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Effects of molecular structure on the rheology and processability of high density polyethylene blow molding… Ariawan, Alfonsius Budi 1998

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EFFECTS OF M O L E C U L A R STRUCTURE ON THE RHEOLOGY AND PROCESSABILITY OF HIGH DENSITY POLYETHYLENE BLOW MOLDING RESINS by A L F O N S I U S B U D I A R I A W A N B . A . S c , The University o f British Columbia, 1996 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R OF A P P L I E D S C I E N C E in T H E F A C U L T Y O F G R A D U A T E S T U D I E S Department of Chemical Engineering We accept this thesis as conforming to the required standard T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A June 1998 © Alfonsius Bud i Ariawan, 1998 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of &lr\emic«.| £ N 6 / W & & P > J 6 The University of British Columbia Vancouver, Canada Date Qu l y C ' DE-6 (2/88) ABSTRACT Resin processability depends heavily on its Theological properties. The molecular structure o f the resin, in turn, influences its rheological behavior. In this work, experiments were conducted using capillary and extensional rheometers, a melt indexer and a blow molder unit to determine the rheological properties and processability o f high density polyethylene blow molding resins. Twenty four commercial resins were analyzed in terms o f their shear flow properties, extensional flow properties, extrudate swell characteristics, and melt strength. The studied samples had varying molecular weight characteristics and were produced using a variety o f technologies. Using the experimental results, correlations between rheological properties and molecular structures were determined. Furthermore, to assess resin processability, pi l low mold (blow molding) experiments were performed. The implications o f rheology on processability (parison sag and weight swell) were then discussed. Additional experiments were also conducted to assess the usefulness of melt index (MI), stress exponent (S.Ex.) and melt flow ratio (MFR) in characterizing rheological properties. It was found that shear viscosity is technology dependent and that it is influenced by the weight average molecular weight (Mw) and polydispersity index (PI). Increasing Mw was found to increase the shear viscosity, while increasing PI by increasing the concentration o f smaller molecules increases the tendency o f the resin to shear thin. The extensional viscosity was also affected by Mw in the same manner. The influence o f PI on extensional viscosity, however, was not apparent. In order to relate the melt strength and temperature sensitivity o f shear viscosity to molecular parameters, resins had to be ii grouped according to the polydispersity index ranges o f PK8, 8<PI<10, and PI>10. Moreover, it was possible to relate melt strength to the Hencky strain obtained from creep experiments. With regard to extrudate swell, it was found that the Z-average molecular weight (Mi) and PI are useful for determining the sensitivity o f the swell to changes in shear rate. Extrudate swell behavior and melt strength are important parameters to be considered during parison formation, as observed during blow molding experiments. Finally, MI, S.Ex., and MFR were found to be technology dependent and are useful only for resin comparisons. i i i T A B L E O F C O N T E N T S Page A B S T R A C T i i L I S T O F F I G U R E S v i L I S T O F T A B L E S x i A C K N O W L E D G E M E N T S x i i 1 I N T R O D U C T I O N 1 1.1 Introduction 1 1.2 Background 2 1.2.1 Polymers 2 1.2.2 Polymer Rheology 8 1.2.3 The Process of B low Molding 18 1.3 Thesis Objectives 23 2 L I T E R A T U R E R E V I E W 24 2.1 Introduction 24 2.2 Rheology 24 2.3 Processability 28 3 E X P E R I M E N T A L E Q U I P M E N T A N D P R O C E D U R E S 32 3.1 Introduction 32 3.2 Experimental Equipment 32 3.2.1 Densimeter 32 3.2.2 Extrusion Plastometer 33 3.2.3 Capillary Rheometer 35 3.2.4 Extensional Rheometer 41 3.2.5 B low Molding Machine 48 3.3 Experimental Samples 50 3.4 Experimental Procedure 51 3.4.1 Shear Properties 51 3.4.2 Extrudate Swell Measurements 53 3.4.3 Melt Index, Stress Exponent and Melt Flow Ratio Determinations 54 3.4.4 Melt Strength Measurements 55 iv 4 R E S U L T S A N D D I S C U S S I O N 62 4.1 Introduction 62 4.2 Rheology 64 4.2.1 Shear Properties 64 4.2.2 Extensional F low Properties 83 4.2.3 Extrudate Swell Characteristics 91 4.3 Processability 101 4.3.1 Melt Strength 101 4.3.2 Sagging and Weight Swell Characteristics 108 4.4 Mel t Index, Stress Exponent, and Mel t F low Ratio 118 4.5 Implications o f Rheological Behaviour on Processability 125 5 C O N C L U S I O N S 130 6 R E C O M M E N D A T I O N S 136 R E F E R E N C E S 138 N O T A T I O N 142 A P P E N D I X A : Time Temperature Superposition Program Code 145 A P P E N D I X B : G P C Analysis Program Code 168 V LIST OF FIGURES Page Figure 1.1 Differential molecular weight distribution for most probable (solid) and log normal (dashed) molecular weight distribution, both with MJMn=2 5 Figure 1.2 Chemical and molecular structures of (a) L o w Density Polyethylene ( L D P E ) , (b) Linear L o w Density Polyethylene ( L L D P E ) , and (c) High Density Polyethylene (HDPE) 8 Figure 1.3 Analogs o f viscoelastic materials: (a) Voigt model, (b) Maxwel l model... 10 Figure 1.4 A schematic diagram of simple shear experiment 12 Figure 1.5 A schematic diagram of simple (uniaxial) extensional experiment 12 Figure 1.6 Time temperature superposition of complex viscosity curves obtained at different temperatures: (a) before superposition, (b) after superposition, (c) the fit o f shift factors to the Arrhenius equation 16 Figure 1.7 Overview of blow molding process 18 Figure 1.8 Parison inflation in the process o f blow molding 18 Figure 1.9 The controlling of parison wall thickness by adjustment o f mandrel position 19 Figure 1.10 Extrudate swell behavior for (a) Newtonian fluid, and (b) polymer melt 21 Figure 2.1 Illustration of parison diameter swell 31 Figure 3.1 A schematic diagram of the Toyoseiki Automatic Densimeter, Mode l D-H100 33 Figure 3.2 A Schematic diagram of the Tinius Olsen Manually Timed Extrusion plastometer 34 Figure 3.3 Pressure profile for a flow in a capillary 39 Figure3.4 A typ i ca lBag leyp lo t 40 Figure 3.5 A schematic diagram of Kayness Capillary Rheometer and die 41 Figure 3.6 Uniaxial or simple extension 42 vi Figure 3.7 A schematic diagram of Rheometric RER-9000 Extensional Rheometer.. 46 Figure 3.8 (a) Molding and (b) gluing accessories for Rheometric RER-9000 Extensional rheometer 47 Figure 3.9 A Schematic diagram o f the sample cutter supplied with RER-9000 extensional Rheometer 48 Figure 3.10 A schematic Diagram of r M P C O B-13 B l o w Molder 49 Figure 3.11 A schematic diagram of the pi l low mold used in the blow molding Experiment 49 Figure 3.12 Differential molecular weight distributions for some o f the resins studied in this work 51 Figure 3.13 Mel t strength measurement using the dead weight method. The value o f melt strength is interpolated from the graph at time equal to 3 minutes 57 Figure 3.14 A schematic diagram of extensional sample before and after cutting 59 Figure 4.1 Reproducibility o f apparent flow curve for resin R, determined at 180°C, 200°C, and 220°C. Data variation at each shear rate is estimated to be less than 5% 64 Figure 4.2 Apparent flow curves for resins with similar polydispersities determined a t 2 0 0 ° C 65 Figure 4.3 Apparent flow curves for resins with similar Mw determined at 200°C 66 Figure 4.4 Predicted apparent viscosity values at 100 s"1 and 200°C as determined using S T A T G R A P H I C S v 2 . 0 . Only resins manufactured using technology 'a' are included in the analysis 68 Figure 4.5 Predicted and observed apparent shear viscosity at 100 s'1 and 200°C 68 Figure 4.6 Apparent shear viscosity curved simulated at constant PI using the regression relationship at each shear rate 70 Figure 4.7 Apparent shear viscosity curved simulated at constant Mw using the regression relationship at each shear rate 70 Figure 4.8 Determination of molecular weight ranges that are critically affecting a certain property of a resin. Slices were made arbitrarily 72 vii Figure 4.9 Correlation coefficients relating various molecular weight ranges to shear viscosity at 5 s"1 and 200°C (a) negative correlation (b) positive correlation. Correlation coefficient (x,y) = COV(x,y)/ax.ay 73 Figure 4.10 (a) Master curve and (b) Arrhenius fit generated by the F O R T R A N program 75 Figure 4.11 Observed and predicted Ea values as obtained from S T A T G R A P H I C S v2.0 (PI>10) 77 Figure 4.12 The effect of PI, Mw, and Mz on temperature sensitivity of shear flow properties. PI and Mw were arbitrarily set to be constant in (a) and (b), respectively 78 Figure 4.13 Hypothetical M W D showing the shift in M„,MW, a n d M z at constant PI. 80 Figure 4.14 Observed and predicted Ea values as obtained from S T A T G R A P H I C S v2.0(8<PI<10) 81 Figure 4.15 Hencky strain as a function of time determined at different stress levels 83 Figure 4.16 The effect o f Mw on Hencky strain 84 Figure 4.17 The effect o f M w on tensile viscosity 85 Figure 4.18 Differential molecular weight distribution for resins E , F , and N 86 Figure 4.19 Effect ofMw on tensile viscosity 86 Figure 4.20 Differential molecular weight distribution for resins K , L , S, H , and O . . 87 Figure 4.21 The effect o f PI on tensile viscosity 88 Figure 4.22 Differential molecular weight distribution for resins L , M , R, E , and C . 88 Figure 4.23 The effect o f PI on Hencky strain at different times (strain rates). A t shorter times, the effect of polydispersity is non-significant 90 Figure 4.24 The effect of large PI on Hencky strain 90 Figure 4.25 Reproducibility of extrudate swell data 92 Figure 4.26 Extrudate swell data for resins Q, J, and P having similar PI 93 Figure 4.27 Extrudate swell data for resins K , L , S, H , and O having similar PI. 94 viii Figure 4.28 Differential molecular weight distribution for resins Q, J, and P 94 Figure 4.29 Differential molecular weight distribution for resins K , L , S, H , and O . . 95 Figure 4.30 Extrudate swell data for resins L , M , R, E , and C having similar Mw 96 Figure 4.31 Differential molecular weight distribution for resins L , M , R, E , and C . 96 Figure 4.32 Sensitivity o f extrudate swell to changes in shear rate. A l l resins are manufactured by technology 'a' (set 1) 98 Figure 4.33 Sensitivity o f extrudate swell to changes in shear rate. A l l resins are manufactured using technology ' a ' (set 2) 98 Figure 4.34 Differential molecular weight distribution for resins T, E , and C 99 Figure 4.35 Differential molecular weight distribution for resins D , U , A , and B 99 Figure 4.36 Sensitivity o f extrudate swell to changes in shear rate. Included in the plot are resins produced from different technologies. One can see a breakdown of pattern 100 Figure 4.37 Determination of melt strength using the melt indexer 102 Figure 4.38 Observed and predicted melt strength values as obtained from S T A T G R A P H I C S (P/>10) 103 Figure 4.39 Observed and predicted values of melt strength as a function of Mw (P>\0)...., 104 Figure 4.40 3-D plot showing the effect of PI and Mw on melt strength. Density value is fixed arbitrarily. Changing the density value would shift the curve upward or downward accordingly. Note that Mw is related to PI and M„, and hence, when considering the plot, it has to be ensured thatM, is reasonable, so thatP/>/0 105 Figure 4.41 Observed and predicted melt strength values as obtained from S T A T G R A P H I C S v2.0 (8<PI<10) 105 Figure 4.42 Correlation coefficients relating various molecular weight ranges to melt strength (PK8) (a) positive correlation (b) negative correlation. Correlation coefficient (x,y) = COV(x,y)/oK.oy 107 Figure 4.43 Variation o f pillow weight with pillow number. Multiple curves indicate replicate runs 109 ix Figure 4.44 Variation in pi l low width for different resins extruded at different drop times 110 Figure 4.45 Extrudate swell profile for resins E , F, and G 112 Figure 4.46 Mel t strength values for resins E , F, and G 112 Figure 4.47 Variation in pi l low weight o f the three resins extruded at different drop times 114 Figure 4.48 Density values for resins E , F, and G 114 Figure 4.49 P i l low weight normalized to the weight o f pi l low number one to show the magnitude of sagging 115 Figure 4.50 Comparison o f parison sag between resins E , F, and G 116 Figure 4.51 Total length and weight as a function o f parison drop time. Although the total melt volume before extrusion was kept the same, the total length and weight o f the parison were not constant, due to the fact that the mold was located at some distance below the die 117 Figure 4.52 Correlating MI to Mw. Comparison should be made for resins with similar PI, or shear thinning properties 119 Figure 4.53 Implication o f MI on shear viscosity curves 121 Figure 4.54 Correlation between melt strength and MI. 122 Figure 4.55 Implication o f S.Ex. on shear viscosity profile 123 Figure 4.56 Correlation between MFT? and S.Ex 124 Figure 4.57 Implication of melt strength on Hencky strain 127 LIST OF TABLES Page Table 1.1 Influence of rheological properties on parison formation 22 Table 3.1 A summary of the molecular characteristics of the H D P E resins that were studied in this work 50 Table 5.1 Summary of conclusions 135 xi A C K N O W L E D G E M E N T S I would like to express my sincere gratitude to my supervisor Dr . Sawas G . Hatzikiriakos, and Dr . Shivendra K . Goyal for their guidance throughout the course o f this work, and to N O V A Chemicals for the sponsorship of this thesis project. I also wish to thank Dr . Phil Edwards, Dr . Charles Russell, Dr . T im Bremner, and Henry Hay o f N O V A Chemicals, and the members of N O V A Chemicals' rheology group, headed by Dr . Joo Teh, for their support, encouragement, and assistance with the rheological equipment at N O V A Research and Technology Center. M y gratitude also goes to the members of the rheology team of the University o f British Columbia (Rheolab) for their academic guidance. Finally, I would like to extend a special note o f thanks to Natural Sciences and Engineering Research Council o f Canada for its financial support, and to others who have in one way or other helped me with the completion of this project. xii To B, C, D, E, andF • This is not the end. This is not even the beginning of the end This is the end of the beginning Winston Churchill xiii Chapter I Introduction 1 INTRODUCTION 1.1 INTRODUCTION The characteristics of most engineering materials are widely known, owing to their vast usage and development over the course o f many years. There has also been much experience with their applications. However, the same cannot be said about plastics. Even though plastics are becoming increasingly important as building materials, there is still not enough accumulated information and experience, making it hard to truly appreciate the application o f plastics as engineering materials. Studies are continuously being performed in order to gain a better understanding o f plastic behavior. Numerous researches are also being conducted in search o f improvements to the characteristics o f plastics as a relatively new class o f engineering materials. B l o w molding is one type o f conversion process that is used to produce plastic products with hollow parts. The process of blow molding is principally governed by the rheological behavior of the resin used. The molecular structure o f the resin, in turn, influences its rheological properties. High density polyethylene is one o f the most common types o f resin used in blow molding. The purpose o f this chapter is to provide a general introduction to polymeric materials and their processing behavior. Background information on polymers and, particularly, high density polyethylene is provided in the next section of the chapter, followed by a general discussion on polymer rheology. In addition, the process o f blow molding and the applicability of polymer rheology to this process are discussed. Finally, the objectives o f this thesis project are described in the last section. l Chapter 1 Introduction 1.2 B A C K G R O U N D 1.2.1 Polymers Over the last fifty years, there has been a particularly great interest in materials made up o f high molecular weight molecules, also known as 'macromolecules'. Polymers are macromolecules consisting o f a large number of repeating units called 'monomers' being joined together to form a pattern, in much the same way as links make up a chain. A 'homopolymer' is a type of polymer in which only one kind of monomer is used to make up the macromolecular chain. I f two or three different kinds o f monomers are used, the products are called 'co-polymers' and 'terpolymers', respectively. In order to have a technological significance, however, the molecular weight of a polymer has to be high enough. Polymers with very high molecular weights are also called 'high polymers'. The chains that make up a polymer are usually not of equal length. Instead, there is a distribution of chain lengths or molecular weights. There are various factors that affect the exact shape o f the distribution, such as the mechanism o f polymerization and the subsequent treatment carried out on the polymer. For example, treating the polymer with solvent may selectively remove low molecular weight fractions. Higher molecular weight components may preferentially be removed by shear induced degradation. Also , either crosslinking or chain scission may be caused by chemical reactions such as oxidation. Since the average molecular weight and the molecular weight distribution o f polymeric chains affect the physical properties o f the bulk polymer, it is therefore important to have some quantitative measures of these parameters. Several molecular weight averages are normally determined for this purpose. 2 Chapter 1 Introduction The weight average molecular weight can be defined by considering the weight fraction Wj(Mi) o f a polymer having molecular weight Mt. The weight average molecular weight, Mw, is then M _ £ A / , - T » , ( M , ) ( U ) Often, the distribution wt(Mi) is expressed as a continuous function w(M)dM. Under these circumstances, the summation in Equation 1.1 is replaced by an integral. Experimentally, Mw may be determined by Gel Permeation Chromatography (GPC) [Dealy and Wissbrun (1995)]. The number average molecular weight is also defined similarly by considering the number fraction rti(Mj) and summing it over Mi. However, nj(Mi) may be related to Wi(Mi) by the expression rii(Mi) = Wi(Mi)/Mi and, hence, the number average molecular weight, M„, can be defined in terms o f weight fraction distribution as (1.2). M„ may be measured experimentally by chemical or spectroscopic end group determinations or by G P C [Dealy and Wissbrun (1995)]. There are also other molecular weight averages which can be defined similarly. Two that are important are also called the Z and Z+l molecular weight averages. These are defined as: 3 Chapter I Introduction ZM f 2 - w , (M f ) (1.3) and E M , 3 • H ' , . ( M < ) (1.4). Figure 1.1 shows two typical molecular weight distribution curves. A s shown in this figure, Mz is weighted heavily towards the high end o f the molecular weight distribution, while M , is weighted towards the low end. The type o f distribution curve depends on the chemistry o f the polymerization reaction and any subsequent treatment. The 'most probable' distribution is described by the equation and is usually true for polymers produced by condensation reactions, such as N y l o n 6-6 and polyethyleneterephthalate (PET). The more complicated 'log-normal' distribution curve is described following the Gaussian normal error curve having the form (1.5) (1.6). 4 Chapter 1 Introduction The two parameters M0 and in Equation 1.6 describe the location of the distribution and its breadth, respectively. This distribution usually holds for polymers produced by polymerization with a heterogeneous catalyst, such as high density polyethylene (HDPE). There are also other types of distribution curves, some even having more than one peak. The latter are called multi-nodal distributions. The molecular weight distribution of a polymer can be described quantitatively by introducing a parameter called the polydispersity index (PI) which is defined as the ratio of Mw to M„. A PI value of unity indicates monodispersity and larger PI values indicate broader molecular weight distributions. .1 MOLECULAR WEIGHT Figure 1.1 Differential molecular weight distribution for most probable (solid) and log normal (dashed) molecular weight distribution, both with MJMn=2. 5 Chapter 1 Introduction Chemically, there are three basic types o f polymerization reactions, namely addition reactions, condensation reactions and a combination o f the two. A n addition reaction may occur simply by external chemical activation o f molecules, resulting in the combination o f these molecules in a chain type reaction. Rearrangement o f atoms within two reacting molecules, or the opening up o f molecules containing any ring atoms may also cause the addition type o f polymerization to occur. This type of polymerization does not result in the formation o f any by-product, unlike the condensation type o f polymerization. In a condensation reaction, chemical union o f two molecules is achieved by splitting out a molecule (a by-product) which is usually small. Normally, this by-product is immediately removed since it may inhibit further polymerization, or appear as an undesirable impurity in the final product. Nylon, phenolics, amino resins and polyester pre-polymers are examples o f products formed through condensation polymerization. The third method o f polymerization involves the combination o f both addition and condensation reactions. Normally, the condensation reaction takes place first to form a relatively small polymer, which is then capable of undergoing addition polymerization. This type o f polymerization reaction is used in the formation o f polyesters and polyurethanes. In actual practice, there are many different ways of carrying out these polymerization reactions. However, most involve one o f the following four general methods o f polymerization, i.e. the polymerization o f monomer in bulk, in solution, in suspension, or in emulsion form. Bulk and solution polymerization techniques are usually used for the formation o f both addition and condensation type polymers, whereas suspension and emulsion techniques are used widely for addition polymerization. Different 6 Chapter 1 Introduction polymerization techniques can result in polymers with different molecular characteristics. The number and size o f the resulting polymer molecules may be significantly affected by the rate o f polymerization, the solvent and the extraneous media involved in each technique, causing this difference in the molecular characteristics. This causes the properties of the polymer to be affected accordingly. Some polymers may be used as they are, i.e. directly from the polymerization process, but most o f them require mixing with additives before they can be made into useful products. Some typical additives are plasticizers, pigments, fillers, lubricants, extenders, antioxidants, and heat and light stabilizers. These physical mixtures o f polymer and additives are called plastics. Therefore, it is important to distinguish between the words polymer and plastic, since they refer to essentially different materials although they are often used as i f they were synonymous. In this work, the rheology and relevant processability o f high density polyethylene (FfDPE) blow molding resins were studied. Polyethylene is one type o f polymer which has found usage in many applications. It is characterized by its toughness, near-zero chemical absorption, excellent chemical resistance, excellent electrical insulating properties, low coefficient o f friction, and ease o f processing. Different types o f polyethylene are classified according to their density. H D P E , for example, has a density range between 0.941 to 0.965 g/cm 3 at 25°C. Polyethylene having different densities differ in their rigidity, heat resistance, chemical resistance, and ability to sustain load. Generally, as density increases, hardness, heat resistance and stiffness increase, while permeability decreases. Figure 1.2 shows schematically the molecular structures o f the 7 Chapter I Introduction different types of polyethylene. Low density polyethylene is characterized by its large number of side branches, which may also be relatively long. Linear low density polyethylene has less branching, while high density polyethylene has very few side branches. "H H" i i - - c - c I I _H H Polyeihylei Figure 1.2 Chemical and molecular structures of (a) Low Density Polyethylene (LDPE), (b) Linear Low Density Polyethylene (LLDPE), and (c) High Density Polyethylene (HDPE). HDPE polymers are highly crystalline and tough materials. Applications for HDPE include blow molded containers for household and industrial chemicals, injection molded items such as crates, housewares, pails, and dunnage containers, and extruded items such as pipes, tubes, and wire insulators. HDPE is also blown into film for packaging and rotationally molded into containers, toys, and sporting goods. Within the density range of HDPE, stiffness, tensile strength, melting point and chemical resistance improve at the high end. However, materials with the highest densities have lower stress crack resistance and lower impact strength at low temperatures. 1.2.2 Polymer Rheology Rheology is defined as the science of material behavior under deformation, due to the presence of external forces. It attempts to understand why a material behaves in a certain way when a force is applied. The response of a deformed material is related to the ^ ^ — r ^ (a) (b) (c) 8 Chapter 1 Introduction deforming force by what is known as a constitutive equation. To many design engineers, rheology is an important science. For example, in order to optimize the design of an extruder, a plastics engineer must know the relationship between the rate of deformation (shear rate) and the response of the polymer, so that the viscosity of the polymer may be determined. In injection molding, the same information is needed for the design of the mold so that the polymer melt will completely fill it with each injection. In blow molding, the processes of parison sag and swell are governed entirely by the rheological properties of the melt. The rheology of polymer melt is made interesting by the fact that polymers are viscoelastic materials, i.e. they exhibit both viscous and elastic properties. The viscous part of the polymer tends to dissipate energy as heat when a force is applied during deformation. The elastic nature of the polymer, however, tends to store this energy and use it to bring the polymer back to its original undeformed state when the force is removed. Due to this elastic response, polymeric materials are said to have 'memory'. This memory can be understood by considering the molecular structure of a polymer. The long chain molecules that make up a polymer cause entanglements to occur at the molecular level, forming a temporary network. The network is only temporary because of Brownian motion, which has a greater effect at higher temperatures. However, in an equilibrium state, there is a most probable molecular configuration. Deforming the polymer melt will alter this molecular configuration, but when the deformation is stopped, the Brownian motion will tend to return the polymer molecules to the equilibrium configuration. 9 Chapter 1 Introduction The rheological response of a polymer may be represented by a model consisting of a combination of springs and dashpots. The spring may be thought of as the elastic component of the material, while the dashpot represents the viscous component. The two simplest models of viscoelasticity are known as the Voigt and Maxwell models and are shown in Figure 1.3. The force in the spring is assumed to be proportional to its elongation and the force in the dashpot is assumed to be proportional to its rate of elongation, reflecting the time dependency of the response of the material. The Voigt model can be used to explain viscoelastic behavior in the presence of a constant force, also called creep behavior. Upon the application of a constant force, the assembly in the Voigt model will not respond immediately due to the presence of the dashpot. The ratio of the proportionality constants of the spring and the dashpot governs this time dependency. Such a response is also called a 'retarded' elastic response. The Maxwell assembly, on the other hand, attempts to explain the stress relaxation phenomena in a viscoelastic material. When the assembly is subjected to a sudden elongation, a stress is F F (a) (b) Figure 1.3 Analogs of viscoelastic materials: (a) Voigt model, (b) Maxwell model. 10 Chapter I Introduction created. The magnitude of this stress, however, decreases with time as the assembly 'relaxes'. An equation that includes an exponentially decaying term can be written to show this behavior. In a real polymeric system, a more complex combination of springs and dashpots is needed to represent its rheological behavior. The simplest type of viscoelastic behavior is called linear viscoelasticity and can be observed at very small and slow deformations. With small deformations, molecules are disturbed from their equilibrium configuration to only a negligible extent, and when the deformations are slow, there is time for the molecules to be brought back to the equilibrium configuration by Brownian motion. Under these conditions, the response of the material is independent of the rate, size and kinetics of the deformation. Linear viscoelasticity, however, is not particularly useful in polymer processing, since, in polymer processing, large and fast deformations are involved. It is nonetheless useful for equilibrium molecular characterizations and resin comparisons (quality control). In a simple shear experiment, material is placed between two parallel plates separated by a distance h, as shown in Figure 1.4. One of the plates is kept stationary, while the other is moved at a constant velocity V. Assuming that no material slippage occurs at the material-plate boundary, the shear strain, y0, can defined as (1.7). The shear rate can then be defined as (1.8). 11 Chapter 1 Introduction In the simple extensional experiment shown in Figure 1.5, a rod - shaped material is subjected to an elongational force at one end. The resulting extensional strain is defined as ea = In (1.9). A X < • Figure 1.4 A schematic diagram of simple shear experiment 0 i Ao Force Figure 1.5 A schematic diagram of simple (uniaxial) extensional experiment Knowing the shear and elongational strain magnitudes, the shear and tensile relaxation moduli can then be defined as 12 Chapter I Introduction (1.10) Mt.<.)