  the creation. Overall, this decreases the degree of  fibrillation  and as a result a maximum in the axial profile  of  E, appears. The flow  in the land region is simple shear with significant slip at the wall (see velocity profiles  in Figure 5.9) and therefore  no additional fibrils  are created. Furthermore, due to significant  slip at the wall, the true shear rate is small; this causes the breakage rate to become negligible; thus E, in the land region remains essentially constant. Figure 5.11 shows the effect  of  the apparent shear rate y A 32Q!nD]) on the structural parameter (degree of  fibrillation)  predicted by the STT and viscoelastic models. Again predicted profiles  from  the two models are qualitatively similar. For small values of  the apparent shear rate, E, increases till the exit of  the conical section and remain constant thereafter  instead of  reaching maximum. Above a certain apparent shear rate value, y A , of  about 55 s"1, the rate of  breakage overcomes the rate of  creation near the exit of  the conical section, resulting the maximum in the axial profile  of  E,. Y A S " 1 Axial Distance (m) Figure 5.11: Axial profiles  of  structural parameter along the centerline of  a conical die having an entrance angle of  60° for  various apparent shear rates indicated in the figure:  thin and thick lines show the structural parameter values from  STT and viscoelastic models respectively. Figure 5.12 shows the variation of  structural parameter with axial distance along the centerline of  dies having various entrance angles. The comparison between the variation of  S, along the centerline from  the viscoelastic simulation and from  the STT simulation is shown again. Viscoelastic simulations predict in general lower values of  E, for  all entrance angles compared to those of  the STT model. The increase in the structural parameter, 2, , with increase of  the entrance angle is due to the increase in the elongational rate. The simulation results also show that the creation of  fibrils  takes place inside the conical region with negligible change in fibrillation  in the die land region. This agrees with the experimental observation by Ariawan (2002), Ariawan et al. (2002a), and Ochoa and Hatzikiriakos (2004). 1.2 1.0 CD 0) 2 0.6 CD CL TO 0.4 o 2 0.2 to o.o :RR = 352:1 along the center line • YA = 5869 s"1 • 1 1 ) 1 1 1 1 2a I : I 8° 15° 30° 6 0 ° 8° 1 1 1 15° 30° 60° 90° f - 1 l i it . — i i v^x i i . . . 0.00 0.02 0.04 0.06 0.08 Axial distance (m) Figure 5.12: Axial profiles  of  structural parameter along the centerline of  conical dies having various entrance angles. Thin and thick lines show the structural parameter values from  STT and viscoelastic models respectively. 5.3.2 Effect  of  Die Entrance Angle Simulations were performed  for  conical dies having RR = 352, L/D = 20 and various entrance angles in the range of  8° < 2a < 90° . The simulated dependence of  the extrusion pressure on die entrance angle is shown in Figure 5.13. Figure 5.13 also shows the comparison between the fitting  from  STT model (Patil et al., 2006) and the predictions from  proposed viscoelastic constitutive model. The agreement between the predicted extrusion pressure and that obtained from  the experimental analysis is overall good. The STT model represents a better fit  to the experimental data in Figure 5.13 and in most other cases that will be presented below. However, it is noted that the parameters of the STT model were fitted  to extrusion experimental results, while the viscoelastic model is a truly predictive one (its parameters were fitted  to rheological data). Entrance Angle, 2a Figure 5.13: The effect  of  die entrance angle on the extrusion pressure: Comparison between experimental results, predictions from  STT and viscoelastic model. The initial decrease in the extrusion pressure with entrance angle in Figure 5.13 is similar to the trend seen in capillary extrusion of  polymer melts and other viscous liquids. This trend can be predicted by using the lubrication approximation assumption (Horrobin and Nedderman, 1998; Selvadurai and Yu, 2006). However, lubrication approximation is only valid for  small entrance angles and use of  this for  larger entrance angles continues to predict decrease of  the extrusion pressure monotonically. In fact,  the extrusion pressure of  PTFE increases significantly  with increase of  entrance angle beyond a certain value 2a « 30°. Such a behaviour is commonly observed in the extrusion of  elastic solids (for example, see Horrobin and Nedderman (1998) and the references  therein). At very small entrance angles PTFE paste behaves mostly as a shear-thinning fluid with little fibrillation  (small value of  ) and this is captured by the present model. The flow  type parameter, \|/, for  small entrance angle is also close to zero and that ensures that the dominant contribution to the stress tensor comes from  the shear-thinning part. As the entrance angle increases, the flow  becomes more extensional and this has an impact on \|/ and subsequently on (both dramatically increasing). The paste now becomes more solidlike and this can be modelled by the elastic strain-hardening term included in the constitutive rheological model of  Eq. (5.3). The dominant contribution at high entrance angles comes from  the strain-hardening term, which causes the significant  increase in the extrusion pressure. It should be mentioned that our primary focus  in modelling the PTFE paste extrusion process was to develop a truly predictive model. As already discussed above, the comparison below between STT and viscoelastic models with experimental results, STT model calculations are closer to experiment simply because its parameters are best fitted to the available PTFE processing data. On the other hand, the parameters of  the viscoelastic model are all determined from  independent rheological experiments and this makes the model a truly predictive one. Figure 5.14 depicts the variation of  the average structural parameter at the exit, ^ e x j l , with the entrance angle. The model predicts increase of  the degree of  fibrillation  with increase of  the entrance angle. This increase in the degree of  fibrillation  can now be related to the tensile strength of  the extrudates, as more fibrils  are expected to increase the dimensional stability of  the extrudates (Ochoa and Hatzikiriakos, 2004). Figure 5.14 also plots the effect  of  the entrance angle on the tensile strengths of  dried extrudates reported by Ochoa and Hatzikiriakos (2004). The experimental tensile strength goes through a minimum with increase of  the entrance angle. This countertuitive behavior of tensile strength with entrance angle is discussed in the previous chapter. 1.0 0.6 * IjjlP 0 4 0.2 0.0 - 0 . 2 CO CL a) 3 Q) Q) c (D 20 40 60 Entrance angle (2a) 80 100 Figure 5.14: The effect  of  die entrance angle on the cross-sectional average structural parameter at the exit, ^ e x i t , (from  Viscoelastic and STT simulations) and the tensile strength of  dried extrudates (from  experiments). 5.3.3 Effect  of  Apparent Shear Rate Simulations were performed  for  various apparent shear rate values for  a conical die having an entrance angle 2a = 30° and L/D -20. The dependence of  the extrusion pressure on apparent shear rate, j A is shown in Figure 5.15, where the agreement between the experimental and simulation results is very good. The viscoelastic model predicts a higher extrusion pressure compared to that of  STT model. The evolution of  the structural parameter with apparent shear rate contributes to the monotonic increase of  the extrusion pressure. Figure 5.16 plots the average structural parameter at the exit ^ e x i t obtained from the viscoelastic and STT models, as a function  of  the apparent shear rate, y A . The simulations predict a small increase in fibrillation  with increase in apparent shear rate, and this effect  saturates quickly. The experimental data of  tensile strength show essentially no effect  of  the apparent shear rate on the tensile strength of  the extruded paste (Figure 5.16). 