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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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E F F E C T S OF M O L E C U L A R STRUCTURE ON T H E R H E O L O G Y AND PROCESSABILITY OF HIGH DENSITY P O L Y E T H Y L E N E BLOW MOLDING RESINS by ALFONSIUS BUDI A R I A W A N B . A . S c , The University o f British Columbia, 1996  A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T OF T H E R E Q U I R E M E N T S FOR T H E D E G R E E OF M A S T E R OF A P P L I E D SCIENCE in T H E F A C U L T Y OF G R A D U A T E STUDIES Department o f 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 OF BRITISH C O L U M B I A June 1998 © Alfonsius B u d i Ariawan, 1998  In  presenting this  degree  at the  thesis  in  partial  fulfilment  University  of  British  Columbia,  of  the  requirements  I agree that the  freely available for reference and study. I further agree that copying  of  department  the  or  understood  that  his  or  her  representatives.  It  permission.  Department of  &lr\emic«.|  £N6/W&&P>J6  The University of British Columbia Vancouver, Canada  DE-6 (2/88)  Quly  C  '  is  advanced  permission for extensive  granted by  by  an  Library shall make it  this thesis for scholarly purposes may be  publication of this thesis for financial gain shall not  Date  for  head of  my  copying  or  be allowed without  my  written  ABSTRACT  Resin processability depends heavily on its Theological properties.  The molecular  structure o f the resin, i n 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. characteristics  and were  The studied samples had varying molecular weight  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, pillow 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 o f 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 (M ) and polydispersity index (PI). Increasing M w  w  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 M  w  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. W i t h 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.  iii  TABLE OF CONTENTS Page ABSTRACT  ii  LIST OF FIGURES  vi  LIST OF TABLES  xi  ACKNOWLEDGEMENTS 1  2  3  xii  INTRODUCTION  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 o f B l o w Molding  18  1.3 Thesis Objectives  23  L I T E R A T U R ER E V I E W  24  2.1 Introduction  24  2.2 Rheology  24  2.3 Processability  28  EXPERIMENTAL EQUIPMENT AND PROCEDURES  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 l o w 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  RESULTS A N D DISCUSSION  62  4.1 Introduction  62  4.2 Rheology  64  4.2.1  Shear Properties  64  4.2.2  Extensional F l o w 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 Melt Index, Stress Exponent, and M e l t F l o w Ratio  118  4.5 Implications o f Rheological Behaviour on Processability  125  5  CONCLUSIONS  130  6  RECOMMENDATIONS  136  REFERENCES  138  NOTATION  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  Figure 1.1 Differential molecular weight distribution for most probable (solid) and log normal (dashed) molecular weight distribution, both with MJM =2 n  Page 5  Figure 1.2 Chemical and molecular structures o f (a) L o w Density Polyethylene  ( L D P E ) , (b) Linear L o w Density Polyethylene ( L L D P E ) , and (c) H i g h Density Polyethylene ( H D P E )  8  Figure 1.3 Analogs o f viscoelastic materials: (a) Voigt model, (b) M a x w e l l model... 10 Figure 1.4 A schematic diagram o f simple shear experiment  12  Figure 1.5 A schematic diagram o f simple (uniaxial) extensional experiment  12  Figure 1.6 Time temperature superposition o f 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 o f blow molding process  18  Figure 1.8 Parison inflation in the process o f blow molding  18  Figure 1.9 The controlling o f 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 o f parison diameter swell  31  Figure 3.1 A schematic diagram o f the Toyoseiki Automatic Densimeter, Model D-H100  33  Figure 3.2 A Schematic diagram o f the Tinius Olsen Manually Timed Extrusion plastometer  34  Figure 3.3 Pressure profile for a flow in a capillary  39  Figure3.4 A t y p i c a l B a g l e y p l o t  40  Figure 3.5 A schematic diagram o f Kayness Capillary Rheometer and die  41  Figure 3.6 Uniaxial or simple extension  42  vi  Figure 3.7 A schematic diagram of Rheometric R E R - 9 0 0 0 Extensional Rheometer.. 46 Figure 3.8 (a) M o l d i n g and (b) gluing accessories for Rheometric R E R - 9 0 0 0 Extensional rheometer  Figure 3.9 A Schematic diagram o f the sample cutter supplied with R E R - 9 0 0 0 extensional Rheometer  Figure 3.10 A schematic Diagram o f  r M P C O  B-13 B l o w Molder  47  48 49  Figure 3.11 A schematic diagram o f the pillow mold used in the blow molding Experiment  Figure 3.12 Differential molecular weight distributions for some o f the resins studied in this work  49  51  Figure 3.13 Melt 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%  Figure 4.2 Apparent flow curves for resins with similar polydispersities determined at200°C  Figure 4.3 Apparent flow curves for resins with similar M determined at 200°C w  64  65 66  Figure 4.4 Predicted apparent viscosity values at 100 s" and 200°C as determined 1  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  Figure 4.5 Predicted and observed apparent shear viscosity at 100 s' and 200°C 1  Figure 4.6 Apparent shear viscosity curved simulated at constant PI using the regression relationship at each shear rate  Figure 4.7 Apparent shear viscosity curved simulated at constant M using the  68 68  70  w  regression relationship at each shear rate  Figure 4.8 Determination o f molecular weight ranges that are critically affecting a certain property o f a resin. Slices were made arbitrarily  vii  70  72  Figure 4.9 Correlation coefficients relating various molecular weight ranges to  shear viscosity at 5 s" and 200°C (a) negative correlation (b) positive correlation. Correlation coefficient (x,y) = COV(x,y)/a .a 1  x  y  Figure 4.10 (a) Master curve and (b) Arrhenius fit generated by the F O R T R A N program  73  75  Figure 4.11 Observed and predicted E values as obtained from S T A T G R A P H I C S a  v2.0 (PI>10)  77  Figure 4.12 The effect of PI, M , and M on temperature sensitivity o f shear flow properties. PI and M were arbitrarily set to be constant in (a) and (b), respectively w  z  w  Figure 4.13 Hypothetical M W D showing the shift in M„,M , a n d M at W  78  z  constant PI.  80  Figure 4.14 Observed and predicted E values as obtained from S T A T G R A P H I C S a  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 M  on Hencky strain  w  84  Figure 4.17 The effect o f M on tensile viscosity  85  Figure 4.18 Differential molecular weight distribution for resins E , F , and N  86  Figure 4.19 Effect ofM  86  w  w  on tensile viscosity  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 o f large PI on Hencky strain  90  Figure 4.25 Reproducibility o f 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 M  96  w  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 Figure 4.36 Sensitivity o f extrudate swell to changes in shear rate. Included in  99  the plot are resins produced from different technologies. One can see a breakdown o f pattern  100  Figure 4.37 Determination o f 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 o f melt strength as a function o f M  w  (P>\0)....,  104  Figure 4.40 3-D plot showing the effect of PI and M on melt strength. Density w  value is fixed arbitrarily. Changing the density value would shift the curve upward or downward accordingly. Note that M is related to PI and M„, and hence, when considering the plot, it has to be ensured thatM, is reasonable, so t h a t P / > / 0 w  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  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)/o .o K  y  Figure 4.43 Variation o f pillow weight with pillow number. Multiple curves indicate replicate runs  ix  107  109  Figure 4.44 Variation in pillow 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 M e l t strength values for resins E , F, and G  112  Figure 4.47 Variation in pillow 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 l o w weight normalized to the weight o f pillow number one to show the magnitude o f sagging  Figure 4.50 Comparison o f parison sag between resins E , F, and G  115 116  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 Figure 4.52 Correlating MI to M . w  117  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 o f melt strength on Hencky strain  127  LIST OF TABLES Page  Table 1.1 Influence of rheological properties on parison formation  Table 3.1 A summary of the molecular characteristics of the H D P E resins that were studied in this work  Table 5.1 Summary of conclusions  22  50  135  xi  ACKNOWLEDGEMENTS  I would like to express my sincere gratitude to my supervisor D r . S a w a s G . Hatzikiriakos, and D r . 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 D r . Phil Edwards, D r . Charles Russell, D r . T i m Bremner, and Henry H a y o f N O V A Chemicals, and the members o f N O V A Chemicals' rheology group, headed by D r . 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 o f the rheology team o f 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 o f 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 o f 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.  E v e n 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 o f blow molding is principally governed by the rheological behavior o f the resin used. influences its rheological properties.  The molecular structure o f the resin, i n turn, H i g h 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 o f the chapter, followed by a general discussion on polymer rheology. In addition, the process o f blow molding and the applicability o f 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 1.2.1  BACKGROUND 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 o f 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 o f polymer in which only one kind o f 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 o f 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 o f equal length. Instead, there is a distribution o f 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. F o r example, treating the polymer w i t h 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 o f these parameters. weight averages are normally determined for this purpose.  2  Several molecular  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 M . t  The weight average molecular  weight, M , is then w  _ £ A / , - T » , ( M , )  M  (  Often, the distribution w (Mi) is expressed as a continuous function w(M)dM. t  these circumstances, Experimentally, M  w  U  )  Under  the summation in Equation 1.1 is replaced by an integral. may be determined by G e l Permeation Chromatography ( G P C )  [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. T w o that are important are also called the Z and Z+l molecular weight averages. These are defined as:  3  Chapter I Introduction  ZM -w,(M ) 2  f  f  (1.3)  and  E M ,  3  (1.4).  •H',.(M ) <  Figure 1.1 shows two typical molecular weight distribution curves. A s shown in this figure, M is weighted heavily towards the high end o f the molecular weight distribution, z  while M , is weighted towards the l o w 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  (1.5)  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  (1.6).  4  Chapter 1 Introduction  The two parameters M and 0  its breadth, respectively.  in Equation 1.6 describe the location of the distribution and 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 M to M„. A PI value of unity indicates monodispersity and larger PI values indicate w  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 MJM =2. n  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 o f 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. polyester  pre-polymers  are  examples  Nylon, phenolics, amino resins and  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 o f 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 o f 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. B u l k 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  6  addition  polymerization.  Different  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 i n each technique, causing this difference in the molecular characteristics.  This causes the  properties o f 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. polymer and additives are called plastics.  These physical mixtures o f  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 w h i c h has found usage i n many applications. chemical  absorption,  It is characterized by its toughness, near-zero  excellent chemical resistance,  excellent  properties, l o w coefficient o f friction, and ease o f processing.  electrical insulating 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 at 25°C. 3  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  I  _H  -  ^ ^ — r ^  c  (a)  (b)  I  H  Polyeihylei  (c)  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  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 disturbedfromtheir 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, y , can defined as 0  (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  e  a  (1.9).  = In  AX  <  •  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.<.)-"% where o(t) and OE(0  a  r  e  <L11)  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)-y  <U2>-  0  The modulus can be separated into the elastic (storage) and viscous (loss) moduli, G' and  G", respectively, i.e.  G = \G*\ = >lG' +G" 2  (1.13)  2  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 C o x - M e r z rule [Dealy and Wissbrun (1995)]:  1  / . '  To  \ \  CO  J  \  CO  )  A l s o , using the generalized M a x w e l 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. parameters indicate the relaxation behavior o f the molecules.  These  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.  F o r large and fast deformations, the theory o f 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 o f 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 o f '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 o f  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* ), the resulting master curve can be accomplished by plotting r\*/ar 1  versus oxar. The shift factor can then be related to temperature by either the Arrhenius type equation  (1.17).  or the W L F (Williams, Landel, and Ferry) equation  (1.18).  Equation 1.17 is valid for temperatures that are at least 100 K above the glass transition temperature, T . The term E in this equation is the flow activation energy, which gives g  a  an indication o f the sensitivity o f material flow to temperature change. For temperatures closer to T , Equation 1.18 is found to be more useful, where the constants C° and C / g  can be determined using any optimization technique. K n o w i n g E , or C° and C / , a?at a  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  15  Chapter I Introduction  10-  2  10-  1  10°  10  Frequency (s ) 1  (b)  16  1  10  2  10  3  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 i n 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 o f 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 o f 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  ^  Closed Mold  Compressed Air  Blowpin  ID]  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 o f 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 i n 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 o f the extruded parison reflects not only the two external forces but also the deformation it experienced in the die.  A t the exit o f the die, polymer melt also tends to swell. This is another manifestation o f melt elasticity. Figure 1.10 illustrates the extrudate swell behavior o f Newtonian and polymeric materials.  The swelling o f fluids (increase o f effective diameter) upon the  emergence from a die is not unique. F o r 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 o f the viscoelasticity o f the polymer melt, there is also a time dependency involved.  In general, most o f the swelling takes  place during the first several seconds o f 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 o f 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  20  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 o f an extrusion process. The distortion is most severe i n 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 o f the resin may be the contributing factors [Ramamurthy (1986), Hatzikiriakos (1994)].  A t the exit o f the die, before the two halves o f 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 o f the sag depends exclusively on the rheological properties o f the polymer melt. T o quantitatively analyze the sagging behavior o f 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.  