-"% <L11) where o(t) and OE(0 a r e the shear and extensional stresses, respectively. In the linear viscoelastic regime, G and E are independent o f deformation size and hence, for a shear type o f deformation, the following equation can be written: o{t) = G{t)-y0 <U2>-The modulus can be separated into the elastic (storage) and viscous (loss) moduli, G' and G", respectively, i.e. G = \G*\ = >lG'2+G"2 (1.13) where G* is the complex modulus. In an oscillatory shear experiment, these moduli can be obtained as functions o f frequency, a. The viscosity o f the material can then be approximated as the complex viscosity using the Cox-Merz rule [Dealy and Wissbrun (1995)]: 1 / . ' \ \ CO J \ CO ) To Also , using the generalized Maxwel l model, the following equations can be derived: 13 Chapter 1 Introduction « G^co-x) G"(<y) = Z ? rrr (1.16) **[i+M)] where G/ and A, are the discrete relaxation modulus and time, respectively. These parameters indicate the relaxation behavior of the molecules. Therefore, by fitting Equations 1.15 and 1.16 to the data obtained from oscillatory shear experiments, the relaxation behavior o f the molecules can be determined as a discrete set o f data consisting o f Gt and Xi values. For large and fast deformations, the theory of linear viscoelasticity is no longer valid. This regime is also called the non-linear viscoelastic regime. There is no general theory that can be used to predict non-linear viscoelastic material behavior, i.e. in this regime, information gained from one type of deformation cannot be used to predict behavior in a different type o f deformation. Material functions such as the damping function are often defined to indicate deviation from viscoelastic behavior. Rheological properties are usually temperature dependent and hence, to obtain a complete picture o f material behavior, experiments must be carried out at many temperatures. Fortunately, it is often found that data taken at several temperatures can be brought together on a single master curve by means of 'time-temperature superposition'. This makes it possible to display on a single curve, corresponding to a particular reference temperature, To, the material behavior covering a much larger range of 14 Chapter 1 Introduction deformation. According to the time-temperature superposition theory, quantities that contain the unit o f time need to be shifted by a shift factor, a?, such that the unit o f time is eliminated. Thus, i f one makes a plot o f the complex viscosity, TJ* (Pa.s) versus the frequency o f shear (s*1), the resulting master curve can be accomplished by plotting r\*/ar versus oxar. The shift factor can then be related to temperature by either the Arrhenius type equation Equation 1.17 is valid for temperatures that are at least 100 K above the glass transition temperature, Tg. The term Ea in this equation is the flow activation energy, which gives an indication o f the sensitivity of material flow to temperature change. For temperatures closer to Tg, Equation 1.18 is found to be more useful, where the constants C° and C / can be determined using any optimization technique. K n o w i n g E a , or C° and C / , a?at any other temperature can be calculated, and, referring back to the master curve, the rheological property at the desired temperature can then be obtained [Dealy and Wissbrun (1995)]. Figure 1.6 illustrates the time-temperature superposition principle. The complex viscosity obtained from oscillatory experiments is plotted as functions o f frequency at different temperatures in Figure 1.6(a). Selecting the temperature o f 150°C (1.17). or the W L F (Williams, Landel, and Ferry) equation (1.18). 15 Chapter I Introduction 10-2 10-1 10° 101 102 103 Frequency (s1) (b) 16 Chapter 1 Introduction (c) Figure 1.6 Time temperature superposition of complex viscosity curves obtained at different temperatures: (a) before superposition, (b) after superposition, (c) the fit of shift factors to the Arrhenius equation as the reference temperature and performing both horizontal and vertical shifts to these data yield the master curve shown in Figure 1.6(b). The scatter observed in the plot at low frequency is due to the experimental difficulties at very low deformation rates. In Figure 1.6(c), the natural logarithmic values of the shift factors are plotted versus the inverse o f temperature according to Equation 1.17. The activation energy o f the polymer can then be determined by calculating the slope of the straight line obtained in Figure 1.6(c). 17 Chapter 1 Introduction 1.2.3 The Process of Blow Molding In blow molding, the polymer is first softened by electrical and mechanical heating in an extrusion barrel. The resulting melt is then extruded through a die head to form a hollow tube called a parison. The thickness of the blow molded part is determined by the thickness of the parison, which is in turn determined by the shape of the die opening and the extrusion speed. Several seconds after the parison is fully extruded, the two halves of the mold close and pinch off the extruded parison which surrounds the blow pin. The blow pin provides the opening for compressed air to enter the parison and inflate the polymer melt so it forms to the contours of the mold. An overview of this process is depicted in Figure 1.7. Figure 1.8 illustrates the molding process, and in Figure 1.9, the method of controlling the wall thickness is shown schematically. The control of wall thickness is a critical part of the process and may be achieved partially by controlling the position of the mandrel in the die head. Lowering the position of the mandrel increases the parison wall thickness and vice versa. Mandrel Extruder Figure 1.7 Overview of blow molding process. 18 Chapter 1 Introduction Parison 11 -Mandrel IA ^ Blowpin ID] Closed Mold Compressed Air Open Mold Pinch Off Figure 1.8 Parison inflation in the process of blow molding. Raised M a n d r e l Lowered M a n d r e l Thinner Wall Thicker Wall Figure 1.9 The controlling of parison wall thickness by adjustment of mandrel position. B l o w molding is an effective way to process hollow parts, and it involves relatively low tooling costs. However, the process requires long cycle times, secondary trimming, and high start-up costs. 19 Chapter I Introduction Rheology is o f central importance in the process of blow molding. During extrusion, the flow details and pressure drop in the barrel and the die are governed principally by the viscous properties o f the polymer melt, and the general flow path is controlled by the geometry o f the extruder and die. When the melt leaves the die, however, its behavior is no longer controlled by any contacting solid walls. Instead, the only external forces acting on the melt are gravity and the blow pressure. The parison behavior in response to these forces is governed entirely by the rheological properties o f the melt. Moreover, due to the elastic nature o f polymer melt, the behavior of the extruded parison reflects not only the two external forces but also the deformation it experienced in the die. A t the exit of the die, polymer melt also tends to swell. This is another manifestation o f melt elasticity. Figure 1.10 illustrates the extrudate swell behavior of Newtonian and polymeric materials. The swelling of fluids (increase of effective diameter) upon the emergence from a die is not unique. For the case o f polymer melts, the swell can be as much as 3 times the original diameter o f the die, i f not more. Generally, extrudate swell increases with increasing shear rate and decreases with increasing length to diameter ratio o f the capillary die. Since this is a manifestation of the viscoelasticity o f the polymer melt, there is also a time dependency involved. In general, most of the swelling takes place during the first several seconds of material emergence. However, sometimes hours are needed before the swell reaches its ultimate value. Extrudate swell may have a significant effect on the shape and weight of the final molded product. When polymer melt exits the die, it may also exhibit 'sharkskin' or 'melt fracture', which are irregularities in the surface o f the extrudate that can affect the surface finish o f 2 0 Chapter 1 Introduction Figure 1.10 Extrudate swell behavior for (a) Newtonian fluid and (b) polymer melt the final blow molded product. This effect occurs above a certain critical stress in the die and is often the factor that limits the rate of an extrusion process. The distortion is most severe in narrow M W D , high viscosity resins. Sometimes, increasing the extrusion temperature or reducing the extrusion speed helps to eliminate this effect, but these actions w i l l increase the cycle time. The detailed origins o f this phenomenon are not fully understood, but it has been thought that the shape and the material o f construction o f the die, and the formulation of the resin may be the contributing factors [Ramamurthy (1986), Hatzikiriakos (1994)]. A t the exit of the die, before the two halves of the mold close, the parison sags under its own weight. In more severe cases, sagging may cause the parison to break off. In other cases, it w i l l cause large variations in thickness and diameter along the parison. The extent of the sag depends exclusively on the rheological properties o f the polymer melt. To quantitatively analyze the sagging behavior of polymer melt, melt strength is 21 Chapter 1 Introduction often defined as the ability o f the melt to counteract sagging. Increasing the temperature w i l l decrease the melt strength o f the polymer melt and hence, increases sagging. Once a molten parison is formed, the way that it inflates is again a reflection o f melt viscoelasticity. The rheological properties that govern this part o f the process are the extensional flow properties. The deformation is not a true uniaxial extension, but the results from uniaxial extension experiments are thought to be useful for determining inflation performance. For example, resins that exhibit 'extension thickening' are thought to be easier to inflate and unlikely to exhibit blow-out (the tendency o f the parison to bulge at the centre during inflation), even i f the inflation pressure is high. 'Extension thinning', on the other hand, is thought to imply unstable inflation and increased likelihood o f blow-outs. The influence o f the various rheological properties on parison formation is summarized in Table 1.1. Table 1.1 Influence of rheological properties on parison formation Property Effect on parison formation High shear rate viscosity Extrusion pressure and/or rate (length, diameter and thickness, melt fracture, curtaining) L o w shear rate viscosity Parison sag (length, diameter and thickness, curtaining) Extrudate swell Parison length, diameter, thickness, curtaining Critical stress Melt fracture "Relaxation" time Sag, stretch orientation 22 Chapter 1 Introduction 1.3 Thesis Objectives In order to produce acceptable blow molded products, it is necessary to understand the rheological properties and the processability of blow molding resins. Rheological properties such as shear and extensional flow properties, extrudate swell, and sensitivity o f viscosity and extrudate swell to changes in deformation rate and temperature are highly influenced by the molecular characteristics o f the resins. Resin processability is also affected by molecular parameters, such as Mw, PI and density. Mel t strength, sagging, and weight swell are parameters which characterize the processability o f blow molding resins. The objectives o f this work were, therefore, to either quantitatively or qualitatively 1. correlate the rheological properties (shear viscosity, extensional strain behavior, extrudate swell, and viscosity and swell sensitivities to changes in temperature and shear rate) o f commercial H D P E blow molding resins to their molecular parameters (M„, Mw, Mz, PI, and density), 2. correlate the processability properties (melt strength, sagging characteristics, and weight swell) o f commercial H D P E blow molding resins to their molecular parameters (M„, Mw, Mz, PI, and density), 3. determine the significance o f the empirically defined parameters: stress exponent, melt index, and melt flow ratio on resin quality control, and 4. correlate processability to rheological properties. 23 Chapter 2 Literature Review 2.0 L I T E R A T U R E R E V I E W 2.1 I N T R O D U C T I O N Numerous publications have been reported on the rheological characterization o f polyethylene and the process o f blow molding. However, most o f these publications pertain either to pure rheological characterization, with emphasis on finding possible explanations for certain observed polymer behaviors, or to the mathematical modeling o f polymer behavior during the blow molding process. Very few attempts have been made to characterize the rheological properties of resins, with the effect o f rheology on resin processability as the ultimate objective. In this chapter, some o f the publications related to this work are reviewed. 2.2 R H E O L O G Y Yoshikawa et al (1990) studied the influence of molecular weight distribution and long chain branching on the viscoelastic behavior of commercial H D P E melts. The two types o f samples studied were produced using different kinds of catalysts. The study found that long chain branching increases the relaxation times in dynamic viscoelastic functions, relaxation modulus and elongational viscosity. It was also found that it is possible to use melt index to predict shear flow behavior. However, differences in shear viscosity corresponding to the stress for which melt index was measured were small, although reasonably large differences were obtained for the melt index values. Larger and more significant differences are observed at much lower stress levels. With regard to extensional viscosity, the authors attributed the large rises in extensional viscosity to long 24 Chapter 2 Literature Review chain branches and the high concentration o f large molecules in the molecular weight distribution. In another study, Yoshikawa et al. (1989) investigated the dependence o f shear viscosity on Mw. The authors found that for H D P E with no long branches, the zero shear viscosity was proportional to M w 3 ' 5 . For H D P E with long branches, the exponent was found to be higher than 3.5. Hence, for branched H D P E , the dependence o f zero shear viscosity on Mw was stronger. The effect o f molecular weight and molecular weight distribution on the viscous and elastic behavior o f other polyethylene resins have also been studied by several workers including Han and Vil lamizar (1978), Shroff and Mitsuzo (1977), Bersted (1976), Mendelson and Finger (1975). These studies found that increasing Mw increases the shear viscosity in the low deformation regime. Above some critical value o f Mw, the zero shear rate viscosity for amorphous and linear polymers has been found to be proportional to MjA. On the other hand, the breadth o f the molecular weight distribution was found to affect the shear sensitivity o f the shear viscosity curves. In general, increasing the polydispersity reduces the shear rate at which shear thinning begins [Dealy and Wissbrun (1995)]. The flow o f polymer depends on the availability o f free volume and thermal energy. Near the glass transition temperature, the availability o f free space becomes the limiting factor. However, at higher temperatures, where there is no lack o f free volume, energy barriers become more significant. In this temperature regime, the activation energy is constant and it reflects the sensitivity o f polymer flow to changes in temperature. It has 25 Chapter 2 Literature Review been found that activation energy is affected by the degree o f branching. For polyethylene, increasing the percentage of long chain branching increases the activation energy significantly [Dealy and Wissbrun (1995)]. For H D P E , typical values o f activation energy at high temperatures fall in the range between 20 kJ/mol to 25 kJ/mol [Van Krevelen (1990)]. Mavridis and Shroff (1992) have carried out a study on the temperature dependence of polyolefin melt rheology. They have developed a unified framework for handling the temperature dependence o f rheological data. Shenoy et al (1983) proposed a simpler method to estimate the polymer flow curve and its dependence on temperature and shear rate, knowing the melt index and glass transition temperature o f the polymer. The authors suggested the use of melt index as a substitute for the shift factor in time temperature superposition theory. Master curves for several types o f polymers were calculated using the suggested method. It is interesting to note that the resulting master curve was a general one, applicable to any grade or class o f polymer, regardless o f the technology used to produce the resins, the shape o f the molecular weight distribution or any other molecular parameters. Wi th regard to the extensional flow properties o f polymers, most previous studies were conducted by subjecting polymer samples to tensile force under constant rate conditions (more detailed descriptions of extensional flow properties are provided in chapter 3). This procedure yields the extensional viscosity o f the material directly. Munstedt and Laun (1981) found that increasing the polydispersity o f L D P E increases the magnitude o f tensile thickening in the extensional viscosity curve. However, it should be noted that the increase in polydispersity was achieved by increasing the concentration o f 26 Chapter 2 Literature Review larger molecules. The authors did not investigate the effect o f increasing the low molecular weight tail o f the distribution. B y increasing Mw, on the other hand, the authors found an increase in the extensional viscosity at low stress levels. The authors also concluded that extensional flow properties of polymers are affected by certain features o f the molecular weight distribution, and that differences too small to be detected by chromatography can affect the extensional viscosity curve. Rauschenberger and Laun (1997) developed a mathematical model to predict isothermal and nonisothermal elongation o f an extruded filament at a given force. In the modeL the pre-strain history o f the material was included. The model was tested with L D P E and was found to predict the melt behavior for uniaxial elongation experiments very well . Considering the process o f blow molding, it is more useful to determine melt properties under constant stress conditions (creep). However, to date, there are not many experimental data published based on such conditions. Extrudate swell is another important rheological property. It has been found that, for many linear polymers, increasing polydispersity increases the ultimate swell o f the polymer extrudate. However, for polymers with very similar molecular weight distributions, extrudate swell profiles can differ significantly. This is thought to be due to the fact that extrudate swell is very sensitive to small amounts of high molecular weight material [Dealy and Wissbrun (1995)]. Koopmans (1988) has first recognized this. Koopmans also noted that it is misleading to use the polydispersity index as a measure of 27 Chapter 2 Literature Review molecular weight distribution, since it may not reflect the contribution o f the high molecular weight components. In another study, Koopmans (1992a, c) determined the effect o f molecular weight distribution on the time dependency of extrudate swell. The author found that the presence o f high molecular weight molecules resulted in a faster initial swell but a smaller maximum swell, which could be due to the cooling o f the extrudate before the maximum swell was reached. The author also concluded that it is important to consider the full molecular weight distribution and not a few average molecular weight values, such as Mw a n d M z , to understand the flow behavior of polydisperse polymers. 2.3 P R O C E S S A B I L I T Y Very little information is available about the melt strength o f polymers. A possible reason includes the inability to generalize such results found for a given polymer to other materials. It has also been found that melt strength is dependent on a number o f parameters and cannot be used to determine the elongational viscosity o f polymer melts [Mantia and Acierno (1985)]. Mantia and Acierno (1985) studied the melt strength properties of H D P E , L L D P E , and L D P E . The authors found that H D P E and L L D P E have a high breaking/stretching ratio and low melt strength values. L D P E , on the other hand, exhibits very large melt strength values but low breaking/stretching ratio values. The authors also found that it is possible to predict melt strength by measuring the melt index o f the polymer. Increasing melt index was found to indicate decreasing melt strength. The breaking/stretching ratio, on the other hand, increases with melt index. 28 Chapter 2 Literature Review In another study, Mantia and Acierno (1983) found that it is possible to correlate melt strength to the product of Mw and polydispersity index for H D P E . The authors also investigated the effect of temperature on melt strength and found a mathematical correlation between melt strength and temperature. Goyal (1994) performed a study on the melt strength o f L L D P E having various molecular weights, densities, comonomer types and molecular weight distributions at several extrusion temperatures. The author found the same implication of melt index on melt strength in that increasing melt index indicated lower melt strength. For L L D P E with short chain branching, it was found that density did not affect melt strength significantly. O n the other hand, for high pressure L D P E (with long chain branching), melt strength was found to increase with a decrease in resin density. The author also found that broadening the molecular weight distribution did not have any significant influence on melt strength. However, a change in the modality o f the distribution can have a major effect on the melt strength behavior of a resin. In addition, non-reactive additives such as slip agent, anti-block and fiuoroelastomer polymer processing aids did not have any effect on the melt strength o f a resin, even though these additives can substantially alter the apparent shear viscosity behavior o f a resin. In the above studies, melt strength was determined by pulling polymer extrudate from a capillary barrel using a pulley at increasing speed. The force required to pull the extrudate was measured and the steady force level or the force at which the extrudate broke was taken as the melt strength value. The extrusion o f the polymer in the barrel was done at some specific conditions. Considering this method o f melt strength 29 Chapter 2 Literature Review measurement, it can be said that the reported melt strength values correspond to high shear rates and temperatures lower than the barrel temperature, since pulling was done on extrudates outside the barrel. Although this may be used to compare the melt strength properties o f resins, the method is not particularly useful for blow molding processes. During parison formation, the polymer melt is subjected to very slow deformation. Hence, melt strength data obtained according to this method do not really reflect the melt strength o f the polymer during parison formation. In this work, another procedure that is more useful for blow molding processes was used (see chapter 3). No similar studies have previously been carried out using this new technique of melt strength measurement. Henze and W u (1973) studied the factors affecting parison diameter swell in a blow molding process (see Figure 2.1 for the definition o f parison diameter swell). The authors found that diameter swell increases with increasing shear rate (decreasing parison residence time). A correlation between the local diameter ratio and the local weight ratio has also been found. The authors found that the diameter swell ratio approximately equals the local weight swell ratio raised to the power of 0.25. In addition, it was found that the swell potential, rate of swell and melt strength strongly influence the diameter swell. A procedure for determining parison flow behavior in regard to swell and drawdown has been described by Sheptak and Beyer (1965). The authors used a pillow mold having multiple pinch off points to determine the weight and diameter distribution of the parison. In addition, photographs were taken at different times during parison formation to 30 Chapter 2 Literature Review determine the relaxation behavior o f the polymer melt. The qualitative analysis used by these authors is adopted in this work. Figure 2.1 Illustration of parison diameter swell Kaylon and Kamal (1986) also conducted a pillow mold study to investigate the relation between capillary extrudate swell and parison swell. The authors found excellent agreement between the area swell values determined on the basis o f capillary and parison swell experiments. In comparing the relation between capillary and parison swell for a number o f H D P E resins, Wilson et al. (1970) found that the relative order o f magnitude o f capillary extrudate swell correlates well with the relative order of parison diameter swell. In another study by Alroldi (1978), the same result was obtained. 31 Chapter 3 Experimental Equipment and Procedures 3 EXPERIMENTAL EQUIPMENT AND PROCEDURES 3.1 INTRODUCTION This chapter describes the experimental equipment and procedures used in the determination o f the rheological properties and the processability o f the H D P E resins. The theory behind each piece o f equipment is discussed accordingly in each subsection. In addition, one section is devoted to the molecular properties o f the resins. 3.2 EXPERIMENTAL EQUIPMENT 3.2.1 Densimeter Density is a measure of the degree of crystallinity in a polymer and is commonly used to classify polyethylenes. A high density polyethylene has greater crystallinity than a low density polyethylene. The frequency, size and type o f side branching on molecules highly influence the density value o f a polymer. Increasing the frequency and the size o f side branching decreases the density and affects the processing behaviour o f the polymer accordingly. In this work, the densities o f H D P E resins were measured using a Toyoseiki Automatic Densimeter, Model D-H100. The densimeter is equipped with an analytical balance and a specimen clamp. A schematic diagram of the densimeter is shown in Figure 3.1. Samples were prepared according to A S T M procedure PE-206 using a Wabash press which is equipped with electric heaters and controlled cooling rate capabilities. B y measuring the sample weight in air and in distilled water by means of an analytical 32 Chapter 3 Experimental Equipment and Procedures balance, and knowing the density o f the distilled water at the time o f weighing, the absolute density o f the resin can be found. The temperature o f the distilled water is kept constant by circulating water of specified temperature around the beaker using a circulator. The actual temperature of the distilled water is determined manually by a thermocouple and the corresponding value o f the distilled water density is obtained as tabulated in the densimeter manual. Computer Unit Figure 3.1 A schematic diagram of the Toyoseiki Automatic Densimeter, Model D-H100. 3.2.2 Extrusion Plastometer Mel t index is an empirically defined parameter that is critically influenced by the physical properties and molecular structure of a polymer. It serves to indicate the flow properties o f a polymer at a particular low shear rate and is usually indicative o f the molecular properties o f the resin. It is measured as the amount o f extruded polymer in a molten state through a die of specified length and diameter under prescribed conditions of 33 Chapter 3 Experimental Equipment and Procedures temperature and pressure. A high melt index implies ease o f flow and hence, low viscosity, at the particular shear rate corresponding to the load condition. In this work, melt indexes were determined using the Tinius Olsen Manually Timed Extrusion Plastometer. A schematic diagram of the equipment is shown in Figure 3.2 Oil-filled thermometer wells Minimum 4 4in Weight (combined with piston to be 2160g) Insulation Barrel of nitride hardened steel Heater Insulation 0-250 0-005 in Insulating plate Jet retaining plate Figure 3.2 A Schematic diagram of the Tinius Olsen Manually Timed Extrusion Plastometer. The major components of the plastometer are the electrically heated barrel, the piston, the weights, and the capillary die. The barrel is made o f steel with an outside diameter o f 50.8 mm and a length of 162 mm. The hole in the barrel is 9.5504 ± 0.0076 mm in diameter and is located 4.8 mm away from the cylinder axis. The electrical heater and insulator are wrapped around the barrel as shown in the figure. A thermometer is placed in the extra hole on top of the barrel to measure the actual temperature of the 34 Chapter 3 Experimental Equipment and Procedures barrel. The piston used in the assembly is made o f steel, 9.4742 ± 0.0076 mm in diameter and 6.35 ± 0.13 mm in length. A t the top end of the piston, an insulating bushing acts as a barrier to heat transfer from the piston to the weight. The combined weight o f the piston and the load is within 5% o f the selected load. The load is placed on the piston as shown in Figure 3.2. When necessary, additional loads are obtained by adding more weights to the top o f the first weight. The die used in this work is made of steel with the dimensions o f 2.09955 ± 0.0051 mm in diameter and 8.000 ± 0.025 mm in length. For a more detailed description of the melt indexer, the reader is referred to the A S T M procedure D 1238-95. It should be noted that small variations in the design and arrangement o f the component parts may create some discrepancies in the measured melt index values. Also , it is important that the piston arrangement be kept as vertical as possible to minimize the friction between the piston and the side of the barrel. A leveling device is supplied with the melt indexer to facilitate this alignment procedure. 3.2.3 Capillary Rheometer Capillary flow has been the most popular method for studying the rheological properties o f liquids. For this particular flow, simple equations can be derived to determine the shear viscosity for both Newtonian and power law fluids. For other types of fluid, where no specific constitutive equation is known to be valid, special computational techniques are required to calculate the shear stress, shear rate and viscosity. 3 5 Chapter 3 Experimental Equipment and Procedures For a steady and fully developed flow o f an incompressible fluid in a tube o f radius R, a force balance can be performed to yield the absolute value o f shear stress at the tube wall , aw: r dP r=R R dP (3.1) 2 ' dZ where dP is the pressure drop over the differential length dZ o f the tube. For a Newtonian fluid, shear stress is related to deformation by: a = Tj'r  ( 3 2 ) where the viscosity 77, is constant at a given temperature. Combining Equations 3.1 and 3.2, the fully developed parabolic velocity profile of Newtonian fluid can be obtained. Knowing the velocity profile, the shear rate at the tube wall can be calculated by differentiating the velocity profile with respect to the radius o f the tube to yield: dV /(Newtonian) = — dr r=fi 4g (3.3) TTR' where Q is the volumetric flow rate. For non-Newtonian fluids, the same derivation procedure cannot be used. The velocity profile is no longer parabolic and a different constitutive equation is needed to determine the viscosity o f the fluid, which is no longer a constant. 36 Chapter 3 Experimental Equipment and Procedures I f a power law model is assumed, the constitutive equation is given by: R (3.4) where K and n are the consistency index and the power law exponent, respectively. Note that the special case o f Newtonian flow behavior is recovered for n=l. It can be shown that the wall shear rate for a power law fluid is given by [Dealy and Wissbrun (1995)]: It is noted that the term in bracket in the above equation is the wall shear rate for a Newtonian fluid (Equation 3.3). However, for non-Newtonian fluid, this term in itself The constants K and n can be determined from the intercept and the slope o f the straight-line plot o f the above equation in log-log co-ordinates. I f no specific constitutive equation is assumed, it is then not possible to calculate the true shear rate at the wall directly, knowing only YA- A special technique which requires pressure drop data for a number of flow rates is needed. This technique makes use of a plot o f log(aw) versus log(yj) that yields a single curve. The true wall shear rate is then (3.5). has no significance. It is referred to as the 'apparent shear rate', JA-Using Equation 3.4 and 3.5, it can be shown that (3.6). 