Thermoplastics Melt Rheology and Processing

THERMOPLASTIC MELT RHEOLOGY AND PROCESSING A.V. SHENOY Advisory Consultant Pune, India

SAINI National Chemical Laboratory Pune, India

Marcel Dekker, Inc. New

York. Basel Hong Kong

Library of Congress Cataloging-in-Publication Data

Shenoy, V. Thermoplastic melt rheology and processingl V. Shenoy, D. R. Saini. p. cm.- (Plastics engineering ;37) Includes bibliographical references andindexes. ISBN 0-8247-9723-X (hardcover : alk. paper) 1. Thermoplistics. 2. Rheology. I. Saini, D. R. 11. Title. 111. Series: Plastics engineering(Marcel Dekker, Inc.) ;37. TP1180.T5S525 1996 668.4'234~20 96-1 8857 CIP

The publisher offers discounts on this book when ordered in bulk quantities. For more information, write toSpecialSalesProfessional Marketing at the address below. This book is printed on acid-free paper. Copyright 0 1996 by Marcel Dekker, Inc. All Rights Reserved. Neither this book norany part may be reproduced or transmitted in any form or by any means, electronicor mechanical, including photocopying, microfilming,and recording, or by any information storage and retrieval system, without permission in writing from the publisher. Marcel Dekker, Inc. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): l 0 9 8 7 6 5 4 3 2 1 PRINTED IN THE UNITEiD STAXES OF AMERICA

Preface

Thermoplastic melt rheology and processing is a subject that is concerned primarily with the responses of molten polymer to various types of deformations experienced during the processingof plastic products. It is of great importance in manufacturing defect-free finished products under optimal processing conditions. The subject, undoubtedly, is an immense one and encompasses many aspects. In the present book, an easy and rather unconventional technique is described for sorting out rheological data and presenting itin a unified manner. More often than not, the theoretical rheologist works with complicated mathematical approaches that are beyond the comprehension of the common processor in practice, and the experimental rheologist works on simplified flow systems or situations that are too ideal and have little pragmatic value. Therefore, the practical processor resorts to simple rules of thumb based on prior experience in order to get answers to processing problems. What we need is a compromise between the complex but accurate mathematical approach and the crude estimates of a simple rule-of-thumb approach. This has been achieved in this book through effective use of the melt flow index (MW. Using a unification technique introduced and developed bythe authors, rheological data on a very large number of polymers are concisely presented. None of the books currently available dealing with rheology of polymer melts has made useof the mastercurve approach to present unified curves for each generic type of polymer; hence, the treatment inthis book represents a distinct departure from the standard mode of presentation.

hr

Preface

The book gives a step-by-step procedure for developing a spreadsheet computer program that rheograms for most thermoplastics canbe obtained atany temperature of interest with reasonable accuracy without the use ofsophisticated rheometers. The developed technique is also useful for analyzing information relating to polymer processing operations. few typical processing parameters have been chosen in order to demonstrate this use. Hence, only some polymer processes such as injection molding, compression molding, calendering, extrusion, and compounding have found a place in this book. Other polymer processes such as blow molding, coextrusion, and thermoforming are considered outside the scope of this book because of the lack of existing correlations between their process parameters and the unification technique. Developing such correlations and including them here would have made it very difficult to keep this book to an appropriate size. Instead, a number of possible correlations of MFI with various other parameters involved in polymer manufacture, product fabrication, and property evaluation have been included. The book is heavily biased toward MFI using the universally accepted unit g/10 min. Expressing the unit as kg/600 S not only would appear unfamiliarbut might lead to confusion as MFI of 1 (in the normal units of g/10 min) would assume a value of 0.001 (in kg/600 S). Polymer processors usually mentally correlate the value of MFI with the polymer grade that they have to choosefor different processes, and most often this value is not accompanied by the units, because it is taken for granted to be g/10 min. Similarly, the test load condition of MFI measurement is normally expressed in kilograms rather than any other units.Hence, although we earnestly considered using SI units following the current trend, the entire book is written in the most familiar units. Readers of this book will probably have different backgrounds and, therefore, an effort has been madeto include in the initial three chapters material necessary for familiarizing oneself with the general concepts relating to thermoplastics, melt rheology, and rheometry. These chapters provide a basic background that the main core of the book can be better digested. The first chapter gives relevant information on thermoplastics and includes brief discussions of polymerization, molecular weight, molecular-weight distribution, polymer classification, polymerblends, and filled andrecycled polymers. It also outlines the practical significance of melt rheology and the links it has with processing. The second chapter deals with the fundamentals of rheology and provides definitionsof all the basic rheological parameters. It dwellson thenonNewtonian character of polymeric melts and explains the various anomalies encountered during the flow of viscoelastic materials. Most of the illustrations are taken from the past works of various researchers in the field of rheology.

Preface

We demonstrate the various viscoelastic phenomena to project a picture of the complexities in the flow of polymeric materials. Chapter presents some of the methods for measuring the flow characteristics of thermoplastic melts. The entire range of viscometers is not given here; we focus on only those rheometers that find direct relevance to thermoplastic melt characterization. The fourth chapter begins with the origin anddefinition of MFI. It then develops the fundamentals of the unification technique based on a strong theoretical rationale. A demonstration of the use of this technique is followed by ample master rheograms of shear viscosity for a long list of common homopolymers, copolymers, blends, PVC formulations, andfilledandrecycled polymers. The fifth chapter upgrades the master rheogram in the low-shear-rate region where coalescence is impaired by differences in the molecular-weight distribution of the polymers. Further, extensions of the unification technique are discussed to establish the parameters for obtaining master curves for other rheological parameters such as normal stress difference, complex viscosity, storage modulus, and extensional viscosity. Chapter gives the rheological models for the master curves developed in Chapters and and enlists the various model parameters that can be put to direct use during mathematical calculations. The seventh chapter outlines the command sequences required to develop a spreadsheet program for obtaining viscosity versus shear-rate curve from the master rheograms presented in Chapter These curves can be obtained at the temperature of interest merely from the appropriate value of Mm without the use of any type of sophisticated rheometer. The eighth chapter demonstrates how the rheological model for the master rheogram can beeffectively used for determining various processing parameters. It is shown that various processing parameters in injection molding, compression molding, calendering, and extrusion can be evaluated merely from the MFI. It is also shown how optimal compoundingconditions can be determined from the master rheograms that blends with good mechanical compatability can be prepared. Chapter includes most of the correlations of MFI with various other parameters that are relevant from the initial polymer manufacturing stage to the finished plastic product stage. The generously presented curves show the sensitivity of MFI to a very wide spectrum of parameters in the field of polymer science and technology. The final chapter makes some concluding remarksand gives suggestionsfor future work that researchers interested in the field of thermoplastic meltrheology will be enticed to provide further insights. few words of caution are

Preface

included to draw attention to the possible sources of error in MFI measurements that must be avoided at all costs if the unified master curves are to be reused. We believe that the material presented in this book has great pragmatic value for the practicing processor and, at the same time, offers exciting research uses for the academician. We will be gratified if our efforts in introducing, developing, and presenting the concepts in this book are found useful by our readers. A. K Shenoy D. R. Saini

Contents

PREFACE OVERVIEW OF THE SUBJECT " E R Certain Relevant Information on Thermoplastics Some Brief Discussion on Melt Rheology and Processing References

FUNDAMENTALS OF POLYMER MELT RHEOLOGY Flow Classification Non-Newtonian Flow Behavior Rheological Models Other Relationships for Shear Viscosity Function References RHEOMETERS FOR POLYMER MELT C H A R A C T E R I ~ O N Rotational Viscometers Capillary Rheometers Extensional Viscometers References FROM MELT FLOW INDEX TO W O G R A M MFI Test

iii

Contents

4.2 Relevant Equations for Rheogram Generation 4.3 Master Rheograms References UPGRADE A N D EXTENSION OF THE UNIFICATION TECHNIQUE 5.1 Upgrading the Viscosity Master Rheogram in the Low Shear Region 5.2 Extending the Unification Technique to Other Rheological Material Functions References

122 138 172

177 177 181 200

6. RHEOLOGICAL MODELS FOR UNIFIED CURVES 6.1 Suggested Rheological Models References

203 206 225

7. MFI123: SPREADSHEET PROGRAM FOR VISCOSITY VERSUS SHEAR RATE CURVES FROM MASTER RHEOGRAMS 7.1 Preparing the Spreadsheet 7.2 Using the Spreadsheet 7.3 Testing the Results References

227 228 234 236 237

8. FROM MASTER RHEOGRAMS TO PROCESSING PARAMETERS 8.1 Injection Molding 8.2 Compression Molding 8.3 Calendering 8.4 Extrusion 8.5 Viscous Heat Dissipation 8.6 Blending and Filling to Form Multicomponent Polymeric Systems References

9. MFI CORRELATIONS WITH OTHER PARAMETERS 9.1 MFI Correlations in Polymer Manufacture 9.2 MFI Correlations in Polymer Product Fabrication 9.3 MFI Correlations in Polymer Product Property Evaluation References

238 238 244 250 264 269 281 307 312 312 328 35 1 379

ix

CONCLUDING REMARKS Comments on the Presented Work Suggestions for Future Work A Few Words Caution GLOSSARY Conditions and Specifications for MFI APPENDIX A APPENDIX B Data Details and Sources for Master Rheograms APPENDIX C Data Details and Sources for Upgradeand Extension Curves APPENDIX D Manufacturerdsuppliers MFI Equipment NOMENCLATURE AUTHOR INDEX SUBJECT INDEX

This Page Intentionally Left Blank

THERMOPIASTIG RHEOLOGY PROCESSING

This Page Intentionally Left Blank

1 of

Subject Matter

1.l CERTAIN RELEVANT INFORMATION ON THERMOPLASTICS An elementary introduction to thermoplastics is given in this section. Readers wishing to get an exposure to a more comprehensive treatment should consult some of the standard references [l-41 on the subject. Thermoplastics are highmolecular-weight organic substances that have been usually synthesized from low-molecular-weight compounds through the process of polymerization.

1.l

.l Polymerization

Broadly speaking, the two possible routes for synthesizing thermoplastics are based on the mechanism of the formation reaction of the polymer, namely, addition reaction and condensation reaction. Each of these can be subcategorized as shownin Table1.1;however, the details of the subcategories are not discussed below, as they are considered to be outside the scope of this book.

In addition polymerization, the reaction is initiated by a free radical which is usually formed due to the decomposition of a relatively unstable component in the reacting species. In this reaction, repeating units add one at a time to the radical chain, thereby causing an increase in the molecular weight as shown in Fig. 1.1. Reasonably high-molecular-weight polymers can be formed in a short

Chapter 1 Table

PolymerizationReaction Qpes Polymerization Reaction

I

Condensation/

Additioflree-Radical

onlinear Ring Linear

Homopolymer Living

Homogenous Solid Heterogenous Ionic

Random

Uniform

Block

Graft

Stereopolymerization/ . Coordination Polymerization

I

HeadHeadSvndiotactic Atactic Isotactic

l

l

I

I

Head

Tail

I

chiral

time by this polymerization. Because the molecular weight plateaus out with time as seenin Fig. 1.1, longer reaction times givelarger yields without affecting the achieved high molecular weight.

B. CondensationPolymerization In condensation polymerization, the reaction takes place between two polyfunctional molecules to produce one larger polyfunctional molecule with the possible

3

Overview

variation of molecular weight with time during addition polymerization.

elimination of a small molecule such aswater. Long reaction times are essential for forming high-molecular-weight polymers by this step reaction, as be seen from Fig. 1.2, and the reaction continues only until one of the reagents is used up almost completely.

1 .2 MolecularWeight The molecular weight (Ml) of a polymer chain is calculated by multiplying the molecular mass of the repeating unit by the number of times it repeats itself, namely, the degree of .polymerization (pl). Thus, M 1

= mlpl

(1.1)

where the molecular mass m, is basically the of the atomic masses of the elements forming the molecule. Certain corrections like those from end groups and branch points are needed to determine the exact molecular weight, but these are often neglected due to their insignificant contribution. The lengths of the polymer chains formed duringthe polymerization process are not the same and, hence, it results in amixture of polymer chainsof different

Chapter

1.2 Variation

molecular weightwith time during condensation polymerization.

molecular weights. The molecular weights of the constituent molecules are very high, lO"-lO'. Most commercial polymers are polydisperse; that is, theycontain molecules of many different molecular weights. There would be NI molecules of molecular weightM,, NZmolecules of molecular weight M,,and on. Thus, it is essential to characterize molecular weight in terns of average molecular weight and molecular-weight distribution.

(a,,)

is bestunderstood The definition of number average molecular weight through an example by working out the when 10 molecules of polymer of 1,000,000 molecular weight are mixed with 65 molecules of the same polymer of 100,000 molecular weight. Because the combined molecular weight of the 75 (10 + 65) molecules is 16,500,000 (10 X 1,000,000 + 65 X the number average molecular weightis (16,500,000/75). In mathematical terms, can be defined as'follows:

a,,

a,, + N3M3 + x?,,= N,M,N +l -NzMz tNz+N3+*.*

N,Mi Z Ni

"

Overview

(aw),

The weight average molecular weight as the name suggests, is based on the weight of polymer rather than the number of molecules asin the case of the number average molecular weight. Thus, if 2 kg of polymer of 1,000,000 molecular weight is mixed with 2 kg of the same polymer of 100,000 molecular weight, then the total of 4 (2 2) kg would have a combined molecular weight of 2,200,000 (2 X 1,000,000 2 X 100,000). Therefore, the weight average molecular weight is 550,000 (2,200,00014). In mathematical terms, can be defined as

+ +

(a,)

based on the higher moment of the distriThe average molecular weight bution is the average molecular weight and is mathematically defined as

a, = X Ni(Mi)3 N,(M~)' D.

+

+

The 1 averagemolecular weight is based onan evenhighermoment of the distribution than the average molecular weight. In mathematical terms, it is written as K + l

=

X

Ni(Mi)4 Ni(Mi)3

1.l.3 Molecular-Weight Distribution

a,,,

The valueof is sensitive to the presence of highermolecular masses, whereas the value of is sensitive to the lower molecularmasses. The ratio of and B,, is often used as a measure ofthe breadth of the molecular-weight distribution (W) and at times is referred to as the polydispersity index. In monodisperse systems, Mw/Mn = 1, whereas in the examples discussedin Sections 1.1.2A and 1.1.2B7 awlan = 2.5(550,000/220,000), indicating that it is a polydisperse system. In addition to the above relationship, MWD is at times expressed as one of the following ratios: M,IMw, MJM,,, M, M,+lla~,or "

"

"

az+llaz, az+ll~,,, "

il?,Mz+,IM,2. An approximate interrelationship between the various expressions has been given by Van Krevelan et al. [5] based on the analysis of data for a number of

Chapter

polymers:

For further reading on molecular weight and its distributions along with the methods of measurement of the same, one could refer to some of the existing monographs [1,2,6-lo]. It should be noted that MWD is not a permanentcharacteristic of the polymer, as the macromolecules are prone to degradation. There are several kinds of degradation: thermal, mechanical, chemical, radiative, and biological. In polymer hydrodynamics, the aspect of prime importance is the rupture of the polymer chains when the polymeric melt is subjected to high shear rates. This changes the MWD of the polymer. This reductionin chain length often reduces the utility of the polymer for a particular application.

1

Polymer Classification

Polymer classification can be done in a number of different ways, as shown in Tables 1.2-1.4. In the following, a brief discussion is given under a variety of headings based on certain chosen characteristics for comparison. polymer can be classified as linear or branched, depending on its structure, and polyethylene serves as a good example becauseit exists with linear as well as branched structures as can be seen fromFig. Based on the pressure (low or high),the reaction temperature and the choice of the catalyst during the polymerization process, polyethylenes withdifferent densities and structures are formed. High-density polyethylene (HDPE) has a linear molecular structure and a density = g/cm3, low-density polyethylene (LDPE)has a branched structure andadensity = 0.92@cm3, whereas linear low-density polyethylene (LLDPE) with a density of @cm3,although branched is significantly different from LDPE due to the absence of secondary branching and thepresence of short branches [ll]. Raising the reaction temperature during polymerization would normally increase the degree of branching of the polymer. Raising the pressure enables a lower temperature of polymerization, and results in the production of polymers with a higher mass due to increases in the number of collisions between active centers and monomers. However, the use of elevated pressures for polymerization should always be consideredin conjunction withthe choice of a catalyst. Thus, polyethylene(PE) which is produced using a pressure of 1000-2000 atm

Overview Table

Polymers Categorized Based on Polymerization Method

Addition

er Polyolefin ylene Polyethylene ythiozyl terephthalate Polypropylene Polybenzimidazole Polybutylene terephthalate Aromatic polyester polycarbonate Acrylic Polyacrylate

Polydihydroxymethyl cyclohexyl terephthalate Polyamide imide Polyimides

Polymethyl methacrylate Photoconducting viiyl Polyvinyl acetate Polyvinyl alcohol Polyvinyl butyrate Polyvinyl formal Polyvinyl ether Polyvinyl pyrolidone Polyvinyl carbazate Polyvinyl chloride Polyvinylidene fluoride

Polyamide Aliphatic polyamides Electroconducting Aromatic polyamides Pyroelectric

Urethane Polyurethane elastomers Ether Polyacetal Polyphenylene oxide Polyethylene glycol Polypropylene glycol Olefin copolymer Styrene acrylonitrile copolymer Acrylonitrile butadiene styrene terpolymer Thermoplastic olefin elastomer Ethyl methacryl acid copolymer

Piezoelectric Light sensitive

Chapter 1 Table 1.3 Polymer Classification Based on Chemical Constituents Polymer

I

endAlloy

I

I

Copolymer/ Homopolymer Terpolymer

A

Carbon

Heterochain Random

Polyester Polyolefin

Uniform

Fluorocarbon Block Polyether

Silicone polymer Diene Polyamide

Graft

Vinyl

Acrylic

Polyaldehyde

Comb

Polyurethane Ladder

Table 1.4 Polymer Classification Based on Structure

m

Physical

I

I

Chemical

Crystalline Semicrystalline Amorphous

I

Extended Folded Chain

c

Network Linear Branched Copolymer Stereopolymer Cross-link

Overview

I LOW PRESSURE HDPE

HIGHPRESSURE LOPE

Low PRESSURE

LLDPE

Figure 1.3 Comparison among the of HDPE, LDPE, and LLDPE. (Reprinted from Ref. 11 with kind permission from Elsevier Science Ltd., Kidlington, K)

for high-pressure LDPE be also obtained using triethylaluminum and titanium chlorides as catalysts at lower pressures to form HDPE. Similarly, LLDPE is prepared by copolymerizing ethylene with a-olefins such as butene-l,hexene1, or octene-l in low-pressure reactions utilizing a transition-metal catalyst of Phillips Petroleum or the Ziegler type. In this way structurally different polyethylenes are obtained as shown in Fig. 1.3. Polymers can also be classified as crystalline, semicrystalline, or amorphous polymers depending on their degree of crystallinity. A crystal is basically an orderly arrangement of atoms in space. Polymers that are able to crystallize under suitable temperature conditions are called crystalline polymers. The pri-

Chapter

mary transition temperature, when a crystalline polymer transforms froma solid to a liquid, is the melting temperature, designated as T,,,. On the other hand, an amorphous polymer does not crystallize under any conditions. The phase transition for this type of polymer occurs from the glassy state to rubbery state at a temperature termed as the glass-transition temperature and often designated as T8' The melting temperature and glass-transition temperature can be found by measuring the specific volume or enthalpy of the polymer as a function of temperature as depicted in Figs. and 1.5. The values of Tg and T,,,for important thermoplasticsare given in Table 1.5. T,basically decreases with decreasing amorphous content and, hence, is at times difficult to detect in highly crystalline polymers. However, most thermoplastics have both Tgand T,,,.This is because it is relatively difficult to get to the extreme case of a completely crystalline polymer with an ideal formation of single crys-

1.4 Determination of T, and T,,, of polymers by measuring specific volume as a function of temperature.

Overview

11

Determination of T, and T,,, polymers by measuring enthalpy as a function of temperature.

tals having the relative arrangement of atoms strictly the same throughout the volume. In fact, deviations from the completely ordered arrangement as well as completely disordered arrangement always exist. Thus, it is the degree of crystallinity that truly determines whether a polymer could be classified as a crystalline, amorphous, or semicrystalline polymer. Some of the important thermoplastics which crystallize are PE,polypropylene (PP), and Nylon-66. Crystallization rates are zero at temperatures lowerthan or equal to Tg and very small at T,. At temperatures approximately halfway between T,,, and TB,they achieve the highest values [12]. Thus, PP with a T, = 165°C and TB= -15°C usually crystallizes in the range of 90-110°C [13]. On the other hand, poly(ethy1ene terephthalate) (PET) crystallizes very slowly when cooled in the T,-Tg range and usually forms glasses under ordinary conditions. However, under conditions of high extensional stresses, PET can be forced to crystallize, whereas polystyrene(PS) and polymethyl methacrylate (PMMA) vit-

Chapter 1 Table 1.5

Information on Selected Homopolymers

structure of

Polyme LDPE HDPE PP PS

T8

Tm

("C)("c) ("c)

repeating unit

-18

115 134 165

100 105

-120 -CH+XICH,-

Tp

Td

220

300 380

("C)

240

260 260

360

160

250

330

180

225

250

40-50

240

290

380

69

260

300

306

150

250

330

410

PVDF

-35

175

300

PP0

208

257

315

85-150

285

220 283

228

330 360

144

325

340

430

340

425

-

170

380 210

PMMA

-cHTcHc.5H5

-2-7

C"CH3

POM PA

Y

Q

-y4cH35-cH

YSI 0

F)

PET

-"c-C-Q-c*4-

Pc

aW-0"

-Q" -Q"-

PPS PES PAS

PEEK

-Q="Or"Q-.%-

-Q"Q+Q

PE1 PAr PVC

4-IfiHC.L

150-187 87

310

550 700

-

216

Overview

13

to form glasses under all conditions. Nylon-6 behaves in a peculiar manner in that it usually supercools to a glass, absorbs moisture, and then crystallizes All the polymers mentioned in the above subsections, fall in the class termed homopolymers. Simply consideringthe repeating units, a polymergets classified as a homopolymer or a copolymer.