F o r 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  Effect on parison formation  Property H i g h 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 b l o w molded products, it is necessary to understand the rheological properties and the processability o f 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 M , w  PI and density.  M e l 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„, M , M , PI, and density), w  z  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„, M , M , PI, and density), w  z  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  INTRODUCTION 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. V e r y few attempts have been made to characterize the rheological properties o f 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  RHEOLOGY Yoshikawa et al (1990) studied the influence o f molecular weight distribution and  long chain branching on the viscoelastic behavior o f commercial H D P E melts. The two types o f samples studied were produced using different kinds o f 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. W i t h 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 M . w  The authors found that for H D P E with no long branches, the zero shear  viscosity was proportional to M  3 w  ' . 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 M  w  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 H a n and Villamizar (1978), Shroff and Mitsuzo (1977), Bersted (1976), Mendelson and Finger (1975).  These studies found that increasing M  w  increases the  shear viscosity in the low deformation regime. Above some critical value o f M , the zero w  shear rate viscosity for amorphous and linear polymers has been found to be proportional to Mj . A  O n 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.  25  It has  Chapter 2 Literature Review  been found that activation energy is affected  by the degree o f branching.  For  polyethylene, increasing the percentage o f long chain branching increases the activation energy significantly [Dealy and Wissbrun (1995)].  F o r 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 o f 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 o f 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.  W i t h 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 o f 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 M , w  on the other hand, the  authors found an increase in the extensional viscosity at l o w stress levels. The authors also concluded that extensional flow properties o f 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 L a u n (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 o f 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 o f  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 o f 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 M  w  2.3  a n d M , to understand the flow behavior o f polydisperse polymers. z  PROCESSABILITY V e r y 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 o f 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 l o w 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 M  w  and polydispersity index for H D P E .  The authors also  investigated the effect o f 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 o f 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 o f 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  29  strength  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 o f 0.25. In addition, it was found that the swell potential, rate o f 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 o f 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  K a y l o n and K a m a l (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 o f parison diameter swell. In another study by Alroldi (1978), the same result was obtained.  31  Chapter 3 Experimental Equipment and Procedures  3 E X P E R I M E N T A L 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 E X P E R I M E N T A L EQUIPMENT 3.2.1 Densimeter Density is a measure o f the degree o f crystallinity in a polymer and is commonly used to classify polyethylenes. A high density polyethylene has greater crystallinity than a l o w 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 o f 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.  By  measuring the sample weight in air and in distilled water by means o f 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 o f specified temperature around the beaker using a circulator.  The actual temperature o f 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 M e l t index is an empirically defined parameter that is critically influenced by the physical properties and molecular structure o f a polymer. It serves to indicate the flow properties o f a polymer at a particular l o w 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 o f specified length and diameter under prescribed conditions o f  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 o f the equipment is shown in Figure 3.2  Weight (combined with piston to be 2160g) Oil-filled thermometer wells  Insulation  Barrel of nitride hardened steel Heater Insulation Minimum 4 4in  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 o f 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 m m and a length o f 162 mm. The hole in the barrel is 9.5504 ± 0.0076 m m 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 o f the barrel to measure the actual temperature o f the  34  Chapter 3 Experimental Equipment and Procedures  barrel.  The piston used in the assembly is made o f steel, 9.4742 ± 0.0076 m m in diameter and 6.35 ± 0.13 m m in length. A t the top end o f 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 o f steel with the dimensions o f 2.09955 ± 0.0051 mm in diameter and 8.000 ± 0.025 mm in length. F o r a more detailed description o f the melt indexer, the reader is referred to the  ASTM  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. A l s o , it is important that the piston arrangement be kept as vertical as possible to minimize the friction between the piston and the side o f 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.  F o r this particular flow, simple equations can be derived to  determine the shear viscosity for both Newtonian and power law fluids. F o r other types o f 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.  35  Chapter 3 Experimental Equipment and Procedures  F o r 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, a : w  r  dP  R r=R  (3.1)  dP  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 o f Newtonian fluid can be obtained. K n o w i n g 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:  /(Newtonian)  dV = — dr  (3.3)  4g r=fi  TTR'  where Q is the volumetric flow rate.  F o r 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)]:  (3.5).  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 has no significance. It is referred to as the 'apparent shear rate', JA-  U s i n g Equation 3.4 and 3.5, it can be shown that  (3.6).  The constants K and n can be determined from the intercept and the slope o f the straightline 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 o f flow rates is needed.  This technique makes use o f a  plot o f log(a ) versus log(yj) that yields a single curve. The true wall shear rate is then w  37  Chapter 3 Experimental Equipment and Procedures  given by  (3.7)  where b is the Rabinowitsch correction given by  (3.8).  tf(logo-J 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 o f 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, i n the barrel and assuming that the pressure at the outlet o f the capillary is equal to the ambient pressure, P . Pd can be related to the force that is driving the piston a  (plunger), F , as: d  (3.9)  nR  2 b  where Rb is the radius o f the barrel. (Pd-Pa)  or, since for melts  Pd  The pressure drop  in Equation 3.1 is then  (-AP ) W  is nearly always much larger than  P, a  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 o f the capillary and the small residual pressure at the exit. Figure 3.3 shows the pressure profile for a flow in a capillary.  38  Chapter 3 Experimental Equipment and Procedures  BARREL  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) i n 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. determining  Another way o f  P d is by making use o f an orifice die with L=0. Using the corrected en  pressure drop, the wall shear stress can then be calculated as  (3.10).  4(L/D)  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)].  39  Chapter 3 Experimental Equipment and Procedures  Figure 3.4  A typical Bagley plot.  In this w o r k , a K a y e n e s s G a l a x y I V C a p i l l a r y R h e o m e t e r ( m o d e l 0 0 5 2 ) w a s used. T h e b a r r e l is e q u i p p e d w i t h a dual z o n e electric heater and an adaptive PID c o n t r o l l e r w i t h an a c c u r a c y o f 0.1°C. 0.005 m m i n diameter.  temperature  T h e barrel is 175 m m i n l e n g t h and 9.55 m m ±  A stepper m o t o r is used to drive the p i s t o n f r o m a speed o f 0.5  m m per m i n u t e to a m a x i m u m speed o f 2 5 0 m m per m i n u t e . F o r c e is measured b y means o f a l o a d c e l l i n s t a l l e d o n t o p o f the p i s t o n driver.  T h e rheometer also comes w i t h data  a n a l y s i s s o f t w a r e and tungsten carbide dies o f different L/D  ratios and diameters.  s c h e m a t i c d i a g r a m o f the rheometer and a die is s h o w n i n F i g u r e 3.5.  40  A  Chapter 3 Experimental Equipment and Procedures  Locking Screw  -  Ok O k 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 o f deformation. therefore necessary for the extensional properties  F o r these reasons, it is  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  O.U)  = L -A 0  0  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  42  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 o f the original length o f the sample, Lo, as the reference length i n the denominator to make the definition o f strain more meaningful, since for materials with fading memory (liquid-like), the significance o f L  0  decreases as the stretching continues.  B y simple integration, the strain for finite deformation can be obtained as  e = In  L ~\  r  (3-13).  2  This is also called the Hencky strain.  The strain rate can be derived from the above  relation as  • _de  dt  _ 1  L  dL  dt  =  d\nL  (3.14).  dt  Realizing that dUdt is the velocity, V, at the end o f the sample, equation 3.14 can hence be written as follows:  f  =  _V_  (3-15).  L  Therefore, i n a constant strain rate experiment, the speed at which the sample is being pulled is controlled according to the instantaneous length o f 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. F o r 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.  (3.16)  where TJE is the extensional viscosity at diminishingly small strain rates and n  0  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 o f 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 o f the material as a function o f time. The strain versus time curve is useful for determining the melt strength o f a polymer melt.  At a  particular time, a resin with lower melt strength will 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. A l s o , constant stress experiments are more representative o f the parison formation stage o f 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  44  Chapter 3 Experimental Equipment and Procedures  differentiating the strain versus time curve to get the strain rates at different times.  A Rheometrics R E R - 9 0 0 0 Extensional Rheometer was used in this work to determine the elongational properties o f the H D P E resins. The rheometer consists o f 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.  T o 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 o f  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  - CUEajN P L A C E ELE  (a)  8AMPI g f»l l i e n  MTO P I Anf=  @  ®  @  mninri niiniini  ®  ®  ®  ULiUllLi^^ If PI n PI ©  ©  n PAPI 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 M o l d i n g Machine T o determine the sagging and swell characteristics o f the blow molding resins under actual run conditions, an Impco B-13 B l o w Molder unit was used. manufactured  The unit is  by Ingersoll-Rand Plastics Machinery L t d . and is equipped with an  Edwards Z o n e - A - M a t i c M o d e l C C - 5 mold chiller unit.  Parison programming is done  through a M o d e l A081-822 programmer supplied by M o o g Incorporated.  Figure 3.10  shows a schematic picture o f the blow molder.  The pillow mold shown in Figure 3.11 was used with the blow molder unit. The mold facilitated the measurements o f sagging and swell characteristics by providing a number o f pillows, molded from different parts o f 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 o f the mold. The width and weight o f each pillow corresponding to the different parts o f the parison could be measured to determine how wall thickness and product weight vary i n the vertical direction.  MAX. w j c c n o N srnotce  Figure 3.10 A schematic Diagram of I M P C O 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 N O V A 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  M  n  M  w  M  z  PI  Density (g/cm) 3  A B C D E F G H 1 J K L M N O P Q R S T U V  w X  a a a a a a a a  •«: •.; • y . :.:=•.:.• • :/:•: •  a  a a a a a  •  .•  9030 9020 9330 9340 9820 10700 9960 12700 17300 10800 15700 15300 11700 13100 i 12500 11500 10600 10300 13200 27800 9310 27000 44000 25100  152000 157100 133400 153600 131400 143500 147300 104800 104100 108900 137300 133400 130800 174900 101200 116000 108000 133700 113000 132300 153500 129000 148000 138400  50  835400 971500 848400 933500 776600 705000 772000 408900 434600 590600 693200 761200 650900 852200 397000 570800 503300 676200 640300 462700 776600 377000 423000 535300  16.83 17.42 14.3 16.44 13.38 13.41 14.79 8.25 6.02 10.1 8.75 8.72 11.18 13.35 8.1 10.09 10.2 12.98 8.6 4.76 16.49 4.8 3.4 5.51  0.9570 0.9567 0.9579 0.9587 0.9627 0.9609 0.9596 0.9586 0.9575 0.9551 0.9550 0.9611 0.9581 0.9603 0.9597 0.9542 0.9548 0.9549 0.9548 0.9393 0.9617 0.9465 0.9560 0.9544  Chapter 3 Experimental Equipment and Procedures  The density o f each resin was determined using the densimeter as described i n 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 o f 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 o f 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 o f 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. U p 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 o f polymers, the effect o f 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 o f shear rate and temperature by manually collecting extrudates at the exit o f 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 o f the capillary run. Absolute care was taken during the collection o f the extrudates to ensure that they were not pinched in anyway. F o r 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 o f the extrudate and D is the diameter o f the capillary die. 0  T o 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 o f the extrudates were also kept to be approximately the same during each cut.  A t high shear rates, melt fracture (surface distortion o f 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. A l s o , 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. procedure was adopted due to its simplicity.  The  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 F l o w Ratio Determinations M e l 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 o f the load used, depending on how these parameters are defined.  The procedure began with the loading o f approximately 5 grams o f 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 o f 360 seconds was allowed before the dead weight load was placed on top o f the piston (a lighter weight may be placed on the piston during the pre-heat period to improve the packing o f the resin in the barrel). The polymer melt would then start to flow out o f the die. After a reasonable length o f 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 o f  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.  T o 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 o f experimental runs were performed.  I2, h and I21 are defined as the weights o f the polymer that flows out o f the barrel i n 10 minutes when dead weights o f 2.16, 6.48, and 21.6 k g are used, respectively. The melt index takes the value o f I2 in the unit o f grams/10 minutes. The stress exponent is calculated as:  log| SEx. =  (3.18)  6.48^ 2.16  and the M e l t F l o w Ratio ( M F R ) is calculated as:  MFR  =^  (3-19)  h  3.4.4 M e l 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 k g was placed on top o f 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 o f extrudate to be obtained. Plotting the log o f time versus the log o f 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 o f the actual set o f 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 o f constant rate experiments, were performed to determine the extensional properties o f the resins, since such experiments are more applicable to the process o f blow molding. A s mentioned earlier, during parison formation, the polymer melt is hung from the die exit for a finite period o f 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 o f 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 o f 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 o f the load to 7000 k g was then made before it was suddenly removed. Following this, an annealing period o f 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.  