37 Chapter 3 Experimental Equipment and Procedures given by (3.7) where b is the Rabinowitsch correction given by tf(logo-J (3.8). This correction term measures the fluid's deviation from Newtonian behaviour. It equals unity for a Newtonian fluid and 1/n for a power-law fluid. A large amount of data is needed for this technique since differentiation is required to determine b. In a capillary rheometer, the shear stress is determined by monitoring the driving pressure, Pd, in the barrel and assuming that the pressure at the outlet o f the capillary is equal to the ambient pressure, Pa. Pd can be related to the force that is driving the piston (plunger), Fd, as: where Rb is the radius of the barrel. The pressure drop (-APW) in Equation 3.1 is then (Pd-Pa) or, since for melts Pd is nearly always much larger than Pa, the pressure drop can simply be replaced by Pd. However, this is not the actual pressure drop that is observed for a fully developed flow in a capillary o f length L. End correction is needed to take into account the large pressure drop at the entrance of the capillary and the small residual pressure at the exit. Figure 3.3 shows the pressure profile for a flow in a capillary. (3.9) nRb2 38 Chapter 3 Experimental Equipment and Procedures B A R R E L L FULLY D E V E L O P E D z Figure 3.3 Pressure profile for a flow in a capillary. The pressure end correction can be determined by the method outlined by Bagley (1931) in which the driving pressure, Pd, is plotted as a function o f the length to diameter ratio (L/D) o f capillaries o f fixed diameter at fixed wall shear rate values. This plot is also referred to as the 'Bagley plot'. A typical Bagley plot is shown in Figure 3.4. The end correction is obtained by extrapolating the plot to L/D=0. Another way o f determining Pend is by making use of an orifice die with L=0. Using the corrected pressure drop, the wall shear stress can then be calculated as In general, the Bagley plot may include some curvature at high L/D ratio. This is due to the dependence o f viscosity on pressure, the effect o f pressure on polymer slip at wall , or viscous heating [Hatzikiriakos and Dealy (1992)]. 4(L/D) (3.10). 39 Chapter 3 Experimental Equipment and Procedures Figure 3.4 A typical Bagley plot. In this w o r k , a Kayeness Ga l axy I V Cap i l l a r y Rheometer (model 0052) was used. The barrel is equipped w i t h a dual zone electric heater and an adaptive PID temperature contro l ler w i t h an accuracy o f 0.1°C. The barrel is 175 m m in length and 9.55 m m ± 0.005 m m in diameter. A stepper motor is used to drive the p is ton f rom a speed o f 0.5 m m per minute to a m a x i m u m speed o f 250 m m per minute. Fo rce is measured by means o f a load ce l l instal led on top o f the piston driver. The rheometer also comes w i t h data analysis software and tungsten carbide dies o f different L/D ratios and diameters. A schematic d iagram o f the rheometer and a die is shown in F igure 3.5. 40 Chapter 3 Experimental Equipment and Procedures Locking Screw - O k Ok Holder IS! Power Switch — Emerceacy Switch Figure 3.5 A schematic diagram of Kayness Capillary Rheometer and die. 3.2.4 Extensional Rheometer Extensional properties o f resins are very important in blow molding processes. During parison formation, the polymer melt is allowed to hang for some time before the two halves o f the mold close and air is blown in. During this period, the polymer is subjected to elongational deformation by its own weight, resulting in parison sag. When air is blown to mold the parison, the polymer melt is also subjected to extensional deformation, in addition to a shearing type of deformation. For these reasons, it is therefore necessary for the extensional properties o f blow molding resins to be determined before their processability can be evaluated. 'Simple extension' or 'uniaxial extension' is the type o f extensional deformation illustrated in Figure 3.6. The deformation can be generated by introducing a tensile force on one end o f a rod - shaped sample, which is fixed at the opposite end. The stretching 41 Chapter 3 Experimental Equipment and Procedures force, F and the length of the sample, L are two of the measurable parameters, which may be functions of time. Assuming that the material is incompressible, however, the following relation on the conservation of volume holds at all times L-A = L0-A0 O.U) where L and A denote the length and the cross sectional area of the sample, respectively, and the subscript indicates the original condition of the sample before stretching. The force applied on the sample in the longitudinal direction can easily be converted to stress by simply dividing it by the cross sectional area. This stress in itself, however, has no rheological significance. A Theologically more meaningful variable is, instead, the normal stress difference which is defined as ou-0-22 where, referring to Figure 3.6, u and 22 indicate the normal directions xj and X2, respectively. Note that the force is positive in thexy direction and, due to symmetry, 022=055. Figure 3.6 Uniaxial or simple extension 4 2 Chapter 3 Experimental Equipment and Procedures The strain experienced by the material during stretching can be defined as de = dL/L (312>-The current length o f the sample, L, is used instead of the original length of the sample, Lo, as the reference length in the denominator to make the definition of strain more meaningful, since for materials with fading memory (liquid-like), the significance o f L0 decreases as the stretching continues. B y simple integration, the strain for finite deformation can be obtained as e = In rL2~\ (3-13). This is also called the Hencky strain. The strain rate can be derived from the above relation as • _de _ 1 dL = d\nL (3.14). dt L dt dt Realizing that dUdt is the velocity, V, at the end of the sample, equation 3.14 can hence be written as follows: V_ L f = _ (3-15). Therefore, in a constant strain rate experiment, the speed at which the sample is being pulled is controlled according to the instantaneous length of the sample, L, so that s can be kept constant. The force required to stretch the sample is measured as a function 43 Chapter 3 Experimental Equipment and Procedures o f time as the length, and hence the stretching velocity, change accordingly. The extensional viscosity o f the material can then be calculated as the ratio o f stress difference to strain rate. For a Newtonian fluid, it can be shown that the extensional viscosity is three times the shear viscosity (also observed experimentally). This is also true for polymeric materials at sufficiently small strain and shear rates, i.e. where TJE is the extensional viscosity at diminishingly small strain rates and n0 is the zero-shear rate viscosity. In a constant stress experiment, however, it is the stretching force that is being controlled according to the cross sectional area of the sample. The cross sectional area can be determined by measuring the length o f the sample as a function o f time and by using Equation 3.11, assuming material incompressibility. The measured quantity in a constant stress experiment is the strain of the material as a function o f time. The strain versus time curve is useful for determining the melt strength of a polymer melt. A t a particular time, a resin with lower melt strength wi l l show a higher strain in the curve. Such a resin w i l l hence lead to more sagging during parison formation. Constant stress experiments are, therefore, more useful for blow molding processes since the results can be related directly to processability parameters such as melt strength and sagging characteristics. Also , constant stress experiments are more representative o f the parison formation stage of the blow molding process. In such experiments, the extensional viscosity o f the material can be calculated as the stress to strain rate ratio by first (3.16) 44 Chapter 3 Experimental Equipment and Procedures differentiating the strain versus time curve to get the strain rates at different times. A Rheometrics RER-9000 Extensional Rheometer was used in this work to determine the elongational properties o f the H D P E resins. The rheometer consists of a control panel, equipped with a plotter, and a dewar in which a cylindrically shaped sample is pulled in a bath o f heated D o w Corning 200 fluid. To ensure temperature uniformity, the fluid is circulated through the dewar by a circulator which also functions as a heater for the fluid. The need to float the sample in the fluid is important to eliminate gravity as a driving force for deformation, which is especially critical when testing less viscous materials. Thus, the fluid is chosen so that its density minimizes the buoyancy effect on the sample during the vertical pull. The rheometer is able to perform extensional runs under constant rate or constant stress conditions and is connected to a computer for automatic data acquisition. The rheometer also comes with a set of accessories for sample making. These accessories include a 15-cavity compression mold, gluing fixture, metal clips and a rotary sample cutter. Figures 3.7 to Figure 3.9 show the schematic diagrams o f the rheometer and its accessories. 45 Chapter 3 Experimental Equipment and Procedures Figure 3.7 A schematic diagram of Rheometric RER-9000 Extensional Rheometer. 46 Chapter 3 Experimental Equipment and Procedures 8AMPI g f»l l i e n MTO P I Anf= (a) - CUEajN P L A C E E L E @ ® ® @ ® m n i n r i niinii ni ® ULiUllLi^^ If PI n PI n PA PI y © © ® © © © (b) Figure 3.8 (a) Molding and (b) gluing accessories for Rheometric RER-9000 Extensional Rheometer. 47 Chapter 3 Experimental Equipment and Procedures Figure 3.9 A Schematic diagram of the sample cutter supplied with RER-9000 Extensional Rheometer. 3.2.5 B l o w Molding Machine To determine the sagging and swell characteristics of the blow molding resins under actual run conditions, an Impco B-13 B l o w Molder unit was used. The unit is manufactured by Ingersoll-Rand Plastics Machinery Ltd . and is equipped with an Edwards Zone-A-Matic Model C C - 5 mold chiller unit. Parison programming is done through a Model A081-822 programmer supplied by M o o g Incorporated. Figure 3.10 shows a schematic picture of the blow molder. The pil low mold shown in Figure 3.11 was used with the blow molder unit. The mold facilitated the measurements of sagging and swell characteristics by providing a number o f pillows, molded from different parts of the parison. This was made possible 48 Chapter 3 Experimental Equipment and Procedures by the multiple pinch-off points that were evenly spaced along the vertical direction of the mold. The width and weight of each pil low corresponding to the different parts o f the parison could be measured to determine how wall thickness and product weight vary in the vertical direction. MAX. wjccnoN srnotce Figure 3.10 A schematic Diagram of IMPCO B-13 Blow Molder I Figure 3.11 A schematic diagram of the pillow mold used in the blow molding Experiment 49 Chapter 3 Experimental Equipment and Procedures 3.3 E X P E R I M E N T A L S A M P L E S Twenty four commercial HDPE blow molding resins were provided by NOVA Chemicals Ltd. to be studied in this work. The molecular characteristics of these resins are summarized in Table 3.1. The technologies used to produce the resins include gas phase, solution, and slurry technology. Due to confidentiality, it is not possible to mention the actual name of the technology used to produce the resins. Table 3.1 A summary of the molecular characteristics of the HDPE resins that were studied in this work. Resin Technology Mn Mw Mz PI Density (g/cm3) A a 9030 152000 835400 16.83 0.9570 B a 9020 157100 971500 17.42 0.9567 C a 9330 133400 848400 14.3 0.9579 D a 9340 153600 933500 16.44 0.9587 E a 9820 131400 776600 13.38 0.9627 F a 10700 143500 705000 13.41 0.9609 G a 9960 147300 772000 14.79 0.9596 H a 12700 104800 408900 8.25 0.9586 1 17300 104100 434600 6.02 0.9575 J 10800 108900 590600 10.1 0.9551 K 15700 137300 693200 8.75 0.9550 L 15300 133400 761200 8.72 0.9611 M 11700 130800 650900 11.18 0.9581 N 13100 174900 852200 13.35 0.9603 O •«: •.; • y . .• i 12500 101200 397000 8.1 0.9597 P :.:=•.:.• • :/:•: • • 11500 116000 570800 10.09 0.9542 Q a 10600 108000 503300 10.2 0.9548 R 10300 133700 676200 12.98 0.9549 S 13200 113000 640300 8.6 0.9548 T a 27800 132300 462700 4.76 0.9393 U a 9310 153500 776600 16.49 0.9617 V a 27000 129000 377000 4.8 0.9465 w a 44000 148000 423000 3.4 0.9560 X a 25100 138400 535300 5.51 0.9544 50 Chapter 3 Experimental Equipment and Procedures The density o f each resin was determined using the densimeter as described in a previous section o f this chapter, and molecular weight distributions were obtained by performing G P C analysis with a polyethylene standard. Figure 3.12 shows a plot o f the differential molecular weight distributions for some of the resins. log MW Figure 3.12 Differential molecular weight distributions for some of the resins studied in this work. 3.4 E X P E R I M E N T A L P R O C E D U R E S 3.4.1 Shear Properties The shear properties of the H D P E resins were determined using the capillary rheometry at three temperatures, namely 180°C, 200°C, and 220°C. These temperatures were chosen to closely match the processing temperature in blow molding processes, 51 Chapter 3 Experimental Equipment and Procedures which is around 190°C. The three temperatures facilitated the calculation o f the activation energy for each resin following the time temperature superposition principle. Each experimental run began with the loading o f resin in pellet form into the heated barrel. After the barrel was filled, the piston was put in place and a pre-heat time of 360 seconds was allowed. Following the pre-heat period, the piston was allowed to travel down the barrel at a preset speed corresponding to a desired shear rate. The speed was maintained until a certain preset distance was reached, at which time the steady state force was measured and recorded through the data acquisition board. The piston was then allowed to travel at the next desired speed until the next preset distance was reached before the speed was changed again. Up to nine speeds or shear rates were allowed for each run involving a one-time sample loading. The speeds and distances were preset through the computer using the software provided with the capillary. The software also allows the interpolation o f up to three shear stress and viscosity values by fitting the experimental data with the appropriate equation, such as a power law model. After each run, the barrel, the piston and the die were cleaned thoroughly with a cloth. In addition, to avoid contamination, flushing was done prior to runs involving a different resin. A die o f L/D = 20 and D = 0.7542 mm was used throughout this work. N o Bagley or Rabinowitch correction was applied to any o f the data obtained for all the resins. This was decided to economize time and material and, since, the data are used for comparison purposes involving the same type of polymers, the effect of these corrections would then be non-significant. Hence, these experiments would yield the apparent, instead o f the 52 Chapter 3 Experimental Equipment and Procedures true values o f wall shear rate and viscosity. 3.4.2 Extrudate Swell Measurements Extrudate swell was determined as a function of shear rate and temperature by manually collecting extrudates at the exit of the die during the capillary experiments. A number o f extrudates were collected corresponding to each piston speed (shear rate) and their diameters were then measured in diameter using a digital caliper to determine the average swell corresponding to the shear rate and the temperature of the capillary run. Absolute care was taken during the collection o f the extrudates to ensure that they were not pinched in anyway. For the same reason, measurements using the caliper were done after the extrudates had cooled off to room temperature to ensure solidity. The extrudate swell was calculated as: Die Swell = ^ _ - l <317> where D is the diameter of the extrudate and D0 is the diameter of the capillary die. To ensure consistency and to minimize the error in diameter measurements due to sagging, the extrudates were cut at approximately the same distance from the die exit each time. The lengths of the extrudates were also kept to be approximately the same during each cut. A t high shear rates, melt fracture (surface distortion of extrudates) may be observed for some resins due to flow instability. In such cases, it was not possible to measure the extrudate diameter. Hence, extrudate swell data for such resins were limited to the lower 53 Chapter 3 Experimental Equipment and Procedures shear rate regimes. Also , since diameter measurements were done when the extrudates were at room temperature, the extrudate swell data do not represent the actual swell properties o f the resins as far as the absolute magnitude is concerned. In spite o f this, these data were still useful for comparison purposes between the different resins. The procedure was adopted due to its simplicity. For a more accurate determination o f extrudate swell properties, much more elaborate procedure and equipment are required [Dealy (1985)]. 3.4.3 Melt Index. Stress Exponent and Melt Flow Ratio Determinations Mel t Index, stress exponent and melt flow ratio are empirically defined parameters that are used mostly by industry for quality control purposes. In this work, these parameters were determined using the extrusion plastometer described in the previous section o f the chapter. The difference in the procedure involved the size of the load used, depending on how these parameters are defined. The procedure began with the loading of approximately 5 grams of resin in pellet form into the heated barrel with the die in place. The temperature was set to 190°C in accordance with A S T M procedure D1238-95. After the barrel was filled, the piston was put in place and the timer was switched on. A pre-heat time of 360 seconds was allowed before the dead weight load was placed on top of the piston (a lighter weight may be placed on the piston during the pre-heat period to improve the packing of the resin in the barrel). The polymer melt would then start to flow out of the die. After a reasonable length of extrudate was observed, a cut was made on the extrudate immediately at the exit o f the die and the timer was reset to zero at the same time. When a reasonable amount of 54 Chapter 3 Experimental Equipment and Procedures the polymer has been forced out o f the barrel, another cut was made on the extrudate and the timer was stopped. From the weight o f the extrudate and the time it took to flow out o f the barrel, various parameters can be calculated. To ensure that the extrusion plastometer was working properly, a test run was done with a standard resin and the results checked before the actual set of experimental runs were performed. I2, h and I21 are defined as the weights o f the polymer that flows out of the barrel in 10 minutes when dead weights of 2.16, 6.48, and 21.6 kg are used, respectively. The melt index takes the value of I2 in the unit of grams/10 minutes. The stress exponent is calculated as: SEx. = (3.18) 6.48^ log| 2.16 and the Mel t F low Ratio ( M F R ) is calculated as: MFR = ^ (3-19) h 3.4.4 Mel t Strength Measurements In this work, melt strength is defined as the maximum weight o f itself that a polymer melt is able to support without breaking for 3 minutes at 190°C and a preset load. A l l measurements were done using the extrusion plastometer described in section 3.2.2. The procedure for melt strength measurements was similar to that described in section 3.4.3. Approximately 5 grams o f resin was pre-heated in the barrel at 190°C for 55 Chapter 3 Experimental Equipment and Procedures 360 seconds before a dead weight o f 18.3 kg was placed on top of the piston. After the piston had traveled a certain distance, a cut was made immediately at the exit o f the die. The piston was allowed to travel further down the barrel until it stopped, at which time the timer was immediately reset. Careful observation was then made on the extrudate at the exit o f the die, which was allowed to hang under its own weight until it broke. A s soon as the extrudate broke, the timer was stopped and the time recorded. The weight o f the extrudate was then determined and the equipment cleaned. The procedure was repeated four times, each time differing in the distance that the piston was allowed to travel before the first cut was made on the extrudate. This allowed a set o f four different measurements o f time and the corresponding weight of extrudate to be obtained. Plotting the log o f time versus the log of extrudate weight yielded a straight line, and interpolating the extrudate weight at 3 minutes enabled the melt strength o f the polymer to be determined. A typical plot is shown in Figure 3.13. T o ensure that the plastometer was working properly, a melt index measurement was performed on a standard resin prior to the running of the actual set of experiments. 56 Chapter 3 Experimental Equipment and Procedures log (Maximum Weight, g) Figure 3.13 Melt strength measurement using the dead weight method. The value of melt strength is interpolated from the graph at time equal to 3 minutes. 3.4.5 Extensional Rheology In this work, constant stress (creep), instead of constant rate experiments, were performed to determine the extensional properties o f the resins, since such experiments are more applicable to the process of blow molding. A s mentioned earlier, during parison formation, the polymer melt is hung from the die exit for a finite period of time before it is molded. During this period, the polymer melt is subjected to a constant downward force due to gravity, and this deformation is more appropriately described as a constant stress rather than a constant rate extensional deformation. The temperature at which the extensional properties were determined was set at 150°C and the stresses used were 7 kPa, 5 kPa, and 3 kPa. The boiling point of the oil in the dewar provided the limit as far as temperature is concerned. 57 Chapter 3 Experimental Equipment and Procedures Approximately 16 grams o f resin were required to mold 15 rod shaped samples for the extensional rheometer. The mold assembly was first heated to 172°C before the resin was introduced. After the resin was put in the mold, a preheat period o f 10 minutes was allowed with the spacer still placed in the assembly. The vacuum was then immediately switched on. After a period o f 10 minutes had elapsed, the spacer was removed and the resin was heated for an additional 10 minutes. The whole assembly was then pressed in a Carver press at a load of approximately 7000 kg and let stand under pressure for 10 minutes. The pressure decreased as the polymer in the mold relaxed. After 10 minutes, the load was brought up again to 7000 kg and the polymer melt was allowed to relax for another period o f 10 minutes. A final adjustment of the load to 7000 kg was then made before it was suddenly removed. Following this, an annealing period of 1.5 hours was allowed with the assembly still heated at 172°C to free the polymer in the mold from any built-in stresses. After 1.5 hours, the heater was switched off. When the whole assembly had cooled off, the sample was removed from the mold, not forgetting to switch off the vacuum beforehand. Figure 3.14 shows a diagram o f the sample before and after cutting. Sample preparation is critical in this procedure since it is important to ensure that the samples contain no air voids. To both ends of each cut sample, etching was then done with a previously stirred mixture o f 2 grams K2Cr207 in 100 grams of 98% H2SO4. A period o f approximately 40 minutes was allowed for the mixture to work on each end o f the sample before it was washed off the sample using distilled water. After the samples were etched, drying was done by placing the samples in the oven at 100°C for approximately 15 minutes. After the samples were completely dried, they were glued to the pulling clips using 5-minute 58 Chapter 3 Experimental Equipment and Procedures epoxy in the gluing fixture. The gluing assembly was then allowed to stand for a day before the samples were removed from the fixture. After the samples were glued to the pulling clips, estimates o f sample volumes at 150°C were determined. This was done by measuring the volume o f each sample at room temperature and using the linear coefficient of expansivity to correct for the change in temperature. The volume at 150°C was needed to determine the cross-sectional area o f the sample at each time so that the pulling force can be changed accordingly to maintain constant stress. Obviously, this is permissible assuming incompressibility and a uniform pull o f the sample. Figure 3.14 A schemaUc diagram of extensional sample before and after cutting. The extensional rheometer was calibrated for length and force according to the equipment manual before each run was conducted. After the calibration and when the temperature of the oil in the dewar had reached 150°C, the sample was quickly mounted 59 Chapter 3 Experimental Equipment and Procedures into the pulling fixture b y first adjusting the fixture to the approximate length of the sample. The high temperature in the dewar would cause the sample to expand and hence, create a compression force on the pulling fixture. To mitigate this force, the fixture was moved upward until the indicated force was close to zero. As soon as this was achieved, a timer was switched on and a pre-heat time of 3 minutes was allowed. After the pre-heat period was over, the pull was started. Data on sample length, stress, and strain as functions of time were then obtained as soon as the sample had been pulled to a maximum preset length, or a time of 300 seconds had elapsed, whichever occurred first. To ensure reproducibility, a number of runs were performed for the same sample at the same experimental conditions. A representative set of data for the particular run, possibly one from the most uniform pull, was then chosen for analysis. 3.4.6 Pillow Mold Experiments The sagging and swell characteristics of resins E, F, and G were determined using the IMPCO B-13 Blow Molder as described in the previous section. The temperatures of the rear and front barrel of the extruder were set to 188°C and 199°C, respectively. The melt temperature was determined to be approximately 215°C and the die head was kept at 205°C. The timer was set to 12 seconds and 4 seconds for air blowing and exhausting, respectively. The screw speed of the extruder was kept constant at 125 RPM. The wall thickness (position of the mandrel) was set arbitrarily and kept constant for all runs. The amount of resin that was extruded for each blowing cycle was also kept the same for each run by setting the shot size constant. The shot size determined how far the ram was set back during the accumulation of polymer melt before extrusion. Since the 60 Chapter 3 Experimental Equipment and Procedures die gap was kept constant, the ram rate determined the shear rate experienced by the polymer melt. The ram rate was determined by the shot size and the parison drop time (i.e. the time required to form the parison, or the time needed to extrude all the resin in the accumulator). Parison drop times of 1, 3 and 5 seconds were used in this work. Prior to each run, flushing was done and the unit was tested with a standard resin. 61 Chapter 4 Results and Discussion 4 R E S U L T S A N D D I S C U S S I O N 4.1 I N T R O D U C T I O N This chapter is divided into four main parts. In the first part, rheological results are presented. The second part o f the chapter then focuses on the effects of molecular parameters on the processability o f H D P E resins. In the third section, the rheological and processing implications o f melt index, stress exponent and melt flow ratio are discussed. Finally, a section is devoted to the qualitative analysis o f the implications o f rheology on processability. Before proceeding to the presentation and discussion of experimental results, a brief comment has to be made with regard to the variety of samples chosen for this work. The main objective o f this work was to study the effect of molecular structure on the rheology and processability o f H D P E blow molding resins using resins having a broad range of molecular parameters, in terms of Mw and the molecular weight distribution. A number o f studies had been done previously, some o f which were partially successful in relating rheology or processability to molecular parameters. However, these studies have been performed only on resins with narrow ranges of molecular parameters. To our knowledge, no study that relates the rheological and processing behavior o f resins to the broad range of molecular parameters has been reported previously. Due to the limitations in catalyst technology and process parameters, however, it is not possible to produce resins with broad ranges o f molecular parameters using only one type o f technology. For example, there is no single catalyst that is capable o f polymerizing a resin having a wide range o f Mw and polydispersity values. For example, 62 Chapter 4 Results and Discussion solution polymerization technology produces resins with low to medium Mw. With a slurry type o f polymerization technology, on the other hand, it is possible to produce resins with somewhat higher Mw and with a polydispersity range between six and ten. The gas phase catalyst technology is capable of producing resins with high Mw and polydispersity values less than five, or more than fourteen, depending on the catalyst used [Goyal (1998)]. Due to this difficulty, it was, therefore, necessary to use commercial resins produced from a number of different technologies in this study. Since commercial resins from different technologies are used, it wi l l then be difficult to systematically determine the effect o f individual parameters on resin flow properties or processability, or to perform factorial design experiments. Often, there are more than two parameters that are different for a given set of resins. Therefore, in this work, multiple regression analysis was used to study the influence of molecular parameters on the rheological and processing behaviors o f the resins, wherever possible. Otherwise, qualitative analyses were conducted to investigate the influence of these variables. However, it should be noted that the objective of performing multiple regression analysis was to obtain general trends on various rheological and processing properties of the resins, with the molecular parameters being the independent variables. The analysis was not intended to produce predictive mathematical models. Hence, in the regression analysis, the dependencies o f various rheological and processing parameters on molecular structure was kept as simple as possible. Also , in some cases, outliers in the data set were not included in the analysis. 63 Chapter 4 Results and Discussion 4.2 R H E O L O G Y 4.2.1 Shear Properties The shear properties of the resins were determined using a capillary rheometer at 180°C, 200°C, and 220°C. The range of shear rate used was from approximately 5 s"1 to 900 s"1. Duplicate runs on some resins indicate excellent reproducibility of results. For example, variations in the viscosity data were found to be much less than 5%. Figure 4.1 shows the apparent shear viscosity plot for one of the resins. 