C. When a single repeating unit such as or B exists in a polymer, it is termed a homopolymer. Thus, a homopolymer is AAAAAAAA or BBBBBBB. For example, when

the result is the homopolymer PE, and when

the result is the homopolymer, PP. number of homopolymers along with some of the important characteristic features are given in Table 1.5. All materials that are based on the ethylene doublebond such as PE, PP, PS, and poly(viny1 chloride) (PVC) are termed commodity plastics. In terms of cost, they are among the cheapest of the thermoplastics. They are produced in the largest quantities and are used in a maximum number of applications. Cellulosics comprise of a class of polymers which include cellulose nitrate, three organic esters, namely, cellulose acetate, cellulose acetate propionate, and cellulose acetate butyrate and one ether, namely, ethyl cellulose. Cellulose acetate film or cellophane finds application in the packaging .industry. Cellulose acetate butyrate is the material used between sheets of glass when making safety glass for the automotive industry. Xanthanated cellulose or rayon is used for making fabrics. Those thermoplastics which have properties superior to olefinics, styrenics, and cellulosics aretermed engineering thermoplastics. Acrylic, acetal, nylon, polyester, and polycarbonate are some of the polymers that fall in this category. Acrylic, mainly PMMA, has optical clarity nearly equal to that of glass and is often used for windowpanes. However, simple solvents tend to attack acrylic and cause material crazing when the built-in molding stresses get released.

Chapter

Acetal is formed by polymerizing the simplest amino acids, namely, formaldehyde. It is strong, tough, and self-lubricating. It can be used in a hot-water environmentwithout significant deterioration of properties or dimensional changes. Polycarbonate is made up of very bulky molecules that have almostno order in the solid state and hence give an amorphous, transparent thermoplastic. Although polycarbonate is not as clear as acrylic, it is about three times tougher and thus finds extensive use in high-impact applications such as headlight and taillight covers for automobiles. For the same reason, it is used for hand-powertool cases with the added advantage that the extensive electrical grounding requirements are eliminated. High-performance engineeringthermoplastics have recently assumed increasing importance due to their exceptional properties at elevated temperatures. A number of such specialty polymers has been introduced into the market for hightemperature applications and examples of some of the outstanding onesare poly phenylene oxide (PPO), poly phenylene sulfide (PPS), polyether sulfone (PES), polyaryl sulfone (PAS), polyether ether ketone (PEEK), polyetherimide (PEI), and polyarylate (PAr). Poly phenylene oxide (PPO), which is obtained from the free-radical, stepgrowth oxidative coupling polymerization of 2,6-xyleno17 has many attractive properties such as high heat distortion and impact strength, chemical stability to mineral and organic acids, and low water absorption. PP0 as such is difficult to process and, hence, the commercial resin (Noryl) is made by blending P P 0 with high-impact polystyrene (HIPS) which serves to reduce the processing temperature. Poly phenylene sulfide (PPS) obtained by the condensation polymerization of p-dichlorobenzene and sodium sulfide has outstanding chemical resistance, good electrical properties, excellent flame retardance, a low coefficient of friction, and a high transparencyto microwave radiation. PPSis principally usedin coating applications. This is done by spraying an aqueous slurry of PPS particles and heating to temperatures above 370°C. Particular gradesof PPS and fdled) can be used in injection and compression molding at temperatures at which PPS particles soften and undergo apparent cross-linking. Principal applications of injection- and compression-molded PPS include cookware, bearings, and pump parts for service in various corrosive environments. The bearings can be used in a nonlubricated form if the material of construction is made of PPS loaded with a little molybdenum disulfide and Teflon. Polyether sulfone (PES) and polyaryl sulfone (PAS) comprise a class of engineering thermoplastics with high thermal, oxidative, and hydrolytic stability and a good resistance to aqueous mineral acids, alkali, salt solutions, oils and greases, whereas polyether ether ketone (PEEK) has attractive properties like good abrasion resistance, low flammability and emission of smoke and toxic

Overview

gases, resistance to radiation andhigh-temperaturesteam,and low water absorption. Polyetherimide (PEI), produced by a novel nitro-displacement reaction involving bisphenol-A,4,4’-methylenedianiline,and 3-nitrophthalic anhydride, has high heat distortion temperature, tensile strength, and modulus. It is generally used in high-performance electrical and electronic parts, microwave appliances, and under-the-hood automotive parts. Polyarylate (PAr), which is produced by the condensation polymerization of bisphenol-A and phthalic acids, is an amorphous, high-clarity polymerwith high heat-distortion temperature, ultraviolet 0 stability, inherent flame retardance, and electrical properties; Typical outdoor applications for PAr include solar collectors, safety devices, construction, and transportation, whereas indoors it fhds application in electronic and electrical hardware. All the specialty polymers highlighted above are only representative examples and the list is by no means exhaustive. Each of the above discussed polymers exhibits enhanced rigidity at high temperatures. This is a consequence of their high glass-transition temperatures and presence of aromatic ring structures in the backbone chain, as can be seen from Table 1.5. When two different monomers are used in the polymerization process, the result is acopolymer. The repeating units and B both exist in the polymerized product and their varying configurations give different types of copolymer:

(i) Random copolymer: AA B BB BBB (ii) Uniform copolymer: AB AB AB AB AB (iii) Block copolymer: BBB AAA BBB AAA B B (iv) Graft copolymer: AAAAAAAAAAAAA B B B B B B B Block copolymers may be arranged in various star arrangements, wherein polymer A radiates from a central point with a specified number of and polymer B is attached to the end of each arm. Copolymerization is often used to alter the properties of homopolymers and to achieve specific performance.For example, theflow behavior of PVC is considerably improved by incorporating vinyl acetate as comonomer. Similarly, the thermal stability of polyoxymethylene is improved considerably by incor-

16

Chapter 1

poration of -CH,-CH,-0 units in the chain, yielding an oxymethyline or acetal copolymer. If either of the comonomers on its own could yield a crystalline homopolymer, then copolymerization can have a very marked effect on properties by inhibiting crystallization. For example, PE crystallinity is decreased by increasing the amount of vinyl acetate content in the copolymer, leading to a softer, tougher product, namely, ethylene-vinyl acetate (EVA). The properties of block copolymers are dependent on the length of the sequences of repeating units or domains. The domains in typical commercial block copolymers of styrene and butadiene are sufficiently long to produce flexible plastics called thermoplastic elastomers. In fact, the copolymer butadiene-styrene is a good example of how the thermoplastic characteristics can be changed by altering the portion of two components of the copolymer. Polybutadiene is a synthetic rubber with a high level of elasticity, whereas polystyrene is a clear brittle plastic which is often used for making disposable containers. A copolymer made with 75% butadiene and 25% polystyrene is styrene butadiene rubber (SBR) with direct applications to carpeting, padding, and seat cushions. On the other hand, a copolymer of 25% butadiene and 75% styrene gives an impact styrene which is often used for the manufacture of equipment cabinets and appliances. Most commercial varieties of high-impact polystyrene (HIPS) are graft copolymers in which the main chain is that of butadiene while styrene forms the branches. Copolymers of styrene with acrylonitrile (SAN) and styrene with maleic anhydride (SMA) are typical examples of uniform alternating copolymers. Copolymers represent an industrially important class of polymeric materials, due to their unique combination of properties such as impact resistance, elasticity, and processibility. Block copolymers, in particular, have great technological importance because of the ability of these materials to form thermoplastic elastomers which can be processed by conventional thermoplastic processing techniques. Readers wishing to know more about copolymers may refer to the excellent monographs [15-21] that are available. Whenever simultaneous polymerization of three monomers takes place, the result is a terpolymer. Thus, the addition of a small amount of acrylonitrile to impact styrene yields a terpolymer called ABS (acrylonitrile-butadiene-styrene), with major applications in plumbing systems and telephones. A B S is much tougher than styrene. Information on some selected copolymers/terpolymers are available in Table 1.6.

E. Liquid-Crystalline Polymers A mention must be made of the copolymers of poly(ethy1ene terephthalate) and p-hydroxybenzoic acid which forms a thermotropic system (liquid-crystalline

Overview

17

Table 1.6 Information on Selected Copolymers ~

~

Structure of repeating unit

Copolymer SAN

CH,-CH-CHZ-CHI I CN

~~

Ts ("C)

Tm ("C)

TP ("C)

110

-

230

300

50

100

185

200

90

175

230

300

-50

200

250

330

Td

("C)

0

SBS

(CH,-CH-CH,-CH=CH-CHJ I

0 ABS

CHZ-CH=CH-CH,-CHZ-CH-CH,-CH I

0

CN Polyester-

Elastomer

I

I

I

e ?

-O-(CH,)4-O-C-Q-C

?

(OCH2CH,CH,CH20),-C-~-c

?

Note: Tg = glass-transition temperature; T, = melting temperature; Tp = highest processing temperature; Td = degradation temperature.

order in the polymer melt) and falls into the category of a different class of polymers known as liquid-crystalline polymers. Such types of polymers have attained immense commercial importance due to the possibility of producing ultrahigh-strength modulus fibers and plastics [22-271. The exceptional physical properties of these uniquely structured systems are a direct consequence of the morphology and orientation induced into the polymers due to the flow history during processing. Therefore, an understanding of the rheology of these systems is extremely important.

1.I .5 Polymer Blends Polymer blends are physical mixtures of polymers and provide a means of combining the useful properties of the constituent components to achieve an economic or property advantage. There are a number of commercial thermoplastic blends such as PPO/PS, ABS/PC, PVCPMMA, and so forth currently in use [28], as summarized in Table 1.7. In polymer blends, the individual polymers are chemically different and do not form covalent bonds as in copolymers. The blends are often characterized by their phase behavior as being either miscible or immiscible. Certain blends are completely miscible and form a single phase, whereas others form domains rich in one polymer dispersed within the matrix of the second polymeric component. The degree of thermodynamic compatibility

DKE

Chapter 1

18

Table

CommercialBlends

Blend ABSPC ABSPVC

cycoloy Chemicals Mobay MC2500 Bayblend Mobay MD6500 Bayblend Abtec Abson 89129 Polyman 509 Kralastic F V J cycovin KAB cycovin KAF

Borg-Warner

Schulman Uniroyal Borg-Warner Chemicals Borg-Warner Chemicals

PVC/acrylic PPO/HIPS ABS/polysulfone SAN/polysulfone CPEPVC

Kydex Noryl Arylon

Ucardel P-4174 Hostalit

Haas Rohm and General Electric Uniroyal Union Carbide American Hoechst

Source: Ref. 28. (Reprinted with permission from DRINcGraw-Hill, Milan, Italy.)

between the individual components of the blend have implications in rheology and processing dueto the possibilities of phase separation with increasing incompatibility. Excellent reviews and a book on the subject of polymer blends may be referred to for a deeper understanding. There are extensive reviews available in literature on the rheology of blends These are well supplemented by the fairly comprehensive chapter on the rheology of two-phase systems by Han The various blends which have received the attention of rheologists to date have been summarized in Table 1.8. discussion on polymer blends cannot be considered complete without a reference to interpenetrating polymer networks (IPNs). These form a separate class of polymer blends of a novel type composed of cross-linked polymers. They are more or less intimate mixtures of two or more distinct cross-linked polymer networks withno covalent bonds betweenthe polymers. Thus, polymer cross-links only with other molecules of polymer A, and polymer B does likewise. In other words, IPNs may be described as combinations of chemically dissimilar polymersin which chains of one are completely andpermanently entangled with those of the other. There are essentially two techniques for the production of IPNS. In the first, which is a sequential technique, a cross-linked polymer is first swollen with a second polymer B or its monomer along withcross-linking agents forpolymer B. Then, polymer B or its monomer is polymerized and cross-linked in situ;

d

Overview Table

Investigations on the Rheological Properties of Polymer Blends

[Ref.]

Blend Jacovic HDPE-LDPE Dobrescu Bersted et al. and Martinez HDPE-PMMA Plochocki PP-HDPE

Alle and Lyngaae-Jorgensen Kasajima et al. Noel and Carley Han

.

Yu

PS-HDPE

Han Han and Kim PS-PP PS-POM PS-Nylon Nylon 12-PMMA Carley PMMA-POM Carley PMMA-PS

Carley Carley

shida6-PE

Crossan and Crossan and Carley

Kasajima Alle and Lyngaae-Jorgensen Thornton et al. Nylon PET-PMMA Source: Ref. 40. (Reprintedwithpermission Switzerland.)

GordonandBreachPublishers,Lausanne,

The second technique consists of combining two linear polymers, prepolymers, or their monomers together with their cross-linking agents. The process involves the simultaneous polymerization and cross-linking of the two polymers. Care must be taken to select the right chemical types of polymers that there is no reaction betweenthem. Because 'the second technique involves simultaneous network generation, IPNs prepared this way are at times referred to as simultaneous interpenetrating networks (SINS). At times, IPNs are formed in which only one of the two constituents is crosslinked and they are then referred to as semi-IPNs. The semi-IPNs are again of two kinds-one in which the polymer synthesized first is cross-linked to give semi-l-IPN and the other in which the second polymer to be polymerized is cross-linked to give semi-ZIPN. For a detailed understanding of IPNs, SINS, andsemi-IPNs, the readers should consult books and review articles on the subject

Chapter 1

1 .6 PVC Formulations Poly(viny1 chloride) (PVC) on its is quite rigid and rather difficult to shape into useful products. However, this polymer has excellent compatibility with a large number of additives and, hence, can be made into formulations. The functional additives in PVC compounds include plasticizers, stabilizers, lubricants, pigments, flame retardants, antioxidants, and forth. Plasticizers are chemicals with small molecules which are frequently added to aid in processing and help the molten material flow better under pressure. W stabilizers screen ultraviolet light to prevent materials from yellowing and cracking when exposed to sunlight. External lubricants aid in the removal of the part from the mold. Internal lubricants aid the flow of the material during early stages of processing. Pigments are added to minimize painting, whereas flame retardants are meant to reduce the fuel value of the material. Antioxidants keep materials from turning yellow on heating. Table 1.9 summarizes some of the commonly used additives in PVC [a]; the number of possible formulations are numerous. The presence of these additives in the polymer melt has a significant effect on its flow behavior [65]. Therefore, a knowledge of the complete melt-flow behavior of a formulation is necessary for proper design of the processing equipment, processoptimization, formulation, and new-product development, as illustrated in Ref. 65.

1 .7

Filled Polymers

Filled polymers are being used increasingly in a number of applications because of the specific advantages they offer. wide variety of fillers has been used, not only for cost reduction but also as reinforcing agents; these are listed in Table 1.10. In general, the particulate-type fillers, such as wood flour, calcium carbonate, clay, and sand, are used as extenders; whereas fibrous fillers (such as wollastonite, glass fibers,Franklinfiber)and platelike fillers (such as mica) represent reinforcing additions that improve the mechanical properties of the base matrix. The presence of the fillers affects the melt rheological characteristics of the system. There is extensive literature on the rheology of filled polymer melt systems [58,66-1341, including comprehensive chaptersin a number of books [135-1381. Thebulk of the literature deals with the rheology of systems in the range of 20% to filler loading by volume. Aspects relating to this loading level have been effectivelyreviewedby Utracki andFisa [105]. Rheology of highly filled polymer melt systems in the range of to 60% filler loading by volume has also attracted sufficient attention due to the extended use of polymers as binders during ceramicand metal processing [125-130,139-1501 as well as for the preparation of functional filler composites [119-122,151-1531. Aspects relating to the rheology of highly filled polymer melt systemshave been reviewed

21

Overview Table 1.9 Commonly Used Additives in PVC Plasticizers acid Stearic lead White Di-Zethylhexyl phthalate Lead carbonates monostearate Glyceryl Di-isooctylphthalate diaminostearate Lead Ethyl sulfates Dialphanol phthalate Dibasic lead phosphate Paraffin waxes Tritolyl phosphate Dibasic stearate leadCalcium stearate Trixylyl phosphate Monobasic lead stearate Barium stearate Dioctyl sebacate silicates Basic lead Unmodified polyesters Barium salts Alcohol-modified polyester Cadmium salts Acid-modified polyester Cadmium soaps Chlorinated paraffins Zinc soaps Di-n-alkyltin mercaptide Di-n-alkyltin dilaurates Dibutyltin dimaleate Epoxy resins Hydroxybenzo phenones Benzotriazoles

Lead stearate

Fillers ~

~~

Titanium dioxide Calcium carbonate Calcined clays Cadmium yellow Talc Cadmium red Carbon black Asbestos poly boron Zinc (pigment Barytes Calcium silicate alkylphosphate grade) Slate dust

Aryl phosphates Chlorinated paraffins Barium metaborate Zinc borate

Zinc antimony poly alkylphosphate

Antistatic Antioxidants

Diphenylolpropane Bisphenols Hydroquinone y derivatives nds rsenic ammonium phenols Alkylated compounds Copper

Fatty amines Fatty amides Phosphate esters

Brominated salicylanilides Mercaptans Quaternary ammonium

salts Poly(oxyethy1eneglycol esters)

Source: Ref. 64. (Reprinted with kind permission from Society of Plastics Engineers, Inc.)

~ellulosics

ig~ns se

ed l i ~ i n bark

Synthetics an meal n

Acrylics ~ylons ~olyesters

Jute od Aour Shell flour Cotton-seed hulls Cotton linters Cork dust Oxides ~ u m oxide ~ i ~ t i m o n ytrioxide Zinc oxide Magnesium oxide uartz iatomaceous earth Tripoli rogel A~rogel

Carbonates

ydroxides ate te

hydroxide urn ~ydroxide

carbonate

Silicates

Sulfates

Carbon

~Calcium silicate a~esium silicate Clay

Calcium sulfate arium sulfate

Carbon black Graphite

Metals po~ders/~bers ~ u m i ~ u m Copper ronze Lead Steel Zinc

iscellaneous ~ a r i u mferrite

Fuller’s earth efs, 66 and 67. ( ~ e p r i n t ~with d kind p e ~ i s s i o nfrom Society of lastics Engineers, Inc. and Gulf Publ~s~ing Go., Houston, Texas.)