70 CO Q- 60 0 CO £ 50 L/D = 20 2a = 30° RR = 352:1 •s LU 40 • Experimental: Isopar M Viscoelastic simulations STT model 30 2000 4000 6000 8000 Apparanet Shear Rate, yA(s ) Figure 5.15: The effect  of  apparent shear rate on the extrusion pressure of  PTFE paste extrusion: Comparison between experimental and simulation results from  viscoelastic and STT model. 0.16 RR = 352:1, LVD = 20, 2a = 30° 0.14 0.12 0.10 0.08 0.06 0.04 0.02 Tensile Strength Viscoelastic model STT model 10 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 CO CL O) c CD i CO 0) w c CD 1 Apparent Shear Rate, yA (s ) Figure 5.16: The effect  of  apparent shear rate on the cross-sectional average structural parameter at the exit, ^ e x i t , (from  viscoelastic and STT simulations) and the tensile strength of  dried extrudates (from  experiments). 5.3.4 Effect  of  Die Reduction Ratio Simulations were performed  for  dies having L / D = 20, 2a = 60° and various reduction ratios in the range of  56 < RR < 352. Figure 5.17 depicts the effect  of  die reduction ratio on the extrusion pressure of  the pastes. The agreement between the simulated and experimental dependence of  extrusion pressure on die reduction ratio is excellent. The extrusion pressure increases with increase in the reduction ratio in a nonlinear fashion, which is captured by the simulated results. Reduction Ratio Figure 5.17: The effect  of  the die reduction ratio, RR, on the extrusion pressure: comparison between experimental and simulation results from  viscoelastic and STT model. Figure 5.18 shows that the tensile strength increases with the reduction ratio and reaches a maximum at RR = 156 . The viscoelastic and STT simulations show a continuous increase in H,exit. 5.3.5 Effect  of  Die Length-to-Diameter Ratio Simulations were performed  for  dies with RR = 352 , 2a = 90° and several L/D ratios in the range of  0 to 40 at the apparent shear rate of  5869 s"1. Figure 5.19 plots the effect  of the length-to-diameter ratio of  the die, L/D, on the extrusion pressure of  paste. The agreement between the simulated and experimental dependence of  extrusion pressure on the L /D ratio is reasonable. 0.6 0.5 0.4 , a 0.3 IJJLP 0 . 2 0.1 0 .0 2a = 60° L/D = 20 f A = 5869 s 1 Viscoelastic model STT model • Tensile strength 50 100 150 200 250 300 350 400 Die Reduction Ratio (RR) D. O) c a) s w O) U> c 0 Figure 5.18: The effect  of  die reduction ratio on the average structural parameter at the exit, £,exjt, (from  STT and viscoelastic simulations) and the tensile strength of  dried extrudates (from  experiments). 140 ro 120 CL c o '</> 3 100 80 LU 60 40 RR = 352:1 2a = 90° YA = 5869 s"1 I i i . i i 1 i ' i i • Experimental: Isopar® M Viscoelastic model STT model 10 20 30 L/D ratio 40 50 Figure 5.19: The effect  of  length-to-diameter ratio (L/D) on the extrusion pressure: Comparison between experimental and simulation results from  viscoelastic and STT model (Patil et al.,  2006). The extrusion pressure increases linearly with increase of  the L/D ratio. Note that most of  the resistance to flow  is due to the conical zone. The pressure needed to extrude the polymer through the conical zone is about 60 MPa compared to an extra of  about 25 MPa needed to extrude it through the straight section of  L /D = 40 of  the capillary die. The viscoelastic model overpredicts the dependence of  extrusion pressure and the difference  between experimental and the corresponding simulations results increases with increase of  L/D ratio. The simulation results also predict an initial increase in the magnitude of with increase in L/D ratio as shown in Figure 5.20. This effect  saturates for  L /D > 10 . A similar trend is shown by the variation of  the tensile strength with L / D ratio (Ochoa and Hatzikiriakos, 2004). The behavior of  E,exit predicted from  STT simulations and viscoelastic simulations is similar. 2.0 Viscoelastic model STT model • Tensile strength (experimental) 1-5 -2a = 90°, RR = 352:1, YA = 5869 s"1 T = 35°C J CL lu/1 0.5 -CD 2 <D C/D '</> c 1 <D 0.0 10 20 LVD 30 40 50 Figure 5.20: The effect  of  die L /D ratio, on average structural parameter at the exit, E,exjt, (from  viscoelastic and STT simulations) and the tensile strength of  dried extrudates (from  experiments). 5.4 Conclusions A viscoelastic rheological constitutive equation proposed for  the paste extrusion of  PTFE, with the total stress comprising a shear-thinning term and a strain-hardening term, is capable of  capturing the physics of  the process as previously documented by experiments (Ochoa and Hatzikiriakos, 2004; Ariawan, 2002; Ariawan et al., 2002a; Ochoa, 2006). The PTFE paste has been treated as a shear-thinning fluid  in the absence of  fibrillation.  The creation of  fibrils  gradually turns the paste to exhibit more strain-hardening behaviour, and this elastic behaviour is captured through a hyperelastic modified  Mooney-Rivlin model. Change in the nature of  the paste from  a fluidlike  (shear-thinning) behaviour to a solidlike (strain-hardening) one, is implemented by the introduction of  a microscopic structural parameter, , that accounts for  fibril  formation and breakage. Previously developed evolution equation has been used for  £, based on kinetic network theories (Patil et al., 2006; Jeyaseelan and Giacomin, 1995; Liu et al, 1984; Hatzikiriakos and Vlassopoulos, 1996). Simulation results were found  to be in excellent agreement with the experimental findings  reported by Ochoa and Hatzikiriakos (2004). Based on this agreement it can be concluded that the proposed constitutive equation is suitable for  modeling the behaviour of  the paste and captures the physics of  the process. 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Olley, P., and P. D. Coates, "An approximation to the KBKZ constitutive equation," J. Non-Newt. Fluid Mech. 69, 239-254 (1997). Olley, P., R. Spares, and P. D. Coates, A method  for  implementing  time-integral constitutive  equations in commercial CFD  packages,  J. Non-Newt. Fluid Mech. 86, 337-357 (1999). Owens, R. G., and Phillips, T. N., Computational  rheology,  Imperial college press, London, 2002. Papanastasiou, A. C., L. E. Scriven, and C. W. Macosko, An integral  constitutive equation for  mixed  flows:  viscoelastic characterization,  J. Rheol. 27, 387-410 (1983). Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, Constitutive  modeling  and  flow simulation of  polytetrafluoroethylene  (PTFE)  paste extrusion, J. Non-Newt. Fluid Mech. 139, 44-53 (2006a). Rivl in, R. S., Large elastic deformations  of  isotropic materials,  Philos. Trans. R. Soc. London, Ser. A 240, 459-490 (1948a). Rivlin, R. S., "Large elastic deformations  of  isotropic materials IV. Further developments of  the general theory," Philos. Trans. R. Soc. London, Ser. A 241, 379-397 (1948b). Rivlin, R. S., Large elastic deformations,  in Rheology,  New York: Academic Press, 1956. Selvadurai, A. P. S., and Q. Yu, Constitutive  modelling  of  a polymeric material  subjected to chemical exposure, Int. J. Plasticity 22, 1089-1122 (2006). Sentmanat, M. L., "A novel device for  characterizing polymer flows  in uniaxial extension," SPE, ANTEC proceedings, 992-996 (2003). Sperati, C. A., Physical Constants  of  Fluoropolymers,  Polymer Handbook, John Wiley and Sons, NY, 1989. Valanis, K. C., and R. F. Landel, The  strain-energy  density  function  of  a hyperelastic material  in terms of  the extension ratios, Arch. Appl. Mech. 64, 136-146 (1967). Viriayuthakorn, M., and Caswell, B., Finite  element simulation of  viscoelastic flow,  J. Non-Newt. Fluid Mech. 6, 245-267 (1980). Yamashita, Y., and Kawabata, S., Approximated  form  of  the strain energy density function  of  carbon-black  filled  rubbers for  industrial  applications,  J. Soc. Rubber Ind. (Jpn) 65(9), 517-528 (1992). Yoshimura, A., and R. K. Prud'Homme, Wall  slip corrections  for  coquette and  parallel disk  viscometers, J. Rheol. 