104 « o u •52 V . TO o •c <0 c c a 103 102 J 1 1 1 1 - 0 " S 0 1 ' • i i i i i 11 - 1 1 1 1 1 1 1 1 _ Resin R - A u D A 0 L/D=20, D=0.7542 m m -- • A o --2 0 a A o • * o g -• T=180°C o ~ 2 o : • T=200°C g T=220°C o 2 o -O T=180°C- Reproducibility Run • T=200°C -Reproducibility Run A 1 1 1 1 1 T=220°C -• I Reproducibility Run i i i i i i 11 i i i i i i i 11 101 102 103 Apparent Wall Shear Rate (s1) Figure 4.1 Reproducibility of apparent flow curves for resin R, determined at 180°C, 200°C, and 220°C. Data variation at each shear rate is estimated to be less than 5%. 64 Chapter 4 Results and Discussion To determine the effect of molecular weight on shear viscosity, the resins were grouped according to their polydispersities. Viscosity plots such as those shown in Figure 4.2 were then analyzed. Previous studies have shown that increasing Mw while keeping the polydispersity constant increases the zero shear rate viscosity, or the viscosity at lower shear rates. However, this is not very apparent in the data obtained using the capillary rheometer. One reason is that the shear rate range does not cover sufficiently low values. To see this effect, experiments may have to be carried out using other equipment such as parallel plate rheometer, which is capable of generating very low deformation rates. Another possible reason is the one mentioned previously. Variations in other molecular parameters, such as Mz, may be affecting the viscosity profiles in different way. Moreover, the number of resins in each polydispersity group is too small for a definite conclusion to be drawn. 104 o o .*» TO O •C </) <«* c SJ 8: 103 102 1 1 1 1 1 1 '-1 ;. * 1 1 i 1 1 1 1 1 J 1 1 1 1 f t 1 1 | _ Constant Pl=13.4, T=200°C -UD=20, D=0.7542 mm I • • • • • • • * -: • Resin E - M w = 131400 • i • • Resin F - M w = 143500 i i i i i i Resin N -1 M w = • 174900 i i i i i t 1 i i • i i i i i i 1 101 10 2 10 3 Apparent Wall Shear Rate (s~1) Figure 4.2 Apparent flow curves for resins with similar polydispersities determined at 200°C. 65 Chapter 4 Results and Discussion Plotting the shear viscosity data for resins with similar molecular weight yields results such as those shown in Figure 4.3. Similarly, it is difficult to extract a reasonable trend from this figure. A s the polydispersity o f a resin is increased, the viscosity curve is expected to be steeper, as has been shown in previous studies. In this work, however, due to the limited number o f resins having similar Mw, such a conclusion cannot be drawn as easily. o u 3. v. «J « •c CO c £ 8: 104 103 \ -102 1 1 I 1 1 1 — • - • v Resin T - PI • Resin L - PI = Resin R - PI Resin E - PI • • i i i i 11 i i i—i—i—i i i j Constant Mw=133000, T=200°C L7D=20, D=0.7542 mm v 4.76 8.72 = 12.98 = 13.38 A V J I I I I I I B V I V -J I I I I I I 101 102 Apparent Wall Shear Rate (s1) 103 Figure 4.3 Apparent flow curves for resins with similarMw determined at 200°C. 66 Chapter 4 Results and Discussion To overcome the problem o f having too few resins in each group having similar polydispersity or Mw, another approach to data analysis was used. Based on the knowledge from previous studies that the viscosity profile is affected mainly by polydispersity and Mw, it was then decided that multivariable regression should be performed on the viscosity data. B y performing such analyses, it w i l l then be no longer necessary that the resins be grouped according to polydispersity or Mw. A statistical software, S T A T G R A P H I C S P L U S v2.0, was used in this work for this purpose. In each analysis, the magnitude o f the apparent shear viscosity at a particular shear rate was regressed with polydispersity and Mw as the independent variables. The result from each regression was statistically analyzed by the software to ensure that the independent variables were in fact significant, and that the correct form o f dependency was used for each independent variable. A good correlation could not be found between shear viscosity and the two independent variables, polydispersity and Mw. However, when the data analyzed were only those for resins manufactured using the same technology (e.g. technology ' a ' in Table 3.1), clear trends were observed. A plot o f the predicted values versus the observed values for one analysis is shown in Figure 4.4. Figure 4.5 shows the predicted and observed viscosity values from the same analysis as a function o f polydispersity. The need to include only the resins produced from the same technology in the analysis implies that different technologies produce resins with non-comparable shear viscosity profiles. Hence, i f molecular parameters are to be used to predict the relative profile of shear viscosity, it is important to make sure that the resins are produced from the same technology. Surprisingly, this finding has not been reported in any previous studies. 67 Chapter 4 Results and Discussion 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 Observed Apparent Viscosity (Pa.s) Figure 4.4 Predicted apparent viscosity values at 100 s"1 and 200°C as determined using STATGRAPHICSv2.0. Only resins manufactured using technology 'a' are included in the analysts. «> o u C £ (0 Q. Q. «t "O o £ QJ </> •o O t3 C 05 "O o u '"5 I 3500 3000 2500 2000 1500 1000 500 i i l i i i i 0 • o o I I I I I I I I I I I I I I I I I I I I I I I I I J Shear Rate = 100 s"1 T=200°C, UD=20, D=0.7542 mm tlA = 39S.5 + 0.0185'M,, - 117.98*PI, r*=88% o Predicted • Observed o o ' i I i i I i i i i I i ' ' i I ' ' ' i I i i i ' I ' ' i ' I ' i ' • I i i i i 2 4 6 8 10 12 14 16 18 20 Polydispersity Index Figure 4.5 Predicted and observed apparent shear viscosity at 100 s"1 and 200°C. 68 Chapter 4 Results and Discussion For the range o f shear rates studied in this work, it was found that increasing Mw at a constant polydispersity tends to increase shear viscosity, while increasing polydispersity at a constant Mw decreases the viscosity. This is the expected trend, considering the molecular properties o f a polymer. Increasing Mw implies that, on average, molecules are longer in size. This means that there are more entanglements between molecules, which results in greater resistance to flow, or higher viscosity. A s far as the effect o f polydispersity is concerned, the trend is consistent with previously published results that polymer melts exhibit steeper viscosity curves with increasing polydispersity (increasing shear thinning behavior) [Dealy and Wissbrun (1995)]. However, at much lower shear rates, viscosity is expected to increase with increasing polydispersity. To see this effect more clearly, Figures 4.6 and 4.7 were prepared using the equations obtained from the regression analysis performed at each shear rate. Arbitrary values o f polydispersity and Mw were used to simulate the plots. In Figure 4.6, it can be seen that increasing Mw at a constant polydispersity increases the shear viscosity, and in Figure 4.7, increasing polydispersity at a constant Mw decreases the shear viscosity. The shear thinning effect o f polydispersity on the viscosity can also be seen in Figure 4.7. It is interesting to note that, in Figure 4.7, although the polydispersity index covers a broad range o f values, the differences in the shear thinning behavior depicted by the simulated viscosity curves are comparatively less obvious. This is due to the way polydispersity index is defined, which can be misleading. B y defining the polydispersity index as the ratio o f M w to M„, only changes in the molecular weight distribution which involve smaller molecules are reflected in the value of the index. Altering the molecular weight distribution by changing the concentration of larger molecules wi l l not be 69 Chapter 4 Results and Discussion Figure 4.6 Apparent shear viscosity curves simulated at constant PI using the regression relationship at each shear rate. Figure 4.7 Apparent shear viscosity curves simulated at constantMw using the regression relationship at each shear rate. 70 Chapter 4 Results and Discussion reflected in the value o f the polydispersity index. Hence, it is often more useful to consider the whole molecular weight distribution curve than to only consider polydispersity index as a measure o f the breadth o f the molecular weight distribution, especially when the distribution curve is skewed. There may be portions of the distribution which are not reflected by the polydispersity index. These may significantly affect the rheology and processability o f a resin. To further determine the effect o f molecular weight distribution ( M W D ) on the shear properties o f the resins, a F O R T R A N program that calculates the normalized areas of slices under the differential M W D curves was written. The objective o f this approach was to determine the portions o f the distribution that affect the shear properties the most. The differential M W D curves obtained from G P C for all resins were first arbitrarily divided into several slices as shown in Figure 4.8. The normalized area corresponding to each slice was then calculated using the 4-panel Adaptive Newton-Cotes numerical integration method (normalization was done by dividing the area o f each slice by the total area under the M W D ) . Splines with fitted ends were used to facilitate this integration (see Appendix A ) . The critical molecular weight range would then be implied by the slice, which, for all resins, has a normalized area that correlates the best with, in this case, the shear viscosity. A l l possible combinations of molecular weight ranges (slice areas) were considered for correlation in the program. Correlating the normalized areas with the shear viscosity o f all resins determined at 5 s"1 and 200°C indicated that there are two portions of the M W D that are relatively critical in affecting the shear viscosity. Figure 4.9 shows 3-D plots of the correlation 71 Chapter 4 Results and Discussion 2 3 4 5 6 7 LogMW Figure 4.8 Determination of molecular weight ranges that are critically affecting a certain property of a resin. Slices were made arbitrarily. coefficient, obtained from correlating the shear viscosity to the normalized areas bounded by the two molecular weight limits, as a function o f the upper and lower molecular weight limits. It can be seen from Figure 4.9(a) that one critical portion is in the molecular weight range of approximately 9,000 to 22,000. In this range of molecular weight, the correlation was found to be negative, i.e. increasing this portion of the distribution results in a decrease of viscosity. This is the lower molecular weight range that is reflected b y M „ . This finding is consistent with the previously described analysis, which shows that lower viscosity is obtained when polydispersity is increased or Mw is decreased. B y increasing the portion of the M W D that lies on the left side of the peak, M„, Mw, and Mz are reduced, while polydispersity is increased (M„ is more significantly decreased than Mw- M W D is broader), and hence, lower viscosity is observed. 72 Chapter 4 Results and Discussion (a) (b) Figure 4.9 Correlation coefficients relating various molecular weight ranges to shear viscosity at 5 s"1 and 200°C (a) negative correlation (b) positive correlatioa Correlation coefficient (x,y) = COV (x,y)/a x.a y 73 Chapter 4 Results and Discussion From Figure 4.9(b), the other critical molecular weight can be determined to range from approximately 140000 to 900000. In this case, the correlation is positive. This molecular weight range lies on the right of the M W D peak. It implies the concentration o f larger molecules and is reflected by Mw and Mz. Increasing this portion of M W D , increases the weight average molecular weight, M w , and hence, higher viscosity is observed. B y doing this, however, the M W D is also broadened and, hence, polydispersity is increased. The previously discussed effect o f polydispersity on the shear thinning behavior of the resins is not observed using this analysis. Therefore, it seems that the observed effect of polydispersity on shear thinning is only true i f changes in polydispersity are made by changing the concentration of smaller molecules. Broadening the M W D by increasing the concentration of larger molecules does not seem to affect shear sensitivity significantly, although it increases the magnitude o f the shear viscosity. Comparing Figure 4.9(a) to Figure 4.9(b), one can see that the variation in the correlation coefficient is stronger in the case o f the lower molecular weight ranges [Figure 4.9(a)]. This implies that shear viscosity is more sensitive to changes in the concentration of smaller molecules. To determine the effect of temperature on shear viscosity, the activation energy term, Ea, in Equation 1.14 was calculated using the time-temperature superposition principle. Resins with greater values of Ea have flow characteristics which are more sensitive to temperature changes. A F O R T R A N program was written to facilitate the shifting o f data and is included in Appendix B . Instead of calculating the shift factor by performing a 74 Chapter 4 Results and Discussion two way shift on the shear viscosity data, the program considers the shear stress data and performs only a horizontal shift. After the master curve data on shear stress and shifted shear rate is determined, the master shear viscosity curve is calculated. The golden search optimization method was used in the program to determine the shift factor that minimizes residuals involved in each shift. In this analysis, a reference temperature of 180°C was used throughout and the values of Ea were found to range from 20 kJ/mol to 28 kJ/mol which are comparable to other reported literature values for HDPE [Van Krevelen (1990)]. Figure 4.10 shows a master curve for one of the resins and the fit of shift factors to the Arrhenius type of equation. 104 h CO to o u .2 5 k. co o </> c 2! CO a 103 102 10° ~i—i—i—i 1 1 1 1 1 1 — i — i — i 1 1 1 1 1 1 — i — i — i 1 1 1 1 1 A m ^  Resin C -1 an°, Tref=180°C,LyD=20, D=0.7542mm j mA -A • T=180°C • T=200°C * T=220°C j i i i i i i 11 i i i i i i 111 i i 11 106 105 £ 101 102 Apparent Wall Shear Rate (s'1) 103 104 (a) 75 Chapter 4 Results and Discussion Cb) Figure 4.10 (a) Master curve and (b) Arrhenius fit generated by the F O R T R A N program. To determine the effect of M W D on the flow sensitivity of the resins, multivariable regression was again performed. The activation energy was set as the dependent variable and M„, Mw, and/or Mz as the independent variables. Using the results for all resins, it was found that no general trend could be extracted. However, if the activation energy data were grouped according to the polydispersity ranges of PI>J0, 8<PKJ0, and PK8, good correlations were obtained. The need to separate the resins into three groups has also been reported by Kazatchkov et al. (1997) in his melt fracture study of LLDPE's. It seems that the resins undergo a change in behavior around the polydispersity bracket of 76 Chapter 4 Results and Discussion eight to ten. A change in the general trend in some properties, or a discontinuity in the general trend seems to happen at approximately this range of polydispersity. For polydispersity greater than ten, it was found that Ea could be related to the molecular parameters with an excellent correlation coefficient. Figure 4.11 shows a plot of the observed and predicted values. In mathematical form, the correlation can be written as Ea = 1.69 + 7.8E - 4 • M„ + 1.02E - 5 • Mz +1.31E6 / Mw (4.1). Observed E a (kJ/mol) Figure 4.11 Observed and predicted Ea values as obtained from STATGRAPHICS v2.0 (PI>10). From the equation, it can be deduced that by increasing M„ and Mz, shear properties become more sensitive to temperature change. The opposite can also be deduced when 77 Chapter 4 Results and Discussion Mw is increased. The trend, however, wi l l be more useful i f it is expressed in terms o f polydispersity index, Mw, and Mz, which are more commonly used to describe a molecular weight distribution. The effect of these variables is summarized in Figure 4.12. In Figure 4.12(a), the polydispersity index is fixed arbitrarily at thirteen, while in Figure 4.12(b), 130,000 is arbitrarily used as the value of Mw. One can see that, at constant polydispersity index, increasing Mw affects the magnitude ofEa non-linearly, but in a relatively non-significant manner. The effect of Mz is more significant: increasing Mz increases the magnitude of Ea. This same trend is also observed at constant Mw, as shown in Figure 4.12(b). The effect o f polydispersity index is also shown in the figure. One can see that the effect o f polydispersity index on Ea is relatively significant, and that increasing the polydispersity index decreases the temperature sensitivity of shear flow. O f course, this is achievable only i f changes are made in the concentration o f smaller molecules, considering the definition o f polydispersity index. (a) 78 Chapter 4 Results and Discussion Figure 4.12 The effect of PI, Mw and Mt on temperature sensitivity of shear flow properties. PI and Mw were arbitrarily set to be constant in (a), and (b), respectively. This analysis is only particularly useful for resins which have skewed molecular weight distributions, as is the case with most of the resins studied in this work. For resins with molecular weight distributions which are describable by Gaussian curves, the individual effect o f molecular parameters is not as obvious. This is true since, for a Gaussian distribution, these variables are very dependent on one another. For example, it is not possible to increase Mw while keeping polydispersity index constant without increasing Mz and decreasing M„ proportionally, as shown in Figure 4.13. Hence, for resins with Gaussian - type molecular weight distributions, the effects of molecular parameters have to be considered collectively. 7 9 Chapter 4 Results and Discussion 2 3 4 5 6 7 logMW Figure 4.13 Hypothetical M W D showing the shift in Mw, and Mz at constant PI. For the polydispersity range between eight to ten, the correlation obtained was Ea = 3 5 . 3 - 1 . 3 1 E - 3 - M „ + 5 . 5 3 E - 5 - M w (4.2). Similarly, a good r-squared statistic (degree of freedom corrected) was obtained in this case. Figure 4.14 shows the observed versus predicted Ea values obtained by using the equation. The analysis indicated that the effect of Mz is statistically non-significant. From the equation, it can be deduced that increasing polydispersity while keeping Mw constant, increases Ea. This, o f course, is true only i f the broadening o f the molecular weight distribution is done by decreasing M„ (or by increasing the concentration of smaller molecules). I f polydispersity is increased by increasing Mz, no significant effect should be observed. From the equation, it can also be deduced that w h e n M w is increased 80 Chapter 4 Results and Discussion at a constant polydispersity, temperature sensitivity is reduced. This is the case since, for polydispersity between eight and ten, the second term, which can be written in terms o f My/PI, dominates the equation. 21 22 23 24 25 26 27 28 Observed E (kJ/mol) Figure 4.14 Observed and predicted Ett values as obtained from STATGRAPHICS v2.0 (8<PI<10). For the lowest polydispersity range (P/<8), no good correlation between Ea and the molecular parameters could be obtained. Moreover, for most resins in this polydispersity range, melt fracture was observed, making it hard for the shear stress curves to be superposed. Hence, no definite trend can be stated. In summary, increasing Mw at a constant polydispersity increases shear viscosity, while increasing polydispersity at a constant Mw increases the shear sensitivity o f the 81 Chapter 4 Results and Discussion viscosity curve. This is true only i f polydispersity is increased by increasing the concentration o f smaller molecules. More generally, it was found that shear viscosity is most strongly affected by molecules in the molecular weight ranges o f approximately 9,000 to 22,000 and 140,000 to 900,000. Decreasing the concentration o f molecules in the lower molecular weight range or increasing the concentration in the higher range has the same effect of increasing the shear viscosity for the shear rate range studied in this work. However, the shear flow properties of the resins are more significantly affected by the concentration o f the smaller molecules. It was also found that the temperature sensitivity o f the viscosity ranged from 20 kJ/mol to 28 kJ/mol. For resins with polydispersity less than eight, no definite conclusion can be drawn with regard to the effect o f molecular parameters on Ea. For resins with polydispersity greater than eight, it was found that, at constant Mw, increasing polydispersity increases Ea with the effect entirely contributed by the increase in the concentration o f smaller molecules. Above a polydispersity of ten, however, the concentration of larger molecules becomes more important. Increasing the concentration of smaller molecules now tends to decrease Ea, while increasing the concentration of larger molecules tends to increase Ea. Comparing the values o f Ea for resins with PK10 and Pl>10, one can see that, generally, the range o f activation energy values for resins with PK10 is smaller than that for resins with PI>10. This is consistent with what is observed in the actual industrial blow molding process, in which resins with broader molecular weight distributions are found to be more sensitive to changes in temperature relative to the resins with smaller polydispersity index values [Goyal (1998)]. It is interesting to note that although different technologies 82 Chapter 4 Results and Discussion produce resins with non-comparable viscosity profiles, the activation energy of these resins can still be correlated regardless of the technology used. 4.2.2 Extensional Flow Properties The extensional properties of the resin were determined through 'constant stress' (creep) experiments at 150°C. Three stress levels were used in each run: 7 kPa, 5 kPa, and 3 kPa. The experiments yielded Hencky strain versus time data, some of which are plotted in Figure 4.15. 2 h o c CD 5: i i i i | I i I i | r Resin J-T=150°C Stress = 3 kPa — Stress = 5 kPa L Stress = 7 kPa ~i i i | i i i i l i i i r / / / ' / -1 1 I 1 1 I I I I I L 50 100 150 200 Time (s) 250 300 Figure 4.15 Hencky strain as a function of time determined at different stress levels. The Hencky strain at a particular time signifies the degree of deformation that the polymer is experiencing when subjected to a particular stress for that period of time. 83 Chapter 4 Results and Discussion Hence, it reflects the melt strength properties of the polymer. For the same stress level, a larger Hencky strain at a particular time implies that the polymer has a lower melt strength. From Figure 4.16, it can be seen that increasing Mw decreases the Hencky strain. This is not surprising since increasing Mw means an increase in molecular entanglements and hence, greater melt strength. 2h-•S t c 1 © a: 1 1 I 1 1 1 1 1 I 1 . 1 1 1 1 Constant Pl=13.4 • • • y , i i i i I i i i i • T=150°C, Stress = 3 kPa A / / \ Necking / -• Resin E = 131400 / ' — - Resin F - M w =143500 / ' / / - Resin N =174900 - / I ^ I i i i I t i i i I i i i i I i i i i I i i i i I i i i i 50 100 150 Time (s) 200 250 300 Figure 4.16 The effect of Mw on Hencky strain. To obtain extensional viscosity data, the strain versus time curves were fitted with suitable polynomial equations, which were then differentiated to obtain the extensional rate data. Knowing the stresses and the extensional rates, viscosities could then be calculated. For one group of resins with similar polydispersities, the extensional 84 Chapter 4 Results and Discussion viscosities are plotted versus strain in Figure 4.17. It can be seen that increasing the molecular weight has a clear effect on the magnitude of the extensional viscosity. Higher molecular weight materials exhibit greater tensile viscosity as expected. A s the molecular weight o f the polymer is increased, more entanglements wi l l occur, resulting in an increase in viscosity as the strain increases. Figure 4.18 shows the molecular weight distributions o f these resins. It can be seen from the figure that it is the concentration of larger molecules which is contributing most to the observed effect (although this is not really reflected by Mz values). The same was also found for other sets o f resins. Figures 4.19 and 4.20 illustrates this finding based on another set o f resins. to" m to o o .<* "TO c o to c 2 5e+6 4e+6 3e+6 2e+6 1e+6 0e+0 _ . . 1 • 1 • i i i | i i • Constant Pl=13.4, T=150°C I • Stress=3 kPa \ — A A • • Resin E-M w = 131400 -A • Resin F-M w = 143500 ~-A Resin N-M w = 174900 1 -— t -- AA A A A • • • • • • • • • • • • • • • • | • i I i i 1 3  2 Hencky Strain Figure 4.17 The effect of Mw on tensile viscosity. 85 Chapter 4 Results and Discussion o 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 I | Cons tan t PI = 13.4 Resin E - M w = 131400 . / ' \ ' ^ Resin F - M w = 143500 ;j /"'<^\ Resin N - M w = 174900 jj/ ill I Hi j / \ '••\ \ logMW Figure 4.18 Differential molecular weight distribution for resin E , F, and N . 3.5e+6 3.0e+6 ^- 2.5e+6 w 8 2.0e+6 3 1- 5 e + 6 c ,o § 1.0e+6 •2 X 5.0e+5 0.0e+0 — | I 1 1 1 Cons tan t Pl=8.5, T=150°C Stress=3 kPa • Resin K - M w = 137300 • Resin L - M w = 133400 Resin S - M w = 113000 V Resin H - M = w 104800 • Resin O - M w = 101200 i t * J i l l_ i i i_ _i i i i_ 0.0 0.5 1.0 Hencky Strain 1.5 Figure 4.19 Effect of Mw on tensile viscosity. 86 Chapter 4 Results and Discussion Figure 4.20 Differential molecular weight distribution for resin K, L, S, H, and O. The relationship between extensional viscosity and zero shear rate viscosity cannot be established using the current set of results, since deformations were not low enough to be in the linear viscoelastic regime. In addition, it was found that the effect o f polydispersity on extensional viscosity is not very clear from this data. Figures 4.21 and 4.22 show the extensional viscosity data for a group o f resins with similar M w , and their molecular weight distribution curves, respectively. It can be seen that the effect of polydispersity depends on the part of the distribution that is broadened. It appears that the extensional viscosity is significantly influenced by the concentration of larger molecules, which is consistent with the study reported by Munstedt and Laun (1981). 87 Chapter 4 Results and Discussion Also , as wi l l be shown later, melt strength or extensional flow property is also affected by resin density. 3e+6 W 2e+6 £ 2e+6 to O o S £ 1e+6 (0 c •2 1e+6 to c l2 5e+5 0e+0 - i 1 1 r T - i 1 1 r 0.8 0.7 0.6 ^ 0.5 5 0.4 0.2 0.1 0.0 Constant ^=133000, T=150°C Stress=3 kPa m • Resin I - PI = 8.72 • Resin M - PI = 11.18 A Resin R - PI = 12.98 * v Resin E - PI = 13.38 • Resin C - PI = 14.3 * A A _ A A ' • • A * . ^ % • * * * • • J A j H y V V V " - - ~ • i • ' _1 1 I L i i i i 0.0 0.5 1.0 Hencky Strain Figure 4.21 The effect of PI on tensile viscosity. 1.5 Constant ^=133000 — Resin L - PI = 8.72 — Resin M - PI = 11.18 — Resin R- PI = 12.98 — Resin E - PI = 13.38 Resin C - PI = 14.3 1 2 3 4 5 6 7 log MW Figure 4.22 Differential molecular weight distribution for resin L, M, R, E, and C. 88 Chapter 4 Results and Discussion It is emphasized that the extensional viscosities in this work were determined using constant stress experiments. Hence, the typical viscosity versus strain rate graph is not useful, since, for the same stress level, the resulting viscosity curves corresponding to different resins w i l l coincide. The viscosity wi l l be proportional to the inverse o f strain rate with the proportionality constant being the stress level. This is obvious from Equation 4.3. Since GE is constant, extensional viscosity wi l l be related to strain rate by the equation r]E=aEle (4.3) regardless o f the rheology o f the resins. B y using the data on Hencky strain, the effect of polydispersity on melt strength was determined. For a group o f resins with essentially constant Mw, the Hencky strains at different times are plotted versus polydispersity. The result for one set o f resins is shown in Figure 4.23. Although, there is a fair amount of scatter in the plot, it can generally be seen that Hencky strain decreases as polydispersity is increased to about nine (the scatter can be attributed to experimental error and the effects of other parameters such as density, as w i l l be discussed later). As polydispersity is increased further beyond nine, the Hencky strain starts to increase. At higher polydispersity, however, the effect dies off as shown in Figure 4.24 for another set of resins. Therefore, in terms of melt strength, it implies that increasing polydispersity up to about nine increases the melt strength o f the resin. Increasing polydispersity beyond nine, however, decreases the melt strength, and at higher polydispersity, the melt strength is essentially not further affected by M W D . 89 Chapter 4 Results and Discussion 4 6 8 10 12 14 16 Polydispersity Figure 4.23 The effect of PI on Hencky strain at different times (strain rates). At shorter times, the effect of polydispersity is non-significant 9.96 s 49.8 s 99.6 s -v— 149 s - • - 199 s .s u «= 1 o a: 16.5 Constant Mw=154000, T=150°C Stress=3 kPa _1 I I L I I I l_ 17.0 Polydispersity 17.5 Figure 4.24 The effect of large PI on Hencky strain. 90 Chapter 4 Results and Discussion This observation again indicates the change in behavior experienced by the resins in the polydispersity range o f about eight to ten, as discussed previously and observed by Kazatchkov et al. (1997). It seems that at this range o f polydispersity, polymer molecules have reached a certain limit in molecular arrangements or entanglements that increasing polydispersity further would only reverse or drastically change its effect on polymer properties. This observation on melt strength is confirmed in section 4.3.1. N o theoretical reason has been found for this observation. In summary, increasing MW was found to increase the tendency o f tensile viscosity. Also , it was found that extensional flow properties o f a polymer can be related to its melt strength by considering the Hencky strain curve as a function o f time in a constant stress experiment. A t a constant MW, the effect of polydispersity on Hencky strain is such that increasing polydispersity up to about nine decreases the strain, while increasing polydispersity further reverses the trend. However, for broadly distributed resins (PI>16), it was found that increasing the polydispersity no longer affects the strain significantly. 