Overview

23

by Shenoy [67]. In addition to the additives mentioned above, there are others such as a variety of inorganic salts, mainly metal halides, which have been used for modifying the structure and properties of Nylon 6 [154]. Metal halides cannot be termed as fillers or extenders because their effects are much more pronounced than those expected of the fillers or extenders. They could be better described as reactive additives because of their capability of interacting with the active amide-group sites along the nylon chain. Besides the interest in the detailed mechanism of polymer-salt interaction, there is growing awarenessabout the influence of the metal halides on the processibility of nylons [155,156]. The melting temperature of the pure polyamide could be depressed by use of salt additives to allow processing well below the conventional processingtemperature [157]. The use of salt additives would thus be advantageous for processing very high melting or thermally unstable polymers [156]. The strong interaction between the salt and the polar nylon chains leads to a substantial increase in the polymer glass-transition temperature, a decrease in the crystallization temperature and rate, and an increase in the melt viscosity. The enhanced melt viscosity, achieved through salt addition, would allow more convenient processing of a low-molecular-weight polymer and would help in raising the melt viscosity of Nylon 6 to an appropriate value for optimum processing [158]. The physical and thermal properties of filled polymers are determined not only by the type of the filler but also by its size, shape, size distribution, and amount. key factor in the use of fillers without adversely affecting the material properties is the stress transfer at the filler-matrix interface. The interfacial adhesion can be substantially enhanced via a coupling agent that adheres well to both the matrix and the filler particles. The most commonly used coupling agents are silanes and organo-titanates. The various surface-treatment agents are listed in Table 1.11.The type and amount of the surface treatment on the filler are additional parameters affecting properties of filled polymeric systems. There are other additives such as those listed in Table 1.12 which are added to alter the properties of the polymer matrix. Material systems manifesting abroad spectrum of properties can thus be obtained with filled polymers by altering the filler shape, filler amount, particle a result, filled polysize, surface modifier amount, and matrix polymer type. mers have been successfully used in a number of consumer and engineering applications. It is worth noting that the most exploited filled polymer systems commercially are PVC formulations (discussed in the prior section). The processes involved in the manufacture of these composite products include compounding operations to prepare a well-dispersed filled system, and conversion processes such as injection molding,profile extrusion, compression molding, and sheet extrusion, followed by stamping or thermoforming. The presence of fillers with or without coupling agents affects the melt-flow behavior of the

Silanes

V i n y l ~ i m e t h osilane ~ Vinyltrietho~silane Vinyl~i(2-methoxyethoxy)silane ~ n y l t r i a c e t osilane ~ Vinyltrichlorosilane C h l o r o p r o p y l t ~ e t h silane o~ ~-~lycidayloxypropyltr~ethoxy silane

A- 150/CV4917 A- 15l / C V 4 9 1 ~ ~M050/2-6082 C~4800/2-6075 CV49OO~C

~ t h a c ~ l o x y p r o p y l t ~ e t h osilane xy

M1

~-~~opropyltriethoxysilane o e t h y l - ~ - a ~ o p r o p y l - ~ i m ~ t hsilane oxy

Titanates

Sulfonyl~ide~nctionalsilane Isopropyltriisostearoyltitanate Isopropyl~i(~octy1pyrophosphato) titanate Titanium di(cumylpheny1ate)oxyacetate T i t a ~ u mdi(dioc~1pyrophos~~ate) oxyacetate T e ~ a o c ~ l o ~ t i t a n di(ditridecy1phosphite) ium Neoa~oxy,tri(doc~lphosphate) titanate eoal~oxy,t~(~-ethylamino-ethylamino) titanate ~icyc1o~dioc~l)pyrophosp~ate titanate

4

4

LICA 44 P2

Zirconates

Neoalky1,trisneodecanoylzirconate Neoalkoxy,trisdodecylbenzenesulfon~lzirconate

LZ 01 LZ 09 LZ 38 LZ 44

M6 6 6 6

ircoalum~ates

7 7

Carboxy ~nctionalz~rcoaluminate

7 M7

M ~ M Mercapto ~nctionalz ~ c o a l u ~ i n a t e olymeric esters alt of unsa~ratedfatty acid Fluor~atedalkyl esters 1 M2 3 4 M5 M6 7 M8 9

OD S or SPM

FC-430, FG-431, FC-740 FC-93, FC-95, FC-99, FC-120

Union Carbide Corporation, Old Ridgebury Road, Danbury, CT 06817 Petrarch Systems Inc., Bartram Road, Bristol, PA 19007 Kay-Fries, Inc., Chemical Div. of Dynamit Nobel of America Inc., 10 Link Drive, Rockleigh NJ 07647 Dow Corning Corporation, Midland, MI 48640 Hercules Incorporated, Hercules Plaza, ~ilmington,DE 19894 Kenrich Petrochemical Inc., 140 East 22nd St., P.O. Box 32, Bayonne, NJ 07002-0032 Cavedon Chemical Co. Inc., ~oonsocket,RI 02895 5 Oak Brook, IL 60521 ~ Y ~ - M a l l e n c k r o USA d ~ , Inc., 1 9 ~ 0 7 Barbizon, C o ~ ~ e r c i Che~icals al Division/3~,223-65E73 Center, St. Paul, MN 55144. e~rintedwith ~ e ~ ~ s s from i o nGulf Publishing Co., Houston, Texas.)

7 8 8 9 M9

Chapter 1

Some Commonly Used Matrix Additives

Table 1.l

Matrix additive type

mica1 examples

polyesters Acid-modified Plasticizers Alcohol-modified polyesters Unmodified polyesters Chlorinated paraffins Dialphenol phthalate Di-2-ethylhexyl phthalate Di-isooctyl phthalate Dioctyl sebacate Tritolyl phosphate Trixylyl phosphate Barium stearate Calcium stearate Lead stearate Ethyl diamino-stearate Glyceryl monostearate Stearic acid Paraffin waxes

Lubricants

Source: Ref. 67. (Reprinted with permission from Gulf Publishing

Co.,

Texas.)

polymer and, hence, a thorough knowledge of the inflicted changes is essential for proper equipment design, process optimization, and troubleshooting.

1.l

Recycled Polymers

Recycling of plastics waste has become a necessity in recent years [159-1611 due to the dwindling nonrenewable oil resources and the desire to minimize environmental litter. The book by Leidner [l621 addresses the problem and gives a broad and detailed overview of various aspects of the recovery of value from this waste. Recycling of plastics waste is undoubtedly essential, but to rnanufacture products with acceptable quality to meet suitable market demands is difficult. During processing, the polymer is subjected to varying degree thermal, mechanical, and oxidative degradation, leading to inferior physical properties in the polymer wastes. For example, it was found [l631 that six passes through a commercial screw extruder at 260°C decreased the weight average molecular weight of a PP sample from 270,000 to approximately 80,000, with a corresponding decrease in the melt viscosity. The physical characteristics of the waste polymer such as the flow behavior, tensile strength, ductility, elongation, impact, gloss, haze, and forth are all different from and generally inferior to those of the virgin polymer [164,165].

Overview

27

In fact, it has been shown [l641 that tensile strength and ductility start droping after the fourth cycle of reprocessing. Similarly, at the end of the fifth cycle of reprocessing, it has been shown [l651 that haze and gloss are considerably affected apart from the rheological characteristics of the polymer. It is, therefore, necessary to provide guidelines to the processors for determining the maximum concentration of polymer waste that can be added to the virgin material during processing, without drastic loss in product properties. Also, the processing conditions for the virgin and recycled material would be different in accordance. with the changes in the melt rheology [166,167].

1.2 SOME BRIEF DISCUSSION ON MELT RHEOLOGY AND PROCESSING 1.2.1 Melt Rheology Thermoplastics, by the very definition of the word, denotes polymericmaterials which can be made to soften and take on new shapes by the application of heat and pressure. In their original raw material form, thermoplastics are available in solid state as chips, granules, or powder. These are melted and reshaped to form various plastic products. For details regarding engineeringaspects and properties of plastics, readers could refer to some of the available handbooks [168-1711. Melt rheology is concerned with the description of the deformation of the material under the influence of stresses. Deformation and flow naturally exist when the thermoplastics are melted and then reformed into solid products of various shapes. All polymer melts are viscoelastic materials; that is, their response to external load lies in varying extent between that of a viscous liquid andan elastic solid. In an ideal viscous liquid, the energy of deformation is dissipated in the form of heat and cannot be recovered just by releasing the external forces; whereas, in an ideal elastic solid, the deformation is fully recovered when the stresses are released. polymer melt represents a cluster of entangled, flexible strings of varying lengths. Molecular weight or the degree of polymerization signifies the length of the string, whereas the molecular-weight distribution signifies the extent of length variation inthecluster. If the ethylene molecule were magnified 100 milliontimes,then its length would be about 1 cm.On the same scale, the polymer molecule of LDPE would be 15 m long, whereas that of ultrahighmolecular-weight polyethylene (UHMWPE) would be about km in length. It is obvious that these chains cannot be found in extended form, but they exist instead in an entangled and twisted state. It is these entanglements that provide the resistance to deformation and, therefore, with increasing molecular weight, the melt viscosity goes up, processibility worsens although, of course, mechanical properties improve. The sensitivity of rheological tests is mainly due to

Chapter 1

chain entanglements resulting in large differences inflow behavior even for small differences in chainlengthor branching. Change indeformation rate shows changes in flow behavior. Rheological measurements are often used as an effective tool for 1. Quality control of raw materials, manufacturing process/final product 2. Predicting material performance

The sensitivity of the rheological properties to structural differences in materials forms a handy aid to the quality control engineer when deciding whether to accept or reject an incoming material. A typical viscosity versus shear rate curve would be like the one shown in Fig. 1.6. There is a Newtonian region in the low shear wherethe viscosity does not change with shear rate. At some critical shear rate, there is a continuous drop off of viscosity with shear rate. The drop-off of viscosity with shear rate would occur sooner if themolecular-weight distribution is widened. This is because the shorter molecular chains are of lower viscosity and cause the vis-

A typical viscosity versus shear rate curve for a thermoplastic melt, showing the effect of physical structural changes duringflow.

Chapter 1

30

Table 1.13 ShapingOperations

Shaping Operations Secondary

Dimensional

Three Dimensional

I Thennoforming

Continuous or Steady Type

Intermittent

l Calendering

Blow- Molding

+

I

Vacuum

Pressure

l

Extrusion spinning Coating

with the function of automatically opening and closing the mold and ejecting the finished product. The schematic diagram for injection molding operation is shown in Fig. 1.7. The process begins by dropping the thermoplastic raw material in the form of powder or pellet from a hopper into the heated chamber of the injection-molding unit where it is melted or softened. On reaching molding temperature, the highly viscous polymer melt is pushed with a hydraulic ram through a small nozzle of the injection unit into a cooled mold. The mold is made up of at least two pieces held together by a mechanical toggle clamp. Once the mold is filled with the melt and solidification begins, then an additional amount of melt is packed into the mold to offset the material shrinkage that occurs duringsolidification. When the melt cools in the mold to a temperature that is suitable for removing the product without distortion, the clamp holding the two mold halves is released and the finished article is ejected. In some injection-molding units, the hydraulic ram is replaced by a rotating screw (Fig. l.%),which transports the polymeric material into a reservoir immediately behind the nozzle. During the injection process, the screw stops rotating and acts as a ram to push the melt into the mold. better understanding

31

Overview

Schematic diagram of conventional injection-molding machines. feed type; (b) screw-feed type. (From Ref. 175.) Figure

of injection moldingcan [158,173-1771.

Ram-

be obtained by referring to existing literature

B. Compression Molding Compression molding consists of forcibly squeezing a preweighed mixture of molding powder or a prepelletized charge into the desired shape of a heated mold cavity, not by injecting it into a closed mold, but by closing one-half of the mold on the other (Fig. 1.8). Heat is conducted from the hot mold walls to the polymer. The platens of the mold have provision for heating and cooling that part removal from the mold can be accomplished. Nearly all thermoplastics can be compression molded and veryfrequently mechanical property evaluations are carried out in- the laboratory using compression-molded specimens. The thermoplastic which is most often compression molded is UHMWPE. Many specialty polymers are also at times compression molded because their molding temperatures are too high to permitprocessing in conventional injection-molding machines. Compression molding has certain advantages over injection molding. The molds are simpler than those used in injection molding, fillers remain relatively undamaged, and there is minimum material wastage. The disadvantage is that the process is slower and there are geometrical limitations with respect to the

Chapter 1

32

Figure 1.8 Schematic diagram of the compression-molding process. (From Ref.

175.)

complexity of the parts that can be molded. More information on compression molding is available in existing literature [176,177].

C. Calendering Calendering is a high-production-rate process for manufacturing plastic sheets and films. It is particularly used in processing rigid as well as plasticized PVC, as it prevents polymer degradation during the manufacturing process. A schematic diagram of the calendering process is shown in Fig. 1.9. The calender usually consists of four rolls which are configured in different ways. The most common configuration used is the inverted L shape. The polymeric material is fed between the nip of the first two rolls and then transferred to the next two nips formed by roll numbers 2, 3, and 3, respectively. The first nip

S

3

l

I

Figure Schematic diagram of an inverted Gtype calendering process. l-calender; 2-embossingcalender;3-thicknessgauges;4-water-cooledtrain,5-windupaccumulator; 6-windup station. (From Ref. 175.)

Ovewiew

helps to control the feed rate while the second and third nip is utilized to set the sheet or film thickness. The temperature, speed, and surface finish of the rolls are the variables for accomplishing good product quality.For further details regarding the calendering process, existing literature [158,173,175-1781 may be reviewed. Extrusion is a process for making continuousplastic objects such astubes, pipes, rods, cables, wires, and a variety of profiles which include filaments, films, and sheets. The process is best described by the very word “extrusion,” which is derived from the Latin words ex meaning and trudere meaning ‘‘to push.” The polymer powder or pellet is melted or softened in a heated barrel and conveyed forward, plasticated, homogenized, and pressurized by a rotating screw under high shear into a metal die having a shape that is similar to the shape of the desired article. The main operating variables are the frequency of screw rotation and the barrel temperature profile which is controlled by thermocouples placed inside the metal barrel wall. Sections of the barrel are at times cooled to remove excessive heat generated by viscous dissipation. The die continuously shapes the melt into the desired form and the product is formed which is infinite in one direction. The molten profile produced is cooled either by air water quench or by running it overchill rolls. Further information on various extrusion processes can be found in existing literature [158,173,175-177,1791821. Thermoforming is the process by which flat polymer sheets or films are shaped into fairly deep-drawn container forms as shown in Fig. 1.10. The process involves heating of the flat stock, which is securely clamped along the perimeter, to a temperature a little above the melting or glass-transition temperature and then applying vacuum or pressure to force the deformable sheet to conform to the mold shape. The polymer used for thermoforming should not be prone to creep, as the unsupported sheet must not sag during the heating process. It is for this reason that the most commonly thermoformed polymers are ABS and HIPS. Formore details regarding thermoforming, the readers are referred to existing literature [158,176].

F: Blow molding is the process by which hollow articles such as bottles are manufactured. A molten parison is first prepared by the process of extrusion and then this preshaped sleeve is trapped between two mold halves and air is blown into it to enable the parison to take the shape of the mold. When the polymer comes in contact with the cold mold, it solidifies, thereby enabling the hished article to be ejected. A schematic diagram of the blow-molding process is shown in

Chapter 1

+++++tt+

w w i

I

l

II

l

iii

iv (a 1

U

+++++++

I

I

ii

I

iii

iv

Figure 1.10 Schematicdiagram of thermoformingoperations.(a)Vacuumforming; @) pressure forming. (From Ref. 176.)

Fig. 1.11. There are a number of variations of the blow-molding process and these can be seen from existing literature [158,176,183]. Transfer moldingis a sort hybrid process between compressionand injection molding. schematic diagram of the process is shown in Fig. 1.12. The polymeric mass is first melted in a separate reservoir which basically is a part of the heated mold. The mold is then closed by a ramwhich causesthe molten polymer

Overview

35

Oercendinp parison

(a

f Inflated and coaling (C

Figure 1.l1 Schematic diagram

blow-molding process. (From Ref. 175.)

to transfer from the reservoir into the mold cavity through runners and gates. The shape of the article to be made is determined by the profile of the mold cavity. Transfer-molding cycles are shorter than those of compression molding and the easy flow of premelted charge allows larger, more intricate parts to be manufactured without much warpage. Thus, one of the major limitations of compression molding are overcome by using transfer molding.

36

Chapter 1

W Figure 1.l2 Schematic diagram of transfer-molding process. (From Ref. 175.)

H. Casting Certain low-viscosity systems such asa polymer dispersed in asufficient amount of plasticizers can be easily cast into a mold having the shape/profile the final finished product. In fact, this is a common method by which highly plasticized PVC is processed to form flexible products.

1. Slush Molding Slush molding is a simple variation of the casting process. The low-viscosity material, such as, plastisol is poured into a hot mold and after a thick casting is formed on the wall of the mold cavity, the excess material is poured In order to ensure uniform coating, the mold is often rotated.

J.

RotationalMolding

Measured quantity of polymer powder is poured into the mold, which is then rotatedabout two axes inaheatedoven to formthefinishedproduct upon cooling.

K. Blending,Compounding,andMixing Blending, compounding, and mixing are processes which do not directly result in any final plastic product, but, nevertheless, they could be included under the category of polymer processing operations. The three terms are often synonymously or interchangeably used, and although various researchers have defined these terms, one is at times faced with the dilemma of terminology In the present case, definitions of the terms are given as applicable to the subject matter and hence excludes any other connotations the terms. Blending, in this context, is defined as a process in which two or more components or ingredients are physically intermingled without engenderingany

Overview

significant change in the physical state of the components [185]. The components in this present case are polymers to form polymer blends. On the other hand, compounding is the term used for those cases wherein polymers are softened, melted,and intermingledwith solid fillers and other liquid additives toform filled polymer systems. Discussionsconcerning the methodandmachinery for compounding are available in existing literature [186-1941.

The word mixing is applied to both the processes of blending and compounding and describes the process of intimate intermingling oftwopolymers or polymers with fillers/additives without any specific restrictions. The important aspect in mixing is to evaluate the quality of mixtures [l951 or the goodness of mixing [196]. The two properties which are useful in adjudging the goodness of mixing are the scale %and intensity ofsegregation. Scale of segregation is a measure of the average separation between regimes comprised of the same componentand may be correlated to the average striation thickness, which is the average distance between like interfaces in a mixture [197]. On the other hand, intensity of segregation is a measure of concentration at any point from the mean concentration [197]. The process of mixing involves the breaking down of the individual components into smaller elements and then dispersion of the elements of one component in the space occupied by the other [198]. In the case of filled systems, the process includes the breaking down of agglomerates, their separation, and segregation, until a h a 1 random distribution is achieved of each component in the system. The purpose of mixing is to attain an acceptable degree of homogeneity or uniformity of composition, assessed through the appropriate scale of scrutiny, which is defined [197,199,200] as the minimum size of the segregation region that would cause the mixture to be imperfect for the intended purpose. Thus, it is reasonable to define the perfect admixture as the state in which no variations in composition or morphology are observed at the relevant closeness of examination. For further reading, readers may refer to the book on polymer mixing technology [l841 and the review article on the mixing of polymers [201].

1.2.3

Rheology and Processing Link

Polymer processing is not an art as looked upon by the entrepreneur, but it is a science relating to the flow and deformation of the polymeric materials, namely, rheology. It is through the various processing operations described in Sec. 1.2.2 that a myriad of products are developed for various applications. However, not all the polymeric material that is processed gets formed into the final product for commercial use. The reason for lower material yield is the inability to maintain product quality, often due to the ignorance of the polymer processor with regard to the process and the raw material characteristics. Some of the common

oblems

Chapter 1

38 Table .l

Processing Problems and Causes

Processing Weld lines PVC pipes Degradation in improper and spider

leg design

Discolorationproducts inThermal PP degradation White flow lines in clear acrylic molded parts Pressure buildup leading to die blowup Thin wall sections/holes in blow-molded PE bottles

Low or nonuniform temperature profile High viscosity and improper die design Selection of wrong grade of polymer

processing problems along with the probable causes are given in Table 1.14. Such are the problems which adversely affect productivity and product quality resulting in economic loss. For maximizing profit, it is important to improve product quality, minimize material waste, reduce production downtime,and optimize the process. All these are within the control of the polymer processor who has a good understanding of the polymer melt rheologyand its relationship to the polymer process. If one takes a look at the possible remedies given in Table 1.15 for rectifying the processing problems described in Table 1.14, then one can easily establish a definitive link between rheology and processing. A number of other examples can be cited to show the rheology-processing link, as discussed below. Based on the profile configuration, the extrusion die head pressures are different [202], as shown in Table 1.16; hence, the flow response of the polymer is different in each of these situations. Extrusion polymer in the form of a continuous profile requires sufficient melt strength that the molten product coming from the hot die in the unsupported form is strong enough to retain its shape until it is cooled to a point of substantial strength. Also, in extrusion, inadequate streamlining of the die may lead to nonlaminar flow resulting in melt fracture. The rapid acceleration of the

Table 1.15 Possible Cures to Processing Problems Proper equipment selectioddesign Adequate insbrumentation/contol Proper material handling (drying/feeding) Quality control on the input raw material Use of proper additives

Overview

Table

39

ExtrusionDieHeadPressure

Profile configuration

/pipe/sheet Cast Monofilament Wire Flat film

(X lo"dyn/cm*)

0.35-1.03 0.69-2.07 1.03-5.52 1.38-4.14

Source: Ref.

surface layers at the die exit may induce defects of the shark skin type. Correct geometry and size of the die is necessary to avoid flow defects. For example, it is known that LDPE melt exhibits large vortices in the die entrance area, whereas HDPE melt does not; hence, care has to be taken to correctly design the die in these two cases. In injection molding, on the other hand, the molten final product is well supported on all sides by the cool mold and does not require high melt strength. However, a propergate design is desirable to avoid jetting flows. Whenhandling low-viscosity materials, a region of shallow flights of the screw in the front is necessary to provide thenecessary resistance to ensure that air is excluded. Similarly, a reversed tapered nozzle, trapezoidal half-round runner, diaphragm gate at the edge, or spring-loaded pin opposite the gate would help as well. The flow direction of the molten polymer determines position and extent of splay mark and weld line, and from an application viewpoint, it also dictates whether the component is subjected to hoop stresses or transverseflongitudinal stresses. Molding conditions do have a considerable effect on the structure and mechanical properties of polymers as can be seen from Table 1.17. Thermoforming,film blowing, and fiber spinning involve theability of a polymer sheet, film, or fiber to withstand high mechanical stresses and permit considerable extension withoutfailure in the semisolid state. The ideal response for a material in the thermoforming processis the one which would allow easy extension at anaverage strain andthenrapid stiffening after that.Inblowmolding process, if the melt is more elastic, it will inflate more uniformly and give a more even product. In the calendering operation, a large-diameter calender is preferred that a higher magnitude of the stretching component is achieved. During processing, monitoring of the viscosity versus shear rate function allows one to obtain a better understanding of how the polymer is being affected during the process. If a problem arises, rheological measurement can determine whether the cause of the problem is in the process or in the material, as illustrated by the last two cases considered in Table 1.14.