T o both ends o f each cut sample, etching was then done with a previously stirred mixture o f 2 grams K2Cr207 in 100 grams o f 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 i n 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 o f expansivity to correct for the change i n 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 o f 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 onefromthe 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  4.1  RESULTS A N D DISCUSSION  INTRODUCTION  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 o f 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 o f experimental results, a brief comment has to be made with regard to the variety o f samples chosen for this work. The main objective o f this work was to study the effect o f molecular structure on the rheology and processability o f H D P E blow molding resins using resins having a broad range o f molecular parameters, in terms o f M  w  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 o f molecular parameters.  T o our  knowledge, no study that relates the rheological and processing behavior o f resins to the broad range o f 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 M  w  62  and polydispersity values. For example,  Chapter 4 Results and Discussion  solution polymerization technology produces resins with low to medium M . w  With a  slurry type o f polymerization technology, on the other hand, it is possible to produce resins with somewhat higher M  w  and with a polydispersity range between six and ten.  The gas phase catalyst technology is capable o f producing resins with high M  w  and  polydispersity values less than five, or more than fourteen, depending on the catalyst used [Goyal (1998)].  D u e to this difficulty, it was, therefore, necessary to use commercial  resins produced from a number o f different technologies in this study. Since commercial resins from different technologies are used, it will 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 o f resins. Therefore, in this work, multiple regression analysis was used to study the influence o f molecular parameters on the rheological and processing behaviors o f the resins, wherever possible.  Otherwise, qualitative analyses  were conducted to investigate the influence o f these variables.  However, it should be  noted that the objective o f performing multiple regression analysis was to obtain general trends on various rheological and processing properties o f 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. A l s o , 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" to 1  900 s" . Duplicate runs on some resins indicate excellent reproducibility of results. For 1  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.  10 J 4  -  1  1 11  -" A S  «  1  '  0 0  0  •  -  o2  3  • •  c  O  c  a  • A  10  2  1  1  1 1 1  1  1 1 11 _  a  o • *  o g  T=180°C T=200°C T=220°C T=180°C- Reproducibility Run T=200°C -Reproducibility Run T=220°C -Reproducibility Run i  • I  1  A  10  1  0  •52  TO o •c <0  1  Resin R L/D=20, D=0.7542 m m -  A  V.  -  11  u  D A  o u  • i i i ii  i  i iii  11  o  2  ~ :  o g o  2  i  i i i i ii  11 3  2  1  -  10  10  10  o  Apparent Wall Shear Rate (s ) 1  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 M while w  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 M , may be affecting the viscosity profiles in z  different way. Moreover, the number of resins in each polydispersity group is too small for a definite conclusion to be drawn.  10  4  1  1  1 1 1 1  '-1  ;. *  1  1  1 1 1 1  J  1  1  1  1  f t  1 1 |_  C o n s t a n t Pl=13.4, T=200°C UD=20, D=0.7542 m m  •  •  o o .*» TO O •C  i  1  I  • • • •  -  • * 10  3 :  </) <«*  c SJ  •  Resin E - M  w  = 131400  •  Resin F - M  w  = 143500  Resin N - M  8: 10  2  i  i i i ii 1  w  •  i  •  •  = 174900  i  i i i i t1  10  10  1  i  i  i  •  i i i i i 1  10  2  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 M , such a conclusion cannot be drawn as w  easily.  10  4  1  1  —  •  I  111  • i i i i 11  i  i  i—i—i—i i i j  Constant M =133000, T=200°C L7D=20, D=0.7542 mm  - •  w  v  o u  3. v.  «J «  v  10 \ -  A V  3  B  V  •c CO  Resin Resin Resin Resin  c £  8: 10  2  I  T - PI • 4.76 L - PI = 8.72 R - PI = 12.98 E - PI •= 13.38 J  I I  I  10  I  V  I  -J  I  10  1  I I  I  I  I I  10  2  3  Apparent Wall Shear Rate (s ) 1  Figure 4.3 Apparent flow curves for resins with similarM determined at 200°C. w  66  Chapter 4 Results and Discussion  T o overcome the problem o f having too few resins in each group having similar polydispersity or M ,  another approach to data analysis was used.  w  Based on the  knowledge from previous studies that the viscosity profile is affected polydispersity and M , w  mainly by  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 M . w  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 M  w  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 M . w  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 o f 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" and 2 0 0 ° C as determined using STATGRAPHICSv2.0. Only resins manufactured using technology 'a' are included in the analysts. 1  3500  i i l 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 I I J  Shear Rate = 100 s"  1  T=200°C, UD=20, D=0.7542 mm  «> o 3000 u C  £  tl = 39S.5 + 0.0185'M,, - 117.98*PI, r*=88% A  2500  0 • o  (0  Q. Q. «t "O  o  Predicted  •  Observed  2000  o  £  QJ </>  •o O t3 C 05  1500  o  o  o  1000  "O  o u  I'"5  ' i  500 2  I  4  i  i  I i  6  i  i  i I  i  8  '  '  i  I  '  10  '  '  i  I  i  12  Polydispersity  i  i  '  I '  14  ' i ' I ' i ' • I i i i i  16  18  20  Index  Figure 4.5 Predicted and observed apparent shear viscosity at 100 s" and 200°C. 1  68  Chapter 4 Results and Discussion  For the range o f shear rates studied in this work, it was found that increasing M  w  at a  constant polydispersity tends to increase shear viscosity, while increasing polydispersity at a constant M  w  decreases the viscosity.  This is the expected trend, considering the  molecular properties o f a polymer. Increasing M  w  implies that, on average, molecules are  longer in size. This means that there are more entanglements between molecules, which results i n 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. T o 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 M  w  were used to simulate the plots. In Figure 4.6, it can be seen that increasing M  w  at a  constant polydispersity increases the shear viscosity, and in Figure 4.7, increasing polydispersity at a constant M  w  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 o f the index. Altering the molecular weight  distribution by changing the concentration  69  o f larger molecules w i l l not be  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 constantM using the regression relationship at each shear rate. w  70  Chapter 4 Results and Discussion reflected in the value o f the polydispersity index. consider the  whole molecular weight  Hence, it is often more useful to  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 o f the  distribution which are not reflected by the polydispersity index. These may significantly affect the rheology and processability o f a resin.  T o 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 o f 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, i n this case, the shear viscosity. A l l possible combinations o f 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" and 200°C indicated that there are two portions o f the M W D that are relatively 1  critical in affecting the shear viscosity. Figure 4.9 shows 3-D plots o f 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 o f approximately 9,000 to 22,000.  In this range o f molecular  weight, the correlation was found to be negative, i.e. increasing this portion o f the distribution results in a decrease o f 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 M  w  is  decreased. B y increasing the portion o f the M W D that lies on the left side o f the peak, M„, M , w  and M  z  are reduced, while polydispersity is increased (M„ is more significantly  decreased than M w  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" and 1  200°C (a) negative correlation (b) positive correlatioa Correlation coefficient (x,y) = COV (x,y)/a .a x  y  73  Chapter 4 Results and Discussion  F r o m 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 o f the M W D peak. It implies the concentration o f larger molecules and is reflected by M  w  and M .  Increasing this portion o f M W D ,  z  increases the weight average molecular weight, M , and hence, higher viscosity is w  observed.  B y doing this, however, the M W D is also broadened  polydispersity is increased.  and,  hence,  The previously discussed effect o f polydispersity on the  shear thinning behavior o f the resins is not observed using this analysis. Therefore, it seems that the observed effect o f polydispersity on shear thinning is only true i f changes in polydispersity are made by changing the concentration o f smaller molecules. Broadening the M W D by increasing the concentration o f 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 o f smaller molecules.  T o determine the effect o f temperature on shear viscosity, the activation energy term, E , in Equation 1.14 was calculated using the time-temperature superposition principle. a  Resins with greater values o f E have flow characteristics which are more sensitive to a  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 o f 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 E were found to range from 20 kJ/mol to a  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.  10  4  h  ~i—i—i—i A  11111  1—i—i—i  11111  Resin C  m^  -1 an°, T =180°C,LyD=20, D=0.7542mm  6  j  ref  CO  10  1—i—i—i 11111  to o u  .5 2 k.  co o  10  m  A  3  10 £ 5  -A  </> c  • •  2!  a CO  T=180°C T=200°C  * 10  2  10°  j  i i i i i i 11 10  i  i i  1  T=220°C i  i  i  i 111  10  i  2  Apparent Wall Shear Rate (s' ) 1  (a)  75  10  11  10  3  4  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„, M , and/or M as the independent variables. Using the results for all resins, it w  z  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 E could be related to the a  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 E = 1.69 + 7.8E - 4 • M„ + 1.02E - 5 • M +1.31E6 / M a  z  (4.1).  w  Observed E (kJ/mol) a  Figure 4.11 Observed and predicted E values as obtainedfromS T A T G R A P H I C S v2.0 (PI>10). a  From the equation, it can be deduced that by increasing M„ and M , shear properties z  become more sensitive to temperature change. The opposite can also be deduced when  77  Chapter 4 Results and Discussion  M  w  is increased.  The trend, however, w i l l be more useful i f it is expressed in terms o f  polydispersity index, M , w  and M , z  molecular weight distribution.  which are more commonly used to describe a  The effect o f 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 o f M . w  constant polydispersity index, increasing M  w  One can see that, at  affects the magnitude ofE non-linearly, but a  in a relatively non-significant manner. The effect o f M is more significant: increasing M z  z  increases the magnitude o f E . This same trend is also observed at constant M , as shown a  w  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 E  a  is relatively significant, and that  increasing the polydispersity index decreases the temperature sensitivity o f 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. were arbitrarily set to be constant in (a), and (b), respectively.  PI and M  w  This analysis is only particularly useful for resins which have skewed molecular weight distributions, as is the case with most o f 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 M  w  increasing M  z  while keeping polydispersity index constant without  and decreasing M„ proportionally, as shown in Figure 4.13.  Hence, for  resins with Gaussian - type molecular weight distributions, the effects o f molecular parameters have to be considered collectively.  79  Chapter 4 Results and Discussion  2  3  4  5  6  7  logMW  Figure 4.13 Hypothetical M W D showing the shift in  M , and M at constant PI. w  z  F o r the polydispersity range between eight to ten, the correlation obtained was  E  a  = 35.3-1.31E-3-M„ + 5.53E-5-M  (4.2).  w  Similarly, a good r-squared statistic (degree o f freedom corrected) was obtained in this case. Figure 4.14 shows the observed versus predicted E values obtained by using the a  equation.  The analysis indicated that the effect o f M  z  is statistically non-significant.  F r o m the equation, it can be deduced that increasing polydispersity while keeping M  w  constant, increases E . a  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 o f smaller molecules). I f polydispersity is increased by increasing M , no significant effect z  should be observed. From the equation, it can also be deduced that w h e n M is increased w  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 E values as obtained from STATGRAPHICS v2.0 (8<PI<10). tt  F o r the lowest polydispersity range (P/<8), no good correlation between E  a  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 M  w  at a constant polydispersity increases shear viscosity,  while increasing polydispersity at a constant M  w  81  increases the shear sensitivity o f the  Chapter 4 Results and Discussion  viscosity curve.  This is true only i f polydispersity is increased by increasing the  concentration o f smaller molecules. M o r e 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 i n the lower molecular weight range or increasing the concentration in the higher range has the same effect o f increasing the shear viscosity for the shear rate range studied in this work. However, the shear flow properties o f 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.  F o r resins with  polydispersity less than eight, no definite conclusion can be drawn with regard to the effect o f molecular parameters on E . a  For resins with polydispersity greater than eight, it  was found that, at constant M , increasing polydispersity increases E w  a  with the effect  entirely contributed by the increase in the concentration o f smaller molecules. Above a polydispersity o f ten, however, the concentration o f larger molecules becomes more important. Increasing the concentration o f smaller molecules now tends to decrease E , a  while increasing the concentration o f larger molecules tends to increase E . a  the values o f E  a  for resins with PK10  and Pl>10,  o f activation energy values for resins with PK10  PI>10.  Comparing  one can see that, generally, the range is smaller than that for resins with  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.  i  i  i  i  |  I  i  I  i  | r  ~i  i  i  |  i  i  i  1  I  I  I  I  I  L  i  l  i  i  i  r  Resin J - T = 1 5 0 ° C Stress = 3 kPa — Stress = 5 kPa Stress = 7 kPa  2 h  L  o c  /  CD  5:  /  / '  /  -1  50  1  100  I  1  150  200  250  300  Time (s)  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 o f the polymer. F o r the same stress level, a larger Hencky strain at a particular time implies that the polymer has a lower melt strength. strain.  From Figure 4.16, it can be seen that increasing M  w  This is not surprising since increasing M  w  decreases the Hencky  means an increase i n molecular  entanglements and hence, greater melt strength.  1  •  1 I  1 1 1 1 1 I 1 . 1 Constant Pl=13.4  1 1  1•  • • y ,  T=150°C, Stress = 3 kPa  2h•S  tc ©  / / 1  / ' /  A  i  \  -• Resin E  i I i i i i  Necking  = 131400  —- Resin F - M =143500 w  - Resin N  / '  a:  /  i i  =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  200  250  300  Time (s)  Figure 4.16 The effect of M on Hencky strain. w  T o 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. calculated.  K n o w i n g the stresses and the extensional rates, viscosities could then be F o r one group o f 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 o f 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 will occur, resulting in an increase i n 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 o f larger molecules which is contributing most to the observed effect (although this is not really reflected by M values). The same was also found for other sets o f resins. Figures z  4.19 and 4.20 illustrates this finding based on another set o f resins.  5e+6 to" m  to o o .<* "TO  c o to c  2  _  .  .  