32(1), 53-67 (1988). CHAPTER 6 An Analytical Flow Model for Polytetrafluoroethylene Paste through Annular Dies3 6.1 Introduction Numerous constitutive models have been developed for  flows  of  viscoelastic materials, such as polymer melts (Larson, 1998), solids under plastic deformations (Hoffman  and Sachs, 1953), and elastic-plastic materials that exhibit strain hardening as in the case of  metal forming  or wire drawing (Davis and Dukso, 1994). Although PTFE paste exhibits strain hardening effects  (Mazur, 1995; Ebnesajjad, 2000; Ariawan et al., 2002a), little work has been devoted to its flow  modeling as an elasto-visco-plastic material4. Even with the available equations, significant  modifications  may still be necessary to improve the model predictions. The empirical equation suggested by Benbow and Bridgwater (1993) cannot predict the effect  of  die entrance angle on the extrusion pressure of  PTFE paste, although it works quite well for  other pasty materials (Benbow et al., 1987; Benbow and Bridgwater, 1993). Due to its empirical nature, modifications  of  any theoretical significance  are also difficult  to incorporate. Also the lubrication approximation used by Benbow and Bridgwater (1993) is valid only for  very small entrance angle, which is not consistent with experimental data of  PTFE paste extrusion for  larger die entrance angle used in rod extrusion (Ariawan et al., 2002a; Dealy and Wissbrun, 1990). An improved analytical model for  orifice  extrusion of viscoplastic materials has recently been proposed (Basterfield  et al., 2005). Due to structure formation  (fibrillation),  strain hardening effects  are obtained at high contraction angles during PTFE flow  and therefore  these models (Benbow and Bridgwater, 1993; Basterfield  et al., 2005) are not suitable for  PTFE paste flow  through cylindrical and annular dies. 3 A version of  this chapter has been accepted for  publication. Patil, P. D., Feng, J. J., and Hatzikiriakos, S. G. (2006) An Analytical Flow Model for  Polytetrafluoroethylene  Paste Through Annular Dies, A.I.Ch.E. J. 52: 4028-4038. The flow  equation suggested by Snelling and Lontz (1960) is able to describe the effects  of  die design and extrusion speed more accurately. However, it does not take into account the frictional  force,  which becomes more important when tapered dies of  small entrance angle are used. Also, the analysis provided by Snelling and Lontz (1960) does not account for  the pressure drop along the capillary length of  the die that follows  the entrance (contraction) region. Ariawan et al. (2002a) have proposed a visco-plastic model to predict the dependence of  extrusion pressure on die geometrical parameters for  rod extrusion. This approximate model successfully  captured the non-monotonic dependence of  extrusion pressure on die entrance angle and other geometrical characteristics of  the cylindrical die. Its derivation is based on the RFH (discussed below in detail), whose validity has been demonstrated experimentally (Ariawan et al., 2002a; Snelling and Lontz, 1960). Although this model does not explicitly predict micromechanical details of the extrudates, it predicts the extrusion pressure very well and therefore  is very useful  in die design (Ariawan et al., 2002a). On the other hand, tube extrusion (annular flow)  is an important process from  the industrial point of  view, which has not been modeled in the past. Therefore  the main objective of  the present work is to generalize the model of Ariawan et al. (2002a) to tube extrusion and to validate it using numerical simulations and experimental data. As will be evident later, the new model is capable of  predicting the processing behavior of  paste flow  during tube extrusion, e.g., the extrusion pressure as a function  of  shear rate and the geometrical characteristics of  the die. The organization of  this chapter is as follows.  First, the validity of  the RFH is examined by performing  flow  simulations based on the rheological constitutive model proposed by Patil et al. (2006) Then a mathematical model is derived for  the case of  an annular die based on the developments of  Snelling and Lontz (1960) and Ariawan et al. (2002a) Since the model involves the same material parameters as Ariawan et al.'s model, these are determined from  experimentally measured extrusion pressure for  rod extrusion. Model predictions of  the dependence of  extrusion pressure on the geometrical characteristics of  the die agree well with experimental data. Finally, a short summary of the results concludes the chapter. 6.2 Validation of  Radial Flow The radial flow  hypothesis assumes that the flow  is along the radial direction in the die (assuming a spherical system of  coordinates as in Figure 6.1a), and points located on virtual spherical surfaces  of  a constant radius r from  the die apex (Figure 6.1) have the same radial velocity (Ariawan et al., 2002a; Snelling and Lontz, 1960). The mathematical form  of  the RFH (Snelling and Lontz, 1960), for  a cylindrical die (Figure 6.1a) can be written as: where Q is the volumetric flow  rate and r is the distance from  the die apex. Based on this hypothesis, the kinematics of  PTFE flow  can be calculated at a given volumetric flow rate. Snelling and Lontz (1960) and Ariawan et al. (2002a) have found  experimentally that the pattern of  deformation  can be described adequately by the RFH in the conical zone of  a tapered cylindrical die (Figure 6.1a); there is no scientific  reason to believe that this would not be true for  an annular die. Significant  slippage exists in the tapered zone of the die (including in annular dies) and this contributes towards the validity of  the RFH. The validity of  the RFH is examined numerically by using the flow  model recently developed by Patil et al. (2006). These authors have proposed a rheological constitutive equation for  PTFE paste that takes into account the continuous change of  the microstructure during flow  through fibril  formation.  It consists of  a shear-thinning and a shear-thickening term with their relative contributions to the stress determined by a structural parameter £ : The structural parameter £ represents the fraction  of  the domains of  the paste that are fibrillated  and takes values of  0 and 1 for  the unfibrillated  and fully  fibrillated  cases respectively, y is the rate of  strain tensor, and r|, and r\2 are the shear-thinning and shear-thickening viscosities that are expressed by a Carreau model (Patil et al., 2006): dr dt 2tc(1 - c o s a ) r 2 2 ' [6.1] T = (1 - S)TI,Y + ^ 2 y [4.3] [6.2] where i = 1 refers  to shear-thinning (n, < 1) and i = 2 refers  to shear-thickening ( n 2 > 1). The values of  parameters r | o i , T|; and A; are the infinite  shear viscosity, the zero shear viscosity, the viscosity and a characteristic relaxation time respectively. h ok — DH H D,„ —H (a) (b) (c) Figure 6.1: Schematic illustration of  the "radial flow"  hypothesis. The hypothesis assumes the existence of  a virtual surface  of  radius r as measured from  the die apex, on which all paste particles moving towards the apex have the same velocity: (a) cylindrical die for  rod extrusion and (b) annular die with inside cylinder of  varying diameter (mandrel pin) for  tube extrusion, and (c) annular die with inside cylinder of  constant diameter (mandrel pin) for  tube extrusion. The evolution of  the structural parameter is described by a first-order  kinetic differential  equation: v-V^ = ayVy - P y S , [4.4] where first  and last term of  the right hand denotes the rate of  creation and breakage of fibrillated  domains in the paste, a and P are dimensionless rate constants for  fibril creation and breakage, both assumed to be 1 in our simulations; \\J  is the flow  type parameter; and y is the magnitude of  the strain rate tensor. The flow  type parameter, v|J  , indicates the relative strength of  straining and rotation in a mixed flow  (Dunlap and Leal, 1987; Fuller et al., 1980; Fuller and Leal, 1980). In the present work, finite  element simulations based on this constitutive model are used to validate the RFH inside the conical section of  the die during rod extrusion and tube extrusion (Figure 6.1). Patil et al. (2006) have shown that predictions of  this model agree very well with macroscopic experimental data of  extrusion pressure as a function  of  flow  rate (shear rate) and geometrical characteristics of  the die. Due to the presence of  lubricant in the paste, significant  slippage occurs at the die walls. This has been determined experimentally (Patil et al., 2006) by establishing a relationship between the slip velocity vs and the wall shear stress <7W using the Mooney analysis, vs = C c w (Mooney, 1931). The simulations in the present study are performed by using the parameters for  a paste studied by Ochoa and Hatzikiriakos (2004) and Patil et al. (2006), with C = 1.92 m/MPa s. All the other model parameters are listed in Table 4.2. Simulations are performed  for  three different  cases sketched in Figure 6.1: (a) cylindrical die, (b) annular die with an axisymmetric inside cylinder of  varying diameter that has the same apex as the outside cylindrical surface  and (c) annular die with an inside cylinder of  constant diameter. The geometry is Figure 6.1(b) is convenient for mathematical development, but that in Figure 6.1(c) is more common in applications. 