Chapter 1

40

Effect of Molding Conditions on Structure and Mechanical Performance of Polyacetal

Table 1

1 65 63

Melt temperature (“C) 96 Mold temperature (“C) Degree of crystallinity 66 (%) 160 Spherulite size (pm)60 Skin thickness (pm) Tensile strength (X lo5 dyn/cm*) Tensile modulus (X lo“dyn/cm2) Impact 2.70 energy (X lo“ dyn/cm2) 2.91

80

188 130 67 20 40 3.54 17.3

2

3 210

60 69 3.48

3.07

Source: Ref. 203. (Reprinted with kind permission from Butteworth-Heinemann journals, Elsevier Science Ltd., Langford Lane, Kidlington OX5 lGB, UK.)

Design a processing equipment, too, can be made on an optimum basis through the understanding of rheology. The compression ratio an extruder screw and the length of the various zones-plasticating, compression, and metering-are all chosen depending onthe rheology of the polymer,being processed. For example, a polyethylene film extrusion line cannot be indiscretely used for HDPE as well as LDPE.Due to the inherent structural differences between the two types polyethylenes, for better film quality and greater production flexibility, the general practice is to choose a HDPE grade that has a viscosity which is 70-100% higher than the LDPE grade at the same temperature. The high viscosity HDPE obviously leads to a greater viscous dissipation during melting and thus needs higher torque for the drive. Therefore, a LDPE line cannot be adapted to process HDPE because of the torque limitation. Further, reducing the shear rate is the only way to reduce heat generation.Therefore, a LDPE line can have a maximumscrew speed as high as 150 rpm, whereas the maximum screw speed of a HDPE extruder has to be 100 rpm or less. A good mold design for controlling product quality has to be done also on the basis of the rheology of the system. For example, the optical clarity in a poly(methy1 methacrylate) product is lost due to thermal degradation. When the melt enters the mold through the gate, it experiences high shear rates, leading to viscous heat generation in localized places whereby the polymer degrades and leaves white flow lines in the product. The viscous heat generation is proportional to viscosity; a proper mold design which could reduce the shear rate or provide localized heating in the high shear region would solve the problem knowing the viscosity versus shear rate function for the polymer. A number of such practical examples can be cited wherein the use of rheology can be made.

Overview

The quality of the solid polymer product, too, can be controlled through rheological measurements by establishing a correlation between viscoelastic data and performance properties like impact strength, service temperature, and thermal stability. Rheology thus acts as a bridge for relating the viscoelastic behavior of polymers to fundamentalcharacteristics such asmolecular weight, molecularweight distribution, andlong-chain branching on one hand,and to practical matters such as processibility and mechanical properties of the finished product on the other. Thus, the subject of polymer-melt rheology has received the unabated attention of a number of research workers. Timeand againthe various aspects dealing with polymer melt rheology and related topics have been reviewed by a number of eminent scientists [67,204-2061. Various books [39,136,137,207-2151 have also been written, each giving a different perspective of this rather complex subject. Nevertheless, there is quite a bit of overlap in the information available in these books, although each book publishedinthe latter years has newer information besides the earlier known facts. The present book is rather unique in that it encompasses a different approach which has not been included in any of the earlier books. Chapters 2and 3, which discuss the fundamentals of polymer melt rheology and rheometry, respectively, undoubtedly include a great deal of information that is available in other books as well [39,136,137,207-2311 because they deal with the basics. The rest of this book contains information not available in any existing books on rheology, especially when one considers Chapters and 5. When it comes to the practical applications of polymer melt rheology to processing, only certain background information is drawn from alreadyexisting books on polymer processing [158,173-182,232-2371. However, again the use of the simple pragmatic approach makes this book unique in all respects.

1. Flory, P. J., Principles of Polymer Chemistry, CornellUniversityPress,Ithaca (1953). 2. Tanford, C., Physical Chemistry of Macromolecules, Wiley, New York (1960). 3. Billmeyer, Jr. F. W., Textbook of Polymer Science, Wiley, New York (1962). 4. Meares, P., Polymers:StructureandBulk Properties, VanNostrand,London (1965). in the 5. VanKrevelan, D. W.,Goedhar, D. J., andHoftijzer,P.J.,Correlations molecular weight distribution of polymers, Polymer, 18, 750-751 (1977). 6. Tager, Physical Chemistry of Polymers, MU Publishers, Moscow (1978). 7. Schultz, J. M., Polymer Materials Science, Prentice-Hall, Englewood Cliffs, NJ (1974), Chap. 6. 8. Morawetz, H., Macromolecules in Solution, 2nd ed., Wiley, New York (1975).

Chapter 1 Collins, E. A., Bares, J., and BillmeyerJr., F. W., Experiments in Polymer Science, Wiley, New York Techniques of Polymer Characterization, Butterworths,London Allen, P.W., Saini, D. R. and Shenoy, A. V., Viscoelastic properties of linear low density polyethylene melts, Eur. Polym. J., Shenoy, A. V. and Saini, D. R., Method to reduce experimentation during crystallization studies done by thermoanalytical techniques, Thermochim. Acta, Spruiell, J. E. and White, J. L., Polym. Eng. Sci., Bankar, V. G., Spruiell, J.E., and White, J. L., Melt spinning of Nylon structure development and mechanical properties of as spunfilaments,J. Appl. Polym. Sci., Aggarwal, S. L. (ed.), Block Polymers, Plenum Press, New York Ester, G. M., Cooper, S. L., and Tobolsky, A. V., Block copolymers and related heterophase elastomers,J. Macromol. Rev. Macromol. Chem., Molau, G. E. (ed.), Colloid and Morphological Behaviour of Block and Graft Copolymers, Plenum Press, New York , Allport, D.C.and James, W. H. (eds.), Block Copolymers, Wiley,NewYork Burke, J. J. and Weiss,Y. (eds.), Block and GraftCopolymers,Syracuse University Press, Syracuse Platzer,N.A.J.(ed.), Copolymers, Polyblends and Composites, Advancesin Chemistry Series American Chemical Society, Washington, DC Noshay, A. and McGrath, J. E. (eds.), Block Copolymers Overview and Critical Survey, Academic Press, New York Beny G. C. and Sroog, C. E. (eds.), Rigid chain polymers: Synthesis and properties, J. Polym. Sci. Polym. Symp. Ciferri, A. and Ward, I. M. (eds.), Ultra-HighModulus Polymers, Applied Science Publishers, London Blumstein, A. (ed.), Liquid Crystaline Order in Polymers, Academic Press, New York Samulski, E. T. and DuPre, D. B., Polymeric liquid crystals,inAdvances in Liquid Crystals, (G. H. Brown, ed.), Academic Press, New York Samulski, E. T., Polymer liquid crystal controlled phase transitions can lead to useful materials, Physics Today, (May Gilbert, R.D. and Patton,P. A., Liquid crystal formation in cellulose and cellulose derivatives, Prog. Polym. Sci. Smeykal, J. P., Modern Plastics Encyclopedia, McGraw-Hill, New York Vol. Krause, S., Polymer-polymer compatibility, in Polymer Blends (D. R. Paul and S. Newman, eds.), Academic Press, New York Vol. Chap. pp. Roovers, J., Polymeric alloys, Methods Exp. Phys., Barlow, J. W. and Paul, D. R., Polymer alloys, Annu. Rev. Mater. Sci.,

Chapter Kasajima, M., Bull. Coll. Eng., Hosei Univ., Ale, N. and Lyngaae-Jorgensen, J., Proc. Vmth Intl. Congr. on Rheol.

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Thomton, B. Viasenor, R. G., and Maxwell,B., The melt elasticity of polymer blends: polystyrene/poly(methyl methacrylate),J. Appl. Polym. Sci., Hayashida, K.,Takahashi, J., and Matsui, M. Some contributions haviour of polymer melts, in Proc. Vth Intl. Congr. on Rheol.

the flow beVol. p.

of fiber-filled poly(ethy1ene Wu, S., Order-disorder transitions in the extrusion terephthalate) and blends,Polym. Eng. Sci., Manson, J. and Sperling,L. H., Polymer Blends andComposites,Plenum Press, New York Sperling, L. H., Interpenetrating polymer networks and related materials, J. Poly. Sci., Macromol. Rev., Sperling, L. H., Devia, N., Manson, J. and Conde, SimultaneousZnterpenetrating Networks Based Castor Oil Elastomers and Polystyrene:A Review an International Program, Symposium SeriesNo. American Chemical Society, Washington, DC pp. Sperling, L. H., Manson, J. Qureshi, S., and Fernandez, M., Tough plastics and reinforced elastomers from renewable resource industrial oils: short review, Z & EC Prod. Res. Dm., Sperling, L. H., Interpenetrating Polymer Networks and Related Materials, Plenum Press, New York Shenoy, V., Saini, D. R., and Nadkarni,V.M. Rheology of poly(viny1 chloride) formulations from melt flow index measurements, J. vinyl Technol. 5, Shah, P. L.,Processing melt rheology, inEncyclopedia of PVC, (L. I. Nass, ed.), Marcel Dekker, New York Vol. Chap. pp. Shenoy, A. V., Saini, D. R., and Nadkami, V.M., Rheograms of filled polymer melts from melt flow index, Polym. Compos., Shenoy, V., Rheology of highly filled polymer melt systems, in Encyclopedia of Fluid Mechanics (N. P. Cheremisinoff, ed.), Gulf Publishing Co., Houston, TX Vol. pp. Zakharenko, N. V., Tolstukhina, F. S., and Bartenev, G. M., Flow of rubber-like polymerwithandwithoutcarbonblack, RubberChem.Technol. Chapman, F. M. andLee, T. S., Effect of talc filler on the melt rheology polypropylene, SPE J., Mills, N.J., The rheology of filled polymers,J. Appl. Polym. Sci.,

of

Nazem, F. and Hill, C. T., Elongational and shear viscosities of bead-filled thermoplastic, Trans. Rheol., 18, Han,C.D.,Rheological properties of calciumcarbonate-filledpolypropylene melts, J. Appl. Polym. Sci.,

Overview

45

White, J. L. and Crowder, J. W., The influence of carbon black on the extrusion characteristics and rheological properties of elastomers: Polybutadiene and butadiene-styrene copolymer,J. Appl. Polym. Minagawa, N. and White, J. L., The influence of titanium dioxide on the rheological extrusion properties of polymer melts, J. Appl. Polym. Boira, M. S. and Chaffey, C. E., Effects of coupling agents on the mechanical and rheological properties ofmica-reinforcedpolypropylene, Polym. Eng. Sci, Bigg, D. M.,Rheologyand wire coating of high atomic number of metal low density polyethylene composites,Polym. Eng. Sci, Kataoka, T.,Kitano, T., Sasahara, M., andNishijima,K.,Viscosityofparticle filled polymer melts, Rheol. Acta, Kataoka, T.,Kitano, T., and Nishimura,T., Utility of parallel-plate plastometer for rheological study of filled polymer melts, Rheol. Acta, Mennig, G. and Hinkelmann, B., Zum Fliessverhalten von Kurzglasfasergefullten Styrol-Acrylonitrile-Copolymeren,Angew Makromol. Chem, Chan, Y., White, J. L., and Oyanagi, Y., Influence of glass fibers on the extrusion andinjectionmoldingcharacteristicsofpolyethyleneandpolystyrenemelts, Polym. Eng. Sci., Han, C. D.,Sandford, C., and Yoo, H. J., Effects of titanate coupling agents on the rheological and mechanical properties of filled polyolefins, Polym. Eng. Chan, Yu, White, J. L., and Oyanagi, Y., A fundamental study of the rheological properties of glass fiber-reinforced polyethylene and polystyrene melts, J. Rheol. Monte, S. J. and Sugeman, G., A new generation of age and water-resistant reinforced plastics, Polym. Plastics Tech. Eng. Lobe, V. M. and White, J. L., An experimental study of the influence of carbon black on the rheological properties of a polystyrene melt, Polym. Eng. Sci., Kataoka, T., Kitano, T., Oyanagi, Y., andSasahara, M., Viscous properties of calcium carbonate filled polymer melts,Rheol. Acta, Kitano, T., Kataoka, T., Nishimura, T., and Sakai, T., Relative viscositiesof polymer melts filled with inorganic fillers, Rheol. Acta, Kitano, T., Nishimura, T., Kataoka, T., and Sakai, T., Correlation of dynamic and steadyflowviscosities offilledpolymersystems, Rheol. Acta, Crowson, R. J., Folkes, M. J., and Bright, P. F., Rheology of short glass fiberreinforced thermoplastics and its applications to injection molding I. Fiber motion and viscosity measurement, Polym. Eng. Crowson, R. J. and Folkes, M. J., Rheology of short glass fiber-reinforced thermoplastics and its application to injection molding 11. The effect of material parameters, Polym. Eng. Sci., Goel, D.C., Effect of polymeric additives on the rheological properties of talcfilled polypropylene, Polym Eng. Sci.,

Chapter 1 Tanaka, H. and White, J. L., Experimental investigationsof shear and elongational flow properties of polystyrene melts reinforced with calcium carbonate, titanium dioxide and carbon black, Polym. Eng. Sci., Czamecki, I. and White, J. L., Shear flow rheological properties, fiber damage and mastication characteristics of aramid-glass and cellulose-fiber reinforced polystyrene melts, J. Appl. Polym. Sci, White, J. L., Czarnecki, I., and Tanaka, H., Experimental studies of the influence of particle and fiber reinforcement on the rheological properties of polymer melts, Rubber Chem. Technol., Hancock, M., Tremayne, P., and Rosevear, J., Fillers in polypropylene11, J. Polym. Sci. Polym. Chem. Ed., Knutsson, B. A., White, J. L., and Abbas, K. A., Rheological and extrusion characteristics ofglass-fiber reinforcedpolycarbonate, J. Appl. Polym. Sci., Hinkelmann, B., Zum Fliessverhalten Kurzglass-fasergefillter Thermoplastchmelzen in Statio-naren und Instationaren Bereich, Rheol. Acta, 20, Kitano, T.,Kataoka, T., and Shirata, T., An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers, Rheol. Acta, Han, C. D., Van der Weghe, T., Shete, P., and Haw, J. R., Effect of coupling agents on the rheological properties, processing and mechanical propertiesof filled polypropylene, Polym. Eng. Sci., Stamhuis, J. F. and Loppe, J. P. A., Rheological determination of polymer-filler affinity, Rheol. Acta, Sharma, Y. N.,Patel, R. D., Dhimmar, I. H., and Bhardwaj, I. S., Studies of the effect of titanate coupling agent on the performance of polypropylene-calcium carbonate composite,J. Appl. Polym. Sci., Nakatsuka, T., Kawasaki, H., Itadani, K., and Yamashita, S., Phosphate coupling agents for calcium carbonate filler, J. Appl. Polym. Sci, Lee, W. M., Abe, D. A., Chipalkatti, M. H., and Liaw, T. F., Rheological properties of particulate-filled linear low density polyethylenes,in Proc. Annu. Conf. Reinf. Plast. Compos. Inst. Soc., Plast. Ind., Vol. Paper p. Juskey, V.P. and Chaffey, C. E., Rheology and tensile propertiesof polypropylene reinforced with glycerol-treated mica, Can. J. Chem. Eng. Hinkelmann, B., Zur Analytischen Beschreibung des Fullstoffein-flusses auf das Fliessverhalten von Kunststoff-Schmelzen, Rheol. Acta, Utracki,L.A.andFisa,B.,Rheology of fiber or flake-filledplastics, Polym. Compos., Bigg, D. M., Rheological analysis of highly loaded polymeric composites filled with nonagglomerating spherical filler particles, Polym. Eng. Sci., Bigg, D. M., Rheological behaviour of highly filled polymer melts, Polym Eng. Sci., Suetsugu, Y. and White, J. L., The infiuence of particle size and surface coating ofcalcium carbonate on the rheological properties of its suspension' in molten polystyrene, J. Appl. Polym. Sci,

Overview Althouse, L. M., Bigg, D. M., and Wong, W. M., Evaluating the effectiveness of filler surface treatments,Plastics Compounding (MarcWApril Shenoy, A. V. and Saini, D. R., Interpretation of flow data for multicomponent polymeric systems, Colloid Polym. Sci., Lem, K. W. and Han, C. D., Rheological behaviour of concentrated suspensions of particulates in unsaturated polyester resin.J. Rheol., Luo, H. L., Han, C. D., and Mijovic, J., Effects of coupling agents in the rheological behaviour and physical mechanical properties of filled nylon J. Appl. Polym. Sci., Bigg, D. M., Complex rheology of highly filled thermoplastic melts, Proc.IX Intl. Congress on Rheology in Mexico, Adv. Rheol., Kitano, T., Kataoka, T., and Nagatsuka, Y., Shear flow rheological properties of Rheol. Acta, vinylon and glass-fiber reinforced polyethylene melts, Kitano, T., Kataoka, T., and Nagatsuka, Y., Dynamic flow properties of vinylon fibreandglassfiberreinforcedpolyethylenemelts, Rheol. Acta, Suetsugu, Y. and White, J. L., A theory of thixotropic plastic viscoelastic fluids with a time-dependent yield surface and its comparison to transient and steady J. Non-Newtonian Fluid state experiments on small particle filled polymer melts, Mech., Hinkelmann, B. and Mennig, G., On the rheological behaviour of filled polymer melts, Chem. Eng. Commun., Bretas, R. E. S. and Powell, R. L., Dynamic and transient rheological properties of glass-filled polymer melts, Rheol. Acta, Saini, D. R., Shenoy, A. V., and Nadkami, V. M., Effect of surface treatment on the rheological and mechanical properties of ferrite filled polymeric systems,Polym. Eng. Sci., Saini, D. R. and Shenoy, A. V., Viscoelastic properties of highly loaded femte filled polymeric systems,Polym. Eng. Sci., Shenoy, A.V. and Saini, D.R., Quantitative estimationof matrix filler interactions in ferrite-filled styrene-isoprene-styrene block copolymer systems, Polym. Compo~., Saini, D. R., Shenoy, A. V., and Nadkami, V. M., Melt rheology of highly loaded ferrite-filled polymer composites, Polym. Compos., Shenoy, A. V. and Saini, D.R., Wollastonite reinforced polypropylene composites: Dynamicandsteadystatemeltflowbehaviour, J. Reinf: Plastics Compos., 5, Mutel, A. T. and Kamal, M. R., Characterization of the rheological behavior of fiber-filled polypropylene melts under steady and oscillatory shear using cone-andplate and rotational parallel plate geometry,Po@. Compos., Sacks, M. D., Khadilkar, C, S., Scheiffele, G. W., Shenoy, A. V., Dow, J. H., and Sheu, R. S., Dispersionandrheologyinceramicprocessing, Adv. Ceram., Dow, J. H., Sacks, M. D., and Shenoy, A. V., Dispersion of ceramic particles in polymer melts, Ceram. Trans. (Ceram. Powder Sci. IZ,A),

48

Chapter Dow, J. H., Sacks, M. D., and Shenoy, A. V., Dispersion of alumina particles in polyethylene melts,Ceram. Trans. (Ceram. PowderSci. ZZl), Takahashi, M., Kihira, H., Suzuki, S., andIshigure, Y., Rheologyof oxidehesin systemsforinjectionmolding, Cerarn.Trans. (Ceram.Powder Sci. N), Edirisinghe, M. J. andEvans, J. R.G.,Rheologyofceramicinjectionmolding formulations, BE Ceram. Trans. J., Edirisinghe, M.J. and Evans, J. R. G., Properties of ceramic injection moulding formulations, Part Melt rheology, J. Mater. Sci., Polinski, A. J., Ryan,M.E., Gupta, R. K, Seshadri, S. G., and Frechette, F. J., Rheological behaviourof filled polymer systemsI. Yield stress and shear-thinning effects, J. Rheol. Polinski, A. J., Ryan,M.E., Gupta, R. K., Seshadri, S. G.,and Frechette, F. J., Rheological behaviourof filled polymeric systems11. The effect of a bimodel size distribution of particulates, J. Rheol., Yanovsky, Yu. G. and Zaikov, G. E., Rheological properties of filled polymers, in Encyclopedia Fluid Mechanics (N. P. CheremisiuoE, ed.), Gulf Publishing Co., Houston, TX Vol. pp. Carreau, P. J., Rheology of filled polymeric systems, in Transport Processes in Bubbles, Drops and Particles (R.P. Chhabra and D. De Kee, eds.) Hemisphere Publishing Corp., New York Chap. pp. Nielsen, L. E., Mechanical Properties Polymers and Composites, Marcel Dekker, New York Vol. Chap. pp. Nielsen, L. E.,Polymer Rheology, Marcel Dekker, New York Chap. pp. Vinogradov, G.V. andMalkin, A.Y., Rheology Polymers, MirPublishers, Moscow Han, C. D., Multiphase Flow in Polymer Processing, Academic Press, New York Whalen, T.J. and Johnson, C. F., Injection molding of ceramics,Am. Ceram. Bull., Mangels, J. A., Fabrication of complex shaped ceramic articlesby slip castingand injection molding, in Progress in Nitrogen Ceramics, (F. L. Riley, ed.), Martinus Nijhoff Pub. Boston, pp. Mangels, J. A. andWilliams, R.M., Injection molding ceramics to high green densities, Am. Ceram. Bull., Mutsuddy, B. C., Injection molding research paves way to ceramic engine parts, Ind. Res. Dm., (July Bandyopadhyay,G.and French, K.W.,Near netshapefabrication anddens%cation of silicon nitride, in Proc. of Workshop in Conservation and Substitution Technology for Critical Metals in Bearings and Related Components Mangels, J. A.andTrela, W., Ceramiccomponents byinjectionmolding, Adv. Ceram., Mutsuddy,B.C., Overviewonorganicbinderforwhitewareceramics, in 39th Pacific Coast Regional Meeting