4.2.3 Extrudate Swell Characteristics The method of extrudate swell measurements used in this work was a simple one. However, there were a number of possible error sources associated with the procedure. Most importantly, since measurements were done after the extrudates had cooled to room temperature, the absolute magnitudes of the swell data obtained were obviously not the actual swell at the processing temperature. Secondly, there was a problem associated with sagging, which was especially pronounced at low shear rates. Sagging produces 91 Chapter 4 Results and Discussion higher swell in the lower portion of the extrudates and makes them thinner near the die. Collection o f extrudates was another source of experimental error. Although extreme care was taken during the collection o f extrudates, some pinching may have happened. This also caused a problem in measuring the actual extrudate diameters using a caliper. However, the results are still useful qualitatively and comparatively. Figure 4.25 shows the reproducibility o f extrudate swell data for one of the resins as a function o f shear rate and temperature. Data at each shear rate was estimated to vary by approximately 5%. r i i i i I i i i i I i i i i I i i i i I i i i i_d 0 200 400 600 800 1000 Shear Rate (s'1) Figure 4.25 Reproducibility of extrudate swell data. To determine the effect of molecular weight on the relative magnitude of swell, the resins were grouped according to their polydispersities. Figures 4.26 and 4.27 show the extrudate swell data for two groups of resins. The molecular weight distributions for 92 Chapter 4 Results and Discussion these resins are plotted in Figures 4.28 and 4.29, respectively. One can see that although the molecular weight distributions o f the resins are very similar, extrudate swell data varies relatively significantly. This shows that extrudate swell is very sensitive to the various parts o f the M W D . Attempts were also made to correlate the swell to various areas under the M W D curve, but it was not possible to determine the molecular weight range that is most critical in its affect on extrudate swell. This observation has also been reported by Koopmans (1988), who finally concluded that polydispersity is not a useful parameter to be used in the analysis o f extrudate swell. However, extrudate swell is expected to increase with the increase o f concentration of larger molecules. This may not show in the plot because o f the possibility of the extrudates cooling before the ultimate swell is reached. 80 T 1 1 1 1 1 1 1 f T I I I J" T 75 r- Constant PI = 10.1 Resin Q - M = 108000 H ^ - Resin P-M = 116000 4 Resin J-M = 108900 ~d 35 0 200 400 600 800 1000 Shear Rate (s'1) Figure 4.26 Extrudate swell data for resins Q, J, and P having similar PI. 93 Chapter 4 Results and Discussion 120 100 -i—i—i—|—i—i—i—i—|—i—i i i i ' 1 ' ' r Constant PI = 8.5 h- T=200°C, L/D=20, D=0.7542 mm 1 to •8 T — i— i—r Resin K-M = 137300 j Resin L-M = 133400 A — Resin S-M w = 113000 v— Resin H-M = 104800 Resin 0 - M w = 101200 _ l I I I 1 1 L . • i i i I ' I I I I 1 1 1 1— 200 400 600 Shear Rate (s'1) 800 1000 Figure 4.27 Extrudate swell data for resin K, L , S, H , and O having similar PI. o 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 Constant PI = 10.1 • • Resin 0 . - 1 ^ = 1 0 8 0 0 0 • - Resin J - M w = 108900 / .. Resin P - M w = 116000 j t \ i _L \ 4 5 logMW Figure 4.28 Differential molecular weight distribution for resin Q, J, and P. 94 Chapter 4 Results and Discussion Constant PI = 8.5 IqgMW Figure 4.29 Differential molecular weight distribution for resins K, L, S, H, and O. Plotting extrudate swell data for resins with similar Mw resulted in the same conclusion. It is not possible to qualitatively correlate extrudate swell with polydispersity, because extrudate swell is very sensitive to the different parts o f the M W D . This is shown in Figures 4.30 and 4.31. Extrudate swell is a manifestation of the elastic property of a polymer. Since the elastic property o f a polymer is very much affected by the pre-shear history experienced during the various parts of resin production, a possible reason for the inability to correlate extrudate swell to molecular parameters could be due to the different pre-shear history experienced by the different resins. This seems to be a valid reason. Previous studies 95 Chapter 4 Results and Discussion 0 200 400 600 800 1000 Shear Rate (s'1) Figure 4.30 Extrudate swell data for resins L , M , R, E , and C having similar Mw. 1 2 3 4 5 6 7 logMW Figure 4.31 Differential molecular weight distribution for resins L, M , R, E , and C. 96 Chapter 4 Results and Discussion have also failed to determine such a correlation, and possibly, the effect o f pre-shear history is more significant than previously thought. Also , differences in polymerization technology may render the resins non-comparable as far as die swell is concerned. The rate and mechanism o f termination o f a polymerization process, and the amount of additives used may differ greatly among the different technologies, and may result in a varying degree o f unsaturation in the resulting polymer. This affects the amount of crosslinking and chain scission in the polymer. Although this variation may not be reflected as being significant in the molecular weight distribution curves, the elastic properties o f the resins may very well be affected significantly. In regard to the sensitivity o f swell to changes in shear, it was found that Mz and polydispersity are useful parameters. Figures 4.32 and 4.33 show the effect of Mz and polydispersity on the slope o f extrudate swell data. In the plots, all curves are normalized linearly by shifting the data so that the swell value at the lowest shear rate coincide. One can see from the figures that at a constant Mw, broadening the molecular weight distribution by increasing the concentration of larger molecules increases the swell sensitivity in the lower shear rate regime and decreases it in the shear rate range of 350 s'1 to 700 s"1. Hence, swell is very sensitive at lower shear rates due to the presence of larger molecules. In the processing range o f shear rates (350 s"1 - 700 s"1), however, the sensitivity tends to decrease with polydispersity. Figures 4.34 and 4.35 show the molecular weight distributions of the two sets of resins. However, this analysis seems to be true only for resins manufactured using the same technology, 'a'. I f resins manufactured using other technologies are included in the analysis, as shown in Figure 4.36, one can see a breakdown in the pattern. Hence, not only does the manufacturing 97 Chapter 4 Results and Discussion technology affect the viscosity profile of a polymer, but it seems to affect its extrudate swell profile as well. 60 sp 1 L UD=20. D=0.7542 mm ^ 50 h T — i — | — i — i — i — i — f — i — i — r — i — | — i i | i | i " ' r Constant M =133000, T=200°C 200 400 Shear Rate (s~1) • Resin T - Mz=462700, Pl= 4.76 • Resin E -171^ =776600, Pl= 13.38 A Resin C -1^=848400, Pl= 14.3 , l t I l l l 1 I I I _1 1 600 800 1000 Figure 432 Sensitivity of extrudate swell to changes in shear rate. All resins are manufactured by technology 'a' (set 1). 60 ^ 50 i <D 40 2 30 s I 2 0 - i — i — i — i — | — i — i — i — i — | — i — i i " | i i | 1 r; Constant Mw=154000, T=200°C $ j L/D=20, D=0.7S42 mm 1 i i • Resin D - M2=933500, Pl=16.44 • Resin U-M2=776600, Pl=16.49 A Resin A - Ml=835400, Pl=16.83 v Resin B - M2=971500, Pl=17.42 j i i i I — i — i — u i i l l I i I ' i i i i L 200 400 600 800 1000 Shear Rate (s") Figure 4.33 Sensitivity of extrudate swell to changes in shear rate. All resins are manufactured using technology 'a' (set 2). 98 Chapter 4 Results and Discussion 0.8 0.7 0.6 ^ 0.5 O) 5 0.4 a \ 0.2 0.1 0.0 — i r Constant ^  = 133000 Resin T-PI=4.76 Resin E - Pl=13.38 Resin C - Pl=14.3 / \ A. \ \ \ ill I ill ill 11 // L_ •\ \ w 1 logMW Figure 4.34 Differential molecular weight distribution for resins T , E , and C. logMW Figure 4.35 Differential molecular weight distribution for resins D, U , A, and B. 99 Chapter 4 Results and Discussion Figure 4.36 Sensitivity of extrudate swell to changes in shear rate. Included in the plot are resins produced from different technologies. One can see a breakdown in the pattern. In summary, it was found that no useful correlation could be made between the relative magnitude o f extrudate swell and various molecular weight parameters. This could be due to the fact that the resins studied have different pre-shear history, and that the effect o f pre-shear history is more significant than that of molecular weight distribution. Also, polymerization technology may influence the degree o f unsaturation and hence, the degree o f crosslinking and chain scission in a polymer melt, which wi l l affect its elastic behavior significantly. This renders the resins produced from different technologies to be non-comparable. However, it was found that the sensitivity o f swell to changes in shear rate correlates well with Mz and polydispersity. Increasing Mz was 100 Chapter 4 Results and Discussion found to increase the shear sensitivity o f extrudate swell at low shear rates. In the processing shear rate range of 350 s"1 to 700 s'1, however, shear sensitivity is affected more significantly by polydispersity. In this range, increasing polydispersity tends to decrease the shear sensitivity o f extrudate swell. It should be reiterated that only resins manufactured using the same technology show this trend. 4.3 PROCESSABBLrrY 4.3.1 Melt Strength In this work, melt strength is defined as the maximum weight that a polymer is able to support for a period o f three minutes without breaking, and was determined using the extrusion plastometer at 190°C. The detailed procedure for the experiment was described in Section 3.4.3. In performing the experiments, it was made sure that the range of data obtained included the three-minute period so that interpolation would be required instead o f extrapolation. A plot o f maximum weight versus maximum time before breaking is shown in Figure 4.37 for some resins. The analysis o f the melt strength properties of the resins studied in this work was based on the extensional properties o f the resins. In this section, the analysis o f the data obtained using the melt indexer is emphasized. To determine the effect of molecular properties on melt strength, it was necessary to group the resins into three groups having PI>J0, 8<PI<10 and PI<8. Without doing so, it was found that no useful correlation could be obtained. This is consistent with the results obtained from the extensional experiments. From the analysis of Hencky strain as 101 Chapter 4 Results and Discussion a function o f polydispersity and time, it was found that the melt strength properties of the resins change in behavior at a polydispersity range of around eight to ten. 1.2 5» 0.2 i i i i | I I i I I l i I I I i l l l | l l I i j I I i i I i i i r~ T=190°C • Resin A • Resin B A Resin C v Resin D ' i i i ' ' ' ' • ' i i L J i i i ' ' ' ' ' ' ' ' ' -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.0 log (Maximum Weight, g) Figure 4.37 Determination of melt strength using the melt indexer. In trying to correlate melt strength directly to M„, Mw, Mz, or polydispersity, a difficulty was encountered associated with the fact that melt strength may not be affected by only a single molecular parameter, but two or more. Hence, S T A T G R A P H I C S was used for the analysis. For resins with PI>10, it was found that melt strength is dependent on Mw, M„ (hence, polydispersity), and density. The model obtained with the best r-squared statistic was Melt Strength = 75.5 + 1.47E - 5 • Mw + 4.68E - 4 • M„ - 84.78 • Density (4.4). 102 Chapter 4 Results and Discussion The plot o f observed versus predicted melt strength values using this model is shown in Figure 4.38. In Figure 4.39, predicted and observed melt strength values as a function o f Mw are shown. It can be seen that the model closely predicts the melt strength values. It should be noted, however, that this trend is valid only for the range o f Mw, M„, polydispersity and density studied in this work. ~0 1 2 3 4 Observed Melt Strength (g@3min) Figure 4.38 Observed and predicted melt strength values obtained from STATGRAPHICS (P/>10). Equation 4.4 implies that melt strength decreases with increasing polydispersity (for PI>10), which is attainable by decreasing Mn. Moreover, by rewriting M„ as MJPI, one may see that for large polydispersity, melt strength becomes essentially independent of the breadth o f the molecular weight distribution. This is consistent with the analysis performed earlier, which was based on the extensional flow properties o f the resins. The equation also shows the effect of M w . B y increasing M w , melt strength tends to increase, 103 Chapter 4 Results and Discussion * | 3 ga "g § 2 l!S 1 1 1 I I I I 1 I I I T — 1 1 1 1 1 1 1 1 1 1 1 1 1 | ' ' 1 ' T=190°C - Melt Strength = 75.5+1.47E-5.Mw+4.68E-4.M„-84.78.Density-• Observed 1^=98% '_ - o Predicted 6 — • o 8 n 1 °. 8 8 8 8 • • 1 1 1 1 1 1 1 1 1 i i i i i • • i ' i i i i i — 1 — i — i — i — i — 1.2e+5 1.3e+5 1.4e+5 1.5e+5 1.6e+5 1.7e+5 1.8e+5 M w Figure 4.39 Observed and predicted values of melt strength as a function of Mw (PI>10). which is as expected due to the increase in entanglements in molecular levels. In addition, it can be seen from the equation that melt strength is also significantly affected by density. Resins with lower density are found to have higher melt strengths, due to the greater branching associated with them. Figure 4.40 shows graphically the effect o f Mw and polydispersity on melt strength. For resins with Pl>10, higher melt strength can, therefore, be achieved by narrowing the molecular weight distribution, by increasing the weight average molecular weight, Mw, or by increasing the degree of branching. For resins with 8<PI<10, the following correlation was obtained: Melt Strength = 50.31 + 4.19 * 1 0 - 5 - M w - 55.08• Density (4-5)-A plot o f the observed and predicted values is shown in Figure 4.41. It can be seen 104 Chapter 4 Results and Discussion Figure 4.40 3-D plot showing the effect of PI and Mw on melt strength. The density value is fixed arbitrarily. Changing the density value would shift the curve upward or downward accordingly. Note that Mw is related to PI and Mn, and hence, when considering the plot, it has to be ensured that Mn is reasonable, so that PI>10. 1 2 3 4 5 Observed Melt Strength (g@3min) Figure 4.41 Observed and predicted melt strength values obtained from STATGRAPHICS v2.0 (8<PKI0). 105 Chapter 4 Results and Discussion that the equation predicts melt strength values very well . The trend implied by the equation is the same as that implied by Equation 4.4, i.e increasing Mw and decreasing density have an increasing effect on the melt strength. M„ (or polydispersity) was found not to have a significant effect for resins with this range of polydispersity. A possible reason is the narrow range of polydispersity associated with this group o f resins. For resins with PK8, it was not possible to obtain good correlation using S T A T G R A P H I C S . Therefore, there may be portions of the molecular weight distribution that are affecting the melt strength, which are not reflected in M„, Mw, or Mz. The F O R T R A N program described earlier was then used to analyze the different portions o f the M W D . The resulting correlation coefficients are plotted as a function o f molecular weight limits in Figure 4.42. It can be seen that melt strength is significantly affected by the molecules in the molecular weight range of 9,000 to 220,000 in a positive manner and by molecules in the molecular weight range of 35,000 to 55,000 in the negative manner. However, the molecular weight range of 35,000 to 55,000 is relatively small and is inclusive in the larger range o f 9,000 to 220,000. Hence, it can be said that melt strength is affected by the molecules in the molecular weight range of 55,000 to 220,000, and that increasing the concentration of molecules in this range increases the melt strength. Therefore, by increasing the polydispersity of a polymer (with PI<8) in the direction o f increasing the concentration of moderate to larger molecules, one may increase its melt strength. It is interesting to note that there is not much difference in the correlation coefficient for various molecular weight ranges, implying that all portions o f the distribution are relatively significant in their effect on melt strength. 106 Chapter 4 Results and Discussion (a) (b) Figure 4.42 Correlation rxiefficients relating various molecular weight ranges to melt strength (PI<8) (a) positive correlation (b) negative correlation. Correlation coefficient (x,y) = COV (x,y)/ax.ay 107 Chapter 4 Results and Discussion In summary, melt strength was found to increase with increasing Mw and decreasing density. Analysis had to be done separately according to the three polydispersity groups of PK8, 8<PI<10, and PI>J0. This is consistent with the results obtained when analyzing the Hencky strain data. For narrowly distributed resins, up to a polydispersity in the approximate range of eight to ten, broadening the molecular weight distribution increases the melt strength. Beyond the critical range o f eight to ten, however, increasing polydispersity decreases the melt strength. For resins with high polydispersity, further broadening the molecular weight distribution has been found to result in insignificant change in melt strength. 4.3.2 Sagging and Weight Swell Characteristics Sagging and weight swell characteristics for three of the resins, E , F, and G , were determined by performing pillow mold experiments. The results show the exact behavior o f the resins under industrial operating conditions. In performing the experiments, five replicate runs were carried out each time, and the average results were determined. Figure 4.43 shows a plot of pi l low weight versus parison number for one o f the resins to demonstrate the reproducibility o f the experiments. The numbering of the pillows is such that pillow number one is that which is nearest to the exit of the die and, hence, extruded last. From the figure, one can see that the reproducibility in these experiments was excellent. During parison formation, uneven wall thickness is attributable to both the swell and sagging characteristics of the resin. However, it is difficult to perform experiments that would differentiate the effect of either factor individually. Although useful results may 108 Chapter 4 Results and Discussion be obtained by doing pil low mold experiments, it is nonetheless not possible to determine i f a certain parison flow behavior is solely due to swell or sagging. The effect o f each factor is equally important and opposite. In the analysis o f the results, it was found that it is easier to relate weight swell and sagging to extrudate swell and melt strength, respectively. It was not possible to relate the results to the molecular parameters due to the limited number of resins used in the pi l low mold experiments. However, knowing how the molecular distribution affects melt strength and extrudate swell, one can always relate molecular distribution to sagging and weight swell. s i 18 p 17 |-16 15 14 13 12 11 10 9 8 7 6 Resin G Drop Time = 1 s Drop Time = 3 s j] Drop Time = 5 s 4 6 8 Pillow Number 10 12 Figure 4.43 Variation of pillow weight with pillow number. Multiple curves indicate replicate runs. 109 Chapter 4 Results and Discussion 11 10 9 —•— Resin E - 1 s —«- - Resin E - 3 s -8 — A — Resin E - 5 s _ —v— Resin F-1 s . Resin F-3s -—•— Resin F - 5 s -—••- Resin G - 1 s —s— Resin G - 3 s - Resin G - 5 s 7 8 10 12 0 2 4 6 Pillow Number Figure 4.44 Variation in pillow width for different resins extruded at different drop times. The results from the pillow mold experiments seemed to indicate that weight swell is proportional to extrudate swell, while sagging is inversely proportional to melt strength, which is obvious. Figure 4.44 shows a plot of pil low width versus parison number for the three resins and for parison drop times of 1 s, 3 s, and 5 s (see Figure 3.11 for definition o f pil low width). As mentioned earlier, decreasing parison drop time means faster parison formation and hence, higher shear rate. One can see that as the parison drop time is increased, the width for each pil low is decreased. This shows the effect of extrudate swell and hence, weight swell which decreases with decreasing shear rate (or increasing drop time). On the other hand, as the pillow number increases, the width of each pi l low also increases, due to parison sag, which causes the lower part of the parison 110 Chapter 4 Results and Discussion to become thicker than the upper part. It is also interesting to observe that each line has curvature, which provides an indication o f the elastic properties of the resins. The pillow which was extruded first would have spent the most time in the die and hence, it showed lower swell (fading memory effect). On the other hand, pillows which were extruded later would show greater swell. However, this does not show on the plot due to parison sag. Instead, it is shown as a curvature. One can see that as parison drop time increases, the curvature becomes less obvious, implying the significance of swell at shorter parison drop times. To relate melt strength and extrudate swell o f each resin to parison sag and weight swell, respectively, it is useful to consider Figures 4.45 and 4.46. In Figure 4.45, extrudate swell data as a function of shear rate are plotted for the three resins. In Figure 4.46, the melt strength data for the three resins are presented. It can be seen that, on average, resin E has the highest extrudate swell, while resin F has the lowest. A s far as melt strength is concerned, resin G has the highest, while resin E has the lowest. Referring back to Figure 4.44, for a parison drop time o f 1 s, it can be seen that resin E has the largest width on average, followed by resin G . This can be understood by considering the extrudate swell properties of the three resins. The short parison drop time causes extrudate swell to be a significant factor and since resin E has the highest swell in general, followed by resin G , this is the trend observed in Figure 4.44. However, the effect o f melt strength starts to show for longer parison drop times. For drop time of 3 s, for example, the curves corresponding to resin E and G are brought closer together. This is because o f the competing effect of melt strength and extrudate swell. Although resin E has the highest swell, it has the lowest melt strength. Resin G, on the other hand, has the highest melt strength. As far as resin F is concerned, it has the lowest swell and moderate 111 Chapter 4 Results and Discussion Figure 4.45 Extrudate swell profile for resin E , F, and G. 1.0 , Resin E Resin F Resin G Resin Name Figure 4.46 Melt strength values for resin E , F, and G. 112 Chapter 4 Results and Discussion melt strength and hence, its position in the plot is still unchanged. This is also true for the parison drop time of 5 s. Resin F still maintains its position in the plot, while the positions of resin E and G have been reversed. Parison sag has become the deciding factor for this drop time, such that resin G , which has the highest melt strength, shows larger pi l low width in general. Figure 4.47 shows a plot of pillow weight versus pillow number. Similar observations can be made. Weight swell and parison sag cause the opposite effects, resulting in the observed behavior. For fast parison drop times, swell dominates and hence, resin E which has the highest extrudate swell property, shows the largest weight for each pillow. A s parison drop time increases, sagging becomes increasingly important and this brings the curve corresponding to resin E , which has the highest swell and lowest melt strength, closer to that corresponding to resin G , which has a lower swell and highest melt strength. Resin F is not affected as far as relative position in the plot is concerned since it has the lowest extrudate swell behavior, but moderate melt strength. It is noted that for a parison drop time of 5 s, the weight for each pillow corresponding to resin E is still larger than that corresponding to resin G due to the difference in density. Figure 4.48 compares the densities of the three resins. It can be seen that resin G has the lowest density, making it possible for it to have larger width and smaller weight for a parison drop time of 5 s. Figure 4.49 shows a plot of pillow weight, normalized to the weight o f the first pillow, versus pi l low number for one of the resins. This plot is useful for slope analysis, which should give an indication of sagging and swell characteristics. It can be seen that 113 Chapter 4 Results and Discussion 2 4 6 8 10 12 Pillow Number Figure 4.47 Variation in pillow weight of the three resins extruded at different drop times. 0.963 Resin E Resin F Resin G Resin Name Figure 4.48 Density values for resin E, F, and G. 114 Chapter 4 Results and Discussion O 5S I ^ r Resin E vO) | — •— Drop Time = 1 s . _ Drop Time = 3 s / .0) \ — A — Drop Time = 5 s / I I i. y i Pillow Number Figure 4.49 Pillow weight normalized to the weight of pillow number one to show the magnitude of sagging. increasing parison drop time increases the slope of the curve. A steeper curve implies greater difference in the weight of the first and the last pillow, and hence, is an indication o f greater sagging, provided no swell occurs. Therefore, for the same resin, having the same extrudate swell property, it can be seen that increasing parison drop time increases sagging. When three resins having different swell properties and melt strengths are compared as in Figure 4.50, the slope of the curves is an indication of both the swell and sagging effects. One can see that for resin E , which has the lowest melt strength and highest swell, the slope is the steepest. Resin G , which has the highest melt strength, however, 115 Chapter 4 Results and Discussion has a curve which is steeper than that for resin F, which has a comparatively lower melt strength. This is due to the greater swell exhibited by resin G . The swell results in a downward force that is greater in the case of resin G . Hence, the difference in weight between the first and the last pil low for resin G is larger than that for resin F . <2 S i © 1 o 5* 1 T T Drop Time = 5 s — R e s i n E Resin F — R e s i n G k — i 1 — X 1 4 5 6 7 Pillow Number Figure 4.50 Comparison of parison sag between resin E , F, and G. Figure 4.51 shows a plot o f total parison length and weight for the three resins versus drop time. Considering the total length of the parison, the general trend is that parison length increases as drop time increases. O f course this is as expected, since longer parison time means more sagging. Also, for shorter drop times, swell dominates and tends to shorten the total parison length. The effect of swell can be seen more clearly by comparing the curves for the three resins. It can be seen that resin E , which has the highest swell, has the shortest parison length, while resin F, which has the lowest swell, 116 Chapter 4 Results and Discussion has the longest parison length. Note that the curves are closer for higher parison drop times, implying the competing effect of melt strength. When parison drop time is increased, sagging becomes more important, and the curves are expected to intersect and be ranked according to melt strength. E I I I I 1 1 119 0 1 2 3 4 5 6 Parison Drop Time (s) Figure 4.51 Total length and weight as a function of parison drop time. Although the total melt volume before extrusion was kept the same, the total length and weight of the parison were not constant, due to the fact that the mold was located at some distance below the die. The competing effect of weight swell and parison sag can also be seen by considering the total parison weight versus drop time, though not as clearly. The same trend is observed when parison drop time is increased. However, resin E having the lowest melt strength and highest swell is shown to have the highest parison weight for all drop times. This is as expected since both lower melt strength and higher swell tend to increase total parison weight. For resin F and G , however, the trend is not as clear. Resin 117 Chapter 4 Results and Discussion G is expected to have higher total parison weight, since it has a higher swell. However, resin G has a higher melt strength compared to resin F. Since the two factors have opposite effects, the two curves become similar. Moreover, resin G has a lower density which should be taken into consideration. From the above discussion, it can be concluded that melt behavior during parison formation is critically affected by melt strength and the swell properties o f the resin. From the pi l low mold experiments, however, it is difficult to determine the individual effects o f these factors. Higher melt strength tends to reduce sagging, while higher swell reduces total parison length. Hence, the two effects are competing in opposite directions. It was also found that parison drop time has a significant effect on parison formation considering the dependence of shear rate on the drop time. A shorter drop time implies higher shear rate and higher swell, but shorter time for parison sag to occur. 4.4 M E L T I N D E X , S T R E S S E X P O N E N T A N D M E L T F L O W R A T I O Although melt index (MI) is not a material function, it is a convenient parameter that is often used by industry for resin comparisons. The main advantage o f using MI is the ease at which it can be determined. It does not require elaborate equipment and can be done frequently and quickly. However, the reproducibility o f MI data is very much dependent on the design o f the equipment and the procedure used. It was mentioned earlier that a small change in the design of the equipment or the procedure may result in a very significant variation of the MI data. Generally, MI is used for the analysis of reaction quality control. It serves to indicate the uniformity o f the flow behavior of a polymer made by a particular process, and may 118 Chapter 4 Results and Discussion also be indicative o f the uniformity o f other properties. The value of M Z is affected by the molecular weight o f the polymer, as is the case with all other rheological and processing parameters. B y performing experiments at different conditions, the stress exponent (S.Ex.) and melt flow ratio (MFR) can also be determined. In this work, an attempt was made to relate molecular, rheological and processing parameters to MI, S.Ex., and MFR, to determine the possibility o f using MI, S.Ex., and MFR as an additional set o f parameters to predict polymer behavior. Figure 4.52 shows the relationship of MI to Mw. N o useful mathematical correlation can be found as can be seen by the large scatter in the plot. However, approximate trends can be observed i f the results are sorted according to polydispersity. Generally, the data 0.9 0.8 0.7 0.6 s S 0.5 ^ 0.4 0.3 0.2 0.1 _ l • | 1 ! I . | •8.25 1 1— I 1 | 1 i i i | i i * Technology 'a "1 1 1 1 | '+ Others -- •13.38 i i i 1 i i TTI I | I '+8.72 III 1 I.I '+12.98 : - "+10.1 '+8 1 ' + 8 - 6 '+6.02 '+10.09 "17.42 * ^ - 7 6 . 5 5 ; ^ 4 . 7 9 r •14.3 M6.49 '+13.351 *3.4 - '+8.75 • I i i i r I i i i i I i i i i I i i 1 t 1.0e+5 1.2e+5 1.4e+5 Mw 1.6e+5 1.8e+5 Figure 4.52 Correlating MI to MW. Comparison should be made for resins with similar PI, or shear thinning properties. 