6.2.1 Cylindrical dies The simulations are first  performed  for  cylindrical dies with entrance angles of  8°, 30°, 60° and 90° for  various values of  the apparent shear rate defined  as, y A =32Q/7iD^, where Q is the volumetric flow  rate and Da is the capillary diameter at the exit. The inlet and outlet diameters of  the conical section are Db = 9.52 x 10 -3 m and Da = 5.08 x 10 -4 m respectively. These are typical die dimensions used in experiments that are presented later. To demonstrate the validity of  the RFH, radial velocity profiles  are plotted versus angle 0 ( - a < 0 < a ) along the virtual peripheral surfaces  at constant radial positions from  the die apex. Figure 6.2 shows representative velocity profiles  at three different radial positions from  the die apex for  a die entrance angle of  30°. This indicates that the velocity variation from  the centerline to die wall is generally small, and in agreement with the RFH that implies flat  velocity profiles.  A small variation in the velocity profile occurs at the inlet to the die which is considered unimportant since the contribution of this part of  the flow  to the overall pressure drop is negligible. Figure 6.2: Velocity profiles  along the spherical surfaces  at radius r = 5.8 x 10 3 m (a), 1.16 x 10~2 m (b), and 1.54x10~2 m (c) for  cylindrical die (0 = 0 corresponds to the centerline). In our model, there is only one radial velocity component, which is computed in a cylindrical coordinate system. Figure 6.3 depicts the percentage variation of  velocity, defined  as the difference between velocity at the centerline and the die wall normalized by the centerline velocity, plotted against the die entrance angle at three radial locations (a), (b) and (c). The radial positions (a), (b) and (c) are given by r = R!sina , where R and a indicates corresponding cylindrical radius and die entrance angle (Fig. la). In Figures 6.3, 6.5 and 6.7 the same three cylindrical locations R = 1.5x10 -3 m, 3xl0~3 m and 4xl0~3 m are used, however, corresponding radial locations (a), (b) and (c) varies with die entrance angle. The variation of  the velocity profile  is only significant  for  (c) near the inlet for  dies of  high entrance angle. This variation decreases rapidly in the downstream direction and in fact  in the middle of  the die ((b) location) becomes insignificant.  At position (b), the velocity variation is 12 % for  dies having an entrance angle of  90°, and 2.1 % for  dies having an entrance angle of  60°. For die entrance angles 2a < 60° (typically used in paste extrusion) the variation is negligible and therefore  the RFH applies. 80 60 o o <D > 40 4— o c o ."I 20 03 > vO 0 0 20 40 60 80 100 120 140 Die entrance angle (2a) Figure 6.3: Percentage variation of  velocity normalized by the centerline velocity, from  the centerline of  the die to the die wall plotted with die entrance angle. The three surfaces  are defined  by the cylindrical radius R = 1.5x10 -3 m (a), 3 x 10"3 m (b) and 4xl0" 3 m (c). To study the effect  of  the apparent shear rate yA and the die reduction ratio RR = T>\/ D \ on the velocity variation along the virtual peripheral surfaces  at constant r from  the die apex, simulations are performed  for  a cylindrical die having an entrance angle of  2a = 30° and a reduction ratio RR = 352, for  apparent shear rate values ranging from  1875 s"1 to 8304 s"1. Near the outlet (position (a)), the percentage velocity variation normalized by the centerline velocity is found  to be 0.015 % and 0.026 % for  the apparent shear rates of  y A = 1875 s"1 and y A = 8304 s"1, respectively. Similarly at position (b), the normalized velocity variations are 0.23 % and 0.33 % for  the apparent shear rates of  y A = 1875 s"1 and y A = 8304 s"1, respectively. Simulations were also performed  for various die reduction ratios ranging from  56 to 567 at the apparent shear rate of  5869 s"1. The percentage velocity variations are 0.058 % and 0.02 % at position (a) for  RR = 56 and 567 respectively. At position (b), they are 0.8 % and 0.24 %. Therefore,  the RFH is more accurate at lower flow  rates and large reduction ratios. 6.2.2 Annular Die with Varying Diameter Mandrel Pin Simulations were also performed  for  annular dies with an axisymmetric inside surface  of varying diameter (mandrel pin) having the same apex with the outside cylindrical surface (Figure 6.1b). The existence of  a single apex produces a die geometry that allows the development of  an analytical flow  model (see section 3) in spherical coordinate (r and 0 define  the entire flow  field).  Simulations were performed  for  various die entrance angles ranging from  8° to 90° at various values of  apparent shear rate, j A . This is defined  as jA =48 Q/n(D a - Dp)2(D a + Dp) where Q is the volumetric flow  rate. The die and mandrel diameters at the inlet are Db = 9.5xlO - 3 m and D m = 3 x l 0 - 3 m respectively, and at the outlet are Da = 5.08xlO -4 m and Dp = l ^ x l O ^ m respectively (Figure 6.1b). The coincidence of  the apex requires D =Dm{D a/D h) . Figure 6.4 depicts representative velocity profiles  along the virtual peripheral surfaces  at three various radial positions from  the die apex for  a die entrance angle of  30° (see inset of  Figure 6.4). These are qualitatively similar to the profiles  in the cylindrical dies (Figure 6.2). Small variations in the velocity profiles  are obtained only at the inlet (c) and they are insignificant  to the overall pressure drop. Figure 6.5 shows the percentage variation of  velocity normalized by the velocity on the mandrel surface,  plotted against the die entrance angle at the same radial locations. E o _o <D > 10 15 Angle (0) Figure 6.4: Velocity profiles  along the spherical surfaces  at radius r = 5.8x10 3 m (a), 1.16x10 -2 m (b), and 1.54xl0"2 m (c) for  an annular die with inside cylinder of  varying diameter (9 = 15° corresponds to the outside wall, 0 = 0 ° does not exist due to the presence of  the internal mandrel pin). 40 60 80 100 Die entarnce angle (2a) 140 Figure 6.5: Percentage variation of  velocity from  the wall of  the varying diameter mandrel pin to the die wall normalized by the velocity at the wall of  the varying diameter cylinder, plotted with die entrance angle at three different  spherical locations (a), (b) and (c) defined  in Figure 6.3. The variation of  velocity is significant  only near the inlet (location (c)) and increases with die entrance angle. However, variations in velocity over the lower portion of  the die that contributes significantly  to the pressure drop are very small for  dies having die entrance angles up to 60° (typically used in extrusion operation). Therefore  the RFH can be used safely  for  dies up to entrance angles of  60°. 6.2.3 Annular Die with Axisymmetric Cylindrical Mandrel Pin Similar simulations were performed  for  an annular dies with an inside cylinder of constant diameter (mandrel pin) as depicted in Figure 6.1c. The die entrance angle ranges from  8° to 90°. The outer diameter of  inlet of  the conical section of  the die is Db = 9.5xl0~3 m, the outer diameter of  outlet is Da = 2xl0~3 m and the diameter of  the axisymmetric constant diameter cylinder is Dp = 1.9xlO -3 m (Figure 6.1c). Although the analytical mathematical model will be derived for  an annular die having a single apex (Figure 6.1b), it can still be used for  annular dies with cylindrical mandrel pins, once the RFH is proven for  this geometry. Figure 6.6 depicts representative velocity profiles  along the virtual peripheral surfaces  at three radial positions from  the die apex for  a die having an entrance angle of 30° (see inset of  Figure 6.6). The results are similar to those discussed before.  