Overview Schurtz, J. F., Methylcellulose polymers as binders for extrusion of ceramics, in 39th Pacific Coast Regional Meeting Edirisinghe, M. J. and Evans, J. R. G., Review: Fabrication of engineering ceramics by injection moulding, I. Materials selection, Znt. J. High Technol. Ceram, Edirisinghe, M. J. and Evans, J. R. G., Review: Fabrication of engineering ceramics by injection moulding, II. Techniques, Znt. J. High Technol. Ceram., Bhattacharya, S. K (ed.), Metal-Filled Polymers: Properties and Applications, Marcel Dekker, New York German, R. M., Powder Injection Molding, Metal Powder Industries Federation, Princeton, NJ Runt, J. and Galgoci, E. C., Polymer/piezoelectric ceramic composites: Polystyrene and poly(methy1 methacrylate) with PZT, J. Appl. Polym. Sci., Runt, J. and Galgoci, E. C., Piezoelectric composites of PZT and some semicrystalline polymers, Mater. Res. Bull., Newnham, R. E. and Runt, J. P., Polymer-piezoelectric ceramic composites, Polym. News, Shenoy, V.,Saini, D. R., and Nadkami,V. M., Rheology of Nylon containing metal halides, J. Mater. Siegmann, and Baraam, Z., Nylon-6 containing metal halide 11. Tensile properties, Polym. Eng. Sci., US.Patent No. @U Pont). Siegmann, andBaraam,Z.,Effect of metalhalidesontheglasstransition temperature of Nylon-6, Mukromol. Chem. Rapid Commun. McKelvey, J. M.,Polymer Processing, Wiley, New York Leidner, J., Recovery of the value from post consumer plastics waste, Po1ym.Plust. Tech. Eng., Smith, H.V., The recycling of mixed thermoplastic waste, Po1ym.-Plast. Tech. Eng., Milgram,J., h overviewof plastics recycling, Po1ym.-Plast.Tech.Eng., Leidner, J., Plastics Wuste: Recovery of Economic Value, Plastic EngineeringSeries Vol. Marcel Dekker, New York Schott, H.and Kaghan, W. S., Extrusion and applied rheology, SPE RE%C Preprints p. Bastida, S., Marieta, T., Equiazobal, J. I., and Nazabal, J., Effects of reprocessing on the nature and properties of styrene acrylonitrile,Eur. Polym. J., Rokudai, M. and Fujiki, T.,Influenceof shear histories on the rheological and processibility of branched polymers, Contemp. Topics Polym. Sci., Shenoy, A. V., Saini, D. R., and Nadkarni, V. M., Estimation of the melt rheology of polymer waste from melt flow index,Polymer,

Chapter Shenoy, A. V. and Saini, D. R., Estimation of melt elasticity of degraded polymer from melt flow index, Polym. Degrad. Stabil. Roff, W.J. and Scott, J. R. (eds.), Handbook of Common Polymers, CRC Press, Boca Raton, FL, Butterworth, London Frados, J.(ed.), Plastics Engineering Handbook, SPE 4th ed., Van Nostrand, Reinhold, New York Saechtling, H., International Plastics Handbook, 2nd ed., Hanser Munich Brandrup, J. and Immergut, E. H.(ed.), Polymer Handbook, 3rd ed., Wiley, New York Bernhardt,E.C.andMcKelvey,J. M., Polymer processing-New engineering (July speciality, Mod. Plastics, Bernhardt, E. C. (ed.), Processing of ThermoplasticMaterials, Van Nostrand Reinhold, New York Rubin, I. I., Injection Molding, Wiley, New York Holmes-Walker, W.A., Polymer Conversion, Halstead, London Tadmor, Z. and Gogos, C. G., Principles of Polymer Processing,Wiley, New York Thorne, J. L., Plastics Process Engineering, Marcel Dekker, New York Eldon, R.A.and Swan,A.D., Calendering of Plastics, Butterworths,London Fisher, E. G., Extrusion of Plastics, Iliffe Books Ltd., London Schenkel, G., Plastics Extrusion Technology and Theory, Iliffe Books Ltd., London Tadmor, Z. and Klein, I., Engineering Principles of Plasticating Extrusion, Van Nostrand Reinhold, New York Richardson, P.N., Introduction to Extrusion, Society of Plastics Engineers Inc., Greenwich, CN Rosato, D. and Rosato, D. V. (eds.), Blow Molding Handbook, Hanser, Munich Matthews, G., Polymer Mixing Technology,Applied Science Publishers, New York Fisher, E. G. and Chard, E. D., Int. Plastics Eng., Eise, K.,Compounding of additives and fillers, AIChE Summer National Meeting Jakopin, S., Compounding of Fillers, AdvancesinChemistrySeries, No. Amer. Chem. Soc. pp. Jakopin, S., Compounding of filledmaterials for automotiveapplications, SPE Mid-Michigan Retec Jakopin, S., Compounding of additives, ANTEC, Stade, K.,Techniques for compounding glass fiber-reinforced thermoplastics, PoEng. Sci., (Jan. Swanborough, A., Plastics CompoundingExtruders, ChemicalEngineering No. Instn. Chem. Engrs. pp. Swanborough, A., Machinery for the production of coloured thermoplastic compounds, Colour Compoundings in Western Europe Conf.

Overview 193. Todd, D. B. and Baumann, D. K., l b i n screw reinforced plastics compounding, Polym. Eng. Sci., 18, 321-325 (1978). 194. Todd, D. B. and Baumann, D. K., Compounding glass into plastics, Chem. Eng. Prog., 78, 65-68 (Jan. 1977). 195. Mohr, W. D., Saxton, R. L., and Jepson, C. H., Mixing in Laminar Flow Systems, Ind. Eng. Chem., 49, 1855-1856 (1957). 196. Danckwerts, P. V., Appl. Sci. Res. A 3 , 279 (1952). 197. Mohr, W.D.,in Processing of Thermoplastic Materials, (E. C.Bernhardt, ed.), Van Nostrand, Reinhold, New York (1959), Chap. 3. 198. Palmgren, H.,Part 1, Part 2 review of the processing conditions in the batch operated internal mixer, Eur: Rubber J., 156, 30-44, 70-83 (May/June 1974). 199. Danckwerts, P.V., Continuous flow system, Chem Eng. Sci., 2, 1-13 (1953). 200. Danckwerts, P. V., Theory of mixtures and mixing, Research, London, 6, 355361 (1953). 201. Hold, P., Mixing of polymers-An overview, Adv. Polym. Technol.. 2, 141-151 (Part 1) 197-228 (Part 2) (1982). 202. Glanvill, A. B., The Plastics Engineering Data Book, Industrial Press, New York (1973), p. 60. 203. Wright, D. G. M., Dunk, R., Bouvart, D., and Autran, M., The effect of crystallinity on the propertiesof injection moulded polypropylene and polyacetals, Polymer, 29, 793-796 (1988). 204. White, J. L., Industrial Rheological Measurements Molten Polymers (K. Walters, ed.), Research Studies Press, Chichester(1980). 205. Akay, G., Rheological propertiesof thermoplastics, inEncyclopedia of Fluid Mechanics (N. Cheremisinoff, ed.), Gulf Publishing Houston, TX (1986), Vol. 1, pp. 1155-1204. 206. Bersted, B. H., Effect of long chain branching on polymer rheology, in Encyclopedia of Fluid Mechanics (N. Cheremisinoff, ed.), Gulf Publishing Co., (1988), Vol. 7, pp. 635-666. 207. Severs, E. T.,Rheology of Polymers, Reinhold Publ., New York (1962). 208. Middlemen, S., The Flow of High Polymers, Interscience, New York (1968). 209. Ferry, J.D., Viicoelastic PropertiesofPolymers, 2nd ed., Wiley, New York(1969). 210. Lenk, R. S., Polymer Rheology,Applied Sciences Publishers London (1978). 211. Brydson, J. A., Flow Properties of Polymer Melts, Godwin, London (1981). 212. Cogswell, F. N., Polymer Melt Rheology, Wiley, New York (1981). 213. Janeschitzkriegl, H., Polymer Melt Rheology and Flow Birefringence, SpringerVerlag, Berlin (1983). 214. Barnes, H. A., Hutton, J. F., and Walters, K, An Introduction to Rheology, Elsevier, Amsterdam (1989). 215. Dealy, J. M. and Wissbrun, K. F., Melt Rheology and Its Role in Plastics Processing, Van Nostrand Reinhold, New York (1990). 216. Lodge, A. S., Elastic Liquids,Academic Press, New York (1964). 217. Coleman, B. D., Markovitz, H., and Noll, W., Mscometric FlowofNon-Newtonian Fluids, Springer-Verlag, New York (1966). 218. Scott Blair, G. W., Elementary Rheology, Academic Press, New York (1969).

52

Chapter

219. Hutton, J. F., Pearson, J. R. A., and Walters, K, Theoretical Rheology, Applied Sciences Publishers, London(1975). 220. Darby, R., VkcoelasticFluids, Marcel Dekker, New York (1976). 221. Hams, J., Rheology and Non-Newtonian Flow, Longman, London (1977). 222. Schowalter, W. R., Mechanics of Non-Newtonian Fluids, Pergamon Press, Oxford (1978). 223. Petrie, C. J. S., Elongational Flows, Pitman, London (1979). 224. Zahorski, S., Mechanics of Vkoelastic Fluia5, MartinusNijhiffPub.,Boston (1981). 225. Tanner, R. I., Engineering Rheology, Clarendon Press, Cambridge (1985). 226. Collyer,A.A.andClegg,D.W., Rheological Measurements, Elsevier Science Publishers, London (1988). 227. Collyer, A. A. and Utracki, L. A. (ed.), Polymer Rheometry and Processing, Elsevier Applied Sciences,New York (1990). 228. Van Wazer, J. R., Lyons, J. W., Kim, K Y., and Colwell, R. F., ficosity and Flow Measurement: LaboratoryHandbook of Rheology, Interscience, NewYork (1963). 229. Walters, K, Rheometry, Chapman & Hall, London (1975). (1979). 230. Whorlow, R. W., Rheological Techniques,Ellis Horwood Wiley, New York 231. Dealy, J. M., Rheometers for Molten Plastics: Practical Guide to Testing and Property Measurement, Van Nostrand Reinhold, New York (1982). 232. Brydson, J. A.andPeacock, D. G.,Principles of Plastics Extrusion, Applied Science Publishers, London (1973). 233. Middlemen, S., Fundamentals of Polymer Processing, McGraw-Hill, New York (1977). 234. Fenner, R. T., Principles of Polymer Processing, Chemical Publishing Co., New York (1980). 235. Astarita, G. and Nicolais, L. (ed.), Polymer Processing and Properties, Plenum Press, New York (1983). 236. Cheremisinoff, N. P., Polymer Mixing and Extrusion Technology,Marcel Dekker, New York (1987). 237. Isayev, A. I., Injection and Compression Molding Fundamentals, Marcel Dekker, New York (1987).

Fundamentals Melt Rheology

Polymer

Rheology is the study of flow by definition, and polymer melt rheology is basically concerned with the description of the deformation of polymermelts under the influence of applied stresses. Molten thermoplastics are viscoelastic materials in the sense that their response to deformation lies in varying extent between that of viscous liquids and elastic solids. In purely viscous liquids, the mechanical energy is dissipated into the systems in the form ofheatand cannot be recovered by releasing the stresses. Ideal solids, on the other hand, deform elastically such that the deformation is reversible and the energy of deformation is fully recoverable when the stresses are released. A particular thermoplastic melt behaves as a viscous liquid or an elastic solid during processing operations depending on the relationship between the time scale of deformation towhich it is subjected and the time required for the timedependent mechanism to respond. The ratio of characteristic time to the scale of deformation is defined as the Deborah number De = &/hs by Reiner [1,2], where h, is the characteristic time and h, is the time scale of deformation. The characteristic time, h,, for any material can always be defined as the time required for the material to reach63.2% or 1 - (lle) of its ultimate retarded elastic response to a step change.IfDe > 1.0, elastic effects are dominant, whereas if De < 0.5, viscous effects prevail. For any values of Deborah numbers other than these two extremes, the materials depict viscoelastic behavior. Thermoplastic meltsdisplay the ability to recoil by virtue of their viscoelastic nature.However,they do not return completely to their original state when 53

Chapter

stretched because of their fading memory. Viscoelasticity allows the material to remember where it came from, but the memory of its recent configurations is always much better than that of its bygonepast, thus lending it the characteristics of a fading memory. Meissner [3] found that a filament of LDPE at 150°C, which is stretched rapidly from 1to 30 cm in length, and then suddenly set free, recovers to a length of 3 cm. Thus, recoil in molten thermoplastics is quite often large to the extent that a recovery of a factor of 10 can occur.

2.1FLOWCLASSIFICATION Flow is classified broadly as shear flowand extensional flow. A catalog of various types of shear flows has been given by Bird et al. [4]. In the present book, the discussion is restricted mainly to simple shear flow that occurs when a fluid is held between two parallel plates. Simple shear flow could be of the steady or unsteady type. Thus, flow is classified here under three headings: 1. SteadySimpleShear Flow 2. Unsteady SimpleShear Flow Extensional Flow

The definitionsof important rheological parameters under each ofthe three headings are given below.

2.1

Steady Simple Shear Flow

Fluid deformation under steady simple shear flow can be aptly described by considering the situation in Fig. 2.1 wherein the fluid is held between two large parallel plates separated by a small gap dx, and sheared as shown. The lower plate is moving at a constant velocity while the upper plate is moving at a constant velocity of + under the action of a force f applied to it. A thin layer of fluid adjacent to each plate moves at the same velocity as the plate, assuming the no-slip condition at the solid boundary. Molecules in the fluid layers between these two plates move at velocities which are intermediate between and + Under steady-state conditions, the forcefrequired to produce the motion becomes constantand is related to the velocity. The velocity profile of the fluid within the gap is given by = -$ dr,, where -$ is a constant.

Shear Rate The velocity gradient written as

which is termed the shear rate -$ can also be

Fundamentals Rheology Melt of Polymer

Figure

Simple shear flow

55

a fluid trapped between two parallel plates.

The term &cl/& represents the deformation of the material and is defined as the shear strain y. Thus, the shear rate is the rate of deformation or the rate of shear strain and is expressed as reciprocal seconds (S-’).

B. Shear Stress and Extra Stress Tensor The force per unit area required to shear the material between two parallel plates is defined as the shear stress T ~ and ~ ,it is basically a function (L,,) of the velocity gradient. Thus, 721

=f =L*@$

The units of shear stress are dynes/cm2 or Newtons/m2 or Pascals. It must be noted that T~~is just one component of the stress and, in principle, there are a number of components of stress that must be specified to completely define the state of stress. For example, a general constitutive equation which describes the mechanics of materials in classical fluid dynamics can be written as

-

T = -p!

+ 7 + qV(tr 7

Chapter

where F@, t) denotes the symmetric-Cauchy stress tensor at position and time t, p@, t) is the pressure in the fluid [? bang the unit tensor], 7 is the extra stress tensor, qvis the volume viscosity, and D is the symmetric part of the gradient tensor of the velocity field V@, t):

Note that Cartesiancoordinates are used; vectors are denoted by single bar above the letter, whereas tensors are denoted by double bars above the letter. If the fluid does not undergo a volume change (i.e., it is density preserving or incompressible), then the mass balance equation, better known as the continuity equation, reduces to

In such cases, the last term on the right-hand side of Eq. (2.3) drops out and the volume viscosity has no role to play. It should also be noted that, for flow of an incompressible fluid, the absolute value of the pressure p has no significance because it is only the pressure differences that are truly relevant. Thus, in essence, the constitutive Equation (2.3) for an incompressible fluid connects only the “extra stress tensor” ? or p7 uniquely with the local motion of the fluid but always leaves the pressure p indeterminate. In steady shearing flow, only a limited number of stress components of the extra stress tensor are necessary to completely define the fluid motion and these are written as follows:

F+

The subscript 1 denotes the direction of flow, the subscript 2 denotes the direction perpendicular to the flow (i.e., the direction along the velocity gradient), and the subscript 3 denotes the neutral direction. The various stress components are shown on a representative cubic volume of the fluid in Fig. 2.2. All the components are not shown in the figure in order to maintain clarity. Note that in steady shearingflow, the stress components 713, 731, and 732 are identically equal to zero. T~~ = T~~is called the shear stress and T ~ T~ ~, and , T~~ are called normal stresses.

C. NormalStressDifference The absolute value of any particular component of normal stress is not measurable and is of no rheological relevance, whereas the values of the normal stress

Fundamentalsof Polymer Melt Rheology

Figure 2.2 Various components T ~ ~ , figure).

components on a representative cubic volume of fluid. (Stress and T~~ T ~have ~ ,not beenT shown ~ ~ ,in order to maintain clarity of the

- and T= - 733do have considerablerheological significance. differences The first is termed the primary normal stress difference; the latter is termed the secondary normal stress difference. Thus,

For most fluids, Nl >> N2 and, thus, the latter is often excluded in rheological discussions. Attempts to determine the value of secondary normal stress difference experimentally have beenmade by several rheologists, but withoutsuccess. It is still a challenge to quantitatively determine this material function. Nevertheless, it is not very important in most hydrodynamic calculations barring, of course, wire coating [5] wherein the secondary normal stress difference helps provide the necessary restoring force for stabilizing the wire position whenever it becomes off-centered.

sa

Chapter

ViscometricFunctions The viscosity function -q (referred to as the steady shear viscosity), the primary and secondary normal stress coefficients +land +2, respectively, are the three viscometric functions which completely determinethe state of stress inany rheologically steady shear flow. They are defined as follows: 712

= 721 =

(2.9) (2.10) (2.11)

Viscosity is the resistance of the material to any irreversible positional change of its volume elements while the normal stress coefficients exemplify the response of the material due to its elasticity or its ability to recover from the deformation.

2.1.2

Unsteady Simple Shear Flow

Unsteady simple shear flow would occur when the stresses involved are time dependent. Small-amplitude oscillatory flow, stress growth, stress relaxation, creep, and constrained recoil are some examples of such types of flow [4]. In the following, small-amplitude oscillatory flow is treatedinsufficient detail, whereas others are described briefly; readers are encouraged to refer to Ref. 4 for more information.

A. Small-AmplitudeOscillatoryFlow Small-amplitudeoscillatory flow is often referred to as dynamic shear flow. Fluid deformation under dynamic simple shear flow can be described by considering the fluid within a small gap dx, between two large parallel plates of which the upper one undergoes small-amplitude oscillations in its plane with a frequency The velocity field within the gap can be given by dvl = i ) dr,, but is not a constant as in steady simple shear. Instead, it varies sinusoidally and is given by = qocos

(2.12)

The shear stress in simple dynamic shear flow is expressed in terms of the amplitude and phase shift functions of the frequency as (2.13)

(2.14) where 8 is the phase angle, qo and are the amplitudes of the strain and stress, respectively, and G', G' are linear viscoelastic material functions, respectively,

Fundamentalsof Polymer Melt Rheology

referred to as the dynamic storage modulus and dynamic loss modulus. (2.15)

7 0 2 1

Dynamic storage modulus: G'(@)= 7 YO

Dynamic loss modulus:

7021

G"(w) = - sin

(2.16)

Another term of importance is the ratio of loss to storage modulus, defined as

G"(0) Loss tangent: -= tan G It is also possible to define a dynamic complex viscosity in terms of G' and G" as follows: Gf'(0) =-

Dynamic viscosity:

(2.18)

W

Imaginary part of the complex viscosity: Complex viscosity function:

G'(w)

=-

(2.19)

W

q*(io) =

-

iq"(o)

(2.20)

In the same manner as above, a complex modulus can be defined as Complex modulus:

G*(io) = G'(w) + iG"(w)

(2.21)

The storage modulus G'(w) and the imaginary part of the complex viscosity, i.e., are to be considered as the elastic contributions to the complex functions. They are both measuresof energy storage.Similarly, the loss modulus G"(w) and the dynamic viscosity q'(w) are the viscous contributions or measures of energy dissipation.