1  •  1  •  i  • 4e+6  —  i  i  |  i  i •  Constant Pl=13.4, T=150°C I Stress=3 kPa \  A A • A  3e+6  •  Resin E - M = 131400 -  •  Resin F - M = 143500 ~-  A  Resin N - M = 174900 1  w  w  w  2e+6  1e+6  —  -  AA A  t  -  A A  0e+0  • 3  • • • • • • • • • •  • i  • • • •| • I  i  i  11  2  Hencky Strain  Figure 4.17 The effect of M on tensile viscosity. w  85  Chapter 4 Results and Discussion  0.7  I  |  C o n s t a n t PI = 13.4  0.6  Resin E - M = 131400  ./'\'^  Resin F - M = 143500  ;j /"'<^\  Resin N - M = 174900  jj/  w  w  0.5  o  w  \  ill  0.4 0.3  I  0.2  Hi  0.1  '••\  \  j/  0.0 logMW Figure 4.18 Differential molecular weight distribution for resin E , F, and N .  3.5e+6  — |  I  ^- 2.5e+6 8 2.0e+6 1-  1  •  Resin K - M  •  Resin L - M Resin S - M  w  c ,o  1  Stress=3 k P a  3.0e+6  3  1  C o n s t a n t Pl=8.5, T=150°C  w  w  w  = 137300 = 133400 = 113000  V  R e s i n H - M = 104800  •  Resin O - M  w w  = 101200  5e+6  it*  § 1.0e+6  •2 X  5.0e+5 0.0e+0 0.0  J  i  l  l_  0.5  i  i  i_  _i  1.0  Hencky Strain  Figure 4.19  Effect of M  w  86  on tensile viscosity.  i  i  i_  1.5  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 o f results, since deformations were not l o w 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 , and their w  molecular weight distribution curves, respectively.  It can be seen that the effect o f  polydispersity depends on the part o f the distribution that is broadened.  It appears that  the extensional viscosity is significantly influenced by the concentration o f larger molecules, which is consistent with the study reported by Munstedt and Laun (1981).  87  Chapter 4 Results and Discussion  A l s o , as w i l l be shown later, melt strength or extensional flow property is also affected by resin density. 3e+6  -i  1  r  1  -i  T  1  r  1  Constant ^=133000, T=150°C Stress=3 kPa  W 2e+6  £  • • A v •  2e+6  to O o  £ S  Resin I - PI = 8.72 Resin M - PI = 11.18 Resin R - PI = 12.98 Resin E - PI = 13.38 Resin C - PI = 14.3 *  m *  1e+6  (0  c  •2  1e+6  to  c  A A  A '  A _  l2 5e+5 •  • A j  0e+0 0.0  •  A * . ^ y  H  i  •  '  •  %  V _1  V 1  V  I  0.5  *  *  L  "  * i  • i  -  • i  J  i  - ~  1.5  1.0  Hencky Strain Figure 4.21 The effect of PI on tensile viscosity.  0.8 0.7  Constant ^=133000  0.6 ^  0.5  5  0.4  —  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  0.2 0.1 0.0  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 w i 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 w i l l be related to strain rate by the equation  r] =a le E  (4.3)  E  regardless o f the rheology o f the resins.  B y using the data on Hencky strain, the effect o f polydispersity on melt strength was determined.  F o r a group o f resins with essentially constant M , the Hencky strains at w  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 o f 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 o f other parameters such as density, as w i l l be discussed later). A s polydispersity is increased further beyond nine, the Hencky strain starts to increase. A t higher polydispersity, however, the effect dies off as shown in Figure 4.24 for another set o f resins. Therefore, in terms o f 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  6  4  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  Constant M =154000, T=150°C  49.8 s  Stress=3 kPa  w  99.6 s -v— 149 s - • - 199 s  .s u «=  1  o  a: _1  16.5  I  I  L  I  I  I  l_  17.0  Polydispersity Figure 4.24 The effect of large PI on Hencky strain.  90  17.5  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 M  W  was found to increase the tendency o f tensile viscosity.  A l s o , 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 M , the effect o f polydispersity on Hencky strain is such that W  increasing polydispersity up to about nine decreases the polydispersity further reverses the trend. (PI>16),  strain, while increasing  However, for broadly distributed  resins  it was found that increasing the polydispersity no longer affects the strain  significantly.  4.2.3 Extrudate Swell Characteristics The method o f extrudate swell measurements used in this work was a simple one. However, there were a number o f possible error sources associated with the procedure. M o s t importantly, since measurements were done after the extrudates had cooled to room temperature, the absolute magnitudes o f 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.  91  Sagging produces  Chapter 4 Results and Discussion  higher swell in the lower portion o f the extrudates and makes them thinner near the die. Collection o f extrudates was another source o f 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 o f the resins as a function o f shear rate and temperature. Data at each shear rate was estimated to vary by approximately 5%.  r  0  i  i  i  i  I  i  200  i  i  i  I  i  400  i i i I i  600  i  i  i  I  i  800  i  i i_d  1000  Shear Rate (s' ) 1  Figure 4.25 Reproducibility of extrudate swell data.  T o determine the effect o f molecular weight on the relative magnitude o f swell, the resins were grouped according to their polydispersities. Figures 4.26 and 4.27 show the extrudate swell data for two groups o f resins.  92  The molecular weight distributions for  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 o f larger molecules. This may not show in the plot because o f the possibility o f the extrudates cooling before the ultimate swell is reached.  80  T  75 r-  1  1  1  1  1  1  1  f  T  I  I  I  J"  T  Constant PI = 10.1  Resin Q - M = 108000  35  0  200  400  H  Resin J - M  = 108900  ~d  ^ - Resin P-M  = 116000  4  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  -i—i—i—|—i—i—i—i—|—i—i  i  i  i  '  '  1  r T—i—i—r  '  Constant PI = 8.5 100 h- T=200°C, L/D=20, D=0.7542 mm  1  to  •8  Resin K-M = 137300 j Resin L-M = 133400 A — Resin S - M = 113000 w  v — Resin H-M = 104800 Resin 0 - M = 101200 w  _l  I  I  I  1  1  •  L.  i  i  400  200  i  I ' 600  I  I  I  I 800  1  1  1  1—  1000  Shear Rate (s' ) 1  Figure 4.27 Extrudate swell data for resin K , L , S, H , and O having similar PI.  0.7 • • Resin 0 . - 1 ^ = 1 0 8 0 0 0  t \  • - Resin J - M = 108900  /  Constant PI = 10.1  0.6  w  0.5  o  0.4 0.3  j  .. Resin P - M = 116000 w  i  0.2 0.1 0.0  \ _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 M  w  conclusion.  It  is  not  possible to  qualitatively correlate  resulted in the same 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 o f the elastic property o f a polymer.  Since the  elastic property o f a polymer is very much affected by the pre-shear history experienced during the various parts o f 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 M . w  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.  A l s o , 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 o f additives used may differ greatly among the different technologies, and may result in a varying degree o f unsaturation in the resulting polymer. crosslinking and chain scission in the polymer.  This affects the amount o f  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 M  z  and  polydispersity are useful parameters. Figures 4.32 and 4.33 show the effect o f M  and  z  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 M , w  broadening the molecular weight  distribution by increasing the concentration o f larger molecules increases the swell sensitivity i n the lower shear rate regime and decreases it in the shear rate range o f 350 s'  1  to 700 s" . Hence, swell is very sensitive at lower shear rates due to the presence o f larger 1  molecules.  In the processing range o f shear rates (350 s" - 700 s" ), however, the 1  sensitivity tends to decrease with polydispersity.  1  Figures 4.34 and 4.35 show the  molecular weight distributions o f the two sets o f 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  T—i—|—i—i—i—i—f—i—i—r—i—|—i  i  |  i  |  i  "  '  r  Constant M =133000, T=200°C sp  1  ^  50  L UD=20. D=0.7542 mm  h  •  Resin T - M =462700, Pl=4.76 Resin E -171^=776600, Pl=13.38 z  •  Resin C-1^=848400, Pl=14.3  A  ,  200  l  I  t  l  l  l  400  I  1  I  I  _1  1  1000  800  600  Shear Rate (s~ ) 1  Figure 432 Sensitivity of extrudate swell to changes in shear rate. A l l resins are manufactured by technology 'a' (set 1).  60  -i—i—i—i—|—i—i—i—i—|—i—i Constant M =154000, T=200°C L/D=20, D=0.7S42 mm  i  " |  i  i  | $j  w  ^  i <D  2 s  I  1  r;  1  i  i  50 40 30 20  •  Resin D - M =933500, Pl=16.44  •  Resin U-M =776600, Pl=16.49  A  Resin A - M=835400, Pl=16.83 l  Resin B - M =971500, Pl=17.42  v  j  i  i  i  i  I—i—i—u  200  i  2  2  2  l  l  I  i  600  400  I  '  i  i  800  i  i  L  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  —i  Constant ^ = 133000  0.7  / \ A. \ \ \  Resin T-PI=4.76  Resin E - Pl=13.38  0.6 ^  r  Resin C - Pl=14.3  ill I ill  0.5  O) 5 0.4 a  ill  \  \  w  11  0.2  //  0.1 0.0  •\  L_  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 producedfromdifferent 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 o f 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 w i 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 M  z  100  and polydispersity.  Increasing M  z  was  Chapter 4 Results and Discussion  found to increase the shear sensitivity o f extrudate swell at l o w shear rates.  In the  processing shear rate range o f 350 s" to 700 s' , however, shear sensitivity is affected 1  1  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 o f 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 o f 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.  T o determine the effect o f molecular properties on melt strength, it was necessary to group the resins into three groups having PI>J0, 8<PI<10 and PI<8. it was found that no useful correlation could be obtained.  Without doing so,  This is consistent with the  results obtained from the extensional experiments. From the analysis o f Hencky strain as  101  Chapter 4 Results and Discussion  a function o f polydispersity and time, it was found that the melt strength properties o f the resins change in behavior at a polydispersity range o f around eight to ten.  1.2  i  i  i  i  |  I  I  i  I  I  l  i  I  I  I  i  l  l  l  |  l  l  I  i  j  • • A v  • 5» 0.2 ' i i i ' ' ' ' -0.6 -0.5 -0.4  ' i i L  -0.3  -0.2  I  I  i  i  I  i  i  i  r~  T=190°C Resin A Resin B Resin C Resin D  J i i i  ' ' ' ' ' ' ' ' '  -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„, M , M , or polydispersity, a w  z  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. F o r resins with PI>10, it was found that melt strength is dependent on M , M„ (hence, polydispersity), and density. w  The model obtained with the best r-  squared statistic was Melt Strength = 75.5 + 1.47E - 5 • M  w  + 4.68E - 4 • M „ - 84.78 • Density  102  (4.4).  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 of M  are shown. It can be seen that the model closely predicts the melt strength values.  w  It should be noted, however, that this trend is valid only for the range o f M , w  M„,  polydispersity and density studied in this work.  ~0  2  1  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 PI>10),  which is attainable by decreasing  M. n  Moreover, by rewriting M„ as  MJPI,  (for one  may see that for large polydispersity, melt strength becomes essentially independent o f 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 o f M . B y increasing M , melt strength tends to increase, w  w  103  Chapter 4 Results and Discussion  1  1  I  I  I  I  I  1  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.M +4.68E-4.M„-84.78.Densityw  * |  ga  3  "g §  -  •  Observed  o  Predicted  1^=98%  '_  6  —  2  •  o  lS !  8  1  1  n  °.  8  • •  1.2e+5  1  1  1  1  1  1.3e+5  1  1  1  i  1  i  1.4e+5  i  i  i  8 8 8 •  1.5e+5  M  •  i  '  i  i  1.6e+5  i  i  i—1—i—i—i—i—  1.7e+5  1.8e+5  w  Figure 4.39 Observed and predicted values of melt strength as a function of M (PI>10). w  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 M  w  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, M , or by increasing the degree o f branching. w  F o r resins with 8<PI<10, the following correlation was obtained:  Melt Strength = 50.31 + 4.19 * 1 0 - M - 5  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 M on melt strength. The density value is fixed arbitrarily. Changing the density value would shift the curve upward or downward accordingly. Note that M is related to PI and M , and hence, when considering the plot, it has to be ensured that M is reasonable, so that PI>10. w  w  n  n  1  2  3  4  5  Observed Melt Strength (g@3min) Figure 4.41 Observed and predicted melt strength values obtained from S T A T G R A P H I C S 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 M  and decreasing  w  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 o f polydispersity. A possible reason is the narrow range o f 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 o f the molecular weight distribution that are affecting the melt strength, which are not reflected in M„, M , w  or M . z  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 o f 9,000 to 220,000 in a positive manner and by molecules in the molecular weight range o f 35,000 to 55,000 in the negative manner. However, the molecular weight range o f 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 o f 55,000 to 220,000, and that increasing the concentration o f molecules in this range increases the melt strength. Therefore, by increasing the polydispersity o f a polymer (with PI<8) in the direction o f increasing the concentration o f 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)/a.a x  y  107  Chapter 4 Results and Discussion  In summary, melt strength was found to increase with increasing M  w  and decreasing  density. Analysis had to be done separately according to the three polydispersity groups o f 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 o f 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.  F o r 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 o f 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 o f pillow weight versus  parison number for one o f the resins to demonstrate the reproducibility o f the experiments. The numbering o f the pillows is such that pillow number one is that which is nearest to the exit o f 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 o f the resin. However, it is difficult to perform experiments that would differentiate the effect o f either factor individually.  108  Although useful results may  Chapter 4 Results and Discussion  be obtained by doing pillow 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 o f resins used in the pillow 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.  18  p  17 |16 15  s i  14 13 12 11 Resin G  10  Drop Time = 1 s Drop Time = 3 s j] Drop Time = 5 s  9 8 7 6  4  6  8  10  12  Pillow Number  Figure 4.43 Variation of pillow weight with pillow number. Multiple curves indicate replicate runs.  109  Chapter 4 Results and Discussion  11  10 —•— —«- —A— —v—  Resin E - 1 s Resin E - 3 s Resin E - 5 s _ Resin F-1 s . Resin F-3s —•— Resin F - 5 s —••- Resin G - 1 s —s— Resin G - 3 s - Resin G - 5 s  9  8  7  0  10  8  6  4  2  -  12  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 o f pillow 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 pillow width). A s 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 pillow is decreased.  This shows the effect o f  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 o f each pillow also increases, due to parison sag, which causes the lower part o f 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 o f 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).  O n 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 o f swell at shorter parison drop times. T o 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 o f 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 o f 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 o f 3 s, for example, the curves corresponding to resin E and G are brought closer together. This is because o f the competing effect o f 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 pillow width in general.  Figure 4.47 shows a plot o f pillow weight versus pillow number. observations can be made.  Similar  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 o f 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 o f 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 o f 5 s.  Figure 4.49 shows a plot o f pillow weight, normalized to the weight o f the first pillow, versus pillow number for one o f the resins. This plot is useful for slope analysis, which should give an indication o f 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  I ^  vO)  .0)  r  Resin E  | .  — • — Drop Time = 1 s _ Drop Time = 3 s  \  — A — Drop Time = 5 s  I I i.  / /  O 5S 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 o f the curve. A steeper curve implies greater difference in the weight o f 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 o f the curves is an indication o f 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 o f resin G . Hence, the difference in weight between the first and the last pillow for resin G is larger than that for resin F .  