Small variation in the velocity profiles  are obtained only at the inlet to the die. Figure 6.7 shows the percentage variation of  velocity normalized by the centerline velocity on the mandrel surface,  plotted against the die entrance angle at the three radial locations. The variation of  velocity is significant  only near the inlet and only for  dies having a large entrance angle. The variation in velocity profile  over most of  the die is very small and increases up to 14.8 % for  a die entrance angle of  90°. This clearly indicates that the "radial flow" hypothesis can be used safely  for  annular dies having an entrance angle of  up to 60°. Comparing Figures 6.5 and 6.7, it seems that the RFH applies better in annular dies having mandrel pin of  constant diameter. The RFH concept has been adopted in deriving an analytical flow  model for  PTFE paste extrusion through annular dies and discussed in the next subsection. 10-: 10"; & o _o CD > 1 0 -4 10"! :2a = 30° RR = 352:1 YA  = 5869 s"1 : Db = 9.52E-3 m Da = 2E-3 m D = 1.93E-3 m !. n, J ' r (a) fCT  f (b) \ —1 1 1 . 1 • • 1 1 . 1 1 . . . . . . 10 15 Angle (9) 20 Figure 6.6: Velocity profiles  along the spherical surfaces  at radius r = 5.8 x 10 3 m (a), 1.16 x 10"2 m (b), and 1.54 x 10~2 m (c) for  an annular die having a mandrel pin of constant diameter. 100 80 o 0 60 0) > 4— o c: 40 o 1 ro 5 20 1 1 1 1 1 1 1 1 1 1 1 1 1 fl . 1 1 . 1 1 • 1 1 Tube extrusion RR = 352:1 Y A =  5869 s"1 1 1 1 1 UllSlZI^? • (CJ / -: \ (bi /[  -• / ( b ) v l ' / / D*R  Oulk-i I 1 1 1 1 1 1 1 1 1 1 1 1 • 1 1 1 1 1 1 1 20 40 60 80 100 Die entrance angle (2a) 120 140 Figure 6.7: Percentage variation of  velocity from  the wall of  the constant diameter mandrel pin to the die wall normalized by the velocity at the wall of  the constant diameter cylinder, plotted with die entrance angle at three spherical locations (a), (b) and (c) defined  in Figure 6.3. 6.3 Mathematical Model Now that the validity of  RFH has been established for  cylindrical and annular dies, an analytical model will be derived to describe annular flow  of  PTFE paste. It is based on the RFH and generalizes the earlier model of  Ariawan et al. (2002a) for  annular dies. 6.3.1 Annular Die without Die Land (L/Da = 0) Consider first  an annular die without the cylindrical die land (Figure 6.1b). Figure 6.8b shows a volume element bounded by the spherical surfaces  of  radius r and (r  + dr)  as measured from  the virtual die apex, and by four  planes at the azimuthal locations of  0, 6+ dO,  (j)  and + dtj).  The RFH implies that this element will flow  towards the die apex, such that its bounding surfaces  remain parallel to those at its previous position. Since the element does not rotate or deviate from  its straight path, this also implies that the stresses acting on the element are purely normal stresses. In fact,  these stresses are principal stresses, with the radial direction and the directions normal to the four  bounding planes as the principal directions: a = 0 0 a 0 II V 0 CT a r 0 0 0 CTg 0 0 0 cta [6.3] It has been further  assumed that a 0 = cr^  to simplify  the mathematics. In reality, the squeezing in the 0 and (j) direction is comparable in magnitude if  not equal. The force balance on the volume element (see Figure 6.8b) in the radial direction gives rise to an equilibrium relation: - Irtr 2 (cos Q - cos a) da r - 4^r(cos Q - cos a)ar dr  + 4;rcrl9r(cos Q - cos a)dr  + Infer 0r{sm  Q + sin a)dr  = 0. [6.4] where Q = tan~ D, V D b \ -tan a and / is the coefficient  of  friction.  By letting B _ /(sin Q + sin a ) 2(cosQ - cos a ) and N/ = o> - o^ and rearranging, we obtain d°r 2 B o r =2{c 9-orXl  + B)_-2N l(\  + B)^ [6.5] dr  r r r The term N/ is similar to the first  normal stress difference  in polymer rheology, except that, in this case, it is for  a non-viscometric flow. - H D„ - • i h-Die (a) Mandre (r+tl rjdnTX^  /= (r + dr)sin 0d(j)\ \ A \ /(r+dOde/V Jr^ in Q' . X . / / (b) N. Figure 6.8: (a) Annular die with varying diameter mandrel pin with volume element and (b) its dimensions in the annular conical zone of  a tapered die according to "radial flow" hypothesis. In order to solve the above differential  equation, a relationship describing the first normal stress difference  for  the solid-liquid (paste) system in question is therefore, required. For an ideally plastic material, Saint-Venant's theory of  plastic flow  gives "N, = <jo at the incipience of  yielding, where cr0 is the initial yield stress of  the material (Saint-Venant, 1870). However, for  a completely plastic flow  to occur within an elasto-visco-plastic material, N] has to sufficiently  exceed a 0 so as to overcome the initial yield stress, the elastic stress and any viscous resistance that may develop during the flow (Hoffman  and Sachs, 1953; Chakrabarty, 1998). The generalized Newton's law for  viscous flow  states that a = r)k and Hooke's law of  elasticity establishes the relationship a = Es, where 77 and E are the viscosity coefficient  and the Young's modulus, respectively, and eand sare the logarithmic strain and strain rate tensors, respectively. Combining the two laws, the following  stress-strain relationship for  a visco-elastic material (Hoffman  and Sachs, 1953): a = Es + rje. [6.6] Using the above relation, the term Ge - crr adopts the form  of <Tg-<Tr=  E ( £ s ~£r)+ Vifg  ~ e r )> [6-7] where sg-sr= s„ - e, and eg-sr = s„ - s, are the maximum strain ymax and the maximum strain rate y i m x , respectively. The term maximum strain was introduced by Ludwik (Ludwik, 1909), who realized that N/  should be a unique function  of  ymm • Ludwik is also credited with the modified  Hooke's law expression that takes the final form  of  a power law equation a = Ce",  [6.8] where C is Young's modulus when n = 1. Due to the presence of  both the liquid and solid phases in the PTFE paste system, it is necessary to consider PTFE paste as an elasto-visco-plastic material. To model its flow,  the expression suggested by Snelling and Lontz (1960) is adopted, which is essentially the Kelvin stress-strain relation Eq. (6.6), with modifications  that are similarly employed in the Ludwik power law model Eq. (6.8) for  the elastic (strain hardening) term, and the generalized power law model for  the viscous resistance term. The resulting expression for  the first  normal stress difference  N/ is then written as follows: V e - V , = C r m J + v y m j " • • [6.9] A more general three-dimensional form  of  Eq. (6.9) can be written by considering the general model for  an elastic solid, such as that used by Rivlin (Rivlin, 1948a; Rivlin, 1956), and for  a viscous fluid,  in terms the invariant functions  of  the strain and strain rate tensors, respectively (Macosko, 1994). However, the objective here is to derive a simple analytical flow  model to be compared with macroscopic extrusion pressure measurements. To account for  the initial yield stress, an additional term may be included on the right hand side of  Eq. (6.9). However, this term is expected to be negligible compared to the other terms, as indicated by the fact  that the initial strength of  the preform  is much weaker than that of  the extrudate. Now, it can be shown that the volume element at a distance rb from  the virtual die apex experiences a maximum strain of ymax ~ £ e ~ £ r = , 3 f dr rb r r, = -31n —. [6.10] r The maximum strain rate can then be expressed as y _ drmax _ , dr The "radial flow"  hypothesis for  annular conical dies of  single apex can be written as: dr  Volumetric  Flow  rate Q [6.12] dt  Area of  Surface  27i(cosQ-cosa)r 2 Hence, the maximum strain rate is (see Eq. 6.11) . 