B. Stress Growth The aim of a stress growth experiment is to observe how the stresses change with time as theyapproach their steady shear flowvalues. This is done by assuming that the fluid sample trapped in asmall gap between two parallel plates is at rest for all times previous to t = 0, implying that there are no stresses in the fluid when steady shear flow is initiated at t = 0. For t > 0 when a constant velocity gradient is imposed, the stress is monitored with respect to time until it reaches steady-state value.

C. StressRelaxation The aim of a stress relaxation experiment is to observe how the stresses decay with time (1)after cessation of steady shear flow or (2) after a sudden shearing displacement. In case 1 the fluid sample trapped in a small gap between two

Chapter

parallel plates is allowed to maintain the constant shear rate that was started long before t = 0 that all the transients during the stress growth period have evened out. Then at t = 0, the flow is stopped suddenly and the decay of the stress is monitored with respect to time until it becomes insignificant or dies out. The stress would relax monotonically to zero and more rapidly as the shear rate in the preceding steady shear flow is increased. In case 2, a constant shear rate lasting only for a brief time interval is imposed. The decay of the stress that is generated by this sudden small displacement is monitored. The stress would decrease monotonically with time. For small shear displacements, the relaxation modulus is known to be independent of shear rate.

D. Creep The aim of a creep experiment is to observe the changes in shear displacement as a function of time expressed in terms of creep compliance, after a constant shear stress has been applied and maintained at that value on a sample trapped in a small gap between two parallel plates. The steady-state compliance J , is defined as - ?hz1. If the driving shear stress T~~ is small enough, then the value of the compliance is independent of the driving shear stress.

E. ConstrainedRecoil The aim of a constrained recoil experiment is to observe the shear displacement in a fluid sample trapped in asmall gap between two parallel plates when driving shear stress is suddenly removed after steady state and then held at zero. The shear rate would then only be a function of time in the recoiling fluid. The ultimate recoil of the fluid at infinite time can be determined in this manner.

2.1.3

Extensional Flow

Problems associated with fiber spinning, film blowing, and the foaming process have indicated that the shear flow material functions discussed earlier are not truly the crucial parameters. This realization led to the study of another type of flow, namely, the extensional flow. Extensional flow differs from both steady and unsteady simple shear flows in that it is a shear-free flow. In such a flow, the volume of a fluid element must remain constant. Extensional flow can be visualized as that occurring when a material is longitudinally stretched as, for example, in fiber spinning.In this case, the extension occursin a single direction and, hence, the related flow is termed uniaxial extensional flow. Extension or stretching of polymer melts takes place in other polymer processes as well, such as film blowing and flat-film extrusion. In such cases, the extension occurs in two directions simultaneously; therefore the flow is referred to as biaxial extensional flow in one case and planar extensional flow in the other. Extensional

Fundamentals Rheology Melt of Polymer

61

flows can orient polymer chains and hence determine the performance and appearance of a product.

A. UniaxialExtension Uniaxial extensional flow may be best visualized as a deformation caused by forces acting in a direction perpendicular to the opposite faces of a cylindrical body, as shown in Fig. 2.3. The velocity field in simple uniaxial extensional flow is given by = &xl,

= -;a,,

(2.22)

=

where C is the uniaxial extensional rate. For such a flow field, the rate of deformation tensor is given as € 0

0

0 -;€

0

0

(2.23)

0

-;€

x2

x, Flgure 2.3 Schematic diagram of a fluid element in uniaxial extensional flow.

Chapter

in which (2.24) The uniaxial extensional rate may be constant or vary in the x1 direction of flow. When i is constant; that is, when the axial velocity is proportional to xl, the resulting flow is steady uniaxial extensional flow. In such a flow situation, a cylindrical rod of length 4 is stretched along its axis according to de dt

- = €4 Integrating this equation for a constant strain rate gives

4 = 4, exp(€t)

(2.26)

From Eq. (2.26), it is evident that extensional flow involves severe deformation because fluid parts are separated exponentially. The dimensions of thefluid elements change drastically in contrast with shearflows where particles in neighboring shearing surfaces separate linearly in time.

B. BiaxialExtension In biaxial extensional flow, too, the dimensions of the fluid elements change drastically, but they change in two directions as against the one direction in uniaxial extensional flow. Thus, biaxial extensional flow can be visualized as a deformation caused by forces acting in two directions perpendicular to the opposite faces of a plate as shown in Fig. 2.4. The velocity field in simple biaxial extensional flow is given by v1 = € g ,

= €g*

v, = -2€g3

(2.27)

where is the biaxial extensional rate. For such aflow field, the rate of deformation tensor is given as

(2.28)

C. PlanarEktension Planar extensional flow is the kind of flow wherethere is no deformation in one direction and the velocity field is represented as v1 = €Sl,

v, =

- €g2,

v, = 0

(2.29)

Fundamentals of Polymer Melt Rheology

Figure

Schematic diagram of a fluid element in biaxial extensional flow.

where is the planar extensional rate. In this case, the rate of deformation tensor is given as

-

D=

kP

0 0

0

0

-ip 0 0

0

Extensive reviews [6-101andamonograph [l11 summarize the literature covering significant aspects of extensional flows in various commercial processes, theoretical treatment for the hydrodynamics of such flows, and different methods of determining material functions such as uniaxial, biaxial, and planar extensional viscosities. The material function of prime importance in extensional flow is the extensional viscosity which is basically a measure of the resistance of the material to flow when stress is applied to extend it. In extensional flow, the diagonal components of T~~are nonzero (i.e., T~~= 0 for i # In the case of uniaxial extension, T~~is the primary stress that can

Chapter

be measured, whereas r Z and T~~are generally equal to the pressure of the environment. Thus, the uniaxial extensional viscosity is definedby

-

=

-

=

(2.31)

By the same token, the biaxial extensional viscosity

- rI1= ~3~ -

=-

qE6

can be defined as (2.32)

T)EB(€B)$

and further, the planar extensional viscosity q E p can be written as rll

- 722 = YEP('&)&

(2.33)

2.2 NON-NEWTONIAN FLOW BEHAVIOR The viscoelastic nature of polymer melts and their pecularities in the viscous as well as elastic response to deformation under applied stresses bring them under the category of non-Newtonian fluids. There is a distinctive difference in flow behavior between Newtonianandnon-Newtonian fluids to an extent that, at times, certain aspects of non-Newtonian flow behavior may seem abnormal or even paradoxical [12-161. An interesting movie about polymer fluid mechanics has been prepared [l71 which clearly depicts certain peculiarities of such fluids. The dramatic differences between the qualitative responses of Newtonian and non-Newtonian fluids grossly affect the industrial and practical applications of them.

Newtonian Fluids

2.2.1

Isaac Newton was the first to propose the basic law of viscosity describing the flow behavior of an ideal liquid as

-

(2.34)

where the constant qois called the Newtonian viscosity. Fluids, whoseflowbehaviorfollows the above constitutive equation are known as Newtonian fluids. Some of the common Newtonianfluids with which most people are familiar are water (qo 1 cP), coffee cream (qo 10 cP), olive oil (qo FJ 10' cP), and honey (qo lo4 cP). For a Newtonian fluid, (2.34) yields the following stress components in simple shear flow:

-

=

=

=

-p

(2.35)

=

=

= qop

(2.36)

All other stress components vanish. According to Eq. (2.35), it can be seen that the three normal stress components are equal. The nonvanishing shear stress T

Fundamentals Rheology Melt of Polymer

65

varies linearly with shear rate and has a proportionality constant qo,which is the shear viscosity of the Newtonian fluid. In general, incompressible Newtonian fluids at constant temperature can be characterized by just two material constants: the shear viscosity qoand the density Once these quantities are measured, the velocity distribution andthe stresses in the fluid can, in principle, be found for any flow situation. In other words, different isothermal experiments on aNewtonianfluidwouldyielda single constant material property, namely, its viscosity. On the other hand, a variety of flow experiments performed on a thermoplastic melt, which is a nonNewtonian fluid, would yield a host of material functions that depend on shear rate, frequency, and time.

Non-Newtonian Non-Newtonian fluids are rheologically complex fluids that exhibit one of the following features:

1. Shear-rate-dependent viscosities in certain shear rate ranges with or without the presence of an accompanying elastic solidlike behavior. 2. Yield stress with or without the presence of shear-rate-dependent viscosities. 3. Time-dependent viscosities at fixed shear rates. The definitions of various typesof non-Newtonian fluids along with examples of common realsystems falling in each categoryare given in Table 2.1. Detailed discussions relating to non-Newtonian fluids are available in a number of books [B-271 as well as other review articles FromTable2.1,itcan beseen thatpolymer melts fall within the nonNewtonian category of pseudoplastic fluids and viscoelastic fluids. In the case that polymer melts are loaded with fillers, their flow behavior would depict pseudoplasticity with yield stress, thixotropy, and viscoelasticity. For pseudoplastic fluids, the shear rate at any given point is solely dependent on the instantaneous shear stress, and the duration of shear does not play any role far as the viscosity is concerned. The shearstress versus shearrate pattern for a pseudoplastic fluid with and without yield stress is shown in Fig. 2.5. In the case of thixotropic fluids, the shear rate is a function of the magnitude and duration of shear as well as a function possibly of the time lapse between consecutive applications of shear stress. The shear-stress pattern with time for such fluids is shown in Fig. 2.6. If the shear stress is measured against shear rate which is steadily increasing from zero to a maximum value and then immediately decreasing steadily to zero, a hysteresis loop is obtained as shownin Fig. 2.7. Viscoelastic fluids have a certain amount of energy stored in the fluids as strain energy, thereby showing a partial elastic recovery upon the removal of a