119 Chapter 4 Results and Discussion show that increasing M w decreases MI. Unfortunately, there are too few resins having similar PI that are produced using the same technology, making it difficult for a definite conclusion to be made. MI is reported as the mass o f a polymer that flows through a specific die under certain conditions over a period of ten minutes. Hence, higher MI values indicate faster flow and lower viscosity. B y increasing M w , MI decreases, and slower flow w i l l result, indicating higher viscosity. This is consistent with the finding discussed earlier o f the effect o f Mw on shear viscosity. Therefore, MI can be used to predict viscosity. The effects o f the molecular weight distribution and density should be reflected in the value o f MI. Figure 4.53 shows the implication o f MI on shear viscosity. Since MI includes the effect o f all molecular parameters, it is no longer necessary to differentiate between the resins according to their molecular weight distributions. The plot shows that increasing MI causes a decrease in shear viscosity, which is expected. A faster flowing polymer should have a higher MI and a lower viscosity. Considering the procedure followed to determine MI, however, it should be noted that MI should only be used to predict shear viscosity at a low deformation rates. Also, in comparing resins using MI values, the difference in MI should be large enough for a difference in shear viscosity curve to be observable. This problem has also been reported by Yoshikawa et al. (1990). Moreover, M I should only be used to compare resins manufactured using the same technology such as those shown in Figure 4.53. This is consistent with the previous analysis o f the 120 Chapter 4 Results and Discussion viscosity profile. It is not possible to predict the shear flow behavior o f resins from different technologies using MI. V 104 o u .52 S V . <tJ o •c to c a a 103 102 —ii i I i i T 1 1 I I I I A A V A V A V A V • Resin W-Ml = 0.21 • Resin X-MI=0.41 A Resin C - Ml=0.53 v Resin E - Ml=0.72 1 1 — i — i i i i i | T=200°C, LVD=20, D=0.7542 mm A V J I I I I I _1 I l — L 1 1 , 1 I I I I I I I I 101 102 103 Apparent Shear Rate (s'1) Figure 4.53 Implication of MI on shear viscosity curves. It was also found that MI could be used to predict melt strength, as long as melt strength is defined and determined according to the procedure described in this work. Figure 4.54 shows the relationship between MI and melt strength. Again, differentiation has to be made with respect to the technology used to produce the resins. One can clearly see a general trend of decreasing melt strength with increasing MI from Figure 4.54. It is noted that the three polydispersity groups required in the analysis o f melt strength are not reflected in this case. Also, it is interesting to note that resins produced using I2l Chapter 4 Results and Discussion technologies other than 'a' have much higher melt strength. This can be attributed to the polydispersities o f these resins, which fall mostly near the critical range o f eight to ten. 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Ml Figure 4.54 Correlation between melt strength and MJ. S.Ex. and MFR are both indicative of the shear thinning property o f a resin. Considering the way the two parameters are calculated, it is apparent that S.Ex. and MFR show the differences in polymer flow under two different pressures. Hence, higher S.Ex. and MFR values indicate stronger shear thinning behavior in the shear viscosity curves. However, it is important to note that only comparisons of resins with similar MI are made, in order to show this. S.Ex. shows the shear thinning behavior in the narrow and smaller range o f deformation, while MFR encompasses a larger range o f deformation. 122 Chapter 4 Results and Discussion It was not possible to find a useful correlation between S.Ex. or MFR and the molecular parameters. However, knowing their significance, it is possible to use S.Ex. and MFR to predict the shear viscosity behavior of resins with similar MI. Figure 4.55 shows this observation. Increasing S.Ex. or MFR results in greater shear sensitivity of a polymer flow. Unfortunately, in this work, there are too few resins that are manufactured using the same technology and have similar MI. In trying to relate S.Ex. or MFR to extrudate swell, no general trend was observed. 1 0 4 ni 55 o o to 103 Q) •C <0 c a Q. 102 I I I 1 1 I I 1 • • • • — i — i i i i i 11 i i i i i i i 11 T=200°C, Ml=0.27 -L/D=20, D=0.7542 mm -• • • • • • • • • • • --• • • • • Resin V -S.Ex.=1.51,MFR=38.08 • Resin U -i i i i i 11 i S.Ex.=1.77,MFR=95.83 i i i i i i 11 i i i i i i i i 1 101 102 Apparent Shear Rate (s~1) 103 Figure 4.55 Implication of S.Ex. on shear viscosity profile. 123 Chapter 4 Results and Discussion Figure 4.56 shows a plot of S.Ex. versus MFR. It can be seen that generally, higher S.Ex. reflects higher MFR. This observation implies the redundancy o f measuring MFR when S.Ex. data are available. 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 S.Ex. Figure 4.56 Correlation between MFR and S.Ex. From this analysis, it can be concluded that MI and S.Ex. are useful for predicting shear flow behavior at low deformation rates. It is not possible to relate MI and S.Ex. to the extensional flow properties due to the different type o f deformation involved. MI can also be used to predict melt strength. As far as extrudate swell is concerned, no useful 124 Chapter 4 Results and Discussion correlation was found. Moreover, MI, S.Ex. and MFR are only useful for comparing similar resins produced using the same technology. 4.5 I M P L I C A T I O N S O F R H E O L O G I C A L B E H A V I O U R O N P R O C E S S A B I L I T Y A s mentioned earlier, the process o f blow molding is governed principally by the rheological behavior o f the resin used. The molecular weight and its distribution, in turn, affect the way the resin behaves Theologically. It has been discussed how the molecular parameters affect the rheology o f blow molding resins. It is now useful to consider the implications of rheology on processability. The shear viscosity of a resin provides information about the ease o f flow o f the material. Hence, information on shear viscosity is particularly useful during the extrusion o f the material through the die head. Higher shear viscosity implies that more power is required to extrude the polymer, i f extrusion is to be done at a particular temperature. Otherwise, the temperature has to be raised to lower the material viscosity. However, by increasing the extrusion, and possibly, the blowing temperature, a longer cycle time w i l l be required to cool the molded product. The sensitivity o f viscosity to temperature and shear rate is also important to maintain consistency in the manufacturing of blow molding products. It may be undesirable to use a polymer that is very sensitive to changes in temperature and shear. Although process variables, such as extrusion speed and barrel temperature, are normally well controlled, small variations due to process noise are inevitable. B y using resins that are very sensitive to small changes in these variables, inconsistency in the final molded 125 Chapter 4 Results and Discussion product w i l l result. On the other hand, i f a resin exhibits large shear thinning behavior, it w i l l have higher viscosity at low shear rates and lower viscosity at the processing shear rates. This is desirable since it means that the resin is easy to extrude through the die but at the same time does not flow as easily during subsequent parison formation when there is a very small rate o f deformation (recall that the zero shear rate viscosity is related to the extensional viscosity). Hence, a trade off has to be made depending on the use and importance o f the molded product. During parison formation, it is important that the polymer melt has sufficient melt strength to counteract sagging. Considering the nature of sagging, the rheological properties applicable in this case would be the extensional flow properties of the resin. It has been shown that the Hencky strain correlates well with melt strength, at least qualitatively. This can again be seen in Figure 4.57, where the strain versus time curves were plotted for several resins having different melt strength values. Hence, it is desirable to use resins that show comparatively lower Hencky strains under creep conditions. Extrudate swell is also important during parison formation. As discussed earlier, increasing extrudate swell indicates an increase in weight swell. For resins with large swell, this means that the downward force pulling on the melt during parison formation is greater, resulting in greater sag. Sagging is, o f course, undesirable since it results in products with uneven wall thickness. 126 Chapter 4 Results and Discussion Time (s) Figure 4.57 Implication of melt strength on Hencky strain. The sensitivity o f swell is probably a more important parameter to consider than the absolute magnitude o f the swell. The magnitude of the swell determines the weight and the wall thickness o f the final product, and this can be controlled by adjusting the position o f the mandrel to an optimum one. Hence, it is useful to be able to predict the swell magnitude. Unfortunately, as has been shown in this work, this is difficult to do. The sensitivity o f swell to small variations in process variables, however, is not as easy to control. Hence, it is important that the swell property of a blow molding resin is relatively insensitive to small changes in shear rate (or temperature). I f a resin is very sensitive to changes in shear rate, for example, the final molded product wi l l not have consistent weight, and this is of course undesirable. 127 Chapter 4 Results and Discussion In conclusion, the optimal design of the different parts o f blow molding process require different resin properties. Considering the complex effect of molecular properties on both the rheology and processability o f H D P E resins, one often has to make choices that involve some trade-off between two or more resin properties. The importance of each o f these properties, on the other hand, is determined by the use of the final product. Hence, it is very difficult to say that a specific resin is the best for a particular application. In the design of a resin, consideration and careful weighing have to be made in regard to its application and economic implications. From the results of this study, however, some useful recommendations can still be made in regard to resin design. Understanding the blow molding process and knowing the general processability requirements for blow molding resins, it can generally be said that, ideally, a blow molding resin should have: - reasonable viscosity, - a viscosity profile that is not very sensitive to temperature and shear rate variations, - an extrudate swell profile that is not very sensitive to temperature and shear rate variations, and - high melt strength. Using the results o f this study, one may conclude that one simple way to produce a blow molding resin with these requirements would be to optimize the concentration o f medium 128 Chapter 4 Results and Discussion sized molecules, represented by Mw. Increasing Mw w i l l increase the melt strength o f the resin, while not affecting the sensitivity of shear viscosity to temperature variations significantly. B y increasing Mz, melt strength wil l also increase, but the resin w i l l become more sensitive to changes in temperature and this is, o f course, undesirable. Increasing Mw w i l l also increase the polydispersity. This may affect the sensitivity o f shear viscosity to variations in shear rate. However, the effect wi l l not be very significant compared to when polydispersity is increased by decreasing M„ (increasing the concentration o f smaller molecules). Moreover, by increasing Mw, and hence, the polydispersity, the extrudate swell profile becomes less sensitive to shear rate variations in the processing range between 350 s"1 to 700 s"1. Therefore, generally, it can be said that a resin with a comparatively high Mw value and low Mz value wi l l process better. A s far as M„ is concerned, it has to be optimized accordingly, depending on Mw and polydispersity. 129 Chapter 5 Conclusions 5 C O N C L U S I O N S Polymer science is still an inexact science, due to the many possible manifestations o f molecular structure, resulting in interesting polymeric behavior. In this work, an attempt was made to correlate the rheology and processability o f H D P E resins to molecular weight and its distribution through the study o f twenty four commercial H D P E blow molding resins. The resins vary greatly in terms of molecular parameters, such as Mw, PI, and density. These are commercial resins produced from a number o f different technologies, since it is not possible to produce resins with broad range o f molecular parameters using a single polymerization technology. However, with this set o f commercial resins produced from different technologies, it was not possible to systematically study the individual effects o f specific molecular parameters. Hence, multiple regression was used as a tool for data analysis, wherever possible. In some cases, however, the molecular parameters are insufficient in describing the molecular weight distribution. In these cases, the actual qualitative comparisons of the distribution curves were performed. The conclusions o f this study are summarized as follows: Shear F low Properties: - The expected effects o f Mw and polydispersity on shear viscosity were confirmed. Increase o f Mw and polydispersity increases the magnitude o f shear viscosity and its tendency to shear thin, respectively. The effect of polydispersity has been found to be mainly due to the changes in the concentration of smaller molecules. Broadening the 130 Chapter 5 Conclusions molecular weight distribution by increasing the concentration o f larger molecules does not seem to affect the shear thinning property o f the resins as significantly. - It was also found that it is only useful to compare the shear properties o f resins produced using the same technology. Different reactors, solvents, catalysts and additives appear to cause the shear properties of the resins to be non-comparable. - The activation energies o f the H D P E flow curves were found to range from 20 kJ/mol to 28 kJ/mol. - It was found that the resins have to be grouped according to their polydispersity index, into those with PI<8, 8<PI<10, and PI>10 in order to obtain good correlations between the flow activation energy and molecular parameters. For narrowly distributed resins, there is no apparent correlation between molecular parameters to Ea. Above the polydispersity o f eight, it was found that increasing polydispersity increases the temperature sensitivity of viscosity, with the effect mainly contributed by the increase in the concentration of smaller molecules. For PI greater than ten, however, the concentration of larger molecules becomes important and increasing it w i l l increase both the polydispersity and Ea. Extensional F l o w Properties: - Increasing Mw expectedly decreases the magnitude o f Hencky strain and increases the extensional viscosity in creep experiments. - Hencky strain can be related to melt strength in an inverse manner, i.e. lower Hencky strain at a constant time indicates higher melt strength. 131 Chapter 5 Conclusions - It was found that increasing polydispersity up to about nine decreases the Hencky strain. For PJ>9, broadening the molecular weight distribution initially increases the Hencky strain. For broadly distributed resins (JPI>16), the effect of polydispersity on Hencky strain becomes insignificant. Extrudate swell: - N o correlation could be found to exist between the molecular weight averages and the shape o f the molecular weight distribution to the magnitude of extrudate swell. This is probably due to the possibility of the effect of pre-shear history being more important than the effect of molecular weight distribution. Since the resins studied in this work may have been subjected to deformations in unknown and different ways, it is possible for resins with the same molecular weight distributions to show very different extrudate swell profiles. - It is also thought that differences in polymerization technology may result in some variations in the degree of unsaturation, which may not be reflected in molecular weight distribution curves, but which may affect the elastic properties of the polymers significantly. - The sensitivity o f extrudate swell, however, was found to correlate well with Mz and polydispersity. Broadening the molecular weight distribution by increasing the concentration o f larger molecules increases shear sensitivity at lower shear rates (5 s"1 to 350 s"1) and decreases it at the higher shear rates (350 s"1 to 700 s"1). 132 Chapter 5 Conclusions Mel t Strength: - Three polydispersity groups have to be identified in order to correlate melt strength to molecular parameters, i.e. resins with PK8, 8<PI<10, and PI>10. - Increasing Mw was found to increase melt strength. - Increasing polydispersity for resins with PI<8 was found to increase melt strength. For resins with higher polydispersity (Pl>10), however, broadening molecular weight distribution decreases the melt strength. - It was also found that density affects melt strength in an inverse manner, i.e. increasing density decreases melt strength. Sagging and Weight Swell Characteristics: - The individual effects of sagging and weight swell are difficult to differentiate in pi l low mold experiments. - Sagging and weight swell are reflected by melt strength and extrudate swell, respectively. Parison drop time has a significant effect on parison quality. For short parison drop times, the effect o f weight swell dominates. For longer parison drop times, the effect o f sagging becomes more important. Sagging and weight swell determine the total parison weight and length. 133 ^ Chapters Conclusions MI. S.Ex.. and MFR: - These parameters are only particularly useful for resin quality control (for the same technology). - It may be possible to correlate MI to Mw for resins of the same technology and similar polydispersity, in which case MI is expected to be inversely proportional t o M w . - MI can be used to predict shear viscosity at low shear rate as long as comparisons are made within the resins produced using the same technology. - MI can also be used to predict melt strength o f resins o f the same technology. - S.Ex. and MFR are indicative of the shear thinning behavior o f a resin. Similarly, these parameters are technology dependent, and in addition, comparisons should only be made for resins with the same MI. - S.Ex. and MFR are essentially useful for the same purpose. In addition, it is possible to determine the molecular weight range that is critically affecting a certain polymer property by correlating the normalized area under the molecular weight distribution for each combination of molecular weight ranges to the desired property. These conclusions are summarized in Table 5.1. 134 § 21S 2i> e l o ro 8£< o 5 |3 CO 5 Chapter 6 Recommendations 6 R E C O M M E N D A T I O N S Based on the experience gained during this study, the following recommendations can be made for future work: - The swell behavior and hence, the sagging and weight characteristics of a parison is governed by the elastic nature o f the resins. B y measuring the elastic recoil properties o f a resin, one may be able to obtain a correlation suitable for predicting extrudate swell. A Mel t Elasticity Indexer can be used for this purpose. I f this proves to be possible, the unit can be implemented onsite in the plant as an easy and convenient way of predicting extrudate swell. - Extrudate swell has been found to depend more strongly on the pre-shear history of the resin than on its molecular characteristics. Since it is important to be able to predict extrudate swell, a study should be conducted on resins produced using the same technology and having the same pre-shear history. B y performing such study, correlation between extrudate swell and molecular parameters should become more apparent. A thorough molecular weight characterization wi l l be very useful in such a study. - It has been found that different polymerization technologies produce resins that may have the same molecular weight distributions but which may differ significantly in terms of rheology and processability. Possible reasons include the type of reactor used (dual or single reactor), the catalyst, the monomers used for polymerization, and various additives. A study should be conducted to investigate the differences in technology and how they affect the rheology and processability o f a resin. 136 Chapter 6 Recommendations Similar experiments may be conducted to investigate the effect o f branching distribution, type o f branching, and additives on the rheology and processability o f H D P E resins. Also , the molecular effects on other physical properties of H D P E , such as environmental stress cracking resistance (ESCR) can be studied. 137 References R E F E R E N C E S Acierno, D . , D . Curto, F . P . L a Mantia, and A . Valenza, Flow Properties of Low Density/Linear Low Density Polyethylenes, Polym. Eng. Sci. , 26 (1), 2 8 - 3 3 (1986) Ajroldi , G . , Determination of Rheological Parameters from Parison Extrusion Experiments, Polym. Eng. 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M . , Polymeric Materials and Processing. Plastics. Elastomers, and Composites. Hanser, N Y , 1990 CEL Inc., Polymers — Polyethylene. Technical Publication, Ont Dealy, J. M . , and K . F . Wissbrun, Melt Rheology and Its Role in Plastics Processing: Theory and Applications. Reinhold, N Y , 1995 Dealy, J. M . , Rheometers for Molten Plastics: A Practical Guide to Testing and Property Measurement. Van Nostrand Reinhold, N Y , 1982 Goyal, S. K . , Influence of Polymer Structure on the Melt Strength Behavior of Polyethylene Resins, A N T E C ' 9 4 , 1 2 3 2 - 8 (1994) Goyal, S. K . , Personal Communication, N O V A Research and Technology Center, 1998 138 References Graessley, W . W. , Viscosity of Entangling Polydisperse Polymers, J. Chem. Phys., 47, 1 9 4 2 - 5 3 (1967) Han, C . D., and C . A. Villamizar, Effects of Molecular Weight Distribution and Long-Chain Branching on the Viscoelastic Properties of High - and Low - Density Polyethylene Melts, J. Appl . Polym. 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Hatzikiriakos, Influence of Molecular Structure on the Rheological and Processing Behavior of Polyethylene Resins, Polym. Eng. Sci. , Accepted for Publication, Dec (1997) Koopmans, R. J., Die Swell - Molecular Structure Model for Linear Polyethylene, J. Polym. Sci. , Part A , 26, 1157 (1988) Koopmans, R. J., Extrudate Swell of High Density Polyethylene. Part II: Time Dependency and Effects of Cooling and Sagging, Polym. Eng. Sci. , 32 (23), 1750 - 4 (1992a) Koopmans, R. J., Extrudate Swell of High Density Polyethylene. Part III: Extrusion Blow Molding Die Geometry Effects, Polym. Eng. Sci., 32 (23), 1755 - 64 (1992b) Koopmans, R. J., Extrudate Swell of High Density Polyethylene. Part I: Aspects of Molecular Structure and Rheological Characterization Methods, Polym. Eng. Sci. , 32(23), 1741 - 9 (1992c) L a Mantia, F . P., and D . Acierno, Influence of the Molecular Structure on the Melt Strength and Extensibility of Poly ethylenes, Polym. Eng. Sci. , 25 (5), 279 - 83 (1985) 139 References La Mantia, F. P., and D. Acierno, Melt Strength and Extensibility of High Density Polyethylene, Plas. Rubber Process. Appl., 5 (2), 183 - 5 (1985) Levy, S., and J. H. Du Bois, Plastics Product Design Engineering Handbook. Van Nostrand Reinhold, NY, 1977 Mavridis, H., and R. N. Shroff, Temperature Dependence of Polyolefin Melt Rheology, Polym. Eng. Sci., 32 (23), 1778-91 (1992) Meissner, J., Development of a Universal Extensional Rheometer for the Uniaxial Extension of Polymer Melts, Trans. Soc. Rheol., 16 (3), 405 - 20 (1972) Mendelson, R. A., and F. L. Finger, High - Density Polyethylene Melt Elasticity: Anomalous Observations on the Effects of Molecular Structure, J. Appl. Polym. Sci., 19(4), 1061-78 (1975) Michael, L. B. (edX Plastics of Engineering Handbook of the Society of the Plastic Industry. Inc.. Van Nostrand Rheinhold, NY, 1991 Munstedt, EL, and H. M. Laun, Elongational Behavior of a Low - Density Polyethylene Melt. Transient Behavior in Constant Stretching Rate and Tensile Creep Experiments. Comparison with Shear Data. Temperature Dependence of the Elongational Properties, Rheol. Acta., 18, 492 (1979) NOVACOR, Density by Means of the Densimeter, Laboratory Test Procedures Manual, 1993 NOVACOR, Determination of the Melt Strength of Polyethylene Using a Rheometric RER-9000 Extensional Rheometer, Laboratory Test Procedures Manual, 1994 Ogorkiewicz, R. M. (ed.), Engineering Properties of Themoplastics. A Collective Work Produced by Imperial Chemical Industries Limited, Plastics Division, Wiley-Interscience, NY, 1970 Progelhof, R.C., and J. L. Throne. Polymer Engineering Principles: Properties. Processes. Testing for Design, Hanser, NY, 1993 Ramamurthy, A. V., Wall Slip in Viscous Fluids and Influence of Materials of Construction, J. Rheol., 30 (2), 337 - 57 (1986) Rauschenberger, V., and H. M. Laun, A Recursive Model for Rheotens Tests, SPE J. Rheol., 41 (3), 719 - 37 (1997) Rosato, D. V., and D. V. Rosato (eds.), Blow Molding Handbook. Hanser, NY, 1989 140 References Saini, D . R., and A . V . Shenoy, Viscoelastic Properties of Linear Low Density Polyethylene Melts, Eur. Polym J., 19 (9), 811 - 16 (1983) Shenoy, A V . , S. Chattopadhyay, and V . M . Nadkarni, From Melt Flow Index to Rheogram, Rheol. Acta, 22, 90 - 101 (1983) Shroff, R. N., and S. Mksuzo, Effect of Molecular Weight and Molecular Weight Distribution on Elasticity of Polymer Melts, Soc. Plast. Eng., Tech. Pap., 23, 285 - 9 (1977) Shroff, R. , A . Prasad, and C. Lee, Effect of Molecular Structure on Rheological and Crystallization Properties of Polyethylenes, J. Polym. Sci . , Part B , 34, 2317 - 33 (1996) Van Krevelen, D . W. , Properties O f Polymers: Their Correlation with Chemical Structure: Their Numerical Extimation and Prediction from Additive Group Contributions. Elsevier, N Y , 1990 Vetterling, W . T., W . H . Press, S. A . Teukolsky, and B . P. Flannery. Numerical Recipes: Example Book f F O R T R A N ) . Cambridge University Press, N Y , 1987 Wilson, N . R. , M . E . Bentley, and B . T. Morgan, How Extrusion Variables Affect Parison Swell, S P E J., 26, 34 - 40 (1970) Winter, H . H . , A Collaborative Study on the Relation Between Film Blowing Performance and Rheological Properties of Two Low - Density and Two High -Density Polyethylene Samples, Macromolec. Div . , Commission on Polymer Characterization and Properties, I U P A C , Pure and Appl . Chem., 55 (6), 943 - 76 (1983) Yoshikawa, K . , N . Toneaki, Y . Moteki , M . Takahashi, and T Masuda, Dynamic Viscoelasticity and Stress Relaxation of Column - Fractionated High Density Polyethylene Melts, J. Soc. Rheol., Japan, 18, 87 - 92 (1990) Yoshikawa, K . , N . Toneaki, Y . Moteki, M . Takahashi, and T. Masuda, Dynamic Viscoelasticity, Stress Relaxation, and Elongational Flow Behavior of High Density Polyethylene Melts, Nippon Reorogii Gaku Kaishi, 18, 80 - 6 (1990) 141 Notation NOTATION a, b, c, d, e, f Polymerization technology A Cross sectional area (m 2) Ao Cross sectional area before deformation (m ) ax Shift factor b Rabinowitsch correction C ° i , C ° 2 W L F constants D Diameter (m) D 0 Diameter before deformation (m) E Tensile relaxation modulus (Pa) Ea Activation energy (kJ/mol) F Force (N) G * Complex modulus (Pa) G Shear relaxation modulus (Pa) G ' Storage modulus (Pa) G " Loss modulus (Pa) G-, Discrete relaxation modulus (Pa) h Distance between two parallel plates (m) I2,16,121 Polymer flow rate under different conditions (g/10 minutes) K Power law constant (Pa.sn) L Length (m) Lo Length before deformation (m) M Molecular weight of monodisperse polymer (kg/kmol) M i Molecular weight of polymer chain / (kg/kmol) M„ Number average molecular weight (kg/kmol) Mo Location parameter in the 'log-normal' distribution curve (kg/{kmol.exp[ln(kmol/kg) 1 / 2]}) M w Weight average molecular weight (kg/kmol) M z Z-average molecular weight (kg/kmol) M z + i (Z+7)-average molecular weight (kg/kmol) 142 Notation n Power law exponent n; Number average o f polymer chain /' P Pressure (Pa) P a Ambient pressure (Pa) Pd Barrel driving pressure (Pa) Pend Pressure end correction (Pa) Q Volumetric flow rate (m3/s) R, r Radius (m) t Time (s) T Temperature (K) To Reference temperature (K) T g Glass transition temperature (K) V Velocity (m/s) w Weight fraction W i Weight fraction o f polymer chain /' X , x Horizontal distance (m), or direction P Scale parameter in the 'log-normal' distribution curve A Quantity change 8 Hencky strain e 0 Extensional Strain magnitude Y A Apparent wall shear rate (s"1) y0 Shear rate (s"1) Yo Shear magnitude Yw True wall shear rate (s"1) T| Shear viscosity (Pa.s) TJ* Complex viscosity (Pa.s) TJE Extensional Viscosity (Pa.s) T| 0 Zero shear rate viscosity (Pa.s) Xi Discrete relaxation time (s) 143 Shear stress (Pa) Extensional stress (Pa) Wal l shear stress (Pa) Frequency of oscillation (s 144 Appendix A: Time Temperature Superposition Program Code A P P E N D I X A : Time Temperature Superposition Program Code 145 Appendix A: Time Temperature Superposition Program Code ********************************************************** * * * Alfonsius Budi Ariawan * * Department of Chemical Engineering * * The Uni v e r s i t y of B r i t i s h Columbia * * Vancouver, B.C. * * * * * * SUMMARY: * * The following program reads i n sets of shear stress data as a function * * of temperature and shear rate. A master curve f o r the s p e c i f i c polymer * * i s then ca l c u l a t e d corresponding to a s p e c i f i e d reference temperature. * * The s h i f t f a c t o r i s then correlated using e i t h e r a simple thermally * * ac t i v a t e d model(Arrhenius), WLF model or both, depending on the desire * * of the user. An option i s also provided to ca l c u l a t e the shear * * v i s c o s i t y of the polymer at a desired temperature and shear rate. * PROGRAM SHIFT IMPLICIT DOUBLE PRECISION (A-H,0-Z) DIMENSION TEMP(20),SRT(20,200),STRESS(20, 200) DIMENSION SD(20,200),NDATA(20) DIMENSION ASRT (500),ASTRESS(500),AT(20) DIMENSION X(20),Y(20),A(2),AGUESS(2),SUMSQ(20) INTEGER CHOICE CHARACTER * 100 FNAME(20) CHARACTER * 100 COMMENT PARAMETER (TOL=2.D-5) COMMON/BLKA/Q(300),R(301) ,S(300) COMMON/BLKC/ALNSRT(300),ALNSTRS(300) COMMON/BLKD/REFS RT(300),REFSTRS(300) COMMON/BLKE/WT(300),MDATA,ITOTAL COMMON/BLKF/TREF EXTERNAL OBJ,AMODEL,DMODEL DATA AGUESS,T,IVTS,JVTS,W,II/-1.744D1,5.16D1,5*0.DO/ DATA Il,IN,Gl,GN/2*3,2*0/ C P r i n t i n g of av a i l a b l e options and some information regarding the C program: PRINT 180 PRINT 190 C Prompting f or input: PRINT 200 READ *, CHOICE PRINT 210 READ *, NSET PRINT 220 READ * OPEN (7,FILE='DATA.TXT',STATUS='OLD') 146 Appendix A: Time Temperature Superposition Program Code READ (7,*) (NDATA(I), I=1,NSET) READ (7,*) (TEMP(I), I=1,NSET) READ (7, ,(A)') (FNAME(I), I=1,NSET) READ (7, ' (A) * ) COMMENT CLOSE (7) PRINT 230 READ *, ID DO 30 I=1,NSET OPEN (7,FILE=FNAME(I),STATUS=1 OLD1) IF (ID.EQ.O) THEN DO 10 J=1,NDATA(I) READ (7,*) SRT(I,J),STRESS(I,J) 10 CONTINUE ELSE DO 20 J=1,NDATA(I) READ (7,*) SRT(I,J),STRESS(I,J),SD(I,J) 20 CONTINUE ENDIF CLOSE (7) 30 CONTINUE C Prompting f o r more information: PRINT 240, NSET READ *, NREF TREF=TEMP(NREF) ITOTAL=NDATA(NREF) PRINT 330 READ *, NVIS IF (NVIS.EQ.l) THEN PRINT 340 READ *, T PRINT 350 READ *, W ENDIF C I n i t i a l i z i n g reference curve: DO 40 J=l,ITOTAL ASRT(J)=SRT(NREF, J) REFSRT(J)=DLOG(SRT(NREF,J)) ASTRESS(J)=STRESS(NREF, J) REFSTRS(J)=DLOG(STRESS(NREF, J) ) C Defining the weight of each data point, i f data are a v a i l a b l e : IF (ID.EQ.l) THEN 147 Appendix A: Time Temperature Superposition Program Code W T ( J ) = 1 / S D ( N R E F , J ) * * 2 E L S E C O t h e r w i s e , t h e s a m e w e i g h t o f 1 . D 0 i s a p p l i e d t o a l l p o i n t s : W T ( J ) = 1 . D 0 E N D I F 40 C O N T I N U E DO 90 I = 1 , N S E T I F ( ( N V I S . E Q . l ) . A N D . ( I V I S . E Q . O ) . A N D . ( T . E Q . T E M P ( I ) ) ) T H E N I V T S = I 11=1 E N D I F C C a l c u l a t i o n o f s h i f t f a c t o r f o r e a c h t e m p e r a t u r e , e x c e p t t h e r e f e r e n c e C t e m p e r a t u r e : I F ( I . E Q . N R E F ) T H E N A T ( I ) = 1 . D 0 S U M S Q ( I ) = 0 . D 0 G O T O 90 E N D I F M D A T A = N D A T A ( I ) DO 50 J = 1 , M D A T A I F ( ( I I . E Q . 