3Q ymax — ^ / „ \ 3 ' [6.13] 2;r(cosi2-cosaJr and the normal stress difference  is: (  r.. N,  =CT,  -CT f l =C 3 In V  V  r J  J + T| 32 2n(cosQ-cosa ) r 3 [6.14] Substituting the above into Eq. (6.5) yields the following  expression: 2B+\ r 3 q m f  1 ^ (3 m + 2 B) 2;r(cosf2-cosa) 3m+2B \r  J • + r2I iC, [6.15] where the constant of  integration, C, is evaluated using the boundary condition a r = a r when r = r : 32 (31 n(r„/r))" ,.2B+l dr r \ _ J 3m+2« [6.16] - (3m + 2B) ^ 2;r(cos Q - cos a) The extrusion pressure can then be calculated using Eqs. (6.5) and (6.9) with r = rh, i.e. PE. -arh=araRR" +2{l  +  Bic , D b 2sina 2B+1 12Qsin3a [6.17] (3m  + 2B) ^ 7t(cosn - cosa)D3h j where crra is the stress at the die exit, RR is the reduction ratio of  larger to smaller cross section area of  annular conical section inlet and outlet respectively, defined as(D2 - D ^ ) , and C, 7, n, m, and / are material constants that have to be determined experimentally. When an orifice  die is used, a r a may be present at the die exit due to a pulling force  during extrudate wind-up or calendering. However, crra is typically negligible and the expression for  extrusion pressure can then be simplified  to = 2(1+ 2?UC DL V 2 B T| 2sin a J 12£>sin3 a , 2B+1 \m (3m + 2 B ) y 7r(cos Q - cos a ) D , 3 h RR 4 [6.18] Numerical integration is required in Eq. (6.18). However, for  a range of  the die reduction ratio of  interest, the following  approximation can be used with reasonable accuracy in Eq. (6.18), allowing an analytical solution to be obtained: where a and b are constant fitting  parameters. 6.3.2 Annular Die with Die Land (L/Da * 0) Additional pressure drop in the die land can be computed using a similar force  balance. The forces  acting on a volume element in the capillary zone is shown in Figure 6.9. A force  balance on the element yields In(rt/r)»  a(jyr)b . [6.19] (p 2a-D2p) [pl  ~ D2p) . _ {a z + dc T 2)n± -azn± - f<r r.n\Da +Dp)dz  , [6.20] b c Conical Zone Capillary Zone fo r7tD dz z = o fa  ,tiD dZ < r 3 da^ ji(DJ - Dp)/4 z = L o" D Figure 6.9: Force balance on volume element in the die capillary zone. or da z 4 /ov {D a + Dp) = 41(D 0 + Dp )f{N i + <x2) [Dl-D]) dz [Dl-D]) [6.21] where Ni is the previously defined  first  normal stress difference,  which is expected to be significant  due to the elastic nature of  PTFE paste. At the end of  the die conical zone (hence, at the entrance of  the die capillary zone), N/  can be calculated using Eq. (6.14) with r = ra. Assuming N/  to be approximately constant throughout the capillary zone of the die, the force  balance becomes d<7  *(D a+Dp)f{N Xa +az) dz [Dl-Dl) [6.22] where •N,=C In {R) + 7] \2Qsm3 aR 3/2 ; r (cosQ-cosa) l ) b J [6.23] Solving Eq. (6.21) and applying the boundary condition crz = azL at z = L, yields G = N i (g4A'-i-iD.+Dj/irt-Dl)  _ j J + a t e4/(z-A)(/J„+/J„)/(/J,;-/J,2,)^  where <J z1 is the stress imposed at the exit of  the die, which is typically negligible or zero. The stress present at the entrance of  the die capillary zone, crzo, is obtained from  Eq. (6.23) with z = 0: )U[D-Di [6.24] By putting 8 = 1 - Dp /  Da in Eq. (24), azo can be written as: o n = N l a ( e ^ J L " D ' - \ ) + o l L e ^ " D ' , [6.25] The total extrusion pressure can be obtained by substituting crra = -crz0 into Eq. (6.16). 6.4 Model Predictions and Comparison with Experiments In this section the dependence of  the extrusion pressure on the apparent shear rate, the die entrance angle ( 2 a ) , and the die reduction ratio (RR) is predicted by using the proposed model (Eq. 6.17), for  cylindrical (Figure 6.1a) and annular dies (Figure 6.1b) with no die land section. Extrusion experiments were performed  with cylindrical dies using two PTFE fine  powder resins supplied by Solvay Solexis of  primary and secondary particle diameters of  0.25 urn and 450-550 fim  respectively and standard specific  gravity of 2.16. The paste was prepared by mixing resins with isoparaffinic  liquid as lubricant ' (Isopar® H) supplied by ExxonMobil Chemicals with properties listed in Table 4.1. The two resins have different  molecular weights and are labelled as resins A and resin B in Table 6.1. The material parameters C, n, rj, m and/in Eq. (6.17) are evaluated by fitting  a single set of  experimental data for  resin A and B in a cylindrical die (see Figure 6.10) (Ariawan et al., 2002a). The dimensions of  the cylindrical die are: Db =9 .5x10 ' 3 m, Da = 5X10"4 m and the die entrance angle was 2a = 60°. The model parameters C, n, rj, m and/are determined by nonlinear dynamic optimization using a Gauss-Newton iterative algorithm that minimizes the difference  between model predictions of  the extrusion pressure and the measured values. The standard deviations for  all these parameters were below 5%. The fitted  values of  the parameters are listed in Table 6.1 for  resins A and resin B. Since these are material parameters independent of  the die geometry, they can be used in predicting extrusion pressure for  cylindrical and annular dies of  different geometry, as long as the "radial flow"  hypothesis is valid. Note the small value of  the friction  factor,  / which implies that the pressure drop in the die land is much smaller compared to that in the conical zone. Typically the pressure drop in the die land can account for  about 5% of  the total pressure drop for  short dies (L/Da=5) to 30% for  long dies (L/Da=40) depending on the type of  resin. Table 6.1: Values of  material constants and coefficient  of  friction  needed in Eq. (6.22) to predict the extrusion pressure for  paste slow in cylindrical and annular dies. Resin C (MPa) n 11 (MPa.s) m f Resin A 1.14x10"' 2.28 1.25 xl0~3 1.21 U x l O " 4 Resin B 8.92 xlO"2 2.13 3.56 xlO"3 1.11 1.12x10 -4 Figure 6.10 plots the extrusion pressure for  resin A and B in rod extrusion. The solid lines indicate model prediction using parameters determined by best fitting  of  the experimental data. The steady-state extrusion pressure generally increases with increase of  the apparent shear rate. Although resin A has a lower viscosity than resin B, it has a larger elastic constant, and the strain-hardening effect  leads to a higher pressure drop than resin B. The dotted lines indicate model predictions for  tube extrusion using the same model parameters. As expected, the extrusion pressure for  annular dies is higher than for cylindrical dies under comparable conditions, owing to the presence of  the additional inside cylinder wall. 80 CD CL ^ 6 0 CD 13 C/5 V) CD i CL c o •So 40 3 20 2000 4000 6000 8000 10000 Apparent Shear Rate, 7A  (S"1) Figure 6.10: The effect  of  apparent shear rate on the extrusion pressure of  PTFE paste for a cylindrical (rod extrusion) and an annular conical die (tube extrusion). The experimental data refer  to rod extrusion using an orifice  die (L/D=0), having RR=352, 2 a =60°. Once the material parameters are determined, the model can be used to predict the effects  of  die geometry on extrusion pressure. Figure 6.11 shows the model predictions for  the dependence of  extrusion pressure on the die reduction ratio for  cylindrical die and annular die with varying diameter mandrel pin (Figure 6.1b) using the parameters for resin A. The reduction ratio of  the die is increased by decreasing the small diameter Da for  the cylindrical die, and both Da and Dp for  the annular die. The nonlinear dependence of  extrusion pressure on the die reduction ratio is clearly seen. A • Experimental (Resin A) Experimental (Resin B) Model (Resin A) Model (Resin B) Model (Resin A) Model (Resin B) RR = 352:1 2a = 60° L/D = 0 • / Tube Extrusion . : Ar"A Rod Extrusion a— i • • — • — Reduction Ratio (RR) Figure 6.11: The effect  of  die reduction ratio on the extrusion pressure of  PTFE paste for a cylindrical (rod extrusion) and an annular die (tube extrusion). Figure 6.12 depicts the model prediction for  the dependence of  the extrusion pressure on the die entrance angle for  cylindrical and annular dies. It can be seen that the extrusion pressure initially decreases, and goes through a minimum until it again increases with increase of  the die entrance angle. In both cylindrical and annular dies, the minimum extrusion pressure is required for  a die with die entrance angle of  around 8°. This value depends on the value of  the material parameters. To further  test the validity of  the proposed mathematical model, experiments were performed  using resin B in an annular die with a cylindrical die land attached. The annular die has an exit diameter of  Da = 6.48 xlCT3 m, a mandrel pin of  diameter D p = 4.7xl0~3 m, a die entrance angle of  2a = 180°, a length to diameter ratio L/Da = 35, and a reduction ratio of  (d 2 - D 2 ) / (D 2 - D 2 ) = 35. Details for  the experimental procedure can be found  in previous publications (Ariawan et al., 2002a; Ochoa and Hatzikiriakos, 2004; Ariawan et al., 2001; Ochoa and Hatzikiriakos, 2005). Figure 6.13 compares the measured steady-state extrusion pressure as a function  of  the apparent shear rate with model predictions using the fitted  values of the various parameters listed in Table 6.1. The agreement between the two is excellent, and indicates that our model is capable of  accurate description of  paste extrusion for  both cylindrical and annular dies. CL (U w w 0 c o '(/) UJ 90 80 70 60 50 40 30 20 Resin A RR = 352:1 L/D = 0 7a = 5869 s"1 Rod Extrusion Tube Extrusion 20 40 60 80 100 Die entrance angle (2a) 120 140 Figure 6.12: The effect  of  die entrance angle (2a) on the extrusion pressure of  PTFE paste for  a cylindrical (rod extrusion) and an annular die (tube extrusion). Figure 6.13: The effect  of  apparent shear rate on the extrusion pressure of  PTFE (resin B) paste for  an annular die (tube extrusion). 