Table 2.1 VariousTypesofNon-NewtonianFluids ~

Fluid type Pseudoplastic

Dilatant

Bingham plastics

Pseudoplastic with a yield stress Thixotropic

Rheopectic

Viscoelastic

~~

~~~~

Definition

Typical examples

These fluids depict a decrease in viscosity with increasing shear rate and, hence, are often referred to as shearthinning fluids. These fluids depict an increase in viscosity with increasing shear rate and, hence, are often referred to as shearthickening fluids. These fluids do not flow unless the stress applied exceeds a certain minimum value referred to as the yield stress and then show a linear shear stress versus shear rate relationship.

Polymer melts Polymer solutions Printing inks Pharmaceutical preparations Blood Wet sand Starch suspensions Gum solutions Aqueous suspension of titanium dioxide Thickened hydrocarbon greases Certain asphalts and bitumen Water suspensions of claylfly ash/metallic oxides Sewage sludges Jellies Tomato ketchup Toothpaste Filled polymer melts Heavy crude oils with high wax content

These fluids have a nonlinear shear stress versus shear rate relationship in addition to the presence of a yield stress. These fluids exhibit a reversible decrease in shear stress with time at a constant rate of shear and fixed temperature. The shear stress, of course, approaches some limiting value. These fluids exhibit a reversible increase in shear stress with time at a constant rate of shear and fixed temperature. At any given shear rate, the shear stress increases to approach an asymptotic maximum value. These fluids possess the added feature of elasticity apart from viscosity. These fluids exhibit process properties which lie in-between those of viscous liquids and elastic solids.

Filled polymer melts Water suspensions of bentonite clays Drilling muds Crude oils Coal-water slurries Yoghurt Salad dressing Mayonnaise Some clay suspensions

Polymer melts Filled polymer melts Polymer solutions

Fundamentals of Polymer Melt Rheology

PSEUOOPLASTIC FLUIDWITH Y I E L D STRESS

T Y PSEUDOPLASTIC F L U ID NEWTONIAN FLUID

k Figure Variation of shear stress versus shear rate for pseudoplastic fluids with and without yield stress.

Figure

Variation of shear stress with time for a thixotropic fluid.

Chapter 2

Figure Variation of shear stress with shear rate (which zero to maximum and brought a thixotropic fluid.

steadily increased from

deforming stress. At every instant during thedeformation process, a viscoelastic fluid ties unsuccessfully to recover completely from the deformed state but lags behind. This lag is a measure of the elasticity or so-called memory of the fluid. Due to the presence of elasticity, viscoelastic fluids show some markedly peculiar steady state and transient flow behavior patterns. Viscoelastic effects become important when there are sudden changes in the flow rate (e.g., during start-up and stopping operations of the polymerprocess),inhigh-shear-rate flows (e.g., in processes like extrusion and injection molding),andinflows where changes in cross section are encountered (e.g., entry into the mold cavity during injection molding). Some of the commonencountered effects due to viscoelasticity are discussed below.

2.2.3

ViscoelasticEffects

When a viscoelastic fluid is stirred with a rod at moderate speeds, the fluid begins to climb up the rod instead of forming a vortex. The centrifugal force, during the stirring process, ties to pull the viscoelastic fluid toward the wall, but the elastic forces induce a recovery to this deformation. When the elastic

Fundamentals Rheology Melt of Polymer

69

forces attain a magnitude greater than the centrifugal forces, the viscoelastic fluid begins to climb up the stirrer rod, as can be seen in Fig. This type of phenomenon is commonly called the Weissenberg Effect, ,as Weissenberg was the first to explain such an effect in terms of the stresses in fluids undergoing a steady shear flow In actuality, this effect was observed earlier by Garner and Nissan

B.

Swell

When a viscoelastic fluid flows through an orifice or a capillary, the diameter of the fluid at the die exit is considerably higher than the diameter of the orifice. This happens because,at the die exit, the viscoelastic fluid partially recovers the deformation it underwent when it was squeezed through the capillary. This type of phenomenon is as extrudate swell, die swell, jet swell, Barus effect, or Merrington effect. Metzner discusses the history of extrudate swell and argues against using the last two terms. review on extrudate swell has been given by Bagley and Schrieber An extrudate diameter (DE)of up to three or four times the orifice diameter (Do)are possible with some polymers. The swell ratio S, (i.e., &/Do) decreases with the increase of tube length because of the fading memory of the viscoelastic fluid to deformation. This implies that if longer and longer tubes are used, S, should ultimately approach unity. But it is known that the limiting value of the swell ratio is greater than unity even as the length-to-diameter ratio of the orifice approaches infinity. The phenomenon of die swell is shown in Fig.

7

6

Figure Weissenberg Effect showing how the viscoelasticfluid climbs up the stirrer rod when stirred at moderate speeds. (From Ref.

.

Chapter

Figure2.9 Extrudate swell effect showinghow the viscoelastic fluid swellsin diameter when it exits from a die (From Ref. 34.)

Theoretical analyses of this phenomenon, for flow in round capillaries, are available in which the most basic of them is built on the freerecovery calculations set down by Lodge [l31 using the theory of Berstein et al. The developed expression for die swell S,,, in which the elastic strain recovery S, is balanced by the shear stresses arising in the die is given by S, = (1

+ ;S:)'" + 0.1

where (2.38)

The above analysis does not include the rearrangementof the stress and velocity fields at the die exit and, consequently, it was found necessary to

Fundamentals Rheology Melt of Polymer

71

empirically modify the die swell expression by including a factor of 0.1 in the above expression. The 0.1 term has been added to improve the fit with data for small values of ( T ~ ~ and the ratio ( T ~ ~ T~~)/I% has been taken to be constant. Later work has shown that die swell depends not only on the recoverable shear strain but also on the ratio of the second to first normal stress difference coefficients as well. The influence of this phenomenon in the thermoplastics industry can hardly be overlooked. The industrial problems involvingextrudate swell are particularly complex and challenging becausethe diameter increase depends not only on the particular type of the thermoplastic but also on operating conditions such as temperature and flow rate.

C. DrawResonance Draw resonance, or surging, is defined as the nonuniformity in the diameter of the extrudate when a polymer melt is stretched at different take-up speeds as it comes out of an orifice. This phenomenon is shown schematically in Fig. 2.10. When the take-up speed is small or when there is no stretching, only die swell is observed, as can be seen from Fig. 2.10a. When take-up speed is higher and the stretched extrudate is solidified by quenching, the contour appears as shown in Fig. 2.10b. Now the draw ratio is defined as the ratio of the linear velocity V of the extrudate settled in the quenching bath to the smallest linear velocity V, in the die swell region. When the draw ratio @R) goes beyond a critical value D&, then the resulting phenomenon is draw resonance as shown in Fig.

Figure Draw resonance effect occurring when polymer melt extruded from an orificeatvarioustake-upspeeds.(a)Extrudatewithoutstretching; (b) extrudatewith stretching DR c DR, andnodrawresonance; (c) extrudatewithStretchingDR > DR, showing draw resonance.

Chapter

2.10~.The theoryof draw resonance has beendevelopedandamethod for calculating the critical draw ratio is also available Once draw resonance occurs, its severity enhances with increasing take-up speed.

D. MeltFracture When a polymer melt flows out of a capillary, a striking phenomenon of the distortion of emerging polymer stream is observed at shear stresses beyond a higher critical value andthis is termedmelt fracture The extrudate distortion is a result of polymer molecules reachingtheir elastic limit of storing energy, thus causing melt fracture as a means of stress relief either at the capillary wall or at the capillary entrance. Another view is that the extrudate distortion is due to differential flow-induced molecular orientation between the extrudate skin holding highly oriented molecules and the core wherein there is no significant molecular orientation. It is, of course, possible that the melt fracture occurs due to a combination of the stress relief theory and the differential flow-induced molecular orientation. A number of other mechanisms have been suggested for melt fiacture. Based on a stick-slip mechanism, it is purported that, above a critical shear stress, the polymer melt experiences intermittent slipping due to a lack of adhesion between the melt and die wall, in order to relieve the excessive deformation energy adsorbed during the flow. The stick-slip mechanism has attracted a lot of attention both theoretically and experimentally. The other school of thought is based on thermodynamic argument, according to which meltfracture can initiate anywhere in the flow field when reduction in the fluid entropy due to molecular orientation reaches a critical value beyond which the second law ofthermodynamics is violatedandflow instability is induced distortion It is important todistinguish between meltfracture, which is a or waviness, and a fine-scale high-frequency surface roughness The latter may commence at output rates below those at which melt fracture is observed and is called matte or mattness. The extreme case of mattness is referred to as shark skin. The distinction between shark skin and melt fracture has been convincingly demonstrated as shown in Fig. 2.11.

E. CapillaryEntryFlowPatterns characteristic flow pattern at the capillary entrance developswhen a polymer melt flows at high shear rates from a cylindrical reservoir through a capillary or die, as shown in Fig. 2.12. The qualitative difference between the capillary entry flows of linear and branched polyethylenes has been convincingly presented by Tordella and discussed by others For linear polymers, the converging flow at the die entry fillsthe available space, whereas for branched polymers there is a large dead space filled by recirculating vortices.

FundamentalsMelt of Polymer

Rheology

73

Figure Difference between the phenomenon of matte and melt fracture (on distorted extrudates of different polymers): (1)rigid polyvinyl chloride, (2) polyethylene, polypropylene, and 5 ) polypropylene viewed from two angles; (6) poly methyl methacrylate; (7) polytetrduoroethylene. (From Ref. 66.)

Vortices are induced by the viscoelastic characteristics of the converging fluid [71,72]. When extensional viscosity values are large, the balance of forces makes radial flow impossible, thereby giving rise to secondary flows in the form of vortices in order to provide stress relief. In fact, polymer melts exhibiting larger extensional viscosities have been observed [71] to exhibit larger vortices and vice versa. The vortexor the.circulating stagnant region encompasses a flow cone which becomes unstable with increasing flow rate and eventually fractures periodically as the flow rate is increased further. When the flow cone fractures, the result is melt fracture and the flow is sustained by the intermittent drawing of the fluid from the recirculating vortices.

Chapter

Figure Capillary entry flow pattern for a branchedpolymershowing cone and the recirculating vortex.

the flow

During the process of calendering, very stable abnormal fringe patterns may appear on the roll surface at regular intervals. Although the exact mechanism for abnormal fringe patterns in calendering is as yet unclear, it is certainly related to the viscoelasticity of the material. Depending on the frequency of roll rotation and clearance of roll nip, its intensity would increase or decrease due to the effect of such changes onthe viscoelastic response of the calendered material. For Newtonian fluids, the pressure measured at the bottom of the pressure hole pMis the same as the true pressure p at the wall. For a viscoelastic fluid, on the other hand, the pressure (p T& measured at the bottom of the pressure hole is always lower than the true pressure (p T ~ at ~ the ) wall, no matter how small the hole. This pressure difference arises because the elastic forces tend to pull the fluid away from the hole and results in the pressure-hole error pH= ( p + T&, - (p T ~ ) .This effect is illustrated in Fig. 2.13. The possible sources of error in the measurement have been considered by Higashitani and Lodge [73] along with a review of published data. The effect of pH has been well substantiated for polymer solutions, but the same is not the case with polymer melts.

+

+

When a viscoelastic fluid is trapped betweentwo parallel plates with one of the plates rotating, then there is a nonzero pressure p due to elasticity which tends to separate the two plates. This effect is illustrated in Fig. 2.14.

Fundamentalsof Polymer Melt Rheology

\ VISCOELASTIC

Figure 2.13 Pressure-holeerror tonian fluid. (From Ref. 23.)

in a viscoelastic fluid; it is absent in a New-

During the siphoning process, when the siphon tube is lifted out of the fluid, a Newtonian fluid wilI stop flowing, whereas a viscoelastic fluid will continue unabated. At times, even of the container filled with a viscoelastic fluid may get siphoned out in this manner. This effect is illustrated in Fig. 2.15. This viscoelastic effect indicates the stability of a stretching filament of fluid with respect to small perturbations in its cross-sectional area.It has definite implications in the fiber spinability of polymer melts. It has been observed that when a polymeric fluid flows in a tube with a sudden contraction, large bubbles of the order of one-sixth one-eighth of thesmall-tube diameter come to asudden stop right at the entrance of the

Figure Parallel-plateseparationoccurs in aviscoelasticfluid;itisabsentin Newtonianfluid.(ReprintedfromRef. 33 withpermissionfromGulfPublishing Houston, Texas.)

a Co.,

2

'hbeless siphoningcan be done for aviscoelasticfluidbutnot Newtonian fluid. (From Ref. 23.)

for a

contraction along the centerline before finally passing through after a hold time of about 1 min. This particular behavior has beentermed the Uebler effect [74,75].

This phenomenon has implications in the productionoffoamed plastics wherein a gas, normally nitrogen, is added to polymers such as PE, PP, and PS during two-phase processing.

2.3 RHEOLOGICALMODELS There have been a number of rheological models proposed for representing the flow behavior of polymer melts and these are readily available in a number of books [M-271 and review 'articles [10,28,29,31,32,76]. The constitutive equations, which relate shear stress or apparentviscosity with shear rate, involve the use of two to five parameters. Many of these constitutive equations are quite cumbersome to use in engineering analyses and hence only a few models are often popular. Only such models are described and discussed in this section.

2.3.1

Models for the Steady Shear Viscosity Function

From the typical viscosity versus shear rate curve for polymer melt shown in Fig. 2.16, it can be seen that in the low-shear-rate range, the melt is essentially Newtonian in flow behavior with a constant apparent viscosity, which at zero shear rate is called the zero-shear viscosity qo.In the medium-shear-rate range, the apparent viscosity q begins to decrease, depicting the shear-thinning characteristic until it stabilizes to a constant value q.. at a considerably high shear rate in the upper Newtonian region. It is quite obvious from this figure that a constitutive equation with about three to four parameters would be necessary to describe the rheological behavior of a polymer melt over the entire shear-rate range. However, when dealing with processingproblems, only certain shear-rate

Fundamentalsof Polymer Melt Rheology

I

Zero-shear limit

log-log

k Figure Typical viscosity versus shear rate curve depicting the method for determining the parameters of the General Rheological Model. (FromRef.

ranges attain significance and hence only portions of the flow curve need to be described by the constitutive equations,thereby requiring fewer parameters. Some such simple flow models are discussed below. This is the most popular and simple two-parameter model originally proposed by Ostwald [77,78] and de Waele and has since then been fully described by Reiner The equation for this model is given as follows:

or =

a

x

where reflects the consistency index of the polymer melt, with higher values representative of more viscous materials, and n is the power-law index giving a measure of the pseudoplasticity, with greater departures from unity showing more pronounced shear-thinning characteristics. The power-law index n basically represents the slope of the versus p curve and n - l is the negative slope of the q versus curve in the medium- to high-shear-rate range.

Chapter

B. EllisModel In this model proposed by Ellis and discussed by Reiner [l],the apparent cosity versus shear rate relationship is given in the following form: =1

(2.41)

+ (T/TIn>""'

where is the shear stress at the viscosity of tqo. The Ellis model is a threeparameter model and has the advantage of exhibiting a limiting viscosity qoin the limit of zero shear rate and shear-thinning viscosity at higher shear rates. a' - 1 is related to the slope of the viscosity versusshear rate curve and describes the shear-thinning behavior. The model is able to fit data in the lowto medium-shear-rate ranges. The model has an added advantage because the ratio q,,/Tu2 constitutes a characteristic time of the fluid and is often considered to be related to the fluid elasticity, an idea which was intuitively asserted by Bird [Sl]. There are certain controversies with regard to the ability of the relaxation time constant of Ellis fluid in truly describing elastic behavior [82], but the simplicity of its constitutive equation more than overrides any deficiencies that this model may have in representing a viscoelastic fluid. In fact, in the years following the controversy, there have been studies [83-851 in support of using the Ellis model to determine the viscoelastic behavioral patterns in dif€erent flow situations.

C. CarreauModel The Carreau model [86] has basically four parameters, namely, qo,qm,A, and

N.The relaxation time A is considered to be the characteristic time available as the inverse of the shear rate at which the shear-thinning behavior begins. N is a measure of the shear-thinning characteristics. Both A and N are considered to be adjustable parameters and the model is written

- q m = (Yo

-

(1 + A' **)-"

(2.42)

In the above form,the Carrean model canbe fitted to the entire viscosity versus shear rate curve. However, such a complete set of data up to q.. is rarely determinable.Hence, the popular form of theCarreanmodelthat is used as the truncated three-parameter model after neglecting q., is given below: 3 = qo(1

+ x2 Y )-N

(2.43)

D. GeneralRheologicalModel This model has been formulated by Shenoy et al. along the lines originally proposed by Churchill and Churchill [88], but with the following differences: 1. The model [87] relates viscosity q with shear rate p, whereas Churchill and

Fundamentals Rheology Melt of Polymer

79

Churchill prefer to work with a model relating viscosity q and shear stress T. The power-law model is used as the limiting behavior for high shear whereas q.. is assumed to be a limiting value as T (or 9 in the latter The model requires four constants, whereas the latter involves five constants in its most general form. The Churchill and Churchill model essentially requires postulating the asymptotic behavior at either large or low shear stress (or rate) because the two limiting values are insufficient information to apply the Churchill and Usagi approach The General Rheological Model was basically developed for master curves of viscosity versus shearrate for polymer melts usingthe melt-flow index (MFI) as a normalizing parameter. It can be written in a general formapplicable to any viscosity versus shear rate curve of a polymer melt simply by putting MFI as unity to give

plot of q versus 9 on a log-log scale for low to medium shear rate yields the limiting zero shear viscosity qoby mere readout as shown in Fig. The functional behavior at large shear rates on the same log-log plot in Fig. being linear defines R and n directly. The slope of the straight line defines n - 1, and R is the value of q when 4 = (provided the point satisfies the power-law equation). The best values of K and n - can be computed easily by regressional analysis of the data at higher shear rates. The exponent P is readily evaluated by determining the point of intersection qo). Thus, of the two limiting solutions corresponding to the point

2.3.2

Model for the Normal Stress Difference Function

The elasticity of polymer melts is manifested through two material functions, namely, the primary normal stress coefficient and the secondary normalstress coefficient The secondary normal stress coefficient is not as well characterized as the primary normal stress coefficient due to its small magnitude. The primary normal stress measurements are themselves difficult and require highly

Chapter

sophisticated equipment to produce reasonably accurate data, and that too in a limited low-shear-rate region. The relative ease in the experimental measurements of viscosity functions renders them amenable to extensive study in comparison to the normal stress functions. Hence,therehavebeen attempts to find methods for the prediction of the normal stress difEerence from the viscosity function. presented a relation between the Abdel-Khalik et al. and Bud et al. steady-state values of the primary normal stress coefficient and the viscosity function as

Wagner too provided a method for the prediction of normal stress difference from shear viscosity data using a strain-dependent single integral constitutive equation of Berstein et al. type as follows:

where m is an adjustable parameter whose value lies between 0.13 and 0.2, thereby defining the upper and lower limits of the predicted normal stress difference curve. The sameset of experimental data taken from theliterature has been used by Abdel-Khalik et al. and Wagner and these have been shown to agree reasonably well with the theoretical predictions. is used for obtaining the relationship between In the following, Eq. primary normal stress coefficient and theshear rate. One of themodels described earlier, namely, the Carreau model, is used for the viscosity function. Thus, from Eqs. and the primary normal stress coefficient can be readily obtained as

Jr,

The above expressionwould be valid within the low to medium shear-rate range (i.e., 0 p 10 Otherexpressions for couldhavebeen obtained by considering the other viscosity functions described in the earlier section. ever, this has not been done because the effective upper limit of shear rate of not greater than for actual primary normal stress experimental measurements lies in a range consistent with only the Carreau model. In case measurements of primary normal stress at higher shear rates ever become possible in the future due to improved sophistication in experimental techniques, it is recommended that from the General Rheological Model be used

Fundamentalsof

Melt Rheology

81

alongwith Eq.(2.47) to obtain anew expression for Suchanexpression would then be valid over a larger shear-rate region ranging from low to high. It has been established [99] that the primary normal stress difference exhibits a strong dependence onmolecular-weight distribution as predicted from the theory of second-order fluids. Thus, the following expressionis known [99] to hold (2.49) where the form for the steady-state compliance J. is given [99-1041 as (2.50)

2.3.3

Model for the Complex Viscosity Function

Earlier investigations of the rheological properties of polymer melts have shown that the data under dynamic conditions can be related to that obtained under steady shear withincertain ranges of shear rates and frequencies. Although there is a considerable amount of literature advocating ananalogy between the steady and dynamic measurements, there is no generally accepted method. The problem which arises in comparing dynamicand steady-state data is in the choice of the appropriate rheological model for calculating the material parameters from the measured values. Of all the diverse methods of correlating dynamic and steadystate data based on theoretical models available in the literature, the empirical method suggested by Cox and Mertz [l051 for relating steady-shear viscosity with the absolute value of complex viscosity q* is still the most attractive of all. According to the Cox-Mertz method, = Irl*l(o) at

=

"

(2.51)

The relationship simply indicates that for prediction purposes, the magnitude of complex viscosity is comparable with that of shear viscosity at equal values of frequency and shear rate. The relationship has been found to hold largely for flexible-chain thermoplastic melts,particularlyin the lower and intermediate ranges of o and Combining Eqs. (2.43) and (2.51), the following expression for the complex viscosity q* based on a modification of the Carreau model is written q* = q$(l

+h2oy

(2.52)

where q$ is the zero-frequency viscosity function. similar expression can be derivedbasedonamodificationof the General Rheological Model [87] by combining Eqs. (2.44) and (2.51).

Chapter 2

2.3.4

Model for the Dynamic Modulus Functions

Relating the steady-state normal stress difference T~~ - T~~ and the dynamic storage modulus G', both characterizing the elasticity of the material or the ability to store the energy of external forces, has invariably been a more difficult task and has been the subject of controversy. Although the shapes of the T~~T~ versus and the G' versus curves are generally similar, a relationship between the two has to bebased on a suitable rheological model derived through an appropriate constitutive equation such as was doneby Pao [106,107], Spriggs [108], Bogue [log], or Meister [110]. The results obtained by Pa0 [106,107] for correlating the dynamic functions and stresses in the steady state are limited to only low-shear or low-frequencydata. The adjustable parameters among the material constants of the Spriggs [108], Bogue [log], and Meister [l101 models make them more adaptable for wide ranges of shear rate or frequency data. The Bogue[l091 and Meister[l101models represent integral-type models,each having a different physical origin in its derivation. The Spriggs [l081 model is ofthe differential type and involves material constants which are simple to determine and also have relevance to molecular parameters. In the present case, the Spriggs model has been chosen for correlating the dynamic and steady-state elastic characteristics. The choice is somewhat arbitrary as each of the models is known to have almost the same capability for prediction [l111 and by no means indicates the superiorityof this model over that of Bogue [l091 and Meister [110]. The dynamic functions that conform to the Spriggs model are expressed as follows: (2.53)

z

G''= To E(Z)

(oh)' j j+ ~ (oh)'

'" -

(2.54)

whereas the steady-state functions are given as (2.55) (2.56) are model parameters and is an arbitrary adjustable where qo,A, and constant expressed in terms of an independent parameter i? as

Fundamentals Rheology Melt of Polymer

a3

comparison of the above equations [Eqs. (2.53)-(2.56)] yields the following: (2.58) or

(2.59)

Thus, it is obvious that the dynamic and steady-state characteristics of a polymeric melt would be equivalent when appropriately shifted by an amount c relative to each other. In order to determine the shift factor the procedure suggested by Spriggs [l081 needs to be followed,namely, of superimposingthe plot of /qoversus on the plot of q'(o)/qoversus W. For example,a value of = 2/3 has been found [l121 to correlate the dynamic and steady-state viscoelastic data a particular grade of linear low-densitypolyethylene (LLDPE) over a wide range of shear rate and frequency. It is to be noted that the Spriggs model predicts a correlation at = o only in the case of e = -1 and hence = 1. This correlation is equivalent to that given by the phenomenological theory of Coleman and Markovitz [l131 at low shear rates and frequencies. Because the deformation at very low frequencies can be considered as nearly a steady-state flow, it is natural to expect that at = 1, a plot of versus j 2 superimposed on a plot of 2G' versus W' (with = o) would give a fit in the low-shear and low-frequency region. In the following can be this region, combining Ms. '(2.49, (2.51),and(2.59), written:

e

G'

1 - 2m'

" "

do

(2.60)

The derivation of the above equation is based on the assumption that = 1. However, even if this assumption is relaxed and # 1, Eq. (2.60) should hold, as m' is merely an adjustable parameter. Thus, the equation may be rewritten in more general termsthrough a new adjustable parameter m" incorporating the multiplying factor 2 in (2.60) as well as any anomalies that are introduced for # 1 as

1 G' 02-m"

" "

do

(2.61)

Chapter

Now, combining Eqs.

and

gives

Normally the adjustable parameter m” taken as a constant should suffice for theoretical fit of the storage modulus versus frequency curves usingEq. However, in certain cases such as in filled polymer melt systems, it becomes necessary to assume m” as an adjustable variable dependent exclusively on the frequency in order to get a good theoretical fit of the experimental data. Shenoy and Saini have chosen the following form based on the shape of the plot of m”“versus

m””=

+

C1

Substituting Eq. into dence of G“ on the frequency Because, by definition,

G” = V(q*w)’

would then give the appropriate depen-

- G”’

the relationship between the dynamic loss modulus G” and frequency easily established.

2.3.5

can be

Model for the ExtensionalViscosity Function

Using a converging flow analysis, Cogswell has shown that a relationship between the extensional viscosity and shear viscosity can be easily derived. In his view, as fluid flows from a reservoir into a die, the streamlines converge and the simple shear flow field gets superimposed by an extensional deformation. He suggested that each component can be studied separately and subsequently added to describe the total effect. The relationships between the various parameters of extensional flow and thoseof shear flow are given by Cogswell

-

” -

tan 8

and

where is the average extensional stress, & the average extensional rate, the shear stress at the die wall, the shear rate at the die wall, qEthe extensional

Fundamentals Rheology Melt of Polymer

85

viscosity, q the shear viscosity,and 0 the half-angleof convergence of the streamlines at the entrance of the die. Cogswell [71,115] has found good agreement between the calculated values of from a convergent-flow analysis and that measured in a steady-state experiment [99] using a constant stress melt tensile rheometer for poly methyl methacrylate (€"MA). Shroff et al. [l161 verified the predictions of the convergent-flow analysis of Cogswell with measured qEvalues fromthe isothermal melt-spinning experimentsof Han and Lamonte [l171 for polyolefins.

2.4

OTHER RELATIONSHIPS FOR SHEAR VISCOSITY FUNCTION

2.4.1 Viscosity-TemperatureRelationships An understanding of the mechanism of polymer melt-flow processes in relation

to the nature and composition of the material can be elucidated by a study of the temperature dependency of melt viscosity. The temperaturesensitivity of the melt viscosity has a profound effect on the choice of processing conditions as well as on the quality of the end product. An increase in temperature sets up thermal motion of the molecules, resulting in their displacement based on the available free motion and the overcoming of forces of intermolecblar interactions. Presently, there are two commonly used expressions to evaluate the temperature dependency of the viscosity-onebasedon free-volume concepts, namely, the equation proposedbyWilliams et al. (W-L-F) [118], and the second, of the Arrhenius type, based on the absolute theory of rate processes as derived by Eyring [119].

W-L-F Equation: (2.68) where q and qsare the viscosities at temperatures T and T, respectively; C: and C; are constants; and T, is the standard reference temperature taken as T, 50°K where T, is the 'glass-transition temperature. Modification of Eq. (2.68) using different constants and different characteristic temperatures have been proposed. But because T, is a practical and easily available parameter, Eq. (2.68) is used preferentially.

+

Arrhenius-Erying Equation: q

""P($)

(2.69)

Chapter

where is the viscosity at temperature T, R is the gas constant, is the frequency term depending on the entropy of activation of flow, and E is taken to be the energy of activation for viscous flow. The temperature dependenceof activation of flow process as defined by In q E=R-= 1/T

2.303 CIC;RP + T - T,)'

(C;

(2.70)

is predicted by the W-L-F equation, and rightly because free volume and its changes with temperature play a dominantrole at temperatures relatively near T,. At temperatures T > T, + 100, the temperature dependence of viscosity is decisively affected by the overcoming of the forces of intermolecular interactions, and hence the curved plot of In versus 1/T approaches linearity in short temperature ranges at around 100°C above the highest glass-transition temperature of the polymer. Normal processing ranges for most polymers, exceptingpolystyrene and polycarbonate(PC), fall withintemperatures (2.69) wouldthen much greater than TB + 100.For all suchsystems, provide a valid and useful means of predicting the viscosity-temperature dependence. E and in Eq. (2.69) vary from polymer to polymer and must be evaluated empirically for each polymer system investigated. Within the narrow width of the processing rangefor each polymer,E can be expectedto befairly constant, to give a single value for each polymer. However, as polymers are non-Newtonian materials, their viscosity at fixed temperature is dependent on shearing stress or shearingrate. Thus, in Eq. (2.69), which normally would be a constant for Newtonian materials, would additionally depend on shear stress or shear rate. Bestul and Belcher [l201 have shown mathematically that in the non-Newtonian (shear-thinning) region of polymer melt flow, a clear differentiation between E at a constant shear stress and E at a constant shear rate must be done. Thus, (2.71) (2.72) where E, > E;, for shear-thinning non-Newtonian viscosity, as can be shown easily both graphically and analytically. A number of authors [121-1241 have shown that E, remains constant over a broad range of temperature and shear stress, whereas E;, does not. Porter and Johnson [124], on evaluating E;, and E, for a variety ofpolymers studied by different authors [125-1311, concluded that E, remained constant from the low-shear Newtonian range up to shear stresses of -lo6 dydcm', well within the non-Newtonian region (see Table 2.2). From

Fundamentalsof Polymer Melt Rheology Flow Activation Energies Found Independent of Shear Stress in the Non-Newtonian Regions for Representative Amorphous Linear, Polymers

Table

EstressShearrange Temp.

e9

(kcaVmo1) Polymer PE Linear

6.3 7.1 5.8-8.2 6.8 921 23 23

PP PS ~

~~~~

150-300 125 150-300 126 150-220 127 163-274128 195-260 129 177-232 204-227131

(X Ref. l@ dyn/cm?

0.13-22.0 8.6 2.2-8.9 3.0-15 4.0-12.0 0.69-13.8 1.2-13

.

130

~

Source: Reproduced in part from Ref. 124.

a fundamental viewpoint, constant shear-rate “activation energies” are incorrect because the Arrhenius equation is a rate equation, and hence holding the rate process constant would make data treatment meaningless. Thus, strictly speaking, the validity of Eq. (2.69) is restricted to only constant shear stress although it has been extensively used for both E, and E?.

2.4.2

Wscosity-PressureRelationship

Thermoplastic meltviscosity also depends on pressure [132-1361. Viscosity generally increases with increasing pressure and can be correlated generally by an equation of the type (2.73)

where and B; are constants and p is the pressure. The pressure reduces free volume and, as aresult,it reduces molecularmobility;however, this effect becomes noticeable only at very high pressures. High pressure raises both TB and T, which also reflectsan increase inviscosity. In general, during most practical situations, thermoplastic melts are assumed incompressiblefor ease and simplification. Carley [l321 concluded that pressure effects are of minor significance in most processing situations, provided the temperature is not too close to transition. However, it is noticed in polymer processing operations that, the combination of high pressure and low temperature tends to promote crystallization [137,138], orientation [139], structural changes [140], productquality enhancements, and, in certain cases, unusual flow and defects [141].

Chapter

2.4.3 Viscosity-Molecular Weight Relationship Experiments have shown thatthe following relationship between zero-shear viscosity and molecular weight holds: qo= &Bw for Bw< Bwe

qo=

&a:5for

> Bwc

where BWc is the critical weight average molecular weight, thought to be the point at which molecular entanglement begins to dominate the rate of slippage of molecules.