1 ) . A N D . ( S R T ( I , J ) . E Q . W ) ) J V I S = J C W o r k i n g i n l o g s c a l e : A L N S R T ( J ) = D L O G ( S R T ( I , J ) ) A L N S T R S ( J ) = D L O G ( S T R E S S ( I , J ) ) 50 C O N T I N U E 11=0 N M = N D A T A ( I ) - 1 C U s i n g c u b i c s p l i n e o f f i t t e d e n d s t o a l l o w i n t e r p o l a t i o n b e t w e e n C p o i n t s : C A L L S P L I N E ( A L N S T R S , A L N S R T , M D A T A , N M , Q , R , S , I 1 , I N , G 1 , G N ) I C H E C K = 0 A T 0 = l . D - 6 D A T = 1 . D - 1 C D e t e r m i n a t i o n o f t h e i n t e r v a l w h i c h b r a c k e t s t h e m i n i m u m v a l u e o f C t h e o b j e c t i v e f u n c t i o n : 60 I F ( O B J ( A T 0 + D A T ) - G T . O B J ( A T O ) ) T H E N I F ( I C H E C K . E Q . l ) T H E N 148 Appendix A: Time Temperature Superposition Program Code A T I =AT 0 - D A T A T F = A T 0+DAT G O T O 70 E N D I F E L S E I C H E C K = 1 E N D I F A T 0 =AT 0+DAT G O T O 60 C U s i n g t h e g o l d e n s e a r c h m e t h o d t o m i n i m i z e t h e o b j e c t i v e f u n c t i o n : 70 C A L L G O L D E N ( A T I , A T O , A T F , O B J , T O L , A T ( I ) , S U M S Q ( I ) ) C S h i f t i n g o f c u r v e s a n d d e f i n i n g t h e s h i f t e d c u r v e a s p a r t o f t h e C n e w r e f e r e n c e c u r v e : DO 80 J = 1 , M D A T A A S R T ( I T 0 T A L + J ) = S R T ( I , J ) * A T ( I ) R E F S R T ( I T O T A L + J ) = D L O G ( A S R T ( I T O T A L + J ) ) A S T R E S S ( I T O T A L + J ) = S T R E S S ( I , J ) R E F S T R S ( I T O T A L + J ) = D L 0 G ( A S T R E S S ( I T O T A L + J ) ) C D e t e r m i n i n g t h e w e i g h t o f e a c h p o i n t o f t h e m a s t e r c u r v e : I F ( I D . E Q . l ) T H E N W T ( I T O T A L + J ) = 1 / S D ( I , J ) * * 2 E L S E W T ( I T O T A L + J ) = l . D 0 E N D I F 8 0 C O N T I N U E I T O T A L = I T O T A L + M D A T A 90 C O N T I N U E C S o r t i n g t h e m a s t e r c u r v e d a t a p o i n t s : C A L L S O R T ( I T O T A L , A S R T , A S T R E S S ) O P E N ( 9 , F I L E = 1 O U T P U T . T X T 1 ) W R I T E ( 9 , * ) COMMENT W R I T E ( 9 , 2 5 0 ) C P r i n t i n g o f o u t p u t : DO 1 0 0 1 = 1 , I T O T A L V I S C O S I T Y = A S T R E S S ( I ) / A S R T ( I ) W R I T E ( 9 , 2 6 0 ) A S R T ( I ) , A S T R E S S ( I ) , V I S C O S I T Y 100 C O N T I N U E W R I T E ( 9 , 2 7 0 ) T R E F DO 1 1 0 I = 1 , N S E T 149 Appendix A: Time Temperature Superposition Program Code W R I T E ( 9 , 2 8 0 ) T E M P ( I ) , A T ( I ) , S U M S Q ( I ) 1 1 0 C O N T I N U E M = N S E T I F ( ( C H O I C E . E Q . l ) . O R . ( C H O I C E . E Q . 3 ) ) T H E N DO 1 2 0 I = 1 , N S E T X ( I ) = 1 / ( T E M P ( I ) + 2 7 3 . 1 5 ) Y ( I ) = D L O G ( A T ( I ) ) 1 2 0 C O N T I N U E C L i n e a r r e g r e s s i o n i f t h e f i r s t a n d t h e s e c o n d m o d e l s a r e b o t h c h o s e n : C A L L L I N R E G ( X , Y , N S E T , C E P T , S L O P E , R S T A T ) E A = S L O P E * 8 . 3 1 4 / 1 0 0 0 E N D I F C I f t h e s i m p l e t h e r m a l l y a c t i v a t e d m o d e l i s c h o s e n , t h e n t h e a c t i v a t i o n C e n e r g y o f t h e p o l y m e r i s r e t u r n e d : I F ( C H O I C E . E Q . 1 ) T H E N W R I T E ( 9 , 2 9 0 ) E A , R S T A T G O T O 1 5 0 E N D I F C I f b o t h t h e f i r s t a n d t h e s e c o n d m o d e l s a r e c h o s e n , t h e n u s e t h e C r e s u l t s o b t a i n e d f r o m t h e f i r s t m o d e l t o o b t a i n m o r e d a t a p o i n t s C t o b e f i t t e d f o r t h e s e c o n d m o d e l : I F ( C H O I C E . E Q . 3 ) T H E N X ( 1 ) = T E M P ( 1 ) + 2 7 3 . 1 5 Y ( l ) = D L O G ( A T ( l ) ) M = 2 * N S E T - 1 DO 1 3 0 I = 1 , N S E T - 1 X ( 2 * 1 + 1 ) = T E M P ( I + 1 ) + 2 7 3 . 1 5 Y ( 2 * 1 + 1 ) = D L O G ( A T ( 1 + 1 ) ) X ( 2 * I ) = ( T E M P ( I ) + T E M P ( I + 1 ) J / 2 + 2 7 3 . 1 5 Y ( 2 * I ) = E A * 1 0 0 0 / 8 . 3 1 4 * ( 1 / X ( 2 * I ) - 1 / ( T R E F + 2 7 3 . 1 5 ) ) 130 C O N T I N U E E L S E I F ( C H O I C E . E Q . 2 ) T H E N C I f W L F ( s e c o n d ) m o d e l i s c h o s e n , t h e n p r e p a r e t h e d a t a p o i n t s C t o b e f i t t e d : DO 1 4 0 1 = 1 , N S E T X ( I ) = T E M P ( I ) + 2 7 3 . 1 5 Y ( I ) = D L O G ( A T ( I ) ) 140 C O N T I N U E E N D I F 150 Appendix A: Time Temperature Superposition Program Code C C a l l i n g t h e s u b r o u t i n e G A U S S N t o d e t e r m i n e t h e b e s t e s t i m a t e o f C u n k n o w n p a r a m e t e r s : C A L L G A U S S N ( X , Y , A G U E S S , M , T O L , A M O D E L , D M O D E L , A , S S Q ) C P r i n t i n g t h e s e c o n d s e t o f o u t p u t : I F ( C H O I C E . E Q . 2 ) W R I T E ( 9 , 3 0 0 ) A ( l ) , A ( 2 ) , S S Q I F ( C H O I C E . E Q . 3 ) T H E N W R I T E ( 9 , 3 1 0 ) E A , R S T A T , A ( 1 ) , A ( 2 ) W R I T E ( 9 , 3 2 0 ) S S Q E N D I F C C a l c u l a t i n g t h e v i s c o s i t y , i f d e s i r e d : 1 5 0 I F ( N V T S . E Q . l ) T H E N C I f v i s c o s i t y a t o n e o f t h e e x p e r i m e n t a l l y f o u n d t e m p e r a t u r e a n d C s h e a r r a t e ( p a r t o f t h e i n p u t ) i s r e q u i r e d , t h e n c a l c u l a t e d i r e c t l y : I F ( ( I V T S . N E . 0 ) . A N D . ( J V I S . N E . 0 ) ) T H E N V I S = S T R E S S ( I V I S , J V I S ) / W G O T O 1 7 0 E N D I F I F ( I V T S . N E . 0 ) T H E N A A T = A T ( I V I S ) G O T O 1 6 0 E N D I F C O t h e r w i s e , f i t c u b i c s p l i n e i n t o t h e m a s t e r c u r v e : C A L L + S P L I N E ( R E F S R T , R E F S T R S , I T O T A L , I T O T A L - 1 , Q , R , S , I I , I N , G l , G N ) C A n d c a l c u l a t e t h e s h i f t f a c t o r c o r r e s p o n d i n g t o t h e t e m p e r a t u r e C u s i n g t h e c h o s e n m o d e l : I F ( C H O I C E . E Q . l ) T H E N A A T = D E X P ( E A / 8 . 3 1 4 * 1 0 0 0 * ( 1 / ( T + 2 7 3 . 1 5 ) - 1 / ( T R E F + 2 7 3 . 1 5 ) ) ) E L S E A A T = D E X P ( A ( l ) * ( T - T R E F ) / ( A ( 2 ) + T - T R E F ) ) E N D I F C C a l c u l a t e t h e s h i f t e d s h e a r r a t e a n d from t h e m a s t e r c u r v e C o b t a i n t h e c o r r e s p o n d i n g G ' a n d G " v a l u e s : 160 W I = D L O G ( W * A A T ) V I S = ( D E X P ( F ( R E F S R T , R E F S T R S , Q , R , S , I T O T A L , WI) ) / W ) 1 7 0 W R I T E ( 9 , 3 6 0 ) T , W , V I S E N D I F C F o r m a t s t a t e m e n t s : 151 Appendix A: Time Temperature Superposition Program Code 1 8 0 F O R M A T ( l X ' T h i s p r o g r a m c r e a t e s a m a s t e r c u r v e f o r a p o l y m e r ' , + ' a t a s p e c i f i e d r e f e r e n c e t e m - 1 / I X , ' p e r a t u r e . T h e ' , + ' v i s c o s i t y o f t h e p o l y m e r a t a d e s i r e d ' , + ' t e m p e r a t u r e a n d s h e a r r a t e ' / I X , ' m a y a l s o b e c a l c u l a t e d . ' ) 1 9 0 F O R M A T ( / / , I X ' P l e a s e c h o o s e o n e o f t h e f o l l o w i n g o p t i o n s t o ' , + ' c o r r e l a t e s h i f t f a c t o r t o t e m - ' / I X , ' p e r a t u r e : 1 ) 2 0 0 F O R M A T ( / I X , ' 1 . S i m p l e T h e r m a l l y A c t i v a t e d M o d e l (a g o o d ' , + ' m o d e l f o r T > ( T g + 1 0 0 ) K ) ' / I X , ' 2 . W L F m o d e l (a ' , + ' g o o d m o d e l f o r T g < T < ( T g + 1 0 0 ) K ) ' / I X , 1 3 . B o t h ' ) 2 1 0 F O R M A T ( / l X ' P l e a s e e n t e r t h e n u m b e r o f a v a i l a b l e d a t a s e t s : ' ) 2 2 0 F O R M A T ( / I X , ' P l e a s e c o p y t h e f i l e : D A T A . T X T i n t o t h e a p p r o p r i a t e ' , + ' d i r e c t o r y c o n s i s t i n g o f 1 / I X , 1 i n f o r m a t i o n o n t h e n u m b e r ' , + ' o f d a t a a v a i l a b l e f o r e a c h d a t a s e t , t h e t e m p e r a t u r e 1 / I X , + ' ( i n C e l s i u s ) c o r r e s p o n d i n g t o e a c h d a t a s e t , a n d t h e ' , + ' f i l e n a m e s i n w h i c h d a t a ' / l X , ' o f e a c h s e t a r e s t o r e d . ' , + ' T h e s e d a t a s h o u l d b e s t o r e d i n c o l u m n s o f 2 ( o r 3) i n ' + / I X , ' a s c e n d i n g o r d e r s o f f r e q u e n c y ( c o l u m n 1 ) . T h e ' , + ' s e c o n d c o l u m n o f t h e s e d a t a ' / I X , ' f i l e s s h o u l d r e f e r t o ' , + ' t h e s h e a r s t r e s s . T h e t h i r d c o l u m n m a y b e a d d e d t o ' / I X , + ' c o r r e s p o n d t o t h e s t a n d a r d d e v i a t i o n s ( i n %). ' , + ' I n c a s e s w h e r e n o s t a n d a r d ' / I X , ' d e v i a t i o n i s a v a i l a b l e , ' , + ' t h e y a r e a s s u m e d t o b e z e r o . ' / / I X , ' P l e a s e p r e s s e n t e r ' , + ' w h e n r e a d y . . . ' ) 2 3 0 F O R M A T ( / I X 1 I s t h e r e t h i r d c o l u m n o n t h e f i l e ' , + ' c : \ D A T A . T X T ? ' / I X , * 0 - n o ' / l X , ' 1 - y e s 1 ) 2 4 0 F O R M A T ( / l X ' W h i c h d a t a s e t d o y o u w a n t t h e m a s t e r c u r v e t o b e ' , + ' r e f e r e n c e d t o ? ( 1 - * , I 2 , ' ) * ) 2 5 0 F O R M A T ( / ' R E S U L T I N G M A S T E R C U R V E D A T A P O I N T S : ' / / 3 X , ' R a t e ( 1 / s ) ' , + 4 X , ' S t r e s s ( P a ) ' , 4 X , ' V i s c o s i t y ( P a . s ) ' / ' ' , 2 6 0 F O R M A T ( F l l . 4 , F 1 4 . 0 , F 1 8 . 0 ) 2 7 0 F O R M A T ( / / ' S U M M A R Y O F R E S U L T S : ' / / 1 T h e m a s t e r c u r v e was ' , + ' c a l c u l a t e d b a s e d o n t h e r e f e r e n c e t e m p e r a t u r e o f ' , + F 6 . 1 , ' C ' / ) 2 8 0 F O R M A T ( ' T h e s h i f t f a c t o r c o r r e s p o n d i n g t o t e m p e r a t u r e o f ' , + F 6 . 1 , ' C i s : ' , F 7 . 4 / ' T h e c o r r e s p o n d i n g v a l u e o f t h e ' , + ' o b j e c t i v e f u n c t i o n i s : ' , F 7 . 4 ) 2 9 0 F O R M A T ( / ' U s i n g s i m p l e t h e r m a l l y a c t i v a t e d m o d e l , E a = ' , F 8 . 4 , + ' k J / m o l ' / ' T h e c o r r e s p o n d i n g r e g r e s s i o n c o r r e l a t i o n ' , + ' c o e f f i c i e n t i s R= * , F 6 . 4 ) 3 0 0 F O R M A T ( / ' U s i n g W L F m o d e l , C 1 = ' , F 8 . 4 , ' , C 2 = ' , F 8 . 4 / ' T h e ' , + ' a s s o c i a t e d s u m o f s q u a r e s o f d i f f e r e n c e s i s ' , F 6 . 4 ) 3 1 0 F O R M A T ( / ' U s i n g b o t h s i m p l e t h e r m a l l y a c t i v a t e d m o d e l a n d W L F ' , + ' m o d e l , ' / ' E A = ' , F 8 . 4 , ' k J / m o l , R= ' , F 6 . 4 , ' , C l = * , F 8 . 4 , + ' , C 2 = ' , F 8 . 4 ) 3 2 0 F O R M A T ( l X ' T h e a s s o c i a t e d s u m o f s q u a r e s o f d i f f e r e n c e s i s ' , F 6 . 4 ) 3 3 0 F O R M A T ( / l X ' D o y o u w a n t t o c a l c u l a t e v i s c o s i t y ? * / I X , ' 0 ' , + ' - n o ' / I X , ' 1 - y e s ' ) 3 4 0 F O R M A T ( I X ' A t w h a t t e m p e r a t u r e ( i n C e l c i u s ) ? ' ) 3 5 0 FORMAT ( l X ' A t w h a t s h e a r r a t e ( i n r a d / s ) ? ' ) 3 6 0 F O R M A T ( / ' T h e v i s c o s i t y o f t h e p o l y m e r a t ' , F 6 . 1 , ' C a n d s h e a r ' , + ' r a t e o f ' , F 6 . 2 , ' r a d / s ' / 1 i s : ' , F 1 0 . 4 , ' P a . s ' ) S T O P END 152 Appendix A: Time Temperature Superposition Program Code C T h e s u b r o u t i n e S P L I N E f i t s a c u b i c p o l y n o m i a l i n t o e v e r y C a d j a c e n t p o i n t s o f a g i v e n s e t o f d a t a t o a l l o w i n t e r p o l a t i o n C b e t w e e n p o i n t s . T h e t w o e n d c o n d i t i o n s u s e d a r e t h o s e o f s p l i n e C o f f i t t e d e n d s . S U B R O U T I N E S P L I N E ( X , Y , N , N M , Q , R , S , I I , I N , G l , G N ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N X ( N ) , Y ( N ) , Q ( N ) , R ( N + l ) , S ( N ) D I M E N S I O N A ( 3 0 1 ) , B { 3 0 1 ) , C ( 3 0 1 ) , D ( 3 0 1 ) D I M E N S I O N H ( 3 0 0 ) I F ( I 1 . E Q . 3 ) T H E N A A = 0 . D O DO 20 1 = 1 , 4 T E R M = Y ( I ) DO 10 J = l , 4 I F ( J . N E . I ) T E R M = T E R M / ( X ( I ) - X ( J ) ) 10 C O N T I N U E A A = A A + T E RM 20 C O N T I N U E E N D I F I F ( I N . E Q . 3 ) T H E N M = N - 3 B B = 0 . D 0 DO 40 I = M , N T E R M = Y ( I ) DO 30 J = M , N I F ( J . N E . I ) T E R M = T E R M / ( X ( I ) - X ( J ) ) 30 C O N T I N U E B B = B B + T E R M 40 C O N T I N U E E N D I F DO 5 0 1 = 1 , N M H ( I ) = X ( I + 1 ) - X ( I ) 50 C O N T I N U E A ( 1 ) = 0 . D 0 I F ( I l . E Q . l ) T H E N B ( 1 ) = 1 . D 0 C ( 1 ) = 0 . D 0 D ( 1 ) = 0 . D 0 E L S E I F ( I 1 . E Q . 2 ) T H E N B ( 1 ) = 2 . D 0 * H ( 1 ) C ( 1 ) = H ( 1 ) D ( 1 ) = 3 . D 0 * { ( Y ( 2 ) - Y ( l ) ) / H ( l ) - G l ) E L S E B ( l ) = - H ( l ) C ( 1 ) = H ( 1 ) D ( 1 ) = 3 . D 0 * H ( 1 ) * H ( 1 ) * A A E N D I F DO 60 1 = 2 , N M I M = I - 1 A ( I ) = H ( I M ) B ( I ) = 2 . D 0 * ( H ( I M ) + H ( I ) ) 153 Appendix A: Time Temperature Superposition Program Code C ( I ) = H ( I ) D ( I ) = 3 . D 0 * ( ( Y ( I + 1 ) - Y ( I ) ) / H ( I ) - ( Y ( I ) - Y ( I M ) ) / H ( I M ) ) 60 C O N T I N U E C ( N ) = 0 . D 0 I F ( I N . E Q . l ) T H E N A ( N ) = 0 . D 0 B ( N ) = 1 . D 0 D ( N ) = O . D O E L S E I F ( I N . E Q . 2 ) T H E N A ( N ) = H ( N M ) B ( N ) = 2 . D 0 * H ( N M ) D ( N ) = - 3 . D 0 * ( ( Y ( N ) - Y ( N M ) ) / H ( N M ) - G N ) E L S E A ( N ) = H ( N M ) B ( N ) = - H ( N M ) D ( N ) = - 3 . D 0 * H ( N M ) * H ( N M ) * B B E N D I F C A L L T D M A ( A , B , C , D , R , N , N M ) DO 7 0 1 = 1 , N M I P = I + 1 Q ( I ) = ( Y ( I P ) - Y ( I ) ) / H ( I ) - H ( I ) * ( 2 . D 0 * R ( I ) + R ( I P ) ) / 3 . D 0 S ( I ) = ( R ( I P ) - R ( I ) ) / ( 3 . D 0 * H ( I ) ) 70 C O N T I N U E R E T U R N END C T h e s u b r o u t i n e T D M A s o l v e s t h e t r i - d i a g o n a l m a t r i x u s i n g T h o m a s C a l g o r i t h m . S U B R O U T I N E T D M A ( A , B , C , D , X , N , N M ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , O - Z ) D I M E N S I O N A ( N ) , B ( N ) , C ( N ) , D ( N ) , X ( N ) , P ( 3 0 1 ) , Q ( 3 0 1 ) C A r g u m e n t l i s t : C A , B , C , D C X C N C NM T h e c o e f f i c i e n t s o f t h e t r i d i a g o n a l s e t S o l u t i o n v e c t o r N u m b e r o f u n k n o w n s N u m b e r o f i n t e r v a l b e t w e e n u n k n o w n s P ( l ) = - C ( l ) / B ( l ) Q ( 1 ) = D ( 1 ) / B ( 1 ) DO 10 1 = 2 , N I M = I - 1 D E N = A ( I ) * P ( I M ) + B ( I ) P ( I ) = - C ( I ) / D E N Q ( I ) = ( D ( I ) - A ( I ) * Q ( I M ) ) / D E N 10 C O N T I N U E X ( N ) = Q ( N ) DO 20 I = N M , 1 , - 1 X ( I ) = P ( I ) * X ( I + 1 ) + Q ( I ) 154 Appendix A: Time Temperature Superposition Program Code 2 0 C O N T I N U E R E T U R N E N D C T h e s u b r o u t i n e G O L D E N u s e s t h e g o l d e n - s e a r c h t e c h n i q u e t o f i n d C t h e m i n i m u m v a l u e o f a c e r t a i n f u n c t i o n , F . T h e s u b r o u t i n e C r e t u r n s t h e m i n i m u m v a l u e o f t h e f u n c t i o n a n d t h e v a l u e o f t h e C i n d e p e n d e n t v a r i a b l e a t w h i c h i t o c c u r s . S U B R O U T I N E G O L D E N ( A X , B X , C X , F , T O L , X M I N , F M I N ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , O - Z ) P A R A M E T E R ( R = 0 . 6 1 8 0 3 3 9 9 , C = l . D O - R ) c A r g u m e n t l i s t : c A X : T h e l o w e r l i m i t o f t h e i n t e r v a l a t w h i c h t h e m i n i m u m o c c u r s c B X : A p o i n t i n t h e i n t e r v a l w h e r e t h e f u n c t i o n e v a l u a t i o n i s c l o w e r t h a n F ( A X ) a n d F ( C X ) c C X T h e u p p e r l i m i t o f t h e i n t e r v a l a t w h i c h t h e m i n i m u m o c c u r s c F T h e f u n c t i o n t o b e m i n i m i z e d c T O L P r e - s p e c i f i e d c o n v e r g e n c e t o l e r a n c e c X M I N T h e p o i n t w h e r e m i n i m u m o c c u r s c F M I N T h e m i n i m u m v a l u e o f t h e f u n c t i o n X 0 = A X X 3 = C X I F ( D A B S ( C X - B X ) . G T . D A B S ( B X - A X ) ) T H E N X 1 = B X X 2 = B X + C * ( C X - B X ) E L S E X 2 = B X X 1 = B X - C * ( B X - A X ) E N D I F F 1 = F ( X 1 ) F 2 = F ( X 2 ) 10 I F ( D A B S ( X 3 - X 0 ) . G T . T O L * ( D A B S ( X I ) + D A B S ( X 2 ) ) ) T H E N I F ( F 2 . L T . F 1 ) T H E N X 0 = X 1 X 1 = X 2 X 2 = R * X 1 + C * X 3 F 1 = F 2 F 2 = F ( X 2 ) E L S E X 3 = X 2 X 2 = X 1 X 1 = R * X 2 + C * X 0 F 2 = F 1 F 1 = F ( X 1 ) E N D I F G O T O 10 E N D I F I F ( F 1 . L T . F 2 ) T H E N 155 Appendix A: Time Temperature Superposition Program Code F M I N = F 1 X M I N = X 1 E L S E F M I N = F 2 X M I N = X 2 E N D I F R E T U R N E N D C T h e s u b r o u t i n e L I N R E G p e r f o r m s a l i n e a r r e g r e s s i o n o f t h e f o r m : C C Y = B * X + A C C w h e r e A i s t h e Y - i n t e r c e p t o f t h e l i n e a r i z e d e q u a t i o n a n d B i s t h e C s l o p e o f t h e l i n e . I n t h i s s u b r o u t i n e , t h e c o r r e l a t i o n c o e f f i c i e n t , C R , i s a l s o c a l c u l a t e d . S U B R O U T I N E L I N R E G ( X , Y , N , A , B , R ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N X ( 1 0 ) , Y ( 1 0 ) C A r g u m e n t l i s t : C X : A v e c t o r o f i n d e p e n d e n t v a r i a b l e C Y : A v e c t o r o f d e p e n d e n t v a r i a b l e C N : N u m b e r o f d a t a p o i n t s C A : Y - i n t e r c e p t o f t h e l i n e a r l i n e C B : T h e s l o p e o f t h e l i n e C R : T h e c o r r e l a t i o n c o e f f i c i e n t o f t h e l i n e a r i z e d f i t X B A R = 0 . D O Y B A R = 0 . D O DO 10 1 = 1 , N X B A R = X B A R + X ( I ) Y B A R = Y B A R + Y ( I ) 10 C O N T I N U E X BAR=X B A R / N Y B A R = Y B A R / N S X = 0 . D 0 S Y = 0 . D 0 S U M X Y = 0 . D O DO 2 0 1 = 1 , N S X = S X + ( X ( I ) - X B A R ) * * 2 S Y = S Y + ( Y ( I ) - Y B A R ) * * 2 S U M X Y = S U M X Y + ( X ( I ) - X B A R ) * ( Y ( I ) - Y B A R ) 20 C O N T I N U E B = S U M X Y / S X A = Y B A R - B * X B A R S X = S Q R T ( S X / ( N - l ) ) S Y = S Q R T ( S Y / ( N - l ) ) 156 Appendix A: Time Temperature Superposition Program Code P S = O . D O DO 30 1 = 1 , N X I = ( X ( I ) - X B A R ) / S X Y I = ( Y ( I ) - Y B A R ) / S Y P S = P S + X I * Y I 30 C O N T I N U E R = P S / ( N - 1 ) R E T U R N E N D C T h e s u b r o u t i n e S O R T s o r t s a s e t o f d a t a i n i n c r e a s i n g o r d e r o f o n e C v a r i a b l e u s i n g t h e h e a p s o r t m e t h o d . T h i s s u b r o u t i n e h a s b e e n w r i t t e n C t o s o r t s e t s o f d a t a o f e x a c t l y t h r e e v a r i a b l e s : S U B R O U T I N E S O R T ( N , R A , R B ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N R A ( N ) , R B ( N ) C A r g u m e n t l i s t : C N : N u m b e r o f d a t a p o i n t s t o b e s o r t e d C R A : T h e f i r s t v a r i a b l e a c c o r d i n g t o w h i c h d a t a w i l l b e s o r t e d C R B : T h e s e c o n d v a r i a b l e o f t h e d a t a s e t L = N / 2 + l I R = N 10 I F ( L . G T . 1 ) T H E N L = L - 1 R R A = R A ( L ) R R B = R B ( L ) E L S E R R A = R A ( I R ) R R B = R B ( I R ) R A ( I R ) = R A ( 1 ) R B ( I R ) = R B ( 1 ) I R = I R - 1 I F ( I R . E Q . 1 ) T H E N R A ( 1 ) = R R A RB ( 1 ) = R R B G O T O 30 E N D I F E N D I F C I = L J = L + L 20 I F ( J . L E . I R ) T H E N I F ( J . L T . I R ) T H E N I F ( R A ( J ) . L T . R A ( J + 1 ) ) J = J + 1 E N D I F 157 Appendix A: Time Temperature Superposition Program Code I F ( R R A . L T . R A ( J ) ) T H E N R A ( I ) = R A ( J ) R B ( I ) = R B ( J ) I = J J = J + J E L S E J = I R + 1 E N D I F G O T O 2 0 E N D I F R A ( I ) = R R A R B ( I ) = R R B G O T O 10 30 R E T U R N E N D C T h e s u b r o u t i n e G A U S S J s o l v e s a s e t o f a l g e b r a i c s i m u l t e n e o u s e q u a t i o n s C i n t h e f o r m o f a n a u g m e n t e d c o e f f i c i e n t m a t r i x u s i n g t h e G a u s s - J o r d a n C e l i m i n a t i o n m e t h o d : S U B R O U T I N E G A U S S J ( A , N , N D R , N D C , X , RNORM, I E R R O R ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , O - Z ) D I M E N S I O N A ( N D R , N D C ) , X ( N ) , B ( 5 0 , 5 1 ) c A r g u m e n t l i s t : c A A n a r r a y o f c o e f f i c i e n t s o f a u g m e n t e d m a t r i x c N N u m b e r o f e q u a t i o n s ( e q u i v a l e n t t o t h e n u m b e r o f u n k n o w n s c N D R N u m b e r o f r o w s f o r ' A ' c N D C N u m b e r o f c o l u m n s f o r ' A ' c X O u t p u t a r r a y i n w h i c h t h e v a l u e s o f t h e s o l u t i o n v e c t o r c a r e s t o r e d c RNORM . M e a s u r e o f r e s i d u a l v e c t o r ( [C] - [A] * [X] } c I E R R O R : A n i n d i c a t i o n o f t h e s u c c e s s o f s o l v i n g t h e e q u a t i o n s c I E R R O R = 1 ==> s u c c e s s f u l c I E R R O R = 2 ==>' f a i l s t o f i n d s o l u t i o n c B e g i n : N P = N + 1 C D e f i n i n g a new w o r k i n g m a t r i x ' B ' : DO 2 0 I = 1 , N DO 10 J = 1 , NP B ( I , J ) = A ( I , J ) 10 C O N T I N U E 20 C O N T I N U E C E l i m i n a t i o n s t e p s t o b e d o n e ' N ' t i m e s : DO 80 K = 1 , N K P = K + 1 158 Appendix A: Time Temperature Superposition Program Code B I G S = O . D O C W h e n K = N , s k i p t h e s e a r c h f o r p i v o t c o e f f i c i e n t : I F (K . E Q . N) G O T O 55 C F o r e a c h r o w b e g i n n i n g f r o m t h e ' K ' t h r o w w h e r e K i s t h e e l i m i n a t i o n C s t e p n u m b e r , DO 40 I = K , N B I G = DABS ( B ( I , K ) ) C F i n d t h e l a r g e s t c o e f f i c i e n t s a l o n g t h e r o w : DO 30 J = K P , N B I G = DMAX1 ( B I G , D A B S ( B ( I , J ) ) ) 3 0 C O N T I N U E C T h e n f i n d t h e s c a l e f a c t o r , S , c o r r e s p o n d i n g t o e a c h r o w ( e q u a t i o n ) C b y d i v i d i n g t h e ( I , I ) t h c o e f f i c i e n t b y t h e l a r g e s t c o e f f i c i e n t C f o r t h a t r o w : S = D A B S ( B ( I , K ) / B I G ) C A m o n g t h e s e e q u a t i o n s ( f r o m t h e K t h t o t h e N t h ) , f i n d t h e b i g g e s t C S c a l e f a c t o r : B I G S = DMAX1 ( B I G S , S ) C T h e b i g g e s t s c a l e f a c t o r s h o u l d c o r r e s p o n d t o t h e p i v o t e q u a t i o n : I F ( B I G S . E Q . S) I P I V O T = I 40 C O N T I N U E C I f t h e p i v o t e q u a t i o n i s n o t t h e s a m e a s t h e e q u a t i o n c o r r e s p o n d i n g t o C t h e e l i m i n a t i o n s t e p n u m b e r , t h e n e x c h a n g e r o w ' K ' w i t h t h e r o w C a t w h i c h t h e p i v o t e q u a t i o n i s l o c a t e d : I F ( I P I V O T . N E . K) T H E N DO 50 J = K , N P T E M P = B ( I P I V O T , J ) B ( I P I V O T , J ) = B ( K , J ) B ( K , J ) = T E M P 50 C O N T I N U E E N D I F C I f a t l e a s t o n e o f t h e d i a g o n a l e l e m e n t s i s z e r o , t h e n r e t u r n t o m a i n C p r o g r a m w i t h ' f a i l ' m e s s a g e : 55 I F ( B ( K , K ) . E Q . O.DO) T H E N I E R R O R = 2 R E T U R N E N D I F 159 Appendix A: Time Temperature Superposition Program Code C O t h e r w i s e , s t a r t t h e e l i m i n a t i o n p r o c e s s , e l i m i n a t i n g c o e f f i c i e n t s C a b o v e a n d b e l o w t h e d i a g o n a l e l e m e n t s s i m u l t a n e o u s l y ( s e e a t t a c h e d C e x p l a n a t i o n s ) : C S t a r t f r o m t h e f i r s t r o w t o t h e l a s t DO 70 I = 1 , N C B u t s k i p p i n g t h e p i v o t r o w : I F ( I . N E . K) T H E N QUOT = B ( I , K ) / B ( K , K ) B ( I , K ) = O . D O DO 60 J = K P , NP B ( I , J ) = B ( I , J ) - QUOT * B ( K , J ) 60 C O N T I N U E E N D I F 70 C O N T I N U E 80 C O N T I N U E C R e t u r n w i t h e r r o r m e s s a g e i f t h e l a s t d i a g o n a l e l e m e n t ( a f t e r t h e C c o m p l e t i o n o f a l l e l i m i n a t i o n s t e p s ) e q u a l s z e r o : I F ( B ( N , N ) . E Q . O . D O ) T H E N I E R R O R = 2 R E T U R N E N D I F C O t h e r w i s e , s t a r t f i n d i n g t h e s o l u t i o n v e c t o r b y d i v i d i n g t h e R . H . S . C c o e f f i c i e n t s b y t h e c o r r e s p o n d i n g e l e m e n t s o f t h e d i a g o n a l m a t r i x : DO 90 I = 1 , N X ( I ) = B ( I , N P ) / B ( I , I ) 90 C O N T I N U E C C a l c u l a t i o n o f n o r m o f r e s i d u a l v e c t o r : R S Q = O . D O DO 1 1 0 I = 1 , N SUM = O . D O DO 100 J = 1 , N SUM = SUM + A ( I , J ) * X ( J ) 100 C O N T I N U E R S Q = RSQ + ( A ( I , N P ) - SUM) * * 2 110 C O N T I N U E RNORM = S Q R T (RSQ) C I f e v e r y t h i n g g o e s w e l l , r e t u r n w i t h a ' s u c c e s s ' m e s s a g e : I E R R O R = 1 R E T U R N E N D 160 Appendix A: Time Temperature Superposition Program Code C T h e s u b r o u t i n e G A U S S N r e t u r n s t h e o p t i m u m v a l u e o f t h e p a r a m e t e r s o f C a m o d e l u s i n g t h e G a u s s - N e w t o n m e t h o d o f o p t i m i z a t i o n b y m i n i m i z i n g C t h e s u m o f s q u a r e s o f d i f f e r e n c e s (up t o a s p e c i f i e d t o l e r a n c e ) : C P l e a s e n o t e t h a t i n t h i s c a s e t h e s u b r o u t i n e h a s b e e n m o d i f i e d t o C s o l v e f o r a m o d e l o f o n l y t w o p a r a m e t e r s s i n c e t h e W L F m o d e l i n v o l v e s C o n l y t w o p a r a m e t e r s . A l s o , t h i s w i l l a l l o w t h e c o n d i t i o n n u m b e r t o b e C c a l c u l a t e d m o r e e a s i l y . S U B R O U T I N E G A U S S N ( X , Y , A G U E S S , M , T O L , A M O D E L , D M O D E L , A , S N E W ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N X ( M ) , Y ( M ) , A ( 2 ) , D A ( 2 ) , A A ( 2 ) , A G U E S S ( 2 ) , A L P H A ( 2 , 3 ) E X T E R N A L A M O D E L , D M O D E L D A T A Z , N / 1 . D 0 , 2 / C A r g u m e n t L i s t : C X : A s e t o f i n d e p e n d e n t v a r i a b l e s C Y : A s e t o f d e p e n d e n t v a r i a b l e s C A G U E S S : A v e c t o r o f i n i t i a l p a r a m e t e r v a l u e s C M : T h e n u m b e r o f a v a i l a b l e d a t a p o i n t s C T O L : A s p e c i f i e d c o n v e r g e n c e c r i t e r i a C A M O D E L : T h e s p e c i f i c m o d e l w h i c h p a r a m e t e r s a r e t o b e e s t i m a t e d C DMODEL : A f u n c t i o n t h a t r e t u r n s t h e d e r i v a t i v e v a l u e o f t h e m o d e l c w i t h r e s p e c t t o t h e p a r a m e t e r s C A : A v e c t o r o f b e s t e s t i m a t e d p a r a m e t e r v a l u e s C SNEW : T h e c o r r e s p o n d i n g s u m o f s q u a r e s o f d i f f e r e n c e s a s c a l c u l a t e d C u s i n g t h e b e s t e s t i m a t e d p a r a m e t e r v a l u e s C I n i t i a l i z a t i o n o f p a r a m e t e r v a l u e s : DO 10 1 = 1 , N A ( I ) = A G U E S S ( I ) 10 C O N T I N U E C C a l c u l a t i o n o f t h e s u m o f s q u a r e s o f d i f f e r e n c e s u s i n g t h e i n i t i a l C v a l u e s o f A ( I ) S O L D = S ( A M O D E L , X , Y , M , A , N ) C S e t t i n g u p t h e a u g m e n t e d c o e f f i c i e n t m a t r i x , A L P H A , t o b e s o l v e d C f o r t h e i n c r e m e n t i n p a r a m e t e r v a l u e s , D A ( I ) : 20 DO 60 1 = 1 , N DO 40 J = 1 , N A L P H A ( I , J ) = 0 . D O DO 30 K = 1 , M A L P H A ( I , J ) = A L P H A ( I , J ) + D M O D E L ( A , X ( K ) , I ) * D M O D E L ( A , X ( K ) , J ) 30 C O N T I N U E I F ( J . N E . I ) A L P H A ( J , I ) = A L P H A ( I , J ) 40 C O N T I N U E 161 Appendix A: Time Temperature Superposition Program Code A L P H A ( I , N + 1 ) = 0 . D 0 DO 50 K=1 ,M A L P H A ( I , N + l ) = A L P H A ( I , N + l ) + + D M O D E L ( A , X ( K ) , I ) * ( Y ( K ) - A M O D E L ( X ( K ) , A ) ) 50 C O N T I N U E 60 C O N T I N U E C C a l c u l a t i o n o f t h e E i g e n v a l u e s o f t h e c o e f f i c i e n t m a t r i x , A L P H A : B = - ( A L P H A ( 1 , 1 ) + A L P H A ( 2 , 2 ) ) C = A L P H A ( 1 , 1 ) * A L P H A ( 2 , 2 ) - A L P H A ( 1 , 2 ) * A L P H A ( 2 , 1 ) E I G E N 1 = ( - B + S Q R T ( B * * 2 - 4 * C ) ) / 2 E I G E N 2 = ( - B - S Q R T ( B * * 2 - 4 * C ) ) / 2 C I n i t i a l i z i n g t h e c o n s t a n t f o r M a r q u a r d t ' s m o d i f i c a t i o n : G A M M A = 0 . D O C C a l c u l a t i o n o f t h e c o n d i t i o n n u m b e r : 70 C O N D = D A B S ( ( D M A X 1 ( E I G E N 1 , E I G E N 2 ) + G A M M A ) / + ( D M I N 1 ( E I G E N 1 , E I G E N 2 ) + G A M M A ) ) C T o p e r f o r m M a r q u a r d t ' s m o d i f i c a t i o n i f t h e c o n d i t i o n n u m b e r i s l a r g e : I F ( C O N D . G T . 1 . D 3 ) T H E N G A M M A = G A M M A + 1 . D l DO 80 1 = 1 , N A L P H A ( I , I ) = A L P H A ( I , I ) + G A M M A 80 C O N T I N U E GOTO 70 E N D I F C C a l l i n g t h e s u b r o u t i n e G A U S S J t o s o l v e f o r D A ( I ) : C A L L G A U S S J ( A L P H A , N , N , N + l , D A , R N O R M , I E R R O R ) C C h e c k i n g f o r s i n g u l a r i t y : I F ( I E R R O R . E Q . 2 ) T H E N P R I N T * , " W R N - P r o g r a m f a i l s t o d e t e r m i n e p a r a m e t e r s ! ! ' S T O P E N D I F E R R = 0 . D O C C h e c k i n g f o r c o n v e r g e n c e : DO 90 1 = 1 , N E R R = D M A X 1 ( E R R , D A B S ( D A ( I ) / A ( I ) ) ) 162 Appendix A: Time Temperature Superposition Program Code 90 C O N T I N U E I F ( E R R . L E . T O L ) G O T O 1 3 0 C U p d a t i n g t h e p a r a m e t e r v a l u e s b y Z * D A ( I ) : 1 0 0 DO 1 1 0 1 = 1 , N A A ( I ) = A ( I ) + ( Z * D A ( I ) ) 1 1 0 C O N T I N U E S N E W = S ( A M O D E L , X , Y , M , A A , N ) C W h e r e Z i s a r e a l n u m b e r t h a t r e d u c e s t h e s u m o f s q u a r e s o f C d i f f e r e n c e s : I F ( S N E W . G T . S O L D ) T H E N Z = Z * 5 . D - 1 G O T O 1 0 0 E L S E DO 1 2 0 1 = 1 , N A ( I ) = A A ( I ) 1 2 0 C O N T I N U E SOLD=SNEW G O T O 2 0 E N D I F 1 3 0 R E T U R N END C T h e f o l l o w i n g f u n c t i o n r e t u r n s t h e v a l u e o f t h e s u m o f s q u a r e s C o f d i f f e r e n c e s b e t w e e n t h e c a l c u l a t e d d a t a a n d t h e d a t a o b t a i n e d C f r o m t h e m o d e l : D O U B L E P R E C I S I O N F U N C T I O N S ( A M O D E L , X , Y , M , A , N ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N X ( M ) , Y ( M ) , A ( N ) S = 0 . D 0 DO 10 K=1 ,M S = S + ( Y ( K ) - A M O D E L ( X ( K ) , A ) )**2 10 C O N T I N U E R E T U R N END C T h e f o l l o w i n g f u n c t i o n r e t u r n s t h e v a l u e o f l n ( A T ( i ) ) a s C c a l c u l a t e d u s i n g t h e W L F m o d e l : D O U B L E P R E C I S I O N F U N C T I O N A M O D E L ( X , A ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N A ( 2 ) C O M M O N / B L K F / T R E F D E L T = X - ( T R E F + 2 7 3 . 15) A M O D E L = A ( l ) * D E L T / ( A ( 2 ) + D E L T ) 163 Appendix A: Time Temperature Superposition Program Code R E T U R N E N D C T h e f o l l o w i n g i s a f u n c t i o n t h a t c a l c u l a t e s t h e d e r i v a t i v e o f t h e C W L F m o d e l w i t h r e s p e c t t o t h e p a r a m e t e r s A ( l ) o r A ( 2 ) : D O U B L E P R E C I S I O N F U N C T I O N D M O D E L ( A , X , I ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N A ( 2 ) C O M M O N / B L K F / T R E F D E L T = X - ( T R E F + 2 7 3 . 1 5 ) G O T O ( 1 0 , 2 0 ) , I 10 D M O D E L = D E L T / ( A ( 2 ) + D E L T ) R E T U R N 20 D M O D E L = - A ( l ) * D E L T / ( ( A ( 2 ) + D E L T ) * * 2 ) R E T U R N E N D C T h e f o l l o w i n g i s a f u n c t i o n t h a t a l l o w s i n t e r p o l a t i o n . b e t w e e n p o i n t s C t o b e d o n e u p o n t h e a v a i l a b i l i t y o f t h e p a r a m e t e r s Q , R , S : D O U B L E P R E C I S I O N F U N C T I O N F ( X , Y , Q , R , S , N , Z) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N X ( 3 0 0 ) , Y ( 3 0 0 ) , Q ( 3 0 0 ) , R ( 3 0 1 ) , S ( 3 0 0 ) I F ( Z . L T . X ( l ) ) T H E N 1=1 W R I T E ( 9 , 2 0 ) Z E L S E I F ( Z . G T . X ( N ) ) T H E N I = N - 1 W R I T E ( 9 , 2 0 ) Z E L S E C T h e b i s e c t i o n m e t h o d i s u s e d t o f i n d t h e l o c a t i o n o f t h e p o i n t o f C i n t e r e s t : 1=1 J = N 10 K = I N T ( ( I + J ) / 2 ) I F ( Z . L T . X ( K ) ) J = K I F ( Z . G E . X ( K ) ) I = K I F ( J . G T . I + 1 ) G O T O 10 E N D I F D X = Z - X ( I ) F = Y ( I ) + D X * ( Q ( I ) + D X * ( R ( I ) + D X * S ( I ) ) ) 20 F O R M A T ( / ' W a r n i n g - ' , D 1 0 . 