6.5 Conclusions Numerical simulations were performed  for  conical and annular dies by using a combined shear thinning and shear thickening rheological constitutive model proposed by Patil et al. (2006) to study the validity of  the RFH during PTFE paste flow.  The numerical results have shown that the "radial flow"  hypothesis is valid for  both cylindrical dies and annular dies having a contraction angle up to 60°. Based on these findings,  a simple flow  model is developed to predict the dependence of  extrusion pressure on the extrusion speed (apparent shear rate) in annular dies. The model considers the paste as an elasto-visco-plastic material that exhibit both strain hardening and viscous resistance effects  during flow.  Comparison with limited experimental data from  both cylindrical and annular dies was found  to validate the usefulness  of  this analytical and approximate but simple model. The model successfully  predicts the dependence of extrusion pressure on apparent shear rate for  tube extrusion using model parameters determined by fitting  data for  rod extrusion. The model was also used to predict the dependence of  extrusion pressure on die geometrical parameters. A final  comment relates to the limitations of  the model. While the extrusion pressure can be predicted well as a function  of  the operating parameters and the geometrical characteristics of  the dies, the material's structure (fraction  of  fibrillated domains), is not explicitly calculated. This limitation is the subject of  a future  study and it would be ideal to have an analytical model that can relate flow  kinematics and structure with the mechanical properties of  the final  extrudates as in the numerical flow  model of Patil et al. (2006) 6.6 Bibliography Ariawan, A. B., S. Ebnesajjad and S.G. Hatzikiriakos, Preforming  Behavior of  PTFE Pastes, Powder Technology 121, 249-258 (2001). Ariawan, A. B., Ebnesajjad, S. and Hatzikiriakos, S. G. Paste extrusion of polytetrafluoroethylene  (PTFE)  fine  powder  resins. Can. Chem. Eng. J., 80, 1153-1165 (2002a). Basterfield,  R. A., Lawrence, C. J., and Adams, M. J., On the interpretation  of  orifice extrusion data  for  visco-plastic materials,  Chem. Eng. Sci. 60, 2599-2607 (2005). Benbow, J. J., Bridgwater, J., The  Influence  of  Formulation  on Extrudate  Structure  and Strength,  Chem. Eng. Sci. 42, 735-766 (1987). Benbow, J. J., Oxley, E. W., and Bridgwater, J. The  extrusion mechanics of  pastes-The influence  of  paste formulation  on extrusion parameters.  Chem. Eng. Sci., 42 (9), 2151-2162,(1987). Benbow, J. J., and J. Bridgwater, Paste Flow  and  Extrusion,  Oxford  University Press, Oxford,  1993. Chakrabarty, J., Theory  of  Plasticity,  Singapore: McGraw-Hill Book Co., 1998. Davis E. A., and Dukos, J., Theory  of  Wire  Drawing, J. Appl. Mech. 11, 193-198 (1994). Dealy, J. M., and K. F. Wissbrun, Melt  Rheology and  its Role in Plastics  Processing -Theory  and  Applications,  Van Nostrand Reinhold, New York, 1990. Dunlap P. N., and L. G. Leal, Dilute polystyrene  solutions in extensional flows birefringence  and  flow  modification,  J. Non-Newt. Fluid Mech. 23, 5-48 (1987). Ebnesajjad Sina, Fluoroplastics,  Vol  1 Non-Melt  Processible  Fluoroplastics,  Plastic Desgin Library. William Andrew Corp, NY, 2000. Fuller, G. G., Rallison, J. M., Schmidt, R. L. and Leal, L. G., The  measurements of velocity gradients  in laminar flow  by homodyne  light-scattering  spectroscopy, J. Fluid. Mech. 100(3), 555-575 (1980). Fuller, G. G., and Leal, L. G., Flow  birefringence  of  dilute  polymer solutions in two-dimensional  flows,  Rheol. Acta 19, 580-600 (1980). Hoffman  O, and Sachs G., Introduction  to the theory of  Plasticity  for  Engineers,  New York: McGraw-Hill Company, 1953. Larson, R., Constitutive  Equations for  Polymer Melts  and  Solutions,  Boston: Butterworths, 1998. Ludwik, P., Elemente  der  Technologischen  Mechanik,  Berlin: Springer-Verlag, 1909. Macosko, C. S., Rheology principles, measurements, and  applications,  VCH publishers, Inc., New York, 244-247 (1994). Mazur, S., Paste Extrusion  of  Poly(tetrafluoroehtylene)  Fine  Powders  in Polymer Powder technology,441-481, Narks, M., Rosenzweig, N., Ed. John Wiley & Sons, 1995. Mooney M., Explicit  Formulas  for  Slip  and  Fluidity,  J. Rheol. 2, 210-222 (1931). Ochoa, I., Hatzikiriakos, S. G. Polytetrafluoroethylene  (PTFE)  paste performing: Viscosity  and  surface  tension effects.  Powder Technology, 146(1-2), 73-83 (2004). Ochoa, I., Hatzikiriakos, S. G. Paste extrusion of  polytetrafluoroethylene  (PTFE): Surface  tension and  viscosity effects..  Powder Technology (2005), 153(2), 108-118. Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, Constitutive  modeling  and  flow simulation of  polytetrafluoroethylene  (PTFE)  paste extrusion, J. Non-Newt. Fluid Mech. 139, 44-53 (2006a). Rivl in, R. S., Large elastic deformations  of  isotropic materials,  Philos. Trans. R. Soc. London, Ser. A 240, 459-490 (1948a). Rivlin, R. S., Large elastic deformations,  in Rheology,  New York: Academic Press, 1956. Saint-Venant, B., Memoire  sur I'etablissement  des  Equations Differentielles  des Mouvements  Interieurs  Operes dans  les Corps  Solides  Ductiles,  Compt. Rend. Acad. Sci. Paris. 70, 473-484 (1870). Snelling, G. R., and J. F. Lontz, Mechanism  of  Lubricant-Extrusion  of  Teflon®  TFE-Tetrafluoroethylene  Resins, J. Appl. Polym. Sci. Ill, 257-265 (1960). CHAPTER 7 Conclusions, Recommendations and Contribution to the Knowledge 7.1 Introduction In this work, a mathematical model to simulate the processing behavior of polytetrafluoroethylene  (PTFE) pastes was developed. Rheological experiments were also performed  to obtain the rheological parameters used in the constitutive modeling. The dependence of  the extrusion pressure on the operating parameters and the geometrical characteristics of  the extrusion die were studied. Mechanical properties of  the final  extrudates were related to the predicted level of  fibrillation  observations. A simple semi-empirical flow  model was also developed to predict the dependence of  extrusion pressure on the extrusion speed (apparent shear rate) in annular dies. 7.2 Conclusions First, a rheological constitutive equation was proposed based on the assumption that the PTFE paste behaves as a shear-thinning fluid  before  the occurrence of fibrillation.  Gradual increase in fibrillation  during flow  turns the paste into a shear-thickening fluid.  Thus, the total stress is expressed as a function  of  a shear-thinning term and a shear-thickening term premultiplied by the structural parameter, . This structural parameter, H, , represents the mass fraction  of  the paste which is fibrillated.  An evolution equation has been developed for  based on the kinetic network theories. This captures the formation,  evolution and breakage of  fibrils.  