Reiner, M., Deformation, Strainand Flow, Wdey-Interscience, New York Reiner, M., The Deborah number, Physics Today, Meissner, J., DehnungsverhaltenvonPolyathylen-Schmelzen, Rheol. Acta, Bird, R. B., Armstrong, R. C., and Hassager, Dynamics of Polymeric Liquids, Vol. l , Fluid Mechanics, Wiley, New York p. Tadmor, Z. and Bird, R. B., Rheological analysis of stabilizing forces in wirecoating dies, Polym. Eng. Sci., Cogswell, F. N., Polymer melt rheology during elongational flow, Appl. Polym. Symp.,

Dealy, J. M.,Extensional flow of non-Newtonian fluids-A review, Polym. Eng. Sci.,

Dealy, J. M., Extensional rheometers for molten polymers: A review,

J. Non-

Newtonian Fluid Mech.,

Hill, J. W. and Cuculo, J. A., Elongational flow behaviour of polymeric fluid, J. Macromol. Sci, Rev. Macromol. Chem.,

Bird, R. B., Polymer kinetic theories and elongational flows,

Chem. Eng. Com-

mm.,

Petrie, C. J. S., Elongational flows, Pitman,London Walker, J., Serious fun with Polyox, SillyPutty,Slime, and Other Non-Newtonian fluids, Sci. Am., Lodge, A. S., Elastic Liquids,Academic Press, New York Fredrickson, A. G., Principles and Applications of Rheology, Prentice-Hall, Englewood Cliffs, NJ Coleman,B.D.,Markowitz,H., and Noll,W., Vicometric Flows of NonNewtonian Fluids, Springer-Verlag, New York Truesdell, C., The Meaning of Viscometry in Fluid Dynamics, Ann. Rev.J?luid Mech., Markovitz, H., Rheological Behaviour of Fluids, Education Services, Inc., Watertown, MA

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Truesdell,C.andNoll,W., TheNon-LinearField of Mechanics, Springer-Verlag, Berlin Skelland, A. H. l?, Non-Newtonian Flow and Heat Transfer, Wiley,NewYork Astarita,G.and Mmcci, G., Principles of Non-Newtonian Fluid Mechanics, McGraw-Hill, London Darby, R., Viscoelastic Fluids, Marcel Dekker, New York Bird, R. B., Armstrong, R. C., and Hassager, O., Dynamics of Polymeric Liquids, Wiley, New York Vols. and Middleman, S., Fundamentals of Polymer Processing, McGraw-Hill, New York Schowalter, W., Mechanics of Non-Newtonian Fluids, Pergamon, Oxford Tanner, R. I., Engineering Rheology, Clarendon Press, Oxford Barnes, H., Hutton,J. F., and Walters,K, Introduction to Rheology, Elsevier, Amsterdam Rosen, M. R., Characterization of non-Newtonian flow,Po1ym.-Plastics Technol. Eng.,

Becker, E., Simple non-Newtonian fluid flows,

Adv. Appl. Mech.,

Schramm,G., Intmduction to Practical Viscometry, GebruderHaakeGmbH, Karlsruhe, Gemany Cheremisinoff, N. P., Properties and concepts of single fluid flows, in Encyclopedia of Fluid Mechanics,Gulf Publishing Co., Houston,TX Vol. Chap. pp. Cheremisinoff,N.P.,Rheologicalcharacterizationandprocessibilitytesting,in Encyclopedia of Fluid Mechanics,Gulf Publishing Houston, TX Vol. Chap. pp. Chung, J. T., Fundamentals of polymer materials, in Encyclopedia of Fluid Mechanics, Gulf Publishing Houston, TX Vol. Chap. pp. Walters, K, Rheometry, Chapman and Hall, London pp. and Weissenberg, K, Rep. Gen. Brit. Rheol. Club, p. Weissenberg, K., Acontinuum theory of rheological phenomena,Nature, Russell, R.J., Ph.D. Thesis, Imperial College, University of London Gamer, F. H. and Nissan, A. H., Rheological properties of high viscosity solution of long molecules, Nature, Metzner, B., Historical comments on stress relaxation following .steady flow through a duct or orifice, Trans. Rheol., Bagley, E.B. and Schrieber, H. P., Elastic effects in polymer extrusions, in RheVol. Chap. pp. ology, (F.R. Eirich, ed.), Academic Press, New York Clegg, P. L., Elastic effects in the extrusion of polyethylene, Rheology in of Elastomers (P. Mason and N. Wookey, eds.), Pergamon Press, New York pp.

Chapter

Nakajima, N. and Shida, M., Trans. Rheol., Bagley, E. B. and Duffey,H. J., Recoverable shear strain and barus effect in polymer extrusion, Trans. Rheol. Graessley, W. W.,Glasscock, S. D., and Crawley, R. L., Die swellin molten polymers, Trans. Rheol., Tanner, R.I., A theory of die swell, J. Polyrn. Sci. A , Pearson, J. R.A. and Trottnow, R., On die swell: Some theoretical results, J. NonNewtonian Fluid Mech., Berstein, B., Kearsley, E., and Zapas, L., A study of stress relaxation with finite Rheol., strain, Trans. Reddy, K R. 'and Tanner, R. I., J. Rheol., Petrie, C. J. S. and Denn, M. M., Instabilities in polymer processing, AIChE J.,

Tordella, J. P., Unstable flow of molten polymers: A second site of melt fracture, in Rheology, (F.R. Eirich, ed.), Academic Press, New York Vol. Tordella, J. P., Fracture in the extrusion of amorphous polymers through capillaries, J. Appl. Phys., Spencer, R. S. and Dillon, R. E., The viscous flow of molten polystyrene,J. Colloid Sci.,

Schreiber, H. P., Bagley, E. B., and Birks, A. M., Filament distortion and die en& angle effects in polyethylene extrusion,J. Appl. Polyrn. Sci, Benbow, J. J., Charley, R. and Lamb, P., Unstable flow of molten polymers, Nature,

Uhland, E., Rheol. Acta, Uhland, E.,Rheol. Acta, Janssen, L. P. B. M., A thermodynamic approach to the understanding of slip phenomena, Rheol. Acta, Okubo, S. and Hori, Y., J. Rheol., Weill, A., About the origin of shark skin, Rheol. Acta, Weill, A., Capillary flowof linear polyethylene melt: Sudden increase of flow rate, J. Non-Newtonian Fluid Mech., Akay, G., Rheology of reinforced thermoplastics and its application to injection molding IV.Transient injection capillary flow and injection molding, Polyrn. Eng. Sci., Akay, G., Unstable capillary flow of reinforced polymer melts,J. Non-Newtonian Fluid Mech., Bersted, B. H., Investigation of the oscillatory flow phenomena in high density polyethylene, J. Appl. Polym. Sci, 28, Ruckenstein, E. andRajora,P.,Onthe no slip boundary condition of hydrodynamics, J. Colloid. Interf Sci., Balmer, J., Entropy and melt fracture,J.Appl. Polym.Sci., Hlavacek, B., Carreau, P.,andSchreiber,H.P.,in ScienceandTechnology of Polymer Processing (N.P. Suh and N. H. Sung, eds.), MIT Press, Cambridge, MA

Benbow, J. J. and Lamb, P., New Aspects of melt fracture,SPE Trans.,

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Oyanagi, Y.,A study of irregular flow behaviour of high density polyethylene, Appl. Polym. Symp. Southern, J. H.and Paul, D. R., Elastic fracture of polystyrene solutions, Polym. Eng. ScL, White, J. L.,Critique on flow patterns in polymer fluids at the entrance of a die and instabilities leading to extrudate distortion, Appl. Polym. Boudreaux, Jr., E. and Cuculo, J. A., Polymer flow instability-A review and analysis, J. Mucromol. Rm. Mucromol. Chem., Cogswell, F.N., Converging flowof polymer meltsin extrusion dies,Polym. Eng. White, J. L. and Kondu, A. J., Ends pressure losses in extrusionof polymer melts through die, J. Appl. Polym. Higashitani, K. and Lodge, A. S., Tram. Rheol., Metzner, A. B., Behaviourof suspended matter in rapidly accelerating viscoelastic fluids: The Uebler effect,AIChE J., Metzner, A. B., Uebler, E. A., and Chan, C. F., Converging flow of viscoelastic materials, AIChE J., Bird, R.B.,UsefulNon-Newtonian models, Ann. Rm. Fluid Mech., Ostwald, W., Ueber die Geschwindigkeitsfunktion der Viskositat Disperser Systeme, I. Kolloid-Z., Ostwald, W., Ueber die Viskositat Kolloider hosungen in Struktur-Laminar und 'Ibrbulezgebiet, Kolloid-Z., De Waele, A., Viscometry and plastometry,J. Oil Color Chem. Assoc., Reiner, M., Deformation and Flow, Lewis Publishers, London Bird, R.B., Experimental tests of generalized Newtonian models containing a zero-shear viscosity and a characteristic time, Can. J. Chem. Eng., Astarita, G., Letter to the Editor commenting on the paper by Bird Can. J. Chem Eng., Carreau, P., De Kee, D. and Daroux, M., An analysis of the viscous behaviourof polymer solutions, Can J. Chem. Eng., Cross, M.M., Relation between viscoelasticity and shear-thinning behaviour in liquids, Rheol. Acta, Chhabra, R. P., Tiu, C., and Uhlherr, P. H. T., Creeping motion of spheres through Ellis model fluids, Rheol. Acta, Carreau, P. J., Rheological equations from molecular network theories, Tram. Rheol. Shenoy, U. V., Bamane, S. V., and Shenoy, A. V., A general rheological .model for polymer melts, in 40th Canadian Chemical Engineering Conference Churchill, S. W. and Churchill, R.W., A general model for the effective viscosity of pseudoplastic and dilatant fluids, Rheol. Acta, Churchill, S. W. and Usagi, R., A general expression for the correlation of rates of transfer and other phenomena, AIChE. J.

Chapter 2

Abdel-Khalik, S. I., Hassager, O., and Bird, R. B., Prediction of melt-elasticity from viscosity data, Polyrn. Eng. Sci., Bird, R. B., Hassager, O., and Abdel-Khalik, S. I., CO-rotational rheological models and the Goddard expansion, AZChE J., Wagner, M. H., Analysis of time-dependent non-linear stress-growth data for shear and elongational flow of a low-density branched polyethylene melt, Rheol. Acta, Wagner, M. H., Prediction of primary stress difference from shear viscosity data using a single integral constitutive equation,Rheol. Acta, Huppler,J.D.,Ashare, E., and Holmes, L. A., Rheological properties of three solutions, PartI. Non-Newtonian viscosity, normal stresses, and complex viscosity, Trans. Soc. Rheol.,

Ashare, E., Ph.D. Thesis, University of Wisconsin, Carreau, P. J., Macdonald, I. F., and Bird, R. B., A non-linear viscoelastic model for polymer solutions and melts-II. Chem. Eng. Sci, 23, Ballenger, T.F., Chen, I. J.,Crowder,J.W.,Hagler, G. E., Bogue,D.C.,and of the White,J. L., Polymermeltflowinstabilitiesinextrusion:investigation mechanism and material and geometric variables,Truns. Soc. Rheol. Chen, I. J. and Bogue, D. C., Tiie-dependent stressin polymer melts and review of viscoelastic theory, Truns. Soc. Rheol., Oda, K, White, J. L., and Clark, E. S., Correlation of normal stresses in polystyrene melt and its implications, Polym. Eng. Sci., Prest Jr., W. M., Viscoelastic properties of blends of entangled polymers, J. Polym. S&,

Onogi, S., Masuda, T., and Kitagawa, K.,Rheological properties of anionic polystyrene I. Dynamic viscoelasticity of narrow distributed polystyrenes,II. Dynamic viscoelasticity of blends of narrow distributed polyethylene, Macromolecules, Mills, N. J. and Nevin, Oscillatory shear measurements on polystyrene melts in the terminal region, J. Polym. Sci., Masuda, T.,Takahashi,M.,andOnogi, S., Steady state compliance of polymer blends, Appl. Polyrn. Symp., Minoshima,W.,White,J. L., and Spruiell, J. E., Experimental investigation of influence of molecular weight distribution on the rheological properties of polypropylene melts, Polym Eng. Sci., Cox, W.P. and Mertz, E. H., Correlation of dynamic and steady flow viscosities, J. Polymer Sci.,

Pao,Y.-H.,Hydrodynamictheory for the flow of a viscoelastic fluid, J. Appl. Phy~., Pao, Y.-H., Theories for the flow of dilute solutions of polymer and of non-diluted liquid polymers, J. Polym. Sci., Spriggs, T.W., A four constant model for viscoelastic fluids. Chem. Eng. Sci., Bogue, D. C., An explicit constitutive equation based on integrated Znd. Eng. Chem. Fundurn.,

history,

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Meister, B. J., An integral constitutive equation based on molecular network theory, Trans. Soc. Rheol., Han, C. D., Rheology in Polymer Processing,Academic Press, New York

p. Saini, D. R. and Shenoy, A. V., Dynamic and steady-state rheological properties of linear-low-density polyethylene melt,Polym Eng. Sci., Coleman, B. D. and Markovitz, H., Normal stress effects in second order fluids, J. Appl. PhyS., Shenoy, A.V. and Saini, D.R., Wollastonite reinforced polypropylene composites: Dynamic and steady state melt flow behaviour, J. Reinforced Plastics Compos.,

Cogswell, F. N., Measuring the extensional viscosities of polymer melts,

Trans.

Soc. Rheol.,

Shroff, R.N., Cancio, L. V., and Shida, M., Trans. Rheol., Han, C. D.and Lamonte, R.R., Trans. Rheol., Williams, M. L., Landel, R. F., and Ferry, J. D., The temperature dependence of relaxation mechanism in amorphous polymer and other glass forming liquids, J. Am. Chem.

Eyring,H.,Viscosity,plasticityanddiffusionasexamples of absolute reaction rates, J. Chem. Phys., B e d , A. B. and Belcher, H. V., Temperature coefficients of non-Newtonian viscosity at fixed shearing stress and at fixed rate of shear,J. Appl. Phys., Philippoff, W. and Gaskins, F. H., Viscosity measurements on molten polyethylene, J. Polym. Sci., Mendelson, R.A., SPE Trans., 5, Meissner, J., The effect oftemperatureontheflowpropertiesoflowdensity polyethylene melt, inProc. Inst. Cong. Rheol. (F. H. Lee, ed.), Interscience, New York Part p. Porter, R.S. and Johnson, J. F., Temperature dependence of polymer viscosity. The influence of shear rate and stress. The influence of polymer composition J. POlym SC~., Mendelson, R. A., Polyethylene melt viscosity: shear rate temperature superposition. A continuum theory of rheological phenomena, Trans. Rheol.,

Ferguson, J., Wright, B. and Haward, R. N., The flow properties of polyethylene whole polymer and fractions,J. Appl. Chem., Sabia, R., On the characterization of non-Newtonian flow,J. Appl. Polym.Sci, Schott, H. and Kaghan,W.

VISCOUS flow of molten polyethylene resins, J. Appl.

Polym. Sci., 5,

Schott, H. and Kaghan, W. S., SPE Trans. Ballman, R.L. and Simon, R. H. M., The Suence of molecular weight distibution on some properties of polystyrene melt, J. Polym Sci.,

Chapter Rudd, J. F., The effect of molecular weight distribution on the rheological properties of polystyrene, J. Polym. Sci., 44, Carley, J. L., Effect of static pressure on polymer melt viscosities, Mod. Plastics, Penwell, R. C.,Porter, R. S., and Middleman, S., Determination of the pressure coefficient and pressure effects in capillary flow, J. Polym. Sci. Cogswell, F.N.,Theinfluence of pressureontheviscosity

of polymermelts,

Plastics Polymer,

Maxwell, B. and Jung,

Hydrostatic pressure effect on polymer melt viscosity,

Mod. Plastics,

Ito, K.,Tsutsui, M., Kasajima, M., and Ojama, T., Capillary flowof polymer melts under hydrostatic pressure, Appl. Polym. Symp., Southern, J. H.and Porter, R. S., The properties of polyethylene crystallized under orientation and pressure effectsof a pressure capillary viscometer,J, Appl. Polym. Sei,

Bludell, D. J., Cogswell, F. N., Holdsworth, P.J.,andWillmouth, F. M., Melting Polymer, behaviour of stress crystallized high density polyethylene, Wasiak, and Ziabicki, Effect of spinning conditions and orientation on the degree of crystallization of nylon and polyester, Appl. Polym. Symp., Spruiell, J. E.andWhite,J. L., Structure development during the melt spinning of fibers, Appl. Polym. Symp., Metzner, P.,Hamieton,C.W.,and Men, E.H.,SPETrans. Baumann,G. F. and Steingiser, S., Rheological measurements on polycarbonate, J. Polym. Sci.,

Fox, T.G., Grateh, S., and Loshach, S., Vicosity Relationships for Polymers in Bulk and in Concentrated Solution in Rheolom, Vol. l (F. R. Eirich, ed.), Amdemic Press, New York Kraus, G. and Gruver, J. L., Rheological properties of cis-polybutadiene,J. Appl. Polym. Sci., Porter, R. S. and Johnson, J. F., The effect of molecular weight and distribution Rheol., on polymer rheology near the entanglement region, Trans.

Rheometers Polymer Melt Characterization

Rheometry is the measuring arm of rheology andits basic function is to quantify the rheological material parameters of practical importance. Rheometry is normally used in the thermoplastics industry to provide rheological information for dEerent purposes at various levels of sophistication, as can be seen fromTable 3.1. Although it is often essential to have the complete rheological characterization of the thermoplastic in use, different sections of the industry do resort to shortcuts based on limited rheological information generated by simple unsophisticated rheometers due to lack of time, finances, and technical personnel. A rheometer is an instrument for measuring the rheological properties and can do one of the following two things: 1. It can apply a deformation modeto the material and measure the subsequent force generated. 2. It can applya force mode to a material and measure the subsequent deformation.

that the forces/deformation The best designs of rheometers use geometries can be reduced by subsequent calculation to stresses and strains, and produce material parameters. Rheometers used for determining the material functions of thermoplastic melts can be divided into two broad categories: (1)rotational type and (2) capillary type. Further, subdivisions are possible and these are shown in Table 3.2. In what follows, only those rheometers whichare popularly used for rheological 95

Chapter

Table 3.1 Use

Rheometry in PolymerIndustry

esult Instrument Department

measurement point Table/calculator Single Simple floor Shop Quality control Development group

cterization er CompleteResearch laboratory

Few readings Complete characterization or one two material functions

Simple Table/calculator Calculator/computer

material functions all

characterization of thermoplastic meltsare described and discussed in detail. For example, although the bob-n-cup rotational viscometer has beenused [l]in the fifties for polyethylene melts, it has not been included in further detail. This is because this geometry is not at all popular for thermoplastic melts studies, although Cogswell [2] did suggest it in the seventies for measuring shear viscosities under conditions of controlled pressure. For a general discussion on rheometry, as applicable to various types of fluids, it is advisable to refer to some of the excellent monographs on this subject

3.1 ROTATIONALVISCOMETERS For thermoplastic melt studies, rotational viscometers with either the cone-nplate or parallel-disk configuration are used. The major advantages of cone-n-plate viscometers are as follows:

1. Constant shear rate throughout the melt sample. 2. Small quantity of sample required for measurement. On the other hand, the chief advantage of the parallel-disk configuration is that it can be used for polymer melts of extremely high viscosity and elasticity. The basic limitation in rotational viscometers is that they are restricted in their useonly to low shear rates for unidirectional shear andlow-frequency oscillations during oscillatory shear. At shear rates greater than a value between 1 and S-' as well as at higher frequencies, a flow instability normally sets in the polymer melt sample, which then begins to emerge out of the gap between the cone-n-plate or parallel disks, thereby giving erroneous results. As a consequence of the above, the measured material functions do not actually conform to thehigher deformation rates which are normally prevalent in polymer processing operations. Commercially available rotational instruments, such as the Rheometrics Mechanical Spectrometer (Rheometrics,Inc.,Piscataway, NJ) and Sangamo

i

I

v I

Code-N-Plate

I

Parallel Disk

I

I

Plunger T~

Screw Extrusion Type

Plunger Type

I

(a) Rheometrks M e c h a ~ c ~(a) ~ o n s ~ t o (a) Han's Slit Spectromet~r" Automatic Rheometer (b) S a n g ~ ~eissenberg o Rheometer" (b) Instron Rheogo~ometer" capillary Rheo~ete~

(a) Waake Rheocord"

(b) Brabender Plasticorder"

Melt Flow I n d¥essa e~r (a)

(b) Ceasta (c)D a v e n ~ ~

Chapter 3

Controls Weissenberg Rheogoniometer (Sangamo Controls, Bognor Regis, UK) can be used for unidirectional rotational shear as well as oscillatory shear and come with interchangeablecone-n-plate/parallel-disk configurations