3 , ' i s o u t s i d e i n t e r p o l a t i o n r a n g e ' / ) R E T U R N E N D 164 Appendix A: Time Temperature Superposition Program Code C T h e f o l l o w i n g i s t h e o b j e c t i v e f u n c t i o n t h a t i s t o b e m i n i m i z e d C t o o b t a i n t h e m a s t e r c u r v e : D O U B L E P R E C I S I O N F U N C T I O N O B J ( X ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) C O M M O N / B L K A / Q ( 3 0 0 ) , R ( 3 0 1 ) , S ( 3 0 0 ) C O M M O N / B L K C / A L N S R T ( 3 0 0 ) , A L N S T R S ( 3 0 0 ) C O M M O N / B L K D / R E F S R T ( 3 0 0 ) , R E F S T R S ( 3 0 0 ) C O M M O N / B L K E / W T ( 3 0 0 ) , M D A T A , I T O T A L O B J = 0 . D O DO 10 1 = 1 , I T O T A L C T h e o b j e c t i v e f u n c t i o n s h o u l d o n l y b e c a l c u l a t e d f o r a c e r t a i n r a n g e C o f s h e a r r a t e v a l u e s , f o r w h i c h t h e r e i s a common s t r e s s v a l u e C b e t w e e n t h e r e f e r e n c e c u r v e a n d t h e c u r v e t h a t i s t o b e s h i f t e d : I F ( ( R E F S T R S ( I ) . G E . A L N S T R S ( 1 ) ) . A N D . + ( R E F S T R S ( I ) . L E . A L N S T R S ( M D A T A ) ) ) T H E N O B J = O B J + ( R E F S R T ( I ) -+ F ( A L N S T R S , A L N S R T , Q , R , S , M D A T A , R E F S T R S ( I ) ) + - D L O G ( X ) ) * * 2 * W T ( I ) E N D I F 10 C O N T I N U E R E T U R N E N D *******************+*************^*****^ S a m p l e I n p u t F i l e : D a t a . t x t : 12 12 12 1 8 0 2 0 0 2 2 0 c : / f o r t r a n / p r o g r a m / p r o j e c t / d a t a l . t x t c : / f o r t r a n / p r o g r a m / p r o j e c t / d a t a 2 . t x t c : / f o r t r a n / p r o g r a m / p r o j e c t / d a t a 3 . t x t R e s i n X , L o t # X X 1 2 3 - 4 5 6 0 0 D a t a l . t x t : 4 . 8 8 7 1 3 6 2 4 8 8 . 0 0 0 0 4 4 9 8 0 1 2 . 0 0 0 5 3 7 8 4 2 0 . 9 4 5 6 7 7 6 1 3 0 . 0 0 0 7 9 6 9 9 6 2 . 8 3 4 1 0 7 7 3 4 9 7 . 7 4 1 1 2 9 5 4 5 165 Appendix A: Time Temperature Superposition Program Code 2 0 2 . 4 6 1 7 3 2 4 4 3 0 0 . 2 1 2 0 2 5 8 3 4 9 5 . 6 9 2 4 3 4 1 0 6 9 8 . 1 5 2 7 4 3 8 0 8 7 2 . 6 9 2 9 4 9 4 9 S a m p l e O u t p u t F i l e : R e s i n X , L o t # X X 1 2 3 - 4 5 6 0 0 R E S U L T I N G M A S T E R C U R V E D A T A P O I N T S : R a t e ( 1 / s ) S t r e s s (Pa) V i s c o s i t y ( P a . s ) 2 . 7 9 0 0 3 1 2 0 2 1 1 1 8 4 3 . 5 4 2 6 3 4 6 1 8 9772 4 . 5 6 7 1 3 8 8 3 5 8 5 0 3 4 . 8 8 7 1 4 0 1 2 9 8 2 1 1 5 . 7 9 9 0 4 3 1 2 8 7 4 3 7 6 . 8 5 0 7 4 6 3 7 9 6 7 7 0 8 . 0 0 0 0 4 9 8 5 6 6232 8 . 6 9 8 5 5 1 5 5 2 5 9 2 7 1 1 . 9 5 7 3 5 8 9 9 0 4 9 3 3 1 2 . 0 0 0 0 5 9 5 1 7 4 9 6 0 1 5 . 1 8 2 6 6 5 4 3 2 4 3 1 0 1 7 . 1 2 6 6 6 8 6 8 8 4 0 1 1 2 0 . 9 4 5 0 7 5 1 3 4 3 5 8 7 2 1 . 7 4 6 3 7 6 3 1 9 3 5 1 0 3 0 . 0 0 0 0 8 7 6 6 8 2 9 2 2 3 5 . 8 7 1 2 9 3 3 7 5 2 6 0 3 4 5 . 5 4 7 0 1 0 3 5 4 3 2 2 7 3 5 5 . 7 9 9 2 1 1 1 9 2 5 2 0 0 5 6 2 . 8 3 4 0 1 1 7 8 2 4 1 8 7 5 7 0 . 8 5 0 3 1 2 3 6 4 6 1 7 4 5 9 7 . 7 4 1 0 1 4 0 4 8 9 1 4 3 7 1 1 5 . 5 8 2 0 1 4 9 8 8 1 1 2 9 6 1 4 6 . 7 5 8 8 1 6 4 9 3 9 1 1 2 3 1 7 1 . 3 8 6 3 1 7 5 4 9 5 1 0 2 3 2 0 2 . 4 6 0 0 1 8 6 1 2 8 919 2 1 7 . 6 1 5 6 1 9 1 9 5 0 882 2 8 2 . 9 8 3 5 2 1 2 2 0 8 7 4 9 3 0 0 . 2 1 0 0 2 1 5 4 6 8 7 1 7 3 5 9 . 3 1 4 8 2 3 1 0 6 9 643 3 9 8 . 5 6 5 5 2 4 1 7 8 0 606 4 9 5 . 6 9 0 0 2 5 8 0 8 0 5 2 0 4 9 8 . 2 0 8 3 2 6 3 0 4 8 527 5 0 6 . 0 7 3 6 2 6 1 7 2 8 517 6 3 2 . 5 9 3 8 2 8 3 3 8 4 447 6 9 8 . 1 5 0 0 2 8 9 6 7 1 414 8 7 2 . 6 9 0 0 3 0 9 3 0 8 354 SUMMARY O F R E S U L T S : 166 Appendix A: Time Temperature Superposition Program Code T h e m a s t e r c u r v e was c a l c u l a t e d b a s e d o n t h e r e f e r e n c e t e m p e r a t u r e o f 1 8 0 . 0 C T h e s h i f t f a c t o r c o r r e s p o n d i n g t o t e m p e r a t u r e o f 1 8 0 . 0 C i s : 1 . 0 0 0 0 T h e c o r r e s p o n d i n g v a l u e o f t h e o b j e c t i v e f u n c t i o n i s : . 0 0 0 0 T h e s h i f t f a c t o r c o r r e s p o n d i n g t o t e m p e r a t u r e o f 2 0 0 . 0 C i s : 0 . 7 2 4 9 T h e c o r r e s p o n d i n g v a l u e o f t h e o b j e c t i v e f u n c t i o n i s : . 0 0 1 9 T h e s h i f t f a c t o r c o r r e s p o n d i n g t o t e m p e r a t u r e o f 2 2 0 . 0 C i s : 0 . 5 7 0 9 T h e c o r r e s p o n d i n g v a l u e o f t h e o b j e c t i v e f u n c t i o n i s : . 0 1 1 3 U s i n g s i m p l e t h e r m a l l y a c t i v a t e d m o d e l , E a = 2 6 . 0 7 6 0 k J / m o l T h e c o r r e s p o n d i n g r e g r e s s i o n c o r r e l a t i o n c o e f f i c i e n t i s R= . 9 9 8 2 167 Appendix B: GPC Analysis Program Code APPENDLX B: G P C Analysis Program Code 168 Appendix B: GPC A nalysis Program Code * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * A l f o n s i u s B u d i A r i a w a n * * D e p a r t m e n t o f C h e m i c a l E n g i n e e r i n g * * T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a * * V a n c o u v e r , B . C . * * * * A p r i l 1 5 , 1998 * * * * S U M M A R Y : * * T h i s p r o g r a m i n p u t s G P C p r o f i l e s o f s e v e r a l p o l y m e r s a n d d i v i d e s * * e a c h p r o f i l e i n t o a n u m b e r o f d i f f e r e n t MW g r o u p s . E a c h p r o f i l e * * i s t h e n f i t t e d t o s p l i n e s o f f i t t e d e n d s a n d t h e a r e a * * c o r r e s p o n d i n g t o e a c h MW g r o u p i s c a l c u l a t e d u s i n g 4 - p a n e l * * N e w t o n - C o t e s m e t h o d . * * * ************************************************************************ PROGRAM G P C I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) D I M E N S I O N A L O G M W ( 8 0 0 ) , C O N C ( 8 0 0 ) D I M E N S I O N X ( 2 0 ) , A R E A ( 3 0 , 2 0 , 21) C H A R A C T E R * 2 0 R E S I N C H A R A C T E R * 40 F N A M E ( 3 0 ) E X T E R N A L F D A T A E P S / l . D - 6 / C R e a d i n g i n f o r m a t i o n f r o m i n p u t f i l e : O P E N ( 3 , F I L E = ' G P C D A T ( a l l ) . T X T ' , S T A T U S = ' O L D 1 ) DO 10 I = 1 , 30 R E A D ( 3 , ' ( A ) ' , ERR=20) F N A M E ( I ) N R E S I N = I 10 C O N T I N U E 20 C L O S E (3) C P r o m p t i n g f o r m o r e i n f o r m a t i o n : P R I N T 1 9 0 R E A D * , A L O W E R P R I N T 2 0 0 R E A D * , U P P E R P R I N T 2 1 0 R E A D * , N NP = N + 1 C W o r k i n g i n L o g s c a l e : X ( l ) = D L O G 1 0 (ALOWER) X ( N P ) = D L O G 1 0 ( U P P E R ) 169 Appendix B: GPC Analysis Program Code C C a l c u l a t i n g t h e w i d t h o f e a c h m o l e c u l a r w e i g h t r a n g e : D E L T A = ( X ( N P ) - X ( l ) ) / N C C a l c u l a t i n g t h e v a r i o u s m o l e c u l a r w e i g h t l i m i t s : DO 30 I = 2 , N I M = I - 1 X ( I ) = X ( I M ) + D E L T A 30 C O N T I N U E C I n i t i a l i z a t i o n o f a r e a u n d e r e a c h s l i c e : DO 50 I = 1 , N R E S I N DO 40 J = 1 , NP A R E A ( I , J , J ) = 0 . D 0 40 C O N T I N U E 50 C O N T I N U E C O p e n i n g t h e o u t p u t f i l e : O P E N ( 5 , F I L E = ' G P C . T X T ' ) W R I T E ( 5 , 1 5 0 ) K = l 60 I F ( K . L E . N ) T H E N KP=K+1 W R I T E ( 5 , 1 7 0 ) W R I T E ( 5 , 1 6 0 ) DO 70 J = K , N W R I T E ( 5 , 2 3 0 ) X ( K ) 70 C O N T I N U E W R I T E ( 5 , 2 3 0 ) W R I T E ( 5 , 1 6 0 ) DO 80 J = K P , NP W R I T E ( 5 , 2 3 0 ) X ( J ) 80 C O N T I N U E 170 A ppendix B: GPC A nalysis Program Code W R I T E ( 5 , 1 8 0 ) DO 1 3 0 J = 1 , N R E S I N O P E N ( 7 , F I L E = F N A M E ( J ) , S T A T U S = ' O L D ' ) R E A D ( 7 , *) R E S I N DO 90 I = 1 , 1 0 0 0 R E A D ( 7 , * , E R R = 1 0 0 ) A L O G M W ( I ) , C O N C ( I ) N D A T A = I 90 C O N T I N U E 100 W R I T E ( 5 , 2 3 0 ) W R I T E ( 5 , 2 2 0 ) R E S I N I F ( K . E Q . l ) T H E N N D A T A M = N D A T A - 1 C F i t t i n g t h e MWD c u r v e w i t h s p l i n e : C A L L S P L I N E (ALOGMW, C O N C , N D A T A , N D A T A M , 3 , 3 , + 0 . D 0 , 0 . D 0 ) C I n t e g r a t i n g t h e f i t t e d c u r v e c o r r e s p o n d i n g t o t h e a p p r o p r i a t e C m o l e c u l a r w e i g h t s l i c e u s i n g 4 - p a n e l N e w t o n - c o t e s m e t h o d : C T o t a l a r e a : C A L L A D N C ( 4 , F , A L O G M W ( 1 ) , A L O G M W ( N D A T A ) , E P S , + T O T A L , N P O I N T ) C C a l c u l a t i o n o f n o r m a l i z e d a r e a : DO 1 1 0 J J = 2 , NP C A L L A D N C ( 4 , F , X ( 1 ) , X ( J J ) , E P S , S U M , N P O I N T ) A R E A ( J , 1 , J J ) = SUM / T O T A L W R I T E ( 5 , 2 3 0 ) A R E A ( J , 1 , J J ) 110 C O N T I N U E E L S E C C a l c u l a t i o n o f o t h e r i n t e r m e d i a t e s l i c e s : DO 1 2 0 J J = K P , NP A R E A ( J , K , J J ) = AREA(J, 1, J J ) -AREA ( J , 1,K) W R I T E ( 5 , 2 3 0 ) A R E A ( J , K , J J ) 1 2 0 C O N T I N U E E N D I F 171 A ppendix B: GPC Analysis Program Code 1 3 0 C O N T I N U E C L O S E (7) K = K + 1 G O T O 60 E N D I F 1 4 0 C L O S E (5) C F O R M A T s t a t e m e n t s : 1 5 0 F O R M A T ( 3 X , ' R e s i n ' , 4 5 X , ' M W L i m i t s ( i n L o g ) ' \ ) 1 6 0 F O R M A T ( 5 X , ' * ' , 5 X , \ ) 1 7 0 F O R M A T ( / F 1 0 . 4 \ ) 1 8 0 F O R M A T ( F 1 0 . 4 / ) 1 9 0 F O R M A T ( l X , ' E n t e r t h e a b s o l u t e l o w e r l i m i t f o r i n t e g r a t i o n : • A ) 2 0 0 F O R M A T ( / I X , " E n t e r t h e a b s o l u t e u p p e r l i m i t f o r i n t e g r a t i o n : ' , \ ) 2 1 0 F O R M A T ( / I X , ' E n t e r t h e n u m b e r o f MW g r o u p s d e s i r e d : ' , \ ) 2 2 0 F O R M A T ( 1 X , A 1 0 , \ ) 2 3 0 F O R M A T ( F 1 0 . 4 , \ ) S U B R O U T I N E S P L I N E ( X , Y , N , N M , I I , I N , G l , GN) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) COMMON / B L K B / X X ( 8 0 1 ) , Y Y ( 8 0 1 ) , N N , NNM COMMON / B L K C / Q ( 8 0 0 ) , R ( 8 0 1 ) , S ( 8 0 0 ) D I M E N S I O N X ( N ) , Y ( N ) , H ( 8 0 0 ) D I M E N S I O N A ( 8 0 1 ) , B ( 8 0 1 ) , C ( 8 0 1 ) , D ( 8 0 1 ) c A r g u m e n t L i s t : c X : A n a r r a y o f i n d e p e n d e n t v a r i a b l e s c y : A n a r r a y o f d e p e n d e n t v a r i a b l e s c N : T o t a l n u m b e r o f a v a i l a b l e d a t a p o i n t s c NM : N u m b e r o f i n t e r v a l s c 11 : B o u n d a r y c o n d i t i o n a t X ( l ) : c 1 . N a t u r a l c 2 . C l a m p e d c 3 . F i t t e d c I N : B o u n d a r y c o n d i t i o n a t X ( n ) c 1 . N a t u r a l c 2 . C l a m p e d c 3 . F i t t e d c G l , G N : D e r i v a t i v e v a l u e s a t X ( l ) a n d X ( n ) , r e s p e c t i v e l y c ( n e e d e d o n l y i f ' c l a m p e d ' s p l i n e i s d e s i r e d ) C A s s i g n i n g dummy v a r i a b l e v a l u e s t o b e u s e d i n COMMON b l o c k : NN = N NNM = NM S T O P E N D C S u b r o u t i n e S P L I N E b e g i n s : 172 Appendix B: GPC A nalysis Program Code DO 5 I = 1 , N X X ( I ) = X ( I ) Y Y ( I ) = Y ( I ) 5 C O N T I N U E C I f ' f i t t e d ' s p l i n e i s d e s i r e d , t h e n u s e L a n g r a n g e C p o l y n o m i a l i n t e r p o l a t i o n m e t h o d t o d e t e r m i n e c o n d i t i o n a t t h e e n d C p o i n t ( s ) : I F ( I I . E Q . 3) T H E N A A = 0 . D O DO 40 I = 1 , 4 T E R M = Y ( I ) DO 30 J = 1 , 4 I F ( J . N E . I ) T E R M = T E R M / ( X ( I ) - X ( J ) ) 30 C O N T I N U E A A = A A + T E R M 40 C O N T I N U E E N D I F I F ( I N . E Q . 3) T H E N M = N - 3 B B = 0 . D 0 DO 60 I = M , N T E R M = Y ( I ) DO 50 J = M , N I F ( J . N E . I ) T E R M = T E R M / ( X ( I ) - X ( J ) ) 50 C O N T I N U E B B = BB + T E R M 60 C O N T I N U E E N D I F C C a l c u l a t i n g i n t e r v a l s i z e : DO 70 I = 1 , NM H ( I ) = X ( I + 1 ) - X ( I ) 70 C O N T I N U E C C a l c u l a t i n g t h e c o e f f i c i e n t s f o r t h e t r i d i a g o n a l s e t : A ( l ) = 0 . D 0 I F ( I I . E Q . 1) T H E N B ( l ) = 1 . D 0 C ( l ) = 0 . D 0 D ( l ) = 0 . D 0 173 Appendix B: GPC Analysis Program Code E L S E I F ( I I . E Q . 2) T H E N B ( l ) = 2 . D O * H ( l ) C ( l ) = H ( l ) D ( l ) = 3 . D O + ( ( Y ( 2 ) - Y ( D ) / H ( l ) - G l ) E L S E B ( l ) = - H ( l ) C ( l ) = H ( l ) D ( l ) = 3 . D O * H ( l ) * H ( l ) * A A E N D I F DO 80 I = 2 , NM I M = I - 1 A ( I ) = H ( I M ) B ( I ) = 2 . D O * C ( I ) = H ( I ) D ( I ) = 3 . D O * 80 C O N T I N U E C ( N ) = 0 . D 0 ( H ( I M ) + H ( I ) ) ( ( Y ( I + 1 ) - Y ( I ) ) / H ( I ) - ( Y ( I ) - Y ( I M ) ) / H ( I M ) ) I F ( I N . E Q . 1) T H E N A ( N ) = O . D O B ( N ) = 1 . D 0 D ( N ) = O . D O E L S E I F ( I N . E Q . 2) T H E N A ( N ) = H (NM) B ( N ) = 2 . D O * H(NM) D ( N ) = - 3 . D O * ( ( Y ( N ) - Y ( N M ) ) / H(NM) - GN) E L S E A ( N ) = H(NM) B ( N ) = - H ( N M ) D ( N ) = - 3 . D O * H(NM) * H(NM) * BB E N D I F C C a l l i n g T h o m a s A l g o r i t h m t o s o l v e f o r t h e t r i d i a g o n a l s e t : C A L L T D M A ( A , B , C , D , R , N , NM) C D e t e r m i n i n g t h e c o e f f i c i e n t s o f t h e c u b i c p o l y n o m i a l s C p a s s i n g t h r o u g h e a c h p a i r o f d a t a p o i n t s : DO 90 I = 1 , NM I P = I + 1 Q ( I ) = ( Y ( I P ) - Y ( I ) ) / H ( I ) - H ( I ) * + ( 2 . D O * R ( I ) + R ( I P ) ) / 3 . D O S ( I ) = ( R ( I P ) - R ( I ) ) / ( 3 . D O * H ( I ) ) 90 C O N T I N U E R E T U R N E N D C S u b r o u t i n e T D M A b e g i n s : C T h i s s u b r o u t i n e s o l v e s t r i d i a g o n a l m a t r i x u s i n g T h o m a s C A l g o r i t h m : S U B R O U T I N E T D M A ( A , B , C , D , X , N , N M ) 174 A ppendix B: GPC A nalysis Program Code I M P L I C I T D O U B L E P R E C I S I O N ( A - H , O - Z ) D I M E N S I O N A ( N ) , B ( N ) , C ( N ) , D ( N ) , X ( N ) , P ( 8 0 1 ) , Q ( 8 0 1 ) P ( l ) = - C ( l ) / B ( l ) Q ( l ) = D ( l ) / B ( l ) DO 10 I = 2 , N I M = I - 1 D E N = A ( I ) * P ( I M ) + B ( I ) P ( I ) = - C ( I ) / D E N Q ( I ) = ( D ( I ) - A ( I ) * Q ( I M ) ) / DEN 10 C O N T I N U E X ( N ) = Q ( N ) DO 2 0 I = N M , 1 , - 1 X ( I ) = P ( I ) * X ( I + 1 ) + Q ( I ) 20 C O N T I N U E R E T U R N E N D C S u b r o u t i n e A D N C b e g i n s : S U B R O U T I N E A D N C ( N , F , A , B , E P S , A R E A , N P O I N T ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , O - Z ) D I M E N S I O N H ( 2 0 ) , T O L ( 2 0 ) , S R ( 2 0 ) , X R ( 2 0 ) D I M E N S I O N F F ( 2 0 , 2 0 ) , C ( 6 ) , B B ( 6 , 7 ) C T h e f o l l o w i n g D A T A s t a t e m e n t d e f i n e s t h e c o n s t a n t s n e e d e d f o r t h e C v a r i o u s i n t e g r a t i o n m e t h o d s : D A T A C , B B / 5 . D - 1 , 0 . 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 D 0 , 3 . 7 5 D - 1 , 4 . 4 4 4 44 44 4 4 4 4 4 4 4 4 D - 2 , + 1 . 7 3 6 1 1 1 1 1 1 1 1 1 1 1 D - 2 , 7 . 1 4 2 8 5 7 1 4 2 8 5 7 1 4 D - 3 , 3 * 1 . D O , 7 . D O , + 1 . 9 D 1 , 4 . 1 D 1 , 1 . D 0 , 4 . D 0 , 3 . D 0 , 3 . 2 D 1 , 7 . 5 D 1 , 2 . 1 6 D 2 , 0 . D 0 , + 1 . D 0 , 3 . D 0 , 1 . 2 D 1 , 5 . 0 D 1 , 2 . 7 D 1 , 2 * 0 . D 0 , 1 . D O , 3 . 2 D 1 , 5 . 0 D 1 , + 2 . 7 2 D 2 , 3 * 0 . D 0 , 7 . D 0 , 7 . 5 D 1 , 2 . 7 D 1 , 4 * 0 . D 0 , 1 . 9 D 1 , 2 . 1 6 D 2 , + 5 * 0 . D 0 , 4 . 1 D 1 / C D e f i n i n g t h e m a x i m u m a d a p t i v e l e v e l a l l o w e d : D A T A M A X L / 2 0 / C A r g u m e n t L i s t : C N : ' P a n e l ' n u m b e r ( e g . N = 2 f o r a d a p t i v e S i m p s o n C r u l e , e t c . ) T h e f u n c t i o n t o b e i n t e g r a t e d L o w e r l i m i t o f i n t e g r a t i o n U p p e r l i m i t o f i n t e g r a t i o n C o n v e r g e n c e c r i t e r i a R e s u l t o f i n t e g r a t i o n N u m b e r o f f u n c t i o n e v a l u a t i o n s C F C A C B C E P S C A R E A C N P O I N T I n i t i a l i z a t i o n o f v a r i a b l e s : A R E A = 0 . D 0 S = 0 . D 0 175 Appendix B: GPC A nalysis Program Code C I n i t i a l n u m b e r o f f u n c t i o n e v a l u a t i o n s : N P O I N T = N + 1 C D e f i n i n g t h e o r i g i n a l i n t e g r a t i o n i n t e r v a l : X I = A X R ( 1 ) = B C D e f i n i n g t h e l e n g t h o f i n t e r v a l a n d t h e c o n v e r g e n c e c r i t e r i a o f C e a c h a d a p t i v e l e v e l , u p t o t h e m a x i m u m l e v e l a l l o w e d : H ( l ) = ( B - A ) / N I F ( M 0 D ( N , 2 ) . E Q . O . D O ) T H E N N R = N + 2 E L S E N R = N + 1 E N D I F I F ( M O D ( I N T ( N / 2 . D O ) , 2 ) . E Q . O . D O ) T H E N RN = 2 * * NR - 4 E L S E RN = 2 * * NR - 6 E N D I F T 0 L ( 1 ) = RN * E P S DO 10 I = 2 , M A X L I M = I - 1 H ( I ) = H ( I M ) / 2 . D O T O L ( I ) = T O L ( I M ) / 2 . D O 10 C O N T I N U E C C a l c u l a t i o n o f a r e a o v e r t h e o r i g i n a l i n t e g r a t i o n i n t e r v a l : NP = N + 1 J = - 1 . D 0 DO 2 0 I = 1 , N P J = J + 2 I F ( I . N E . NP) T H E N F F ( 1 , J ) = F ( A + ( 1 - 1 ) * H ( l ) ) E L S E F F ( 1 , J ) = F ( B ) E N D I F S = S + B B ( N , I ) * F F ( 1 , J ) 20 C O N T I N U E S = S * C ( N ) * H ( l ) C D e f i n i n g t h e f i r s t l e v e l : L = 1 C U p d a t i n g t h e n u m b e r o f f u n c t i o n e v a l u a t i o n s : .176 A ppendix B: GPC A nalysis Program Code 30 N P O I N T = N P O I N T + N DO 40 I = 1 , N F F ( L , 2 * I ) = F ( X 1 + ( I * 2 - 1 ) * H ( L ) / 2 . D 0 ) 40 C O N T I N U E C C a l c u l a t i n g t h e a r e a s u n d e r t h e t w o new i n t e g r a t i o n i n t e r v a l s C ( e a c h i n t e r v a l b e i n g h a l f t h e s i z e o f t h e p r e v i o u s i n t e g r a t i o n C i n t e r v a l ) : S L = 0 . D 0 S R ( L ) = 0 . D 0 DO 50 I = 1 , NP S L = S L + B B ( N , I ) * F F ( L , I ) S R ( L ) = S R ( L ) + B B ( N , I ) * F F ( L , I + N ) 50 C O N T I N U E S L = S L * H ( L ) * C ( N ) / 2 . D O S R ( L ) = S R ( L ) * H ( L ) * C ( N ) / 2 . D O C I f t h e s u m o f t h e a r e a s a r e s i g n i f i c a n t l y d i f f e r e n t f r o m t h e a r e a C c a l c u l a t e d f o r t h e p r e v i o u s i n t e r v a l s i z e . . . . I F ( D A B S ( S L + S R ( L ) - S ) . G T . T O L ( L ) ) T H E N L M = L L = L + 1 C . . . a n d t h e l e v e l i s s t i l l s m a l l e r t h a n t h e m a x i m u m p o s s i b l e l e v e l , I F ( L . L E . M A X L ) T H E N C m o v e t o t h e l e f t h a l f o f t h e i n t e g r a t i o n i n t e r v a l , i n c r e a s e t h e C i n t e g r a t i o n l e v e l a n d d e f i n e a new i n t e g r a t i o n i n t e r v a l : S = S L DO 60 I = 1 , NP F F ( L , 2 * 1 - 1 ) = F F ( L M , I ) 60 C O N T I N U E X R ( L ) = X I + N * H ( L ) G O T O 30 E L S E C O t h e r w i s e , i n t e g r a t i o n h a s f a i l e d : W R I T E ( 5 , 9 0 ) X I R E T U R N E N D I F E L S E 177 Appendix B: GPC Analysis Program Code C I f t h e s u m o f t h e a r e a s a r e c l o s e d t o t h e a r e a c a l c u l a t e d C f o r t o t h e p r e v i o u s i n t e g r a l i n t e r v a l , t h e n i n c o r p o r a t e C t h e s u m i n t o t h e t o t a l a r e a . . . A R E A = A R E A + S L + S R ( L ) X I = X I + N * H ( L ) C m o v e t o t h e r i g h t a n d f i n d t h e c o r r e c t l e v e l t o g o t o : DO 80 I = L , 1 , - 1 C I f t h e d i f f e r e n c e b e t w e e n X I a n d X R ( L ) i s c l o s e d t o z e r o ( o r C s m a l l e r t h a n h a l f o f t h e m i n i m u m i n t e r v a l s i z e ) , t h e n t h e c o r r e c t C l e v e l h a s b e e n f o u n d : I F ( D A B S ( X l - X R ( I ) ) . L T . H ( M A X L ) / 2 . D 0 ) T H E N L = I C I f t h e f i r s t l e v e l i s f o u n d , t h e n i n t e g r a t i o n i s d o n e : I F ( I i . E Q . 1) R E T U R N L M = L - 1 S = S R ( L M ) DO 70 J = 1 , NP F F ( L , 2 * J - 1 ) = F F ( L M , J + N ) 7 0 C O N T I N U E C O t h e r w i s e , c o n t i n u e . . . G O T O 30 E N D I F 80 C O N T I N U E E N D I F C F O R M A T s t a t e m e n t : 90 F O R M A T ( / I X , " W A R N I N G - I n t e g r a t i o n f a i l s b e y o n d x = " , D 1 0 . 3 ) R E T U R N E N D C T h e f o l l o w i n g f u n c t i o n r e t u r n s a n i n t e r p o l a t e d v a l u e , k n o w i n g C t h e s p l i n e c o e f f i c i e n t s : D O U B L E P R E C I S I O N F U N C T I O N F ( Z ) I M P L I C I T D O U B L E P R E C I S I O N ( A - H , 0 - Z ) COMMON / B L K B / X ( 8 0 1 ) , Y ( 8 0 1 ) , N , NM COMMON / B L K C / Q ( 8 0 0 ) , R ( 8 0 1 ) , S ( 8 0 0 ) C P r i n t w a r n i n g m e s s a g e i f u s e r t r i e s t o i n t e r p o l a t e o u t s i d e t h e C i n t e r p o l a t i o n r e g i o n : 178 Appendix B: GPC Analysis Program Code I F (Z . L T . X ( l ) ) T H E N 1 = 1 P R I N T 2 0 , Z E L S E I F ( Z . G T . X ( N ) ) T H E N I = NM P R I N T 2 0 , Z E L S E C O t h e r w i s e , u s e b i s e c t i o n m e t h o d t o d e t e r m i n e t h e l o c a t i o n o f t h e C p o i n t o f i n t e r e s t : 1 = 1 J = N 10 K = I N T { ( I + J ) / 2) I F (Z . L T . X ( K ) ) J = K I F (Z . G E . X ( K ) ) I = K I F ( J . G T . 1+1) GOTO 10 E N D I F C C a l c u l a t e t h e i n t e r p o l a t e d v a l u e u s i n g c u b i c s p l i n e c o e f f i c i e n t s : DX = Z - X ( I ) F = Y ( I ) + DX * ( Q ( I ) + DX * ( R ( I ) + DX * S(I))) 2 0 F O R M A T ( / ' W a r n i n g - ' , D 1 0 . 3 , ' i s o u t s i d e i n t e r p o l a t i o n r a n g e ' / ) R E T U R N E N D * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * S a m p l e I n p u t F i l e : G P C D A T ( a l l ) . t x t : a . t x t b . t x t c . t x t ( C o n t ' d t i l l x . t x t ) a . t x t : R e s i n A 6 . 6 6 2 8 9 0 . 0 0 0 2 6 6 . 6 5 4 1 2 0 . 0 0 0 5 3 6 . 6 4 5 3 7 0 . 0 0 0 7 2 6 . 6 3 6 6 5 0 . 0 0 1 1 3 ( C o n t ' d t i l l e n d ) 179 Appendix B: GPC Analysis Program Code O u t p u t F i l e : Resin * 3.9542 3.9542 * 4.1542 4.3542 A 0.078 0.1763 B 0.0833 0.185 C 0.082 0.1887 D 0.0813 0.1828 E 0.0919 0.2042 F 0.0805 0.1856 G 0.0834 0.19 H 0.0798 0.1804 I 0.0789 0.1923 J 0.0699 0.1657 K 0.0745 0.1716 L 0.0795 0.1836 M 0.0931 0.2056 N 0.0796 0.1776 0 0.0823 0.1862 P 0.0685 0.1617 R 0.0866 0.1847 S 0.0701 0.1677 T 0.0576 0.1491 U 0.0717 0.159 V 0.0426 0.1126 W 0.0322 0.0912 X 0.0578 0.1514 * 4.1542 4.1542 * 4.3542 4.5542 A 0.0984 0.2154 B 0.1017 0.2188 C 0.1068 0.2347 D 0.1015 0.2203 E 0.1122 0.2359 F 0.1051 0.2286 G 0.1067 0.229 H 0.1006 0.2198 1 0.1134 0.2528 J 0.0958 0.2177 K 0.0971 0.2152 L 0.1041 0.2287 M 0.1125 0.2353 N 0.098 0.2089 0 0.1039 0.2261 P 0.0932 0.2117 R 0.0981 0.2026 S 0.0976 0.222 T 0.0915 0.2214 U 0.0873 0.191 V 0.0701 0.181 W 0.059 0.1618 X 0.0936 0.2277 * 4.3542 4.3542 * 4.5542 4.7542 A 0.117 0.2364 B 0.1171 0.2336 C 0.1279 0.2576 D 0.1189 0.2384 E 0.1237 0.2409 F 0.1235 0.2456 G 0.1224 0.2417 H 0.1191 0.2432 1 0.1394 0.2804 J 0.1219 0.2562 MW L i m i t ( i n Log) 3. 9542 3. 9542 3. 9542 3. 9542 4. 5542 4 . 7542 4. 9542 5. 1542 0. 2934 0. 4127 0. 5187 0. 5998 0 .302 0. 4185 C .519 0. 5942 0. 3167 0. 4463 0. 5592 0. 6426 0. 3016 0. 4212 0. 5271 0. 6089 0. 3279 0 .445 0. 5442 0. 6177 0. 3091 0. 4312 0. 5369 0. 6178 0. 3124 0. 4317 0. 5343 0. 6123 0. 2996 0. 4236 0. 5429 0. 6437 0. 3318 0. 4728 0. 6009 0. 7026 0. 2876 0. 4219 0. 5532 0. 6605 0. 2897 0 .416 0. 5371 0. 6364 0. 3082 0. 4364 0. 5552 0. 6487 0. 3284 0. 4432 0. 5402 0. 6103 0. 2885 0. 3995 0. 5011 0. 5826 0. 3084 0. 4312 0. 5448 0. 6333 0. 2801 0. 4101 0. 5383 0. 6456 0. 2892 0. 3916 0. 4886 0. 5719 0. 2922 0. 4288 0 5627 0. 6727 0.27 9 0. 4285 0.574 0.693 0. 2627 0. 3753 0 4846 0. 5767 0. 2236 0. 3723 0 5394 0. 6856 0.194 0. 3424 0 5185 0. 6772 0. 2855 0. 4394 0 5837 0. 6951 4. 1542 4 . 1542 4 1542 4 . 1542 4. 7542 4 . 9542 5 1542 5 3542 0. 3348 0. 4407 0 5218 0 5816 0 3352 0. 4 357 0 5109 0 5649 0 3644 0. 4773 0 5606 0 6178 0 3399 0. 4 4 58 0 5276 0 5872 0 3531 0 4522 0 5258 0 5804 0 3507 0 4564 0 537 3 0 6001 0 3484 0 4509 0 5289 0 5879 0 3438 0 4631 0 5639 0 6413 0 3939 0 5219 0 6236 0 6995 0. 352 0 4833 0 5906 0 6678 0 3416 0 4626 0.562 0 6374 0 3569 0 4756 0 .5692 0 6372 0 3501 0.447 0 .5172 0 5698 0.32 0 4215 0.503 0 5671 0 3489 0 4625 0.551 0 6208 0 3416 0 4698 0 .5771 0 6576 0.305 0.4 02 0 .4854 0 5547 0 .3586 0 4926 0 .6026 0.682 0 .3709 0 5164 0 .6354 0 .7246 0 .3036 0 .4129 0.505 0 .5769 0 .3298 0 .4968 0.643 0 .7524 0 .3102 0 . 4863 0.645 0 .7659 0 .3815 0 .5259 0 .6373 0 .7177 4 .3542 4 . 3542 4 .3542 4 . 3542 4 . 9542 5 .1542 5 . 3542 5 . 5542 0 . 3424 0 .4234 0 . 4832 0 .5312 0.334 0 . 4092 0 .4633 0 .5054 0 .3705 0 .4539 0.511 0.552 0 .3443 0 .4261 0 . 4857 0 .5314 0.34 0 .4136 0 . 4682 0 .5131 0 .3513 0 .4322 0.495 0 .5491 0 .3442 0 .4223 0 .4812 0 .5304 0 .3625 0 .4633 0 .54 07 0 . 5983 0 .4085 0 .5102 0 .5861 0 .6396 0 .3875 0 . 4949 0 .5721 0.624 3. 9542 3. 9542 3. 9542 3. 9542 5. 3542 5. 5542 5. 7542 5. 9542 0. 6595 0. 7075 0. 7476 0. 7804 0. 6482 0. 6904 C .725 0. 7548 0. 6998 0. 7407 0. 7709 0. 7944 0. 6685 0. 7142 0.75 0. 7786 0. 6723 0. 7172 0. 7536 C .781 0. 6806 0. 7347 0. 7792 0. 8128 0. 6712 0. 7204 0. 7617 0. 7938 0. 7211 0. 7787 0. 8152 0. 8345 0. 7784 0.832 0. 8644 0. 8817 0. 7377 0. 7896 0. 8194 0. 8356 0. 7119 0. 7689 0. 8082 0. 8343 0. 7167 0. 7678 0. 8042 0. 8302 C .663 0. 7068 0. 7456 C .779 0. 64 67 0. 7008 0. 7465 0. 7858 0. 7031 0. 7585 0. 7973 0. 8177 0. 7261 0.783 0. 8172 0.836 0. 6413 0. 6981 0. 7398 0. 7689 0. 7521 0. 8054 0 8358 0.852 0. 7822 0. 8469 0. 8908 0. 9196 0. 6485 0. 7056 0. 7489 0. 7812 0. 7949 0. 87 0.915 0. 9381 0. 7981 0 8816 0 9336 0. 9623 0. 7755 0 8356 0 8798 0. 9112 4 . 1542 4 1542 4 1542 5 5542 5 7542 5 9542 0 6296 0 6697 0 7024 0 6071 0 6417 0 6715 0 6588 0.689 0 7124 0 6329 0 6687 0 6973 0 6253 0 6617 0.689 0 6542 0 6988 0 7324 0 6371 0 6783 0 7105 0 6989 0 7355 0 7547 0.753 0 7855 0 8028 0 7197 0 7495 0 7657 0 6944 0 7337 0 7598 0 6883 0 7247 0 .7507 0 6137 0 6524 0 .6859 0 6212 0 6669 0 .7063 0 6762 0.715 0 .7354 0 7145 0 .7487 0 .7675 0 6116 0 .6533 0 .6823 0 .7353 0 .7657 0 .7819 0 .7893 0 .8332 0.862 0.634 0 .6772 0 .7095 0 .8275 0 .8724 0 .8956 0 .8495 0 . 9015 0 . 9301 0 .7778 0.822 0 .8534 4 .3542 4 .3542 5 .7542 5 . 9542 0 .5713 0.604 0 . 5401 0 .5698 0 .5822 0 .6057 0 .5672 0 .5959 0 .5494 0 .5768 0 .5937 0 .6272 0 .5717 0 .6038 0 . 6348 0 .6541 0 .6721 0 .6894 0 .6538 0.67 180 Appendix B: GPC Analysis Program Code K 0.118 0.2444 0.3655 L 0.1245 0.2527 0.3715 M 0.1228 0.2376 0.3345 N 0.1109 0.2219 0.3235 0 0.1222 0.245 0.3586 P 0.1185 0.2484 0.3766 R 0.1045 0.2069 0.3039 S 0.1244 0.2611 0.395 T 0.13 0.2794 0.425 U 0.1038 0.2163 0.3256 V 0.111 0.2597 0.4267 W 0.1028 0.2512 0.4273 X 0.1342 0.288 0.4324 * 4.5542 4.5542 4.5542 * 4.7542 4.9542 5.1542 A 0.1193 0.2253 0.3064 B 0.1165 0.2169 0.2922 C 0.1297 0.2426 0.3259 D 0.1195 0.2254 0.3072 E 0.1172 0.2163 0.2899 F 0.1221 0.2278 0.3087 G 0.1193 0.2219 0.2999 H 0.124 0.2434 0.3441 1 0.141 0.2691 0.3708 J 0.1343 0.2656 0.373 K 0.1264 0.2474 0.3468 L 0.1282 0.247 0.3405 M 0.1148 0.2117 0.2819 N 0.1111 0.2126 0.2941 0 0.1228 0.2364 0.3248 P 0.1299 0.2581 0.3654 R 0.1024 0.1994 0.2828 S 0.1366 0.2706 0.3806 T 0.1495 0.295 0.414 U 0.1126 0.2219 0.314 V 0.1487 0.3158 0.462 W 0.1484 0.3245 0.4832 X 0.1538 0.2982 0.4096 * 4.7542 4.7542 4.7542 * 4.9542 5.1542 5.3542 A 0.106 0.1871 0.2468 B 0.1004 0.1757 0.2297 C 0.1129 0.1963 0.2534 D 0.1059 0.1877 0.2473 E 0.0991 0.1727 0.2273 F 0.1057 0.1866 0.2494 G 0.1025 0.1806 0.2395 H 0.1194 0.2201 0.2975 1 0.1281 0.2298 0.3056 J 0.1313 0.2386 0.3158 K 0.1211 0.2204 0.2958 L 0.1188 0.2123 0.2803 M 0.0969 0.1671 0.2197 N 0.1016 0.183 0.2472 O 0.1136 0.202 0.2718 P 0.1282 0.2355 0.316 R 0.097 0.1804 0.2497 S 0.134 0.244 0.3234 T 0.1455 0.2645 0.3537 U 0.1093 0.2014 0.2732 V 0.1671 0.3133 0.4226 W 0.1761 0.3348 0.4557 X 0.1444 0.2558 0.3361 4.9542 4.9542 4.9542 5.1542 5.3542 5.5542 0.4648 0. 5403 0. 5973 0 6366 0 6627 0.465 0. 5331 0. 5842 0 6206 0 6465 0.4047 0. 4573 0. 5012 0 5399 0 5734 0.4049 0. 4691 0. 5232 0 5689 0 6082 0.4471 0. 5169 0. 5723 0 6111 0 6315 0.4839 0. 5644 0. 6213 0 6555 0 6743 0.3872 0. 4566 0. 5134 0 5551 0 5842 0.505 0 5844 0. 6377 0 6681 0 6843 0.5439 0. 6331 0. 6978 0 7417 0 7705 0.4178 0. 4896 0. 54 67 0.59 0 6222 0.573 0 6823 0. 7574 0 8024 0 8255 0.586 0 7069 0. 7904 0 8424 0 8711 0.5438 0 6241 0. 6842 0 7284 0 7598 4.5542 4 5542 4 . 5542 4 5542 5.3542 5 5542 5. 7542 5 9542 0.3661 0 4142 0. 4543 0.487 0.3462 0 3883 0.423 0 4528 0.3831 0 4241 0. 4543 0 4777 0.3669 0 4125 0. 4484 0. 477 0.3445 0 3894 0. 4257 0 4531 0.3715 0 4256 0. 4702 0 5037 0.3588 0.408 0. 4493 0 4814 0.4216 0 4792 0. 5157 0.535 0.4 4 67 0 5002 0. 5327 0.55 0.4502 0.502 0. 5319 0 5481 0.4222 0 4792 0. 5185 0 5446 0.4085 0 4597 0.496 0.522 0.3345 0 3784 0 4171 0 4506 0.3582 0 4123 0. 458 0 4973 0.3946 0.45 0. 4888 0 5092 0.4459 0 5029 0.537 0 5558 0.3521 0 4089 0 4507 0 4797 0.46 0 5133 0 5437 0 5599 0.5032 0 5678 0 6118 0 .6406 0.3858 0 4429 0 4862 0 .5185 0.5713 0 6465 0 6914 0 .7145 0.6041 0 6877 0 7396 0 7683 0.49 0.55 0 5942 0 .6256 4.7542 4 7542 4 7542 5.5542 5 7542 5 9542 0.2948 0 3349 0 3676 0.2718 0 3065 0 3363 0.2944 0 3246 0 3481 0.293 0 3288 0 3575 0.2722 0 3086 0.336 0.3035 0.348 0 3816 0.2887 0.33 0 3621 0.3551 0 .3917 0 4109 0.3592 0 .3916 0 4089 0.3677 0 . 3975 0 4137 0.3529 0 .3921 0 4183 0.3315 0 .3678 0 3938 0.2636 0 .3023 0 3358 0.3013 0.347 0 3863 0.3272 0 .366 0 3864 0.3729 0 . 4071 0 4259 0.3065 0 . 3482 0 3773 0.3767 0 . 407 0 4232 0.4184 0 . 4623 0 .4911 0.3303 0 . 3736 0 . 4059 0.4977 0 . 5427 0 .5658 0.5393 0 .5912 0 .6199 0.3962 0 . 4404 0 .4718 4.9542 4.9542 5.7542 5.9542 181 Appendix B: GPC Analysis Program Code A 0.0811 0 1408 0 1888 0 2289 0 2616 B 0.0752 0 1293 0 1714 0 2061 0 2358 C 0.0834 0 1405 0 1815 0 2117 0 2352 D 0.0818 0 1414 0 1871 0 2229 0 2516 E 0.0736 0 1282 0 1731 0 2094 0 2368 F 0.0809 0 1437 0 1978 0 2423 0 2759 G 0.078 0.137 0 1862 0 2274 0 2596 H 0.1007 0. 1782 0 2358 0 2723 0 2916 I 0.1017 0 1776 0 2311 0 2636 0 2809 J 0.1073 0. 1845 0 2364 0 2662 0 2824 K 0.0993 0. 1748 0 2318 0 2711 0 2972 L 0.0935 0. 1616 0 2127 0.249 0.275 M 0.0702 0. 1228 0 1666 0 2054 0 2389 N 0.0815 0. 1456 0 1997 0 2454 0 2847 0 0.0885 0. 1582 0 2137 0 2524 0 2728 P 0.1073 0. 1878 0 2447 0 2789 0 2977 R 0.0833 0. 1527 0 2095 0 2512 0 2803 S 0.11 0. 1894 0. 2427 0 2731 0 2893 T 0.1189 0. 2082 0 2728 0 3168 0 3456 U 0.0921 0. 1639 0.221 0 2643 0. 2966 V 0.1462 0. 2555 0 3307 0 3756 0. 3988 W 0.1587 0. 2796 0 3632 0. 4151 0. 4438 X 0.1114 0. 1918 0. 2518 0.296 0. 3275 5.1542 5. 1542 5. 1542 5. 1542 * 5.3542 5. 5542 5 7542 5. 9542 A 0.0597 0. 1078 0 1479 0. 1806 B 0.054 0. 0961 0 1308 0. 1606 C 0.0572 0. 0981 0 1283 0. 1518 D 0.0596 0. 1053 0 1411 0. 1697 E 0.0546 0. 0995 0. 1359 0. 1633 F 0.0627 0. 1169 0. 1614 0.195 G 0.0589 0. 1081 0. 1494 0. 1815 H 0.0774 0.135 0. 1716 0. 1908 I 0.0759 0. 1294 0. 1619 0. 1792 J 0.0772 0. 1291 0. 1589 0. 1751 K 0.0755 0. 1325 0. 1717 0. 1979 L 0.0681 0. 1192 0. 1555 0. 1815 M 0.0526 0. 0965 0. 1352 0. 1687 N 0.0642 0. 1182 0. 1639 0. 2033 O 0.0698 0. 1252 0.164 0. 1844 P 0.0805 0. 1374 0. 1716 0. 1904 R 0.0693 0. 1262 0. 1679 0. 1969 S 0.0794 0. 1327 0. 1631 0. 1793 T 0.0892 0. 1539 0 1978 0. 2266 U 0.0718 0. 1289 0 1722 0. 2045 V 0.1093 0. 1845 0. 2294 0. 2525 W 0.1209 0. 2045 0. 2564 0. 2851 X 0.0804 0. 1404 0. 1846 0. 2161 *• 5.3542 5. 3542 5. 3542 * 5.5542 5. 7542 5. 9542 A 0.048 0. 0881 0 1208 B 0.0421 0. 0768 0. 1066 C 0.041 0. 0712 0. 0946 D 0.0457 0. 0815 0. 1101 E 0.0449 0. 0813 0. 1086 F 0.0541 0. 0987 0 1323 G 0.0492 0. 0905 0. 1226 H 0.0576 0. 0941 0 1134 I 0.0536 0. 086 0 1033 J 0.0519 0. 0817 0 0979 K 0.057 0. 0963 0 1224 L 0.0511 0. 0875 0 1135 M 0.0438 0. 0826 0 1161 N 0.0541 0. 0998 0 1391 0 0.0554 0. 0942 0 1146 P 0.0569 0. 0911 0 1099 R 0.0568 0. 0986 0 1276 182 s T U V W X A B C D E F G H I J K L M N 0 P R S T U V w X A B C D E F G H I J K L M N O P R S T U V w X 0.0533 0.0646 0.0571 0.0751 0.0836 0.0601 5.5542 5.7542 0.0837 0.1086 0.1004 0.1201 0.1355 0.1043 5.5542 5.9542 0. 0. 0. 0. 0. 0. 0. 0. 0.0401 0.0347 0.0302 0.0358 0.0364 0.0446 0.0413 0.0365 0325 0298 0393 .0364 0388 0457 0388 .0342 .0.0417 0.0304 0.0439 0.0433 0.0449 0.052 0.0442 5.7542 5.9542 0.0327 0.0298 0.0235 0.0286 0.0274 0.0336 0.0321 0.0193 0.0173 0.0162 0.0261 0.026 0.0335 0.0393 0.0204 0.0188 0.029 0.0162 0.0288 0.0323 0.0231 0.0286 0.0314 0. 0. 0. 0. 0. 0. 0. 0728 064 5 0537 064 4 0637 0781 0734 0.0558 0.0498 0.046 0.0654 0.0623 0.0722 0.085 0.0592 0.053 0.0707 0.0466 0.0727 0.0756 0.0681 0.0806 0.0756 0.0999 0.1374 0.1327 0.1432 0.1642 0.1357 183 

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