The relation between the slip velocity and wall stress was determined by performing  several capillary extrusion experiments (Mooney, 1931). This was used as a wall boundary condition in the finite  element simulation. Finite element simulations of  PTFE paste extrusion were presented in order to predict the dependence of  extrusion pressure on apparent shear rate, die reduction ratio, die L/D ratio and die entrance angle. Simulation results were found  to be in excellent agreement with the experimental findings  reported by Ochoa and Hatzikiriakos (2004). Based on this agreement it can be concluded that the proposed constitutive equation is suitable for  modeling the flow  behaviour of  the paste. The average exit structural parameter, was related to the tensile strength of  the pastes. The weakness of  the STT model was that its parameters were freely  fitting  parameters obtained by iteratively reducing the error between the simulated and experimental dependence of  extrusion pressure on apparent shear rate y A and geometrical characteristics of  the die. In chaper 5, a fully  predictive flow  model was developed where PTFE paste was treated as a viscoelastic fluid.  First the PTFE paste has been treated as a shear-thinning fluid  before  the occurrence of  fibrillation.  The formation  of  fibrils  gradually turn the paste to exhibit more strain-hardening behaviour, and this elastic behaviour was captured through a hyperelastic modified  Mooney-Rivlin model. Change in the nature of  the paste from  a fluidlike  (shear-thinning) behaviour to a solidlike (strain-hardening) one, was implemented by the introduction of  a microscopic structural parameter, . The model predictions were compared with the experimental results reported by Ochoa and Hatzikiriakos (2004) as well as with the simulation results from  STT model. The viscoelastic model proposed in this work is superior to the STT model because it is fully predictive and describes the physics of  the process quite well. The predicted effects  of  the die geometrical parameter and operating condition on the £,exit are generally in agreement with the observed ones on the tensile strength. The flow  inside a conical die can be modeled by using the "radial flow" hypothesis (RFH) (Snelling and Lontz, 1960). The validity of  the RFH was confirmed  by comparing radial velocity profiles  generated from  RFH with that the STT model proposed by Patil et al., (2006). The numerical results have shown that the RFH is valid for  both cylindrical dies and annular dies having a contraction angle of  up to 60°. Based on these findings,  a simple flow  model was developed to predict the dependence of extrusion pressure on the extrusion speed (apparent shear rate) in annular dies. The model considers the paste as an elasto-visco-plastic material that exhibits both strain hardening and viscous resistance effects  during flow.  Comparison with limited experimental data from  both cylindrical and annular dies was found  to validate the usefulness  of  this analytical, approximate model which is simple to use. The model successfully  predicts the dependence of  extrusion pressure on the apparent shear rate for  tube extrusion using model parameters determined by fitting  data for  rod extrusion. The model was also used to predict the dependence of  extrusion pressure on die geometrical parameters. 7.3 Contributions to Knowledge Several contributions to knowledge have resulted from  this research work. These are identified  as follows. 1. A simple shear-thinning and shear-thickening model was developed, able to capture PTFE paste flow  quite well. The phenomenon of  fibrillation  was never considered before  in the modelling of  PTFE paste flow.  However, the current model captures the fibrillation  mechanism and its effect  on the rheology of  PTFE paste during extrusion. 2. The slip behavior of  the PTFE paste in capillary dies was studied and it was related experimentally to the wall shear stress. 3. A kinetic equation for  the mechanism and dynamics of  fibrillation  was developed for  the first  time and it was demonstrated that it describes the process well. 4. The proposed viscoelastic model in this work describes the rheological behavior of  the paste through a shear-thinning and strain-hardening term in the total stress tensor premultiplied by . Also the model parameters used in this model are obtained from shear and extensional rheometrical testing. The flow  model developed was able to capture the processing behavior of  PTFE paste through the dependence of  extrusion pressure on operating conditions and die geometrical characteristics. 5. An approximate analytical model was developed based on the "radial flow hypothesis" (RFH) to predict the extrusion pressure in cylindrical and annular dies.The model considers the paste as an elasto-visco-plastic material that exhibit both strain hardening and viscous resistance effects  during flow.  Comparison with limited experimental data from  both cylindrical and annular dies was found  to validate the usefulness  of  this analytical and approximate but simple model. Overall this work has contributed to the understanding of  modeling aspects of  the PTFE paste extrusion. Undoubtedly, more in-depth modelling still needs to be performed in the future  in order to completely unravel the complexities of  the process. However, many of  the findings  in this work have provided the foundation  towards performing properly the microscopic modelling of  the process. The results from  the present study can be utilized in industrial applications i.e to design extrusion dies. Finally, many of  the modeling techniques employed here are novel, and can be used in other studies involving modeling of  viscoelsatic fluid  influenced  by the inherent microscopic structure developed during processing. 7.4 Recommendations for  Future Work Several important aspects of  PTFE paste extrusion are yet to be studied. These are recommended below, as possible objectives for  future  research work. 1. In the present study it was assumed that all the PTFE particles remain spherical during the processing. However, due to high shear and elongational flow  field  inside the converging die, particles do change their shape from  spherical to ellipsoidal. This needs to be taken into account in the modelling 2. The fact  that all the fibrils  are oriented in the direction of  flow,  can be modeled by treating single fibril  as a vector and then mathematically expressing the convection and diffusion  of  vectors by using the well known Fokker-Planck equation. 3. An approximate analytical model proposed in the present work to predict the processing behavior of  PTFE, can be modified  by incorporating the structural parameter in the total stress. The kinetic equation for  the evolution of  can also be integrated exactly to result an analytical expression. 4. In order to make the rheological study of  PTFE pastes more complete, the effects of  other variables, such as resin particle size and its distribution should be investigated in the future.  It is noted that the effect  of  particle size needs to be considered in the modeling. 5. The flow  of  PTFE pastes through a hyperbolic die and a crosshead die for  a wire coating process are also interesting and commercially useful.  The modelling study in the present work can be extended to predict the processing behavior of  PTFE through these dies. 6. The viscoelastic model proposed in the present study is not PTFE paste specific.  It can be extended to model other fluids  that show inherent microstructural development during processing their processing. 7.5 Bibliography Mooney, M., Explicit  Formulas  for  Slip  and  Fluidity,  J. Rheol. 2, 210-222 (1931). Ochoa, I., Hatzikiriakos, S. G. Polytetrafluoroethylene  (PTFE)  paste performing: Viscosity  and  surface  tension effects.  Powder Technology, 146(1-2), 73-83 (2004). Patil, P. D., J. J. Feng, and S. G. Hatzikiriakos, Constitutive  modeling  and  flow simulation of  polytetrafluoroethylene  (PTFE)  paste extrusion, J. Non-Newt. Fluid Mech. 139, 44-53 (2006). Snelling, G. R., and J. F. Lontz, Mechanism  of  Lubricant-Extrusion  of  Teflon®  TFE-Tetrafluoroethylene  Resins, J. Appl. Polym. Sci. Ill, 257-265 (1960). 

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