T  Drop Time = 5 s —Resin E Resin F —Resin G  <2  S  i ©  T  X  1  1  o 5* k — i 1  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 o f the parison, the general trend is that parison length increases as drop time increases. parison time means more sagging.  O f course this is as expected, since longer  Also, for shorter drop times, swell dominates and  tends to shorten the total parison length. The effect o f 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 o f 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 0  I 1  I 2  I 3  I 4  1 5  1 119 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 o f 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. F r o m the pillow 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 o f 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. done frequently and quickly.  It does not require elaborate equipment and can be  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 o f the equipment or the procedure may result in a very significant variation o f the MI data.  Generally, MI is used for the analysis o f reaction quality control. It serves to indicate the uniformity o f the flow behavior o f 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 o f M Z is affected by the molecular weight o f the polymer, as is the case with all other rheological and processing parameters. exponent (S.Ex.)  B y performing experiments at different conditions, the stress  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 o f MI to M . N o useful mathematical correlation w  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  _l  •  |  1  !  I  .  |  1  1—  I  1  |  i  1  *  0.8  ^  i  Technology 'a  '+ Others  i i  -  I.I i i i  -  1  •13.38  0.5 0.4 0.3  -  r  1  '+8.72  I I I  TTI  s S  i  I  0.6  |  | I  0.7  i  i  1 1 1 |  •8.25  "1  0.9  '+12.98  :  "+10.1 '+8 1 '+6.02  '  -  + 8  "17.42 6  '+10.09  *^-76 .  0.1  5  ; ^  4  . 7 9 M6.49  '+13.351  *3.4  0.2  5  •14.3  '+8.75  • I  i  i  i  r  1.0e+5  I  i  1.2e+5  i  i  i  I  i  1.4e+5  i  i  i  I  i  1.6e+5  i  1  t  1.8e+5  Mw  Figure 4.52 Correlating MI to M . Comparison should be made for resins with similar PI, or shear thinning properties. W  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 o f ten minutes. Hence, higher MI values indicate faster flow and lower viscosity. B y increasing M , MI decreases, and slower flow w i l l result, w  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 l o w 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.  —ii  V  10  T  i I i i  1  1  I  I  1  I I  1—i—i  i  i i i |  T=200°C,  LVD=20, D=0.7542 m m  4  A A V  o u  A  V  <tJ  A  V  .52 S V.  A  V  o 10 •c to c  • • A v  a a  10  2  A  V  3  J  I  I  Resin W-Ml = 0.21 Resin X-MI=0.41 Resin C - Ml=0.53 Resin E - Ml=0.72 _1  I I I  I  l — L  11,1  10  101  I  I  I  I  I  I  I I  10  2  3  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 o f 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  S.Ex.  and MFR  Correlation between melt strength and MJ.  are both indicative o f 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 o f 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.  I  I  •  I  1  1 0 4  ni  1 I  • •  I  1  •  —  • •  55 o o  i  —  i  i  i  i  i  11  i  •  • •  •  •  • •  -  Q.  10  2  i  i  i  i  i  i  11  -  • • •  Resin V -S.Ex.=1.51,MFR=38.08 Resin U -S.Ex.=1.77,MFR=95.83  • •  a  i  •  3  c  i  •  to 10  <0  i  T=200°C, Ml=0.27 L/D=20, D=0.7542 mm -  •  Q) •C  i  i  11  i  i  i  i  i  10  i  i  11  i  i  i  i  i  i  2  Apparent Shear Rate (s~ ) 1  Figure 4.55 Implication of S.Ex. on shear viscosity profile.  123  i  1 10 i  3  10  1  Chapter 4 Results and Discussion  Figure 4.56 shows a plot o f S.Ex. versus MFR. S.Ex. reflects higher MFR.  It can be seen that generally, higher  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.  F r o m 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.  A s 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 PROCESSABILITY 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 o f rheology on processability.  The shear viscosity o f 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  o f 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. O n 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 o f sagging, the rheological  properties applicable in this case would be the extensional flow properties o f 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.  A s discussed earlier,  increasing extrudate swell indicates an increase in weight swell.  F o r 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 o f 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 o f 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 will not have consistent weight, and this is o f course undesirable.  127  Chapter 4 Results and Discussion  In conclusion, the optimal design o f the different parts o f blow molding process require different resin properties. Considering the complex effect o f 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 o f  each o f these properties, on the other hand, is determined by the use o f the final product. Hence, it is very difficult to say that a specific resin is the best for a particular application. In the design o f a resin, consideration and careful weighing have to be made in regard to its application and economic implications.  F r o m the results o f this study, however, some useful recommendations can still be made i n 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 M . w  Increasing M  w i l l increase the melt strength o f the  w  resin, while not affecting the sensitivity o f shear viscosity to temperature variations significantly.  B y increasing M , z  melt strength will also increase, but the resin w i l l  become more sensitive to changes in temperature and this is, o f course, undesirable. Increasing M  w  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 w i l l not be very significant compared to when polydispersity is increased concentration o f smaller molecules).  by decreasing M„ (increasing  Moreover, by increasing M , w  the  and hence, the  polydispersity, the extrudate swell profile becomes less sensitive to shear rate variations in the processing range between 350 s" to 700 s" . 1  that a resin with a comparatively high M  w  1  Therefore, generally, it can be said  value and low M value w i l l process better. A s z  far as M„ is concerned, it has to be optimized accordingly, depending on M  w  polydispersity.  129  and  Chapter 5 Conclusions  5  CONCLUSIONS  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 o f molecular parameters, such as M , PI, and density. These are commercial resins produced from a number o f different w  technologies, since it is not possible to produce resins with broad range o f molecular parameters using a single polymerization technology. commercial resins produced  from different  However, with this set o f  technologies,  it was not possible  systematically study the individual effects o f specific molecular parameters. multiple regression was used as a tool for data analysis, wherever possible.  to  Hence, In some  cases, however, the molecular parameters are insufficient in describing the molecular weight distribution. In these cases, the actual qualitative comparisons o f the distribution curves were performed.  The conclusions o f this study are summarized as follows:  Shear F l o w Properties: -  The expected effects o f M  w  Increase o f M  w  and polydispersity on shear viscosity were confirmed.  and polydispersity increases the magnitude o f shear viscosity and its  tendency to shear thin, respectively. The effect o f polydispersity has been found to be mainly due to the changes in the concentration o f 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 o f 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 E . a  Above the polydispersity o f eight, it was found that increasing  polydispersity increases the temperature sensitivity o f viscosity, with the  effect  mainly contributed by the increase in the concentration o f smaller molecules. F o r PI greater than ten, however, the concentration o f larger molecules becomes important and increasing it w i l l increase both the polydispersity and E . a  Extensional F l o w Properties: -  Increasing M  w  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. F o r PJ>9, broadening the molecular weight distribution initially increases the Hencky strain. F o r broadly distributed resins (JPI>16), the effect o f 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 o f extrudate swell. This is probably due to the possibility o f the effect o f pre-shear history being more important than the effect o f molecular weight distribution. Since the resins studied in this w o r k 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 o f unsaturation, which may not be reflected in molecular weight distribution curves, but which may affect the elastic properties o f the polymers significantly.  -  The sensitivity o f extrudate swell, however, was found to correlate well with M  z  polydispersity.  and  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" ) and decreases it at the higher shear rates (350 s" to 700 s" ). 1  1  132  1  Chapter 5 Conclusions  M e l 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 M  -  Increasing polydispersity for resins with PI<8 was found to increase melt strength.  w  was found to increase melt strength.  F o r 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 o f sagging and weight swell are difficult to differentiate in pillow 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. F o r 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 M  w  for resins o f the same technology and similar  polydispersity, in w h i c h 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 o f 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 o f molecular weight ranges to the desired property.  These conclusions are summarized in Table 5.1.  134  el  o ro  8£<  o  5  § 21S  |3  2i>  CO 5  Chapter 6 Recommendations  6 RECOMMENDATIONS 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 o f 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 M e l 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 o f predicting extrudate swell.  -  Extrudate swell has been found to depend more strongly on the pre-shear history o f 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 w i 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 o f rheology and processability.  Possible reasons include the type o f 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. A l s o , the molecular effects on other physical properties o f H D P E , such as environmental stress cracking resistance ( E S C R ) can be studied.  137  References  REFERENCES Acierno, D . , D . Curto, F . P . L a Mantia, and A . Valenza, Flow Properties of Low Density/Linear Low Density Polyethylenes, Polym. E n g . Sci., 26 (1), 2 8 - 3 3 (1986) Ajroldi, G . , Determination of Rheological Parameters from Parison Extrusion Experiments, Polym. Eng. Sci., 18 (10), 742 - 9 (1978) A S T M , Standard Test Method for Flow Rates of Thermoplastics by Extrusion Plastometer, D1238, 1995 Bagley, E . B . , End Corrections in the Capillary Flow of Polyethylene, J. A p p l . Phys., 28, 624 (1957) Bersted, B . H . , A Model Relating the Elastic Properties of High - Density Polyethylene Melts to the Molecular Weight Distribution, I. A p p l . Polym. Sci., 20 (10), 2705 - 14 (1976) Bethea, R . M . , B . S. Duran, and T. L . Boullion. Statistical Methods for Engineers and Scientists. 2 ed., Marcel Dekker, Inc., N Y , 1985 n d  Bremner, T., Personal Communication, N O V A Research and Technology Center, 1998 Cannady, J . L . , Blow Molding - New Developments, Seminar Presented b y Program Division, Technomic Publishing Company Inc., P A , 1988 Carella, J. M . , J. T. Gotro, and W . W . Graessley, Thermorheological Effects of Long Chain Branching in Entangled Polymer Melts, M a c r o m o l e c , 19, 659 - 67 (1986) Charrier, J . M . , Polymeric Materials Composites. Hanser, N Y , 1990  and Processing.  Plastics. Elastomers, and  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. V a n 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 1 9 4 2 - 5 3 (1967)  Polymers, J. Chem. Phys., 47,  Han, C . D., and C . A. Villamizar, Effects of Molecular Weight Distribution and LongChain Branching on the Viscoelastic Properties of High - and Low - Density Polyethylene Melts, J. A p p l . Polym. Sci., 22 (6), 1677 - 700 (1978) Hatzikiriakos, S. G . , and J. M . Dealy, Wall Slip of Molten High Density Polyethylene. Capillary Rheometer Studies, J. Rheol., 36 (4), 703-741 (1992) Hatzikiriakos, S. G . , The Onset of Wall Slip and Sharkskin Melt Fracture in Flow, P o l y m . E n g . Sci., 34 (19), 1441 - 9 (1994)  II.  Capillary  Henze, E. D., and W . C . L . W u , Variables Affecting Parison Diameter Swell and Their Correlation with Rheological Parameters, Polym. Eng. Sci., 13 (2), 1 5 3 - 9 (1973) K a m a l , M . R . , and K . T. Nguyen, Analysis of the Blow Molding Process: Review and Recent Developments, in Polymer Rheology and Processing, A . A . Collyer, and L . A . Utracki (eds.), Elsevier Applied Science, N Y , 1990 K a y l o n , D . M , and M . R., Kamal, An Experimental Investigation of Capillary Extrudate Swell in Relation to Parison Swell Behavior in Blow Molding, Polym. Eng. Sci., 26 (7), 508 - 16 (1986) Kazatchkov, I. B . , N . Bohnet, S. K . Goyal, and S. G . Hatzikiriakos, Influence of Molecular Structure on the Rheological and Processing Behavior of Polyethylene Resins, P o l y m . E n g . Sci., Accepted for Publication, D e c (1997) Koopmans, R . J., Die Swell - Molecular Polym. Sci., Part A , 26, 1157 (1988)  Structure Model for Linear Polyethylene,  J.  Koopmans, R . J., Extrudate Swell of High Density Polyethylene. Part II: Time Dependency and Effects of Cooling and Sagging, Polym. E n g . 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  Polyethylene, Plas. Rubber Process. Appl., 5 (2), 183 - 5 (1985)  of High  Density  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  Polym. Eng. Sci., 32 (23), 1778-91 (1992)  Meissner, J., Development  of a Universal  Extensional  Rheometer for the  Rheology,  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, WileyInterscience, 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 Polyethylene Melts, Eur. Polym J., 19 (9), 811 - 16 (1983)  Low  Density  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. M k s u z o , 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. S c i . , Part B , 34, 2317 - 33 (1996) V a n 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 B o o k 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. D i v . , Commission on Polymer Characterization and Properties, I U P A C , Pure and A p p l . 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 )  Ao  Cross sectional area before deformation (m )  ax  Shift factor  b  Rabinowitsch correction  2  C°i, C°  2  D D  W L F constants Diameter (m) Diameter before deformation (m)  0  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.s )  L  Length (m)  Lo  Length before deformation (m)  M  Molecular weight o f monodisperse polymer (kg/kmol)  Mi  Molecular weight o f polymer chain / (kg/kmol)  M„  Number average molecular weight (kg/kmol)  Mo  Location parameter in the 'log-normal' distribution curve  n  (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 +i z  (Z+7)-average molecular weight (kg/kmol)  142  Notation  n  Power law exponent  n;  Number average o f polymer chain /'  P  Pressure (Pa)  P  Ambient pressure (Pa)  a  Pd  Barrel driving pressure (Pa)  Pend  Pressure end correction (Pa)  Q  Volumetric flow rate (m /s)  R, r  Radius (m)  t  Time (s)  T  Temperature ( K )  To  Reference temperature ( K )  T  Glass transition temperature ( K )  3  g  V  Velocity (m/s)  w  Weight fraction  Wi  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  Extensional Strain magnitude  0  YA  Apparent wall shear rate (s" )  y  Shear rate (s" )  1  1  0  Yo  Shear magnitude  Yw  True wall shear rate (s" )  T|  Shear viscosity (Pa.s)  TJ*  Complex viscosity (Pa.s)  TJE  Extensional Viscosity (Pa.s)  T|  Zero shear rate viscosity (Pa.s)  Xi  0  1  Discrete relaxation time (s)  143  Shear stress (Pa) Extensional stress (Pa) W a l l shear stress (Pa) Frequency o f 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  ********************************************************** * * A l f o n s i u s Budi Ariawan * Department o f Chemical E n g i n e e r i n g * The U n i v e r s i t y o f B r i t i s h Columbia * Vancouver, B.C.  * *  * * * * * * * *  P r i n t i n g o f a v a i l a b l e o p t i o n s and some i n f o r m a t i o n r e g a r d i n g t h e program: PRINT 180 PRINT 190  C  * *  SUMMARY: The f o l l o w i n g program reads i n s e t s o f shear s t r e s s d a t a as a f u n c t i o n o f temperature and shear r a t e . A master curve f o r t h e s p e c i f i c polymer i s t h e n c a l c u l a t e d c o r r e s p o n d i n g t o a s p e c i f i e d r e f e r e n c e temperature. The s h i f t f a c t o r i s then c o r r e l a t e d u s i n g e i t h e r a simple t h e r m a l l y a c t i v a t e d m o d e l ( A r r h e n i u s ) , WLF model o r both, depending on t h e d e s i r e o f t h e u s e r . An o p t i o n i s a l s o p r o v i d e d t o c a l c u l a t e t h e shear v i s c o s i t y o f t h e polymer a t a d e s i r e d temperature and shear r a t e .  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 C  * * * * *  Prompting f o r i n p u t : PRINT 200 READ *, CHOICE PRINT 210 READ *, NSET PRINT 220 READ * OPEN  (7,FILE='DATA.TXT',STATUS='OLD')  146  * * * * * * * *  Appendix A:  READ READ READ READ  Time Temperature Superposition Program Code  (7,*) (NDATA(I), I=1,NSET) (7,*) (TEMP(I), I=1,NSET) ( 7 , ( A ) ' ) (FNAME(I), I=1,NSET) (7, ' (A) * ) COMMENT ,  CLOSE (7) PRINT 230 READ *, ID DO 30 I=1,NSET OPEN  (7,FILE=FNAME(I),STATUS= OLD ) 1  1  IF  (ID.EQ.O) THEN DO 10 J=1,NDATA(I) READ (7,*) SRT(I,J),STRESS(I,J) CONTINUE ELSE DO 20 J=1,NDATA(I) READ (7,*) SRT(I,J),STRESS(I,J),SD(I,J) CONTINUE ENDIF  10  20  CLOSE (7) 30  CONTINUE  C  Prompting f o r more i n f o r m a t i o n : PRINT 240, NSET READ *, NREF TREF=TEMP(NREF) ITOTAL=NDATA(NREF) PRINT 330 READ *, NVIS IF  (NVIS.EQ.l) PRINT 340 READ *, T PRINT 350 READ *, W ENDIF  C  Initializing  THEN  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  D e f i n i n g t h e weight o f each data p o i n t , i f d a t a a r e a v a i l a b l e : IF  (ID.EQ.l) THEN  147  Appendix A: Time Temperature Superposition Program Code  WT(J)=1/SD(NREF,J)**2 ELSE C  Otherwise,  the  same  weight  of  1.D0  is  applied  to  a l l  points:  WT(J)=1.D0 ENDIF 40  CONTINUE DO  90  I=1,NSET  IF  ((NVIS.EQ.l) .AND. (IVIS.EQ.O) .AND. (T.EQ.TEMP(I) ) )  THEN  IVTS=I 11=1 ENDIF C C  Calculation  of  shift  factor  for  each  temperature,  except  the  reference  temperature: IF  (I.EQ.NREF)  THEN  AT(I)=1.D0 SUMSQ(I)=0.D0 GOTO  90  ENDIF MDATA=NDATA (I) DO  C  50  J=1,MDATA  IF  ((II.EQ.1).AND.(SRT(I,J).EQ.W))  Working  in  log  JVIS=J  scale:  ALNSRT(J)=DLOG(SRT(I,J)) ALNSTRS(J)=DLOG(STRESS(I,J)) 50  CONTINUE 11=0 NM=NDATA(I)-1  C C  Using  cubic  spline  of  fitted  ends  to  allow  interpolation  between  points: CALL  SPLINE(ALNSTRS,ALNSRT,MDATA,NM,Q,R,S,I1,IN,G1,GN)  ICHECK=0 AT0=l.D-6 DAT=1.D-1 C  Determination  C  the  60  objective IF  of  the  interval  which  (OBJ(AT0+DAT)-GT.OBJ(ATO)) IF  brackets  function:  (ICHECK.EQ.l)  THEN  THEN  148  the  minimum v a l u e  of  Appendix A: Time Temperature Superposition Program Code  A T I =AT 0 - DAT A T F=AT 0+DAT GOTO  70  ENDIF ELSE ICHECK=1 ENDIF A T 0=AT 0+DAT GOTO C  Using  70  60  the  CALL  golden  GOLDEN  C  Shifting  C  new r e f e r e n c e DO  of  80  search  method  to  minimize  the  objective  (ATI,ATO,ATF,OBJ,TOL,AT(I),SUMSQ(I))  curves  and defining  the  shifted  curve  as  part  curve:  J=1,MDATA  ASRT(IT0TAL+J)=SRT(I,J)*AT(I) REFSRT(ITOTAL+J)=DLOG(ASRT(ITOTAL+J)) ASTRESS(ITOTAL+J)=STRESS(I,J) REFSTRS(ITOTAL+J)=DL0G(ASTRESS(ITOTAL+J)) C  Determining IF  the  weight  (ID.EQ.l)  of  each  point  of  the  master  THEN  WT(ITOTAL+J)=1/SD(I,J)**2 ELSE WT(ITOTAL+J)=l.D0 ENDIF 80  CONTINUE ITOTAL=ITOTAL+MDATA  90  CONTINUE  C  Sorting  C  the  master  curve  data  CALL  SORT(ITOTAL,ASRT,ASTRESS)  OPEN  (9,FILE=  WRITE  (9,*)  WRITE  (9,250)  Printing DO  100  1  points:  OUT P U T . T X T ) 1  COMMENT  of  output:  1=1,ITOTAL  VISCOSITY=ASTRESS(I)/ASRT(I) WRITE 100  (9,260)  ASRT(I),ASTRESS(I),VISCOSITY  CONTINUE WRITE DO  110  (9,270)  function:  TREF  I=1,NSET  149  curve:  of  the  Appendix A: Time Temperature Superposition Program Code  WRITE 110  (9,280)  TEMP(I),AT(I),SUMSQ(I)  CONTINUE M=NSET IF  ((CHOICE.EQ.l).OR.(CHOICE.EQ.3)) DO  120  THEN  I=1,NSET  X(I)=1/(TEMP(I)+273.15) Y(I)=DLOG(AT(I)) 120 C  CONTINUE Linear  regression  i f  the  first  and  the  second  models  are  both  chosen:  CALL LINREG(X,Y,NSET,CEPT,SLOPE,RSTAT) EA=SLOPE*8.314/1000 ENDIF C  If  C  energy IF  the  simple  thermally  of  polymer  the  (CHOICE.EQ.1) WRITE GOTO  activated  is  model  is  chosen,  then  the  activation  returned:  THEN  (9,290)  EA,RSTAT  150  ENDIF C  If  C  results  C  to IF  both be  the  f i r s t  obtained  fitted  for  (CHOICE.EQ.3)  and  the  second  from  the  first  the  second  models model  are  to  chosen,  obtain  then  more  use  data  model:  THEN  X(1)=TEMP(1)+273.15 Y(l)=DLOG(AT(l)) M=2*NSET-1 DO  130  I=1,NSET-1  X(2*1+1)=TEMP(I+1)+273.15 Y (2*1+1)=DLOG(AT(1+1)) X(2*I)=(TEMP(I)+TEMP(I+1)J/2+273.15 Y(2*I)=EA*1000/8.314*(1/X(2*I)-1/(TREF+273.15)) 130  CONTINUE ELSEIF  (CHOICE.EQ.2)  C  If  WLF (second)  C  to  be DO  THEN  model  is  chosen,  then  fitted: 140  1=1,NSET  X(I)=TEMP(I)+273.15 Y(I)=DLOG(AT(I)) 140  CONTINUE ENDIF  150  prepare  the  data  the  points  points  Appendix A: Time Temperature Superposition Program Code  C  Calling  C  unknown CALL  C  the  IF  GAUSSN  to  determine  the  best  estimate  of  GAUSSN(X,Y,AGUESS,M,TOL,AMODEL,DMODEL,A,SSQ)  Printing IF  subroutine  parameters:  the  second  set  (CHOICE.EQ.2)  WRITE  (CHOICE.EQ.3)  THEN  WRITE  (9,310)  WRITE  (9,320)  of  output:  (9,300)  A(l),A(2),SSQ  EA,RSTAT,A(1),A(2) SSQ  ENDIF C  Calculating  150  IF  C  I f  C  shear  the  (NVTS.EQ.l) viscosity rate  IF  viscosity,  i f  desired:  THEN at  one  (part  of  of  the  the  experimentally  input)  is  found  required,  ((IVTS.NE.0).AND.(JVIS.NE.0))  then  temperature calculate  and directly:  THEN  VIS=STRESS(IVIS,JVIS)/W GOTO  170  ENDIF IF  (IVTS.NE.0)  THEN  AAT=AT(IVIS) GOTO  160  ENDIF C  Otherwise,  f i t  cubic  spline  into  the  master  curve:  CALL +  SPLINE(REFSRT,REFSTRS,ITOTAL,ITOTAL-1,Q, R,S,II,IN,Gl,GN)  C  And calculate  C  using  the  IF  the  chosen  shift  factor  corresponding  to  the  temperature  model:  (CHOICE.EQ.l)  THEN  AAT=DEXP(EA/8.314*1000*(1/(T+273.15)-1/(TREF+273.15))) ELSE AAT=DEXP(A(l)*(T-TREF)/(A(2)+T-TREF)) ENDIF C  Calculate  C  obtain  160  the  the  shifted  shear  corresponding  rate  the  master  WRITE  (9,360)  R , S , I T O T A L , WI) ) / W )  T,W,VIS  ENDIF C  from  values:  WI=DLOG(W*AAT) VIS=(DEXP(F(REFSRT,REFSTRS,Q,  170  and  G' and G"  Format  statements:  151  curve  Appendix A: Time Temperature Superposition Program Code  180  FORMAT  (lX'This 'at  +  'viscosity  +  'temperature  190  FORMAT +  200  a  program creates  +  specified of  'correlate FORMAT  the  'model  +  'good  choose  shift  (/IX,'1.  +  shear  T >  model  FORMAT  (/lX'Please  220  FORMAT  (/IX,'Please  +  'directory  +  'of  data  one  Tg < T  'These  +  /IX,'ascending  orders  +  'second  of  +  'the  +  'correspond  +  'In  +  'they  + 230  'when FORMAT  + 240  column  be  FORMAT  stress. to  FORMAT  260  FORMAT FORMAT  to  be  there  1  third  (/lX'Which  data  -  set  to?  (/'RESULTING  ('The  do  'objective  MASTER  you  (/'Using  on  factor  simple  +  'coefficient  kJ/mol'/'The  (/'Using  DATA  of  2  (or  1).  be  the 3)  in  The  ',  refer  added  (in  %). is  ',  stored.  should  file -  ', '  to  ',  to'/IX,  ', available,  ',  press  enter  ',  curve  to  ',  ',  yes ) 1  the  master  be  POINTS:'//3X,'Rate (Pa.s)'/'  (/'Using  corresponding  (1/s)', ',  both  +  ',  C2=  was  ', of  ',  to  temperature value  of  of  ',  the  ',  is:',F7.4)  R=  activated  model,  regression  Ea=  ',F8.4,  correlation  ',  *,F6.4) C1=',F8.4,', squares  simple  of  C2=',F8.4/'The  differences  thermally  ',F8.4,'  activated  kJ/mol,  R=  is model  ',F6.4,',  ',  ',F6.4) a n d WLF ' ,  Cl=  *,F8.4,  ',F8.4)  320  FORMAT  (lX'The  associated  330  FORMAT  (/lX'Do  you  no'/IX,'1  want -  sum  to  squares  of  differences  viscosity?*/IX,'0  is  ',F6.4)  ',  yes')  FORMAT  (IX'At  what  temperature  350  FORMAT  (lX'At  what  shear  viscosity of  of  calculate  340  (/'The  curve  temperature  corresponding  thermally  sum o f  'model,'/'EA=  'rate  The master  1  reference  corresponding  is  +  ' -  the  WLF m o d e l ,  'associated  FORMAT  the  want  CURVE  function  '  +  and  are  (1-*,I2,')*)  C i s : ' , F 7 . 4 / ' T h e  +  360  on  n o ' / l X , ' 1  (Pa)',4X,'Viscosity  shift  +  +  ',  ',  C'/)  F6.1,'  FORMAT  set,  may  number 1  (column  column  the  temperature /IX,  set  columns  sets:') appropriate  (Fll.4,F14.0,F18.0)  +  310  data  each  deviations  column  F6.1,'  FORMAT  on  the  ',  ',  ready...')  (/IX Is  +  +  the  zero.'//IX,'Please  OF R E S U L T S : ' / /  300  ',  Both')  standard'/IX,'deviation  based  FORMAT  each  in  third  'calculated  290  to  set,  frequency  + FORMAT  into  data'/IX,'files  (//'SUMMARY  280  data  standard  no  assumed  4X,'Stress  270  of  to  good (a  data  information  1  stored  The  the  where  'referenced  +  each  these  (a  available  d a t a ' / l X , ' o f  'c:\DATA.TXT?'/IX,*0  + 250  should  Model  WLF model  DATA.TXT  1  +  are  of  of /IX,  which  ',  ',  calculated.')  options  1  corresponding  i n  be  following  (Tg+100)K)'/IX, 3.  file:  Celsius)  cases  <  the  'file  The  ',  also  Activated  copy  '(in  polymer  1  number  +  a  tem-'/IX,'perature: )  the  +  shear  to  for  for  desired  the  enter  available  data  a  of  Thermally  consisting  names  at  (Tg+100)K)'/IX,'2.  for  210  curve  1  rate'/IX,'may  factor  Simple  for  master  tem- /IX,'perature.  polymer  and  (//,IX'Please  a  reference  ',F6.2,'  rate of  the  rad/s  (in (in  Celcius)?') rad/s)?')  polymer ' /  1  at  ',F6.1,'  is:',F10.4,  STOP END  152  '  C and  Pa.s')  shear  ',  Appendix A: Time Temperature Superposition Program Code  C  The  C  adjacent  subroutine  C  between  C  of  SPLINE  points  of  points.  fitted  The  fits given two  a  cubic  set  end  of  polynomial data  conditions  to  used  SPLINE  DOUBLE  DIMENSION  (A-H,0-Z)  A(301),B{301),C(301),D(301)  DIMENSION  H(300)  (I1.EQ.3)  THEN  AA=0.DO DO  20  1=1,4  TERM=Y(I) DO 10  10  J=l,4  IF  (J.NE.I)  TERM=TERM/(X(I)-X(J))  CONTINUE A A = A A + T E RM  20  CONTINUE ENDIF IF  (IN.EQ.3)  THEN  M=N-3 BB=0.D0 DO  40  I=M,N  TERM=Y(I) DO 30  30  J=M,N  IF  (J.NE.I)  TERM=TERM/(X(I)-X(J))  CONTINUE BB=BB+TERM  40  CONTINUE ENDIF DO 5 0  1=1,NM  H(I)=X(I+1)-X(I) 50  CONTINUE A(1)=0.D0 IF  (Il.EQ.l)  THEN  B(1)=1.D0 C(1)=0.D0 D(1)=0.D0 ELSEIF  (I1.EQ.2)  THEN  B(1)=2.D0*H(1) C(1)=H(1) D(1)=3.D0*{(Y(2)-Y(l))/H(l)-Gl) ELSE B(l)=-H(l) C(1)=H(1) D(1)=3.D0*H(1)*H(1)*AA ENDIF DO  60  are  (X,Y,N,NM,Q,R,S,II,IN,Gl,GN)  PRECISION  DIMENSION X ( N ) , Y ( N ) , Q ( N ) , R ( N + l ) , S ( N )  IF  into  allow  ends.  SUBROUTINE IMPLICIT  a  1=2,NM  IM=I-1 A(I)=H(IM) B(I)=2.D0*(H(IM)+H(I))  153  every  interpolation those  of  spline  Appendix A: Time Temperature Superposition Program Code  C(I)=H(I) D(I)=3.D0*((Y(I+1)-Y(I))/H(I)-(Y(I)-Y(IM))/H(IM) 60  )  CONTINUE C(N)=0.D0 IF  (IN.EQ.l)  THEN  A(N)=0.D0 B(N)=1.D0 D(N)=O.DO ELSEIF  (IN.EQ.2)  THEN  A(N)=H(NM) B(N)=2.D0*H(NM) D(N)=-3.D0*((Y(N)-Y(NM))/H(NM)-GN) ELSE A(N)=H(NM) B(N)=-H(NM) D(N)=-3.D0*H(NM)*H(NM)*BB ENDIF CALL  TDMA  DO 7 0  (A,B,C,D,R,N,NM)  1=1,NM  IP=I+1 Q(I) = (Y(IP)-Y(I))/H(I)-H(I)*(2.D0*R(I)+R(IP)  )/3.D0  S(I)=(R(IP)-R(I))/(3.D0*H(I)) 70  CONTINUE RETURN END  C  The subroutine  C  algorithm. SUBROUTINE IMPLICIT DIMENSION  TDMA s o l v e s  the  tri-diagonal  ( A - H ,O - Z )  A(N),B(N),C(N),D(N),X(N),P(301),Q(301)  Argument  C  A, B,C,D  The  C  X  Solution  C  N  Number  of  unknowns  C  NM  Number  of  interval  list: coefficients  of  the  between  P(l)=-C(l)/B(l)  1=2,N  IM=I-1 DEN=A(I)*P(IM)+B(I) P(I)=-C(I)/DEN Q(I)=(D(I)-A(I)*Q(IM))/DEN 10  CONTINUE X(N)=Q(N) DO 2 0  tridiagonal  vector  Q(1)=D(1)/B(1) 10  using  TDMA(A,B,C,D,X,N,NM)  DOUBLE PRECISION  C  DO  matrix  I=NM,1,-1  X(I)=P(I)*X(I+1)+Q(I)  154  unknowns  set  Thomas  Appendix A: Time Temperature Superposition Program Code  20  CONTINUE RETURN END  C  The  subroutine  C  the  minimum v a l u e  C  returns  C  independent  the  Argument  uses a  variable  the  certain  at  of  golden-search  technique  function,  The  the  which  F.  function  i t  and  the  The  BX  :  A  PRECISION  lower  point  lower  limit  in  than  the  The  upper  F  The  function  of  the  interval  interval  limit to  where  of  the  be  minimized  Pre-specified  XMIN  The  point  FMIN  The  minimum v a l u e  interval  convergence  where  at  the  which  minimum of  at  which  tolerance occurs  the  function  X3=CX (DABS(CX-BX).GT.DABS(BX-AX))  THEN  X1=BX X2=BX+C*(CX-BX) ELSE X2=BX X1=BX-C*(BX-AX) ENDIF F1=F(X1) F2=F(X2) (DABS(X3-X0).GT.TOL*(DABS(XI)+DABS(X2))) (F2.LT.F1)  THEN  X0=X1 X1=X2 X2=R*X1+C*X3 F1=F2 F2=F(X2) ELSE X3=X2 X2=X1 X1=R*X2+C*X0 F2=F1 F1=F(X1) ENDIF 10  ENDIF IF  (F1.LT.F2)  the  function  minimum  evaluation  occurs is  and F(CX)  TOL  GOTO  the  (A-H, O-Z)  F(AX)  CX  IF  of  l i s t :  :  IF  value  occurs.  X0=AX  10  find  (R=0.61803399,C=l.DO-R)  AX  IF  to  subroutine  GOLDEN(AX,BX,CX,F,TOL,XMIN,FMIN)  DOUBLE  PARAMETER  c c c c c c c c c  of  minimum v a l u e  SUBROUTINE IMPLICIT  GOLDEN  THEN  155  THEN  the  minimum  occurs  Appendix A: Time Temperature Superposition Program Code  FMIN=F1 XMIN=X1 ELSE FMIN=F2 XMIN=X2 ENDIF RETURN END C  The  subroutine  LINREG performs  a  linear  regression  of  the  equation  and  form:  C C  Y =  B * X + A  C C  where  A  C  slope  of  C  R,  also  is  is  the  the  In  this  of  the  linearized  subroutine,  (A-H,0-Z)  Argument  C  X  :  A vector  of  independent  C  Y  :  A vector  of  dependent  C  N  :  Number  C  A  :  Y-intercept  C  B  :  The  slope  C  R  :  The  correlation  l i s t :  of  data of  of  variable variable  points the  the  linear  line  line coefficient  of  XBAR=0.DO YBAR=0.DO 1=1, N  XBAR=XBAR+X(I) YBAR=YBAR+Y (I) 10  CONTINUE X BAR=X B A R / N YBAR=YBAR/N SX=0.D0 SY=0.D0 SUMXY=0.DO DO 2 0  1=1,N  SX=SX+(X(I)-XBAR)**2 SY=SY+(Y(I)-YBAR)**2 SUMXY=SUMXY+(X(I)-XBAR)*(Y(I)-YBAR) 20  B is  CONTINUE B=SUMXY/SX A= Y B A R - B * X BAR SX=SQRT(SX/(N-l)) SY=SQRT(SY/(N-l))  156  the  linearized  the  coefficient,  X(10),Y(10)  C  10  correlation  LINREG(X,Y,N,A,B,R)  DOUBLE PRECISION  DIMENSION  DO  the  calculated.  SUBROUTINE IMPLICIT  Y-intercept  line.  fit  Appendix A: Time Temperature Superposition Program Code  PS=O.DO DO 3 0  1=1,N  XI=(X(I)-XBAR)/SX YI=(Y(I)-YBAR)/SY PS=PS+XI*YI 30  CONTINUE R=PS/(N-1) RETURN END  C  The  C  variable  subroutine  C  to  sort  using sets  SUBROUTINE IMPLICIT  SORT the  of  sorts heap  data  of  a  sort  set  of  data  method.  exactly  three  RA(N),  C  Argument  C  N  :  Number  C C  RA RB  : :  The The  PRECISION  order has  of  been  one written  variables:  (A-H,0-Z)  RB(N)  list: of  data  points  to  be  IR=N IF(L.GT.1)THEN L=L-1 RRA=RA(L) RRB=RB(L) ELSE RRA=RA(IR) RRB=RB(IR) RA(IR) =RA(1) RB(IR)=RB(1) IR=IR-1 IF(IR.EQ.1)THEN RA(1)=RRA RB ( 1 ) = R R B GOTO  sorted  f i r s t v a r i a b l e according to second v a r i a b l e of the data  L=N/2+l  30  ENDIF ENDIF C I=L J=L+L 20  increasing subroutine  SORT(N,RA,RB)  DOUBLE  DIMENSION  10  in  This  IF(J.LE.IR)THEN IF(J.LT.IR)THEN IF(RA(J).LT.RA(J+1))J=J+1 ENDIF  157  which set  data  will  be  sorted  Appendix A: Time Temperature Superposition Program Code  IF(RRA.LT.RA(J))THEN RA(I)=RA(J) RB(I)=RB(J) I=J J=J+J ELSE J=IR+1 ENDIF GOTO  20  ENDIF RA(I)=RRA RB(I)=RRB GOTO 30  10  RETURN END  C  The  C  i n  subroutine  C  elimination  the  form  SUBROUTINE IMPLICIT  c c c c c c c c c c c c  an  solves  augmented  a  set  of  algebraic  coefficient  GAUSSJ  DOUBLE  (A,  N,  NDR, N D C , X ,  PRECISION  A(NDR,NDC),  Argument  l i s t : array  X(N),  An  Number  of  equations  NDR  Number  of  rows  NDC  Number  of  columns  X  Output  array  are  of  coefficients  .  Measure of  IERROR  :  An indication  RNORM,  for  i n  of  a  DO  20  I  DO  10  =  of  =  1  ==>  IERROR  =  2  ==>'  new 1,  for  'A'  which  the  values  the  vector  (  success  10  to  find  working  matrix  ' B ' :  1, =  NP A(I,J)  CONTINUE  20  CONTINUE  C  Elimination DO  80  K =  KP  =  1,  K +  matrix  the  of  number o f  the  steps  to  be  [C] of  -  [A]  solving  successful fails  N  =  B(I,J)  to  unknowns  'A'  residual  IERROR  J  IERROR)  augmented  (equivalent  Begin:  Defining  equations  Gauss-Jordan  solution  vector  stored  RNORM  1  the  B(50,51)  N  N +  simulteneous using  (A-H, O-Z)  A  =  matrix  method:  DIMENSION  NP C  GAUSSJ  of  done  'N'  times:  N 1  158  solution  *  [X] the  } equations  Appendix A: Time Temperature Superposition Program Code  BIGS C  = O.DO  When K = IF  N,  (K  C  For each  C  step  . E Q . N) row  40  I  BIG Find  the  search  GOTO  for  pivot  coefficient:  55  beginning  = =  K,  from  the  ' K ' t h row  where  K is  the  elimination  DABS  30  J  BIG 30  N (B(I,K))  largest  DO  coefficients  = =  along  the  row:  KP, N DMAX1  (BIG,  DABS(B(I,J)))  CONTINUE  C  Then  C  by  C  for  find  the  dividing that S  =  DABS  C  Among t h e s e Scale  (B(I,K)  equations  =  biggest IF  40  factor,  (I,I)th  S,  corresponding  coefficient  to  each  by  the  largest  Kth to  the  Nth),  row  (equation)  coefficient  /  BIG)  (from  the  find  the  biggest  factor: BIGS  The  scale  the  row:  C  C  the  number,  DO  C  skip  DMAX1 scale  (BIGS  (BIGS,S) factor  . E Q . S)  should  IPIVOT  =  correspond  to  the  pivot  equation:  I  CONTINUE  C  If  C  the  C  at  the  pivot  equation  elimination which IF  the  step  pivot  not  equation  (IPIVOT  . N E . K)  DO  J  =  K,  TEMP  =  B(IPIVOT,J)  50  B(K,J)  =  the  same  then is  as  the  exchange  equation  row  corresponding  'K' with  the  to  row  located:  THEN  NP  B(IPIVOT,J) 50  is  number,  =  B(K,J)  TEMP  CONTINUE ENDIF  C  If  C  program with  55  at  IF  least  one  (B(K,K) IERROR  of  ' f a i l '  the  diagonal  . E Q . O.DO) =  elements  message: THEN  2  RETURN ENDIF  159  is  zero,  then  return  to  main  Appendix A: Time Temperature Superposition Program Code  C  Otherwise,  C  above  C C  below  the  elimination  the  diagonal  process,  elements  eliminating  simultaneously  coefficients (see  attached  explanations): Start DO  C  start  and  But  from  the  70  =  I  1,  skipping IF  first  pivot  . N E . K)  QUOT  =  last  THEN /  B(K,K)  = O.DO  60  J  =  K P , NP  B(I,J) 60  the  row:  B(I,K)  B(I,K) DO  to  N  the  (I  row  =  B(I,J)  -  QUOT  *  B(K,J)  CONTINUE ENDIF  70  CONTINUE  80  CONTINUE  C  Return  C  completion IF  with  error of  (B(N,N)  message  a l l  . E Q . O.DO)  IERROR  =  i f  the  elimination  last  steps)  diagonal equals  element  (after  the  zero:  THEN  2  RETURN ENDIF C  Otherwise,  C  coefficients  by  DO  N  90  start  finding the  I  =  1,  X(I)  =  B(I,NP)  90  CONTINUE  C  Calculation RSQ DO  of  /  the  solution  corresponding  vector  elements  of  by  dividing  the  B(I,I)  norm of  residual  vector:  = O.DO 110  I  =  1,  N  SUM = O . D O DO  100  J  SUM = 100  N *  X(J)  =  RSQ +  (A(I,NP)  -  SUM)  **  2  CONTINUE RNORM =  C  1,  SUM + A ( I , J )  CONTINUE RSQ  110  =  If  SQRT  everything  IERROR  =  (RSQ) goes  well,  return  with  1  RETURN END  160  a  'success'  the  diagonal  message:  R.H.S.  matrix:  Appendix A: Time Temperature Superposition Program Code  C  The  C  a model  subroutine using  C  the  of  sum  C  Please  C  solve  C  only  C  calculated  two  that  a  returns  of  in  model  this  of  optimum value  method  differences case  only  parameters. more  the  Gauss-Newton  squares  note for  GAUSSN the  the  two  Also,  (up  parameters by  to  a  tolerance):  specified  subroutine will  the  optimization  parameters  this  of  of  has  since  allow  been  the  the  of  minimizing  modified  WLF model  condition  to involves  number  to  be  easily.  SUBROUTINE 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 ) IMPLICIT  DOUBLE PRECISION  DIMENSION  (A-H,0-Z)  X(M),Y(M),A(2),DA(2),AA(2),AGUESS(2),ALPHA(2,3)  EXTERNAL AMODEL,DMODEL DATA  Z,N/1.D0,2/  C  Argument  C  X  :  List: A  set  of  independent  C  Y  :  A  set  of  dependent  C  AGUESS  :  A vector  C  M  :  The  C  TOL  :  A  C  AMODEL  :  The  C  DMODEL  :  A  c  function  A  :  A vector  SNEW  :  The  Initialization 10  of  the of  data  to  the  best  best  parameters the  are  derivative  to  be  value  estimated of  the  model  parameters  estimated sum  of  parameter  squares  estimated  parameter  values points  criteria  which  returns  corresponding  using  DO  model  that  respect  C  parameter  available  convergence  specific  C  C  i n i t i a l of  specified  with  C  of  number  variables variables  of  parameter  values differences  as  calculated  values  values:  1=1,N  A(I)=AGUESS(I) 10  CONTINUE  C  Calculation  C  values  of  of  the  sum  of  squares  of  differences  using  the  i n i t i a l  A(I)  SOLD=S(AMODEL,X,Y,M,A,N) C  Setting  up  C  for  increment  20  DO 6 0  the  DO  the  augmented in  coefficient  parameter  matrix,  values,  ALPHA,  to  be  solved  DA(I):  1=1,N 40  J=1,N  ALPHA(I,J)=0.DO DO  30  K=1,M  ALPHA(I,J)=ALPHA(I,J)+DMODEL(A,X(K),I)*DMODEL(A,X(K),J) 30  CONTINUE IF  40  (J.NE.I)  ALPHA(J,I)=ALPHA(I,  CONTINUE  161  J)  Appendix A: Time Temperature Superposition Program Code  ALPHA DO  50  (I,N+1)=0.D0  K=1,M  ALPHA(I,N+l)=ALPHA(I,N+l)+ +  DMODEL(A,X(K),I)*(Y(K)-AMODEL(X(K),A))  50  CONTINUE  60  CONTINUE  C  Calculation  of  the  Eigenvalues  of  the  coefficient  matrix,  ALPHA:  B=-(ALPHA(1,1)+ALPHA(2,2)) C=ALPHA(1,1) *ALPHA(2,2)-ALPHA(1,2)  *ALPHA(2,1)  EIGEN1=(-B+SQRT(B**2-4*C))/2 EIGEN2=(-B-SQRT(B**2-4*C))/2 C  Initializing  the  constant  for Marquardt's  modification:  GAMMA=0.DO C  Calculation  70  the  condition  number:  COND=DABS((DMAX1(EIGEN1,EIGEN2)+GAMMA)/ +  C  of  (DMIN1(EIGEN1,EIGEN2)+GAMMA)) To perform IF  Marquardt's  (COND.GT.1.D3)  modification  i f  the  solve  for  condition  number  THEN  GAMMA=GAMMA+1.Dl DO  80  1=1,N  A L P H A ( I , I ) =ALPHA ( I , I)+GAMMA 80  CONTINUE GOTO  70  ENDIF C  Calling  the  subroutine  GAUSSJ to  DA(I):  CALL GAUSSJ(ALPHA,N,N,N+l,DA,RNORM,IERROR) C  Checking IF  for  singularity:  (IERROR.EQ.2)  THEN  PRINT*,"WRN  Program  -  fails  to  determine  STOP ENDIF ERR=0.DO C  Checking DO  90  for  convergence:  1=1,N  ERR=DMAX1(ERR,DABS(DA(I)/A(I)))  162  parameters!!'  is  large:  Appendix A: Time Temperature Superposition Program Code  90  CONTINUE IF  (ERR.LE.TOL)  C  Updating  100  DO 1 1 0  the  GOTO  130  parameter  values  by  Z *  DA(I):  1=1,N  AA(I)=A(I)+(Z*DA(I) ) 110  CONTINUE SNEW=S(AMODEL,X,Y,M,AA,N)  C C  Where  Z is  a  real  number  that  reduces  the  sum  of  squares  of  differences: IF  (SNEW.GT.SOLD)  THEN  Z=Z*5.D-1 GOTO  100  ELSE DO  120  1=1,N  A(I)=AA(I) 120  CONTINUE SOLD=SNEW GOTO  20  ENDIF 130  RETURN END  C  The  C  of  following  C  from  function  differences the  DOUBLE  between  the  value  calculated  of  data  the and  sum the  of  squares  data  obtained  model:  PRECISION  IMPLICIT  returns the  FUNCTION  S(AMODEL,X,Y,M,A,N)  DOUBLE PRECISION  DIMENSION  (A-H,0-Z)  X(M),Y(M),A(N)  S=0.D0 DO  10  K=1,M  S=S+(Y(K)-AMODEL(X(K) ,A) 10  )**2  CONTINUE RETURN END  C  The  C  calculated  following  DOUBLE  function  using  PRECISION  IMPLICIT DIMENSION  the  returns  the  value  WLF m o d e l :  FUNCTION  AMODEL(X,A)  DOUBLE PRECISION  (A-H,0-Z)  A(2)  COMMON/BLKF/TREF D E L T = X - ( T R E F + 2 7 3 . 15) AMODEL=A(l)*DELT/(A(2)+DELT)  163  of  ln(AT(i))  as  Appendix A: Time Temperature Superposition Program Code  RETURN END C  The  C  WLF model  following  DOUBLE  is  with  a  PRECISION  IMPLICIT  DOUBLE  DIMENSION  function  respect  to  FUNCTION  that  the  calculates  parameters  DMODEL(A,X,  PRECISION  the  A(l)  derivative  or  of  the  A(2):  I)  (A-H,0-Z)  A(2)  COMMON/BLKF/TREF DELT=X-(TREF+273.15) GOTO 10  (10,20),  I  DMODEL=DELT/(A(2)+DELT) RETURN  20  DMODEL=-A(l)*DELT/((A(2)+DELT)**2) RETURN END  C  The  C  to  following be  DOUBLE  done  a  DOUBLE  DIMENSION  function  the  PRECISION  IMPLICIT  IF  is  upon  that  availability  FUNCTION PRECISION  allows  i n t e r p o l a t i o n .between  of  parameters  F(X,Y,Q,R,S,N,  Q,  R,  points  S:  Z)  (A-H,0-Z)  X(300),Y(300),Q(300),R(301)  (Z.LT.X(l))  the  , S(300)  THEN  1=1 WRITE ELSEIF  (9,20)  Z  (Z.GT.X(N))  THEN  I=N-1 WRITE  (9,20)  Z  ELSE C C  The  bisection  method  is  used  to  find  the  location  of  the  point  interest: 1=1 J=N  10  K=INT((I+J)/2) IF  (Z.LT.X(K))  IF  (Z.GE.X(K))  IF  (J.GT.I+1)  J=K I=K GOTO  10  ENDIF DX=Z-X(I) F=Y(I)+DX*(Q(I)+DX*(R(I)+DX*S(I))) 20  FORMAT  (/'Warning  -  ',D10.3,'  is  outside  RETURN END  164  interpolation  range'/)  of  Appendix A: Time Temperature Superposition Program Code  C  The following  is  C  to  master  obtain  DOUBLE  the  PRECISION  IMPLICIT  the  objective  function  that  is  to  be  minimized  curve:  FUNCTION  OBJ(X)  DOUBLE PRECISION  COMMON/BLKA/Q(300),R(301)  (A-H,0-Z) ,S(300)  COMMON/BLKC/ALNSRT(300),ALNSTRS(300) COMMON/BLKD/REFSRT(300),REFSTRS(300) COMMON/BLKE/WT(300),MDATA,ITOTAL OBJ=0.DO DO  10  1=1,ITOTAL  C  The objective  function  C  of  values,  C  between  shear  IF  rate the  reference  should  only  for which curve  be  there  and the  calculated is  curve  a  for  common that  is  ((REFSTRS(I).GE.ALNSTRS(1)).AND.  +  (REFSTRS(I).LE.ALNSTRS(MDATA)))  THEN  OBJ=OBJ+(REFSRT(I)+  F(ALNSTRS,ALNSRT,Q,R,S,MDATA,REFSTRS(I))  +  -DLOG(X))**2*WT(I) ENDIF  10  CONTINUE RETURN END  *******************+*************^*****^ Sample  Input  File:  Data.txt: 12 12 12 180 200 220 c:/fortran/program/project/datal.txt c:/fortran/program/project/data2.txt c:/fortran/program/proj ect/data3.txt Resin  X,  Lot#  XX123-45600  Datal.txt: 4. 8871  36248  8.0000  44980  12.000  53784  20.945  67761  30.000  79699  62.834  107734  97.741  129545  165  a  certain  stress to  be  range  value shifted:  Appendix A: Time Temperature Superposition Program Code  202.46  173244  300.21  202583  495.69  243410  698.15  274380  872.69  294949  Sample Resin  Output X,  Lot#  RESULTING Rate  File: XX123-45600  MASTER  (1/s)  CURVE Stress  DATA (Pa)  POINTS: Viscosity  (Pa.s)  2.7900  31202  11184  3.5426  34618  9772  4.5671  38835  8503  4.8871  40129  8211  5.7990  43128  7437  6.8507  46379  6770  8.0000  49856  6232  8.6985  51552  5927  11.9573  58990  4933  12.0000  59517  4960  15.1826  65432  4310  17.1266  68688  4011  20.9450  75134  3587  21.7463  76319  3510  30.0000  87668  2922  35.8712  93375  2603  45.5470  103543  2273  55.7992  111925  2005  62.8340  117824  1875  70.8503  123646  1745  97.7410  140489  1437  115.5820  149881  1296  146.7588  164939  1123  171.3863  175495  1023  202.4600  186128  919  217.6156  191950  882  282.9835  212208  749  300.2100  215468  717  359.3148  231069  643  398.5655  241780  606  495.6900  258080  520  498.2083  263048  527  506.0736  261728  517  632.5938  283384  447  698.1500  289671  414  872.6900  309308  354  SUMMARY O F R E S U L T S :  166  Appendix A: Time Temperature Superposition Program Code  The  master  curve  The  shift  The  corresponding  factor  The  shift  The  corresponding  The  shift  The  corresponding  Using The  factor factor  simple  was  calculated  corresponding value  the  corresponding value  of  the  corresponding value  thermally  corresponding  of  of  the  based to to  model,  of of  function Ea=  correlation  of  function  temperature  objective  reference  function  temperature  objective to  the  temperature  objective  activated  regression  on  180.0 is: is:  1.0000  C is:  0.7249  .0019  220.0 is:  26.0760  C is: .0000  200.0  coefficient  167  temperature  C is:  0.5709  .0113 kJ/mol is  R=  .9982  of  180.0  C  Appendix B: GPC Analysis Program Code  APPENDLX B: G P C Analysis Program Code  168  Appendix B: GPC A nalysis Program Code  ********************************************* *  *  *  Alfonsius  *  Department  *  The U n i v e r s i t y  *  of  Budi  Ariawan  Chemical of  British  Vancouver,  *  Engineering  *  Columbia  *  B.C.  *  *  *  *  April  *  15,  1998  *  SUMMARY:  *  This  program inputs  GPC p r o f i l e s  *  each  profile  number  *  is  *  corresponding  *  Newton-Cotes  *  * *  then  into  fitted  a  to to  splines each  of of  of  several  different fitted  MW g r o u p  is  ends  polymers  MW g r o u p s . and  the  calculated  divides  *  Each  profile  *  area  using  method.  and  4-panel  * * *  * * ************************************************************************ PROGRAM G P C IMPLICIT  DOUBLE  PRECISION  DIMENSION ALOGMW(800), DIMENSION  X(20),  CHARACTER  *  20  RESIN  *  40  FNAME(30)  CHARACTER EXTERNAL DATA C  DO  EPS /  l.D-6  I  =  READ  CONTINUE  20  CLOSE  C  Prompting  1,  30 ERR=20)  FNAME(I)  I  for  more  information:  200 *,  PRINT  UPPER 210  *,  NP  N +  Working  X(NP)  STATUS='OLD  ALOWER  READ  X(l)  =  file:  190 *,  PRINT  C  from input  (3)  PRINT  =  /  (3,'(A)',  10  READ  21)  FILE='GPCDAT(all) .TXT',  NRESIN  READ  AREA(30,20,  information  (3,  10  0-Z)  F  Reading OPEN  (A-H,  CONC(800)  =  N 1 i n  Log  DLOG10 =  DLOG10  scale: (ALOWER) (UPPER)  169  1  )  Appendix B: GPC Analysis Program Code  C  Calculating DELTA  C  =  the  (X(NP)  Calculating DO  I  = 2,  IM  =  I  X(l))  -  C  Initialization =  DO  40  1, J  of  molecular  weight  DELTA  area  =  1,  under  each  slice:  NP =  0.D0  CONTINUE  50  CONTINUE  C  Opening OPEN(5, WRITE  the  output  file:  FILE='GPC.TXT') (5,150)  K=l 60  IF  (K.LE.N)  THEN  KP=K+1 WRITE  (5,170)  WRITE  (5,160)  DO  70  J  =  WRITE 70  X(K)  CONTINUE WRITE  (5,230)  WRITE  (5,160)  DO  80  J  = K P , NP  WRITE 80  K, N (5,230)  (5,230)  weight  range:  N  NRESIN  AREA(I,J,J) 40  molecular  1  CONTINUE  I  each  /  various  = X(IM) +  50  of  N  30  DO  -  the  30  X(I)  width  X(J)  CONTINUE  170  limits:  A ppendix B: GPC A nalysis Program Code  WRITE DO  (5,180)  130  J  =  1,  NRESIN  OPEN  (7,  FILE=FNAME(J),  READ  (7,  *)  DO  90  I  =  1,  1000  READ ( 7 , * , E R R = 1 0 0 ) NDATA = I 90  WRITE  (5,230)  WRITE  (5,220)  IF  CONC(I)  (K.EQ.l)  RESIN  THEN  NDATAM = NDATA C  Fitting  the  MWD c u r v e  CALL  C  Integrating  C  molecular  SPLINE  C  Total  the  -  1  with  spline:  (ALOGMW, C O N C , 0.D0, 0.D0)  +  NDATA,  fitted  curve  corresponding  slice  using  4-panel  weight  NDATAM,  to  the  Newton-cotes  3,  C A L L ADNC  Calculation DO  of  (4, F , ALOGMW(1), TOTAL, NPOINT)  normalized  110  J J = 2,  appropriate method:  CALL  ALOGMW(NDATA), E P S ,  area:  NP  ADNC(4,F,X(1),X(JJ),EPS,SUM,NPOINT)  AREA(J,1,JJ) WRITE  (5,230)  =  SUM /  TOTAL  AREA(J,1,JJ)  CONTINUE ELSE  C  Calculation DO  of 120  other J J =  intermediate  WRITE  slices:  K P , NP  AREA(J,K,JJ)  120  3,  area:  +  110  ALOGMW(I),  CONTINUE  100  C  STATUS='OLD')  RESIN  (5,230)  = AREA(J, 1 , J J ) AREA(J,K,JJ)  CONTINUE ENDIF  171  -AREA(J,1,K)  A ppendix B: GPC Analysis Program Code  130  CONTINUE CLOSE  (7)  K = K + 1 GOTO  60  ENDIF 140  CLOSE  (5)  C  FORMAT  statements:  150  FORMAT  (3X,  160  FORMAT  170  FORMAT  (5X,'*',5X,\) (/F10.4\)  180  FORMAT  (F10.4/)  190  FORMAT  (lX,'Enter  200  FORMAT  (/IX,"Enter  the  absolute  210  FORMAT  (/IX,'Enter  the  number o f  220  FORMAT  (1X,A10,\)  230  FORMAT  (F10.4,\)  'Resin',45X,'MW  the  Limits  absolute  (in  lower  Log)'\)  limit  upper  for  limit  MW g r o u p s  integration:  for  integration:  desired:  ',\)  STOP END C  Subroutine  SPLINE  begins:  SUBROUTINE  SPLINE  (X,  IMPLICIT  Y,  N,  DOUBLE PRECISION  COMMON  /BLKB/  XX(801),  COMMON  /BLKC/  Q(800),  DIMENSION  X(N),  DIMENSION  A(801),  NM, I I ,  YY(801), R(801),  Y(N),  N N , NNM  H(800)  B(801),  C(801),  Argument  List:  X  :  An  array  of  independent  y  :  An  array  of  dependent  C  A s s i g n i n g dummy v a r i a b l e  :  Total  NM  :  Number  11  :  Boundary  IN  Gl,  :  GN:  number o f  NNM  =  variables  variables  available  condition  Natural  2.  Clamped  3.  Fitted  Boundary  D(801)  data  points  intervals  1.  condition  1.  Natural  2.  Clamped  3.  Fitted  Derivative (needed  NN  of  G l , GN)  S(800)  c c c c c c c c c c c c c c c  N  IN,  (A-H,0-Z)  values  only  i f  at  X(l):  at  X(n)  and X ( n ) ,  respectively  'clamped'  at  X(l)  spline  desired)  values  be  to  N = NM  172  used  is in  COMMON  block:  • A ) ',\)  Appendix B: GPC A nalysis Program Code  DO  5  5  I  =  1,  N  XX(I)  =  X(I)  YY(I)  =  Y(I)  CONTINUE  C  If  C  polynomial  'fitted'  C  point(s): IF  (II AA  . E Q . 3) =  DO  spline  desired,  then  method  to  use  Langrange  determine  condition  THEN  0.DO I  =  1,  TERM  40  =  Y(I)  DO  30  J  IF 30  is  interpolation  4  =  (J  1,  4  . N E . I)  TERM  =  TERM  /  (X(I)-X(J))  CONTINUE AA  40  = AA +  TERM  CONTINUE ENDIF IF  (IN M  . E Q . 3)  =  BB  N =  DO  THEN  3  0.D0 I  =  M, N  TERM  60  =  Y(I)  DO  50  J  IF 50  = M, N  (J.NE.I)  TERM  =  TERM  /  (X(I)-X(J))  CONTINUE BB  60  =  BB +  TERM  CONTINUE ENDIF  C  Calculating DO  70  I  =  1,  H(I)  =  X(I+1)  70  CONTINUE  C  Calculating A(l) IF  interval  = (II B(l)  size:  NM  the  -  X(I)  coefficients  for  the  0.D0 . E Q . 1) =  THEN  1.D0  C(l)  =  0.D0  D(l)  =  0.D0  173  tridiagonal  set:  at  the  end  Appendix B: GPC Analysis Program Code  ELSEIF  (II  . E Q . 2)  B(l)  =  2.DO  C(l)  =  H(l)  D(l)  =  3.DO  THEN  *  H(l)  +  ((Y(2)-Y(D)  /  H(l)  -  Gl)  ELSE B(l)  =  -H(l)  C(l)  =  H(l)  D(l)  =  3.DO  *  H(l)  * H(l)  * AA  ENDIF DO  80  80  I  =  2,  IM  =  I  -  NM  A(I)  =  H(IM)  B(I)  =  2.DO  C(I)  =  H(I)  D(I)  =  3.DO  1 *  (H(IM)+H(I))  *  ( (Y(I+1)-Y(I))/H(I)  -  (Y(I)-Y(IM))/H(IM))  CONTINUE C(N)=0.D0 IF  (IN  . E Q . 1)  A(N)  = O.DO  B(N)  =  D(N)  = O.DO  ELSEIF  THEN  1.D0  (IN  . E Q . 2)  A(N)  =  H (NM)  B(N)  =  2.DO  D(N)  =  -3.DO  THEN  * H(NM) *  ((Y(N)-Y(NM))  /  H(NM)  *  BB  -  GN)  ELSE A(N)  = H(NM)  B(N)  =  -H(NM)  D(N)  =  -3.DO  *  H(NM)  *  H(NM)  ENDIF C  C a l l i n g Thomas CALL  TDMA  (A,  A l g o r i t h m to B,  C  Determining  C  passing  through  DO  D,  R,  N,  coefficients  of  each  data  pair  of  90  I  =  1,  =  I  +  =  (Y(IP)-Y(I))  /  H(I)  (2.DO  +  R(IP))  /  (3.DO  Q(I) S(I)  =  for  the  tridiagonal  NM)  IP + 90  the  C,  solve  the  cubic  polynomials  points:  NM 1 *  R(I)  (R(IP)-R(I))  -  H(I) /  *  3.DO  *  H(I))  CONTINUE RETURN END  C  Subroutine  C  This  C  Algorithm:  TDMA  subroutine  SUBROUTINE  TDMA  begins: solves  tridiagonal  matrix  (A, B , C , D , X , N , N M )  174  using  Thomas  set:  A ppendix B: GPC A nalysis Program Code  IMPLICIT  DOUBLE PRECISION  DIMENSION P(l)  =  -C(l)  Q(l)  =  D(l)  DO  10  /  I  =  2,  I  -  B(N),  (A-H, O-Z)  C(N),  D(N),  X(N),  P(801),  Q(801)  B(l)  /  IM = DEN  10  A(N),  B(l) N  1  = A(I)  *  P(IM)  +  P(I)  =  -C(I)  /  DEN  Q(I)  =  (D(I)  -  A(I)  B(I) *  Q(IM))  /  DEN  CONTINUE X(N) DO  =  20  Q(N) I  X(I) 20  =  N M , 1,  =  P(I)  -1  *  X(I+1)  +  Q(I)  CONTINUE RETURN END  C  S u b r o u t i n e ADNC SUBROUTINE IMPLICIT  begins:  ADNC(N,F,A,B,EPS,AREA,NPOINT)  DOUBLE PRECISION  DIMENSION  H(20),  DIMENSION  FF(20,20),  C  The  C  various  following  TOL(20), C(6),  (A-H, O-Z) SR(20),  DATA s t a t e m e n t  integration  XR(20)  BB(6,7) defines  the  constants  DATA 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 +  for  +  1.9D1,4.1D1,1.D0,4.D0,3.D0,3.2D1,7.5D1,2.16D2,0.D0,  +  1.D0,3.D0,1.2D1,5.0D1,2.7D1,2*0.D0,1.DO,3.2D1,5.0D1,  +  2.72D2,3*0.D0,7.D0,7.5D1,2.7D1,4*0.D0,  1.9D1,2.16D2,  5*0.D0,4.1D1/ Defining DATA MAXL  C  Argument  C  N  :  C  the  maximum a d a p t i v e  level  allowed:  N =  for  /20/ List: 'Panel' rule,  number  (eg.  2  etc.)  C  F  The  C  A  Lower  limit  of  integration  C  B  Upper  limit  of  integration  C  EPS  Convergence  C  AREA  Result  of  integration  C  NPOINT  Number  of  function  Initialization AREA S  =  =  function  of  the  4 44 44 4 4 4 4 4 4 4 4 D - 2 ,  1.73611111111111D-2,7.14285714285714D-3,3*1.DO,7.DO,  + C  needed  methods:  to  be  integrated  criteria evaluations  variables:  0.D0  0.D0  175  adaptive  Simpson  Appendix B: GPC A nalysis Program Code  C  I n i t i a l  number  NPOINT C  function  =  N +  1  Defining  the  original  XI  evaluations:  integration  interval:  A  =  XR(1) C C  of  =  B  D e f i n i n g the l e n g t h o f i n t e r v a l and the convergence c r i t e r i a each adaptive l e v e l , up t o t h e maximum l e v e l allowed: H(l) IF  =  (B-A)  /  (M0D(N,2) NR  N  . E Q . O.DO)  =  N +  2  =  N  1  THEN  ELSE NR  +  ENDIF IF  (MOD(INT(N/2.DO),2) RN  =  2  **  NR -  4  =  2  **  NR -  6  . E Q . O.DO)  THEN  ELSE RN ENDIF T0L(1) DO  =  RN *  10  I  =  2,  IM  =  I  -  =  H(IM)  H(I)  TOL(I) CONTINUE  C  Calculation  J  =  N  =  /  2.DO  TOL(IM)  /  of  over  the  (1-1)  *  area  2.DO  original  integration  1  -1.D0  DO 2 0 J IF  +  MAXL 1  =  10  NP  EPS  I =  (I  = J  1, +  NP  2  . N E . NP)  FF(1,  J)  =  THEN F(A +  H(l))  ELSE FF(1,  J)  =  F(B)  +  BB(N,I)  ENDIF S 20  S C  S  =  S  *  Defining L  C  =  *  FF(1,J)  CONTINUE  =  C(N)  *  H(l)  the  first  the  number  level:  1  Updating  of  function  evaluations:  .176  interval:  of  A ppendix B: GPC A nalysis Program Code  30  NPOINT DO  40 C  40  = I  NPOINT 1,  N  FF(L,2*I)  =  =  F(X1+(I*2-1)*H(L)/2.D0)  CONTINUE C a l c u l a t i n g the  C  (each  C  interval): SL  =  interval  DO  areas being  under half  the  the  two  size  new of  =  previous  intervals integration  0.D0  50  I  =  SL  =  SL +  SR(L)  1, =  NP BB(N,I)  SR(L)  +  *  FF(L,I)  BB(N,I)  *  FF(L,I+N)  CONTINUE SL  =  SL *  SR(L)  =  If  C  calculated IF  the  H(L)  SR(L)  C  sum  * *  of  C(N) H(L)  the  for  L  are  previous  /  2.DO  significantly interval  .GT. TOL(L))  different  from  the  area  s i z e . . . .  THEN  L L +  1  C  ...and  the  level  C  move  C  integration  (L to  DO  is  s t i l l  . L E . MAXL) the  S  =  left level  smaller  than  of  and  the  define  integration a  new  I  =  1,  NP =  F F ( L M , I)  CONTINUE XR(L)  =  GOTO  30  XI  + N * H(L)  ELSE Otherwise,  integration  WRITE  (5,90)  maximum p o s s i b l e  has  interval,  integration  SL 60  the  level,  THEN  half  FF(L,2*1-1) 60  2.DO C(N)  areas  the  =  IF  / *  (DABS(SL+SR(L)-S) LM =  C  integration  the  0.D0  SR(L)  50  + N  failed:  XI  RETURN ENDIF ELSE  177  increase  interval:  the  Appendix B: GPC Analysis Program Code  C  If  C  for  the to  sum o f  C  the  sum  the  the  into  the  AREA  = AREA  XI  XI  C  =  move DO the  80  areas  previous  are  total  +  I  the  =  right  L,  1,  difference  and  s m a l l e r than h a l f of the l e v e l has been found:  =  level  DO  is  (Ii . EQ.  LM = S  find  the  correct  level  to  go  to:  and XR(L) i s  closed  to  minimum i n t e r v a l  XI  size),  then  . L T . H(MAXL)/2.D0)  zero the  (or correct  THEN  I  f i r s t IF  =  L -  found,  1)  then  integration  is  done:  RETURN  1  SR(LM) 70  J  =  FF(L,  C  between  (DABS(Xl-XR(I)) L  70  calculated  incorporate  -1  If  the  area  then  SL + SR(L)  C C  If  the  area...  C  C  to  interval,  + N * H(L) to  IF  closed  integral  1,  NP  2*J-1)  =  FF(LM,J+N)  CONTINUE Otherwise,  continue...  GOTO  30  ENDIF 80  CONTINUE ENDIF  C  FORMAT  90  FORMAT  statement: (/IX,"WARNING  -  Integration  fails  beyond  x  =",D10.3)  RETURN END C  The  following  C  the  spline  DOUBLE  PRECISION  IMPLICIT  C C  function  returns  an  interpolated  value,  knowing  coefficients:  DOUBLE  FUNCTION PRECISION  F(Z) (A-H,0-Z)  COMMON  /BLKB/  X(801),  Y(801),  COMMON  /BLKC/  Q(800),  R(801),  S(800)  i f  tries  P r i n t warning message interpolation region:  user  N , NM  178  to  interpolate  outside  the  Appendix B: GPC Analysis Program Code  IF  (Z 1  . L T .X(l)) =  PRINT ELSEIF I  THEN  1 20,  (Z.  Z  GT. X(N))  THEN  = NM  PRINT  20,  Z  ELSE C C  Otherwise, use bisection point of interest:  10  1  =  J  = N  K =  method  to  determine  the  location  of  the  1 INT {(I+J)  /  2)  IF  (Z  . L T .  X(K) )  J  = K  IF  (Z  . G E .  X(K)) I  = K  IF  (J  . G T . 1+1)  GOTO  10  ENDIF C  Calculate DX  =  Z -  the  interpolated  value  using  cubic  spline  coefficients:  X(I)  F = Y ( I ) + DX * ( Q ( I ) + DX * ( R ( I ) + DX * S ( I ) ) ) 20  FORMAT  (/'Warning  -  ',D10.3,'  is  outside  interpolation  RETURN END *************************************************** Sample  Input  File:  GPCDAT(all).txt: a. t x t b. txt c. t x t  (Cont'd  t i l l  x.txt)  a.txt: Resin  A  6.66289  0.00026  6.65412  0.00053  6.64537  0.00072  6.63665  0.00113  (Cont'd  t i l l  end)  179  range'/)  Appendix B: GPC Analysis Program Code  Output  File:  Resin  MW  Limit(in  Log)  *  3.9542  3.9542  *  4.1542  4.3542  3. 9542 4. 5542  3. 9542 4 .7542  3. 9542 4. 9542  3. 9542 5. 1542  3. 9542 5. 3542  3. 9542 5. 5542  3. 9542 5. 7542  3. 9542 5. 9542  A B C D E F G H I J K L M N 0 P R S T U V W X  0.078 0.0833 0.082 0.0813 0.0919 0.0805 0.0834 0.0798 0.0789 0.0699 0.0745 0.0795 0.0931 0.0796 0.0823 0.0685 0.0866 0.0701 0.0576 0.0717 0.0426 0.0322 0.0578  0.1763 0.185 0.1887 0.1828 0.2042 0.1856 0.19 0.1804 0.1923 0.1657 0.1716 0.1836 0.2056 0.1776 0.1862 0.1617 0.1847 0.1677 0.1491 0.159 0.1126 0.0912 0.1514  0. 2934 0 .302 0. 3167 0. 3016 0. 3279 0. 3091 0. 3124 0. 2996 0. 3318 0. 2876 0. 2897 0. 3082 0. 3284 0. 2885 0. 3084 0. 2801 0. 2892 0. 2922 0.27 9 0. 2627 0. 2236 0.194 0. 2855  0. 4127 0. 4185 0. 4463 0. 4212 0 .445 0. 4312 0. 4317 0. 4236 0. 4728 0. 4219 0 .416 0. 4364 0. 4432 0. 3995 0. 4312 0. 4101 0. 3916 0. 4288 0. 4285 0. 3753 0. 3723 0. 3424 0. 4394  0. 5187 C .519 0. 5592 0. 5271 0. 5442 0. 5369 0. 5343 0. 5429 0. 6009 0. 5532 0. 5371 0. 5552 0. 5402 0. 5011 0. 5448 0. 5383 0. 4886 0 5627 0.574 0 4846 0 5394 0 5185 0 5837  0. 5998 0. 5942 0. 6426 0. 6089 0. 6177 0. 6178 0. 6123 0. 6437 0. 7026 0. 6605 0. 6364 0. 6487 0. 6103 0. 5826 0. 6333 0. 6456 0. 5719 0. 6727 0.693 0. 5767 0. 6856 0. 6772 0. 6951  0. 6595 0. 6482 0. 6998 0. 6685 0. 6723 0. 6806 0. 6712 0. 7211 0. 7784 0. 7377 0. 7119 0. 7167 C .663 0. 64 67 0. 7031 0. 7261 0. 6413 0. 7521 0. 7822 0. 6485 0. 7949 0. 7981 0. 7755  0. 7075 0. 6904 0. 7407 0. 7142 0. 7172 0. 7347 0. 7204 0. 7787 0.832 0. 7896 0. 7689 0. 7678 0. 7068 0. 7008 0. 7585 0.783 0. 6981 0. 8054 0. 8469 0. 7056 0. 87 0 8816 0 8356  0. 7476 C .725 0. 7709 0.75 0. 7536 0. 7792 0. 7617 0. 8152 0. 8644 0. 8194 0. 8082 0. 8042 0. 7456 0. 7465 0. 7973 0. 8172 0. 7398 0 8358 0. 8908 0. 7489 0.915 0 9336 0 8798  0. 0. 0. 0.  *  4.1542  4.1542  *  4.3542  4.5542  4. 1542 4. 7542  4 .1542 4 .9542  4 1542 5 1542  4 .1542 5 3542  4 .1542 5 5542  4 1542 5 7542  4 1542 5 9542  A B C D E F G H 1 J K L M N 0 P R S T U  0.0984 0.1017 0.1068 0.1015 0.1122 0.1051 0.1067 0.1006 0.1134 0.0958 0.0971 0.1041 0.1125 0.098 0.1039 0.0932 0.0981 0.0976 0.0915 0.0873  0.2154 0.2188 0.2347 0.2203 0.2359 0.2286 0.229 0.2198 0.2528 0.2177 0.2152 0.2287 0.2353 0.2089 0.2261 0.2117 0.2026 0.222 0.2214 0.191  0. 3348 0 3352 0 3644 0 3399 0 3531 0 3507 0 3484 0 3438 0 3939 0. 352 0 3416 0 3569 0 3501 0.32 0 3489 0 3416 0.305 0 .3586 0 .3709 0 .3036 0 .3298 0 .3102 0 .3815  0. 4407 0. 4 357 0. 4773 0. 4 4 58 0 4522 0 4564 0 4509 0 4631 0 5219 0 4833 0 4626 0 4756 0.447 0 4215 0 4625 0 4698 0.4 02 0 4926 0 5164 0 .4129 0 .4968 0 . 4863 0 .5259  0 0 0 0 0 0 0 0 0 0  5218 5109 5606 5276 5258 537 3 5289 5639 6236 5906 0.562 0 .5692 0 .5172 0.503 0.551 0 .5771 0 .4854 0 .6026 0 .6354 0.505 0.643 0.645 0 .6373  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  5816 5649 6178 5872 5804 6001 5879 6413 6995 6678 6374 6372 5698 5671 6208 6576 5547 0.682 0 .7246 0 .5769 0 .7524 0 .7659 0 .7177  0 0 0 0 0 0 0 0  6296 6071 6588 6329 6253 6542 6371 6989 0.753 0 7197 0 6944 0 6883 0 6137 0 6212 0 6762 0 7145 0 6116 0 .7353 0 .7893 0.634 0 .8275 0 .8495 0 .7778  0 6697 0 6417 0.689 0 6687 0 6617 0 6988 0 6783 0 7355 0 7855 0 7495 0 7337 0 7247 0 6524 0 6669 0.715 0 .7487 0 .6533 0 .7657 0 .8332 0 .6772 0 .8724 0 . 9015 0.822  0 0 0 0  *  4.3542  4.3542  *  4.5542  4.7542  4 .3542 4 . 9542  4 . 3542 5 .1542  4 .3542 5 . 3542  4 . 3542 5 . 5542  4 .3542 5 .7542  4 .3542 5 . 9542  A B C D E F G H 1 J  0.117 0.1171 0.1279 0.1189 0.1237 0.1235 0.1224 0.1191 0.1394 0.1219  0.2364 0.2336 0.2576 0.2384 0.2409 0.2456 0.2417 0.2432 0.2804 0.2562  0 . 3424 0.334 0 .3705 0 .3443 0.34 0 .3513 0 .3442 0 .3625 0 .4085 0 .3875  0 .4234 0 . 4092 0 .4539 0 .4261 0 .4136 0 .4322 0 .4223 0 .4633 0 .5102 0 . 4949  0 . 4832 0 .4633 0.511 0 . 4857 0 . 4682 0.495 0 .4812 0 .54 07 0 .5861 0 .5721  0 .5312 0 .5054 0.552 0 .5314 0 .5131 0 .5491 0 .5304 0 . 5983 0 .6396 0.624  0 .5713 0 . 5401 0 .5822 0 .5672 0 .5494 0 .5937 0 .5717 0 . 6348 0 .6721 0 .6538  0.604 0 .5698 0 .6057 0 .5959 0 .5768 0 .6272 0 .6038 0 .6541 0 .6894 0.67  V W X  0.0701 0.059 0.0936  0.181 0.1618 0.2277  180  7024 6715 7124 6973 0.689 0 7324 0 7105 0 7547 0 8028 0 7657 0 7598 0 .7507 0 .6859 0 .7063 0 .7354 0 .7675 0 .6823 0 .7819 0.862 0 .7095 0 .8956 0 . 9301 0 .8534  7804 7548 7944 7786 C .781 0. 8128 0. 7938 0. 8345 0. 8817 0. 8356 0. 8343 0. 8302 C .779 0. 7858 0. 8177 0.836 0. 7689 0.852 0. 9196 0. 7812 0. 9381 0. 9623 0. 9112  Appendix B: GPC Analysis Program Code  K L M N 0 P R S T U V W X  0.118 0.1245 0.1228 0.1109 0.1222 0.1185 0.1045 0.1244 0.13 0.1038 0.111 0.1028 0.1342  0.2444 0.2527 0.2376 0.2219 0.245 0.2484 0.2069 0.2611 0.2794 0.2163 0.2597 0.2512 0.288  *  4.5542  *  4.7542  A B C D E F G H 1 J K L M N 0 P R S T U V W X  0.1193 0.1165 0.1297 0.1195 0.1172 0.1221 0.1193 0.124 0.141 0.1343 0.1264 0.1282 0.1148 0.1111 0.1228 0.1299 0.1024 0.1366 0.1495 0.1126 0.1487 0.1484 0.1538  *  4.7542  *  4.9542  A B C D E F G H 1 J K L M N O P R S T U  0.106 0.1004 0.1129 0.1059 0.0991 0.1057 0.1025 0.1194 0.1281 0.1313 0.1211 0.1188 0.0969 0.1016 0.1136 0.1282 0.097 0.134 0.1455 0.1093  0.1871 0.1757 0.1963 0.1877 0.1727 0.1866 0.1806 0.2201 0.2298 0.2386 0.2204 0.2123 0.1671 0.183 0.202 0.2355 0.1804 0.244 0.2645 0.2014  4.9542 5.1542  4.9542 5.3542  V W X  0.1671 0.1761 0.1444  0.3655 0.3715 0.3345 0.3235 0.3586 0.3766 0.3039 0.395 0.425 0.3256 0.4267 0.4273 0.4324  0.4648 0.465 0.4047 0.4049 0.4471 0.4839 0.3872 0.505 0.5439 0.4178 0.573 0.586 0.5438  0. 0. 0. 0. 0. 0. 0. 0 0. 0. 0 0 0  5403 5331 4573 4691 5169 5644 4566 5844 6331 4896 6823 7069 6241  0. 5973 0. 5842 0. 5012 0. 5232 0. 5723 0. 6213 0. 5134 0. 6377 0. 6978 0. 54 67 0. 7574 0. 7904 0. 6842  0 0 0 0 0 0 0 0 0  4.5542  4.5542  4.9542  5.1542  4.5542 5.3542  4 5542 5 5542  4 .5542 5. 7542  4 5542 5 9542  0.2253 0.2169 0.2426 0.2254 0.2163 0.2278 0.2219 0.2434 0.2691 0.2656 0.2474 0.247 0.2117 0.2126 0.2364 0.2581 0.1994 0.2706 0.295 0.2219 0.3158 0.3245 0.2982  0.3064 0.2922 0.3259 0.3072 0.2899 0.3087 0.2999 0.3441 0.3708 0.373 0.3468 0.3405 0.2819 0.2941 0.3248 0.3654 0.2828 0.3806 0.414 0.314 0.462 0.4832 0.4096  0.3661 0.3462 0.3831 0.3669 0.3445 0.3715 0.3588 0.4216 0.4 4 67 0.4502 0.4222 0.4085 0.3345 0.3582 0.3946 0.4459 0.3521 0.46 0.5032 0.3858 0.5713 0.6041 0.49  0 0 0 0 0 0  4142 3883 4241 4125 3894 4256 0.408 0 4792 0 5002 0.502 0 4792 0 4597 0 3784 0 4123 0.45 0 5029 0 4089 0 5133 0 5678 0 4429 0 6465 0 6877 0.55  0. 4543 0.423 0. 4543 0. 4484 0. 4257 0. 4702 0. 4493 0. 5157 0. 5327 0. 5319 0. 5185 0.496 0 4171 0. 458 0. 4888 0.537 0 4507 0 5437 0 6118 0 4862 0 6914 0 7396 0 5942  0.487 0 4528 0 4777 0. 477 0 4531 0 5037 0 4814 0.535 0.55 0 5481 0 5446 0.522 0 4506 0 4973 0 5092 0 5558 0 4797 0 5599 0 .6406 0 .5185 0 .7145 0 7683 0 .6256  4.7542  4.7542  5.1542  5.3542  4.7542 5.5542  4 7542 5 7542  4 7542 5 9542  0.2468 0.2297 0.2534 0.2473 0.2273 0.2494 0.2395 0.2975 0.3056 0.3158 0.2958 0.2803 0.2197 0.2472 0.2718 0.316 0.2497 0.3234 0.3537 0.2732  0 0  0.4226 0.4557 0.3361  0.2948 0.2718 0.2944 0.293 0.2722 0.3035 0.2887 0.3551 0.3592 0.3677 0.3529 0.3315 0.2636 0.3013 0.3272 0.3729 0.3065 0.3767 0.4184 0.3303 0.4977 0.5393 0.3962  3349 3065 0 3246 0 3288 0 3086 0.348 0.33 0 .3917 0 .3916 0 . 3975 0 .3921 0 .3678 0 .3023 0.347 0.366 0 . 4071 0 . 3482 0 . 407 0 . 4623 0 . 3736 0 . 5427 0 .5912 0 . 4404  0 3676 0 3363 0 3481 0 3575 0.336 0 3816 0 3621 0 4109 0 4089 0 4137 0 4183 0 3938 0 3358 0 3863 0 3864 0 4259 0 3773 0 4232 0 .4911 0 . 4059 0 .5658 0 .6199 0 .4718  4.9542 5.5542  4.9542 5.7542  4.9542 5.9542  0.3133 0.3348 0.2558  181  6366 6206 5399 5689 6111 6555 5551 6681 7417 0.59 0 8024 0 8424 0 7284  0 0 0 0 0 0 0 0 0 0 0 0 0  6627 6465 5734 6082 6315 6743 5842 6843 7705 6222 8255 8711 7598  Appendix B: GPC Analysis Program Code  A B C D E F G H I J K L M N 0 P R S T U V W X  0.0811 0.0752 0.0834 0.0818 0.0736 0.0809 0.078 0.1007 0.1017 0.1073 0.0993 0.0935 0.0702 0.0815 0.0885 0.1073 0.0833 0.11 0.1189 0.0921 0.1462 0.1587 0.1114  0 1408 0 1293 0 1405 0 1414 0 1282 0 1437 0.137 0. 1782 0 1776 0. 1845 0. 1748 0. 1616 0. 1228 0. 1456 0. 1582 0. 1878 0. 1527 0. 1894 0. 2082 0. 1639 0. 2555 0. 2796 0. 1918  0 1888 0 1714 0 1815 0 1871 0 1731 0 1978 0 1862 0 2358 0 2311 0 2364 0 2318 0 2127 0 1666 0 1997 0 2137 0 2447 0 2095 0. 2427 0 2728 0.221 0 3307 0 3632 0. 2518  0 2289 0 2061 0 2117 0 2229 0 2094 0 2423 0 2274 0 2723 0 2636 0 2662 0 2711 0.249 0 2054 0 2454 0 2524 0 2789 0 2512 0 2731 0 3168 0 2643 0 3756 0. 4151 0.296  *  5.1542 5.3542  5. 1542 5. 5542  5. 1542 5 7542  5. 1542 5. 9542  A B C D E F G H I J K L M N O P R S T U V W X  0.0597 0.054 0.0572 0.0596 0.0546 0.0627 0.0589 0.0774 0.0759 0.0772 0.0755 0.0681 0.0526 0.0642 0.0698 0.0805 0.0693 0.0794 0.0892 0.0718 0.1093 0.1209 0.0804  0. 1078 0. 0961 0. 0981 0. 1053 0. 0995 0. 1169 0. 1081 0.135 0. 1294 0. 1291 0. 1325 0. 1192 0. 0965 0. 1182 0. 1252 0. 1374 0. 1262 0. 1327 0. 1539 0. 1289 0. 1845 0. 2045 0. 1404  0 1479 0 1308 0 1283 0 1411 0. 1359 0. 1614 0. 1494 0. 1716 0. 1619 0. 1589 0. 1717 0. 1555 0. 1352 0. 1639 0.164 0. 1716 0. 1679 0. 1631 0 1978 0 1722 0. 2294 0. 2564 0. 1846  0. 1806 0. 1606 0. 1518 0. 1697 0. 1633 0.195 0. 1815 0. 1908 0. 1792 0. 1751 0. 1979 0. 1815 0. 1687 0. 2033 0. 1844 0. 1904 0. 1969 0. 1793 0. 2266 0. 2045 0. 2525 0. 2851 0. 2161  *•  *  5.3542 5.5542  5. 3542 5. 7542  5. 3542 5. 9542  A B C D E F G H I J K L M N 0 P R  0.048 0.0421 0.041 0.0457 0.0449 0.0541 0.0492 0.0576 0.0536 0.0519 0.057 0.0511 0.0438 0.0541 0.0554 0.0569 0.0568  0. 0881 0. 0768 0. 0712 0. 0815 0. 0813 0. 0987 0. 0905 0. 0941 0. 086 0. 0817 0. 0963 0. 0875 0. 0826 0. 0998 0. 0942 0. 0911 0. 0986  0 0. 0. 0. 0. 0 0. 0 0 0 0 0 0 0 0 0 0  0 2616 0 2358 0 2352 0 2516 0 2368 0 2759 0 2596 0 2916 0 2809 0 2824 0 2972 0.275 0 2389 0 2847 0 2728 0 2977 0 2803 0 2893 0 3456 0. 2966 0. 3988 0. 4438 0. 3275  1208 1066 0946 1101 1086 1323 1226 1134 1033 0979 1224 1135 1161 1391 1146 1099 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  0.0533 0.0646 0.0571 0.0751 0.0836 0.0601  0.0837 0.1086 0.1004 0.1201 0.1355 0.1043  5.5542 5.7542  5.5542 5.9542  0.0999 0.1374 0.1327 0.1432 0.1642 0.1357  0.0401 0. 0728 0.0347 0. 064 5 0.0302 0. 0537 0.0358 0. 064 4 0.0364 0. 0637 0.0446 0. 0781 0.0413 0. 0734 0.0365 0.0558 0..0325 0.0498 0..0298 0.046 0..0393 0.0654 0..0364 0.0623 0..0388 0.0722 0..0457 0.085 0..0388 0.0592 0..0342 0.053 .0.0417 0.0707 0.0304 0.0466 0.0439 0.0727 0.0433 0.0756 0.0449 0.0681 0.052 0.0806 0.0442 0.0756 5.7542 5.9542  A B C D E F G H I J  K L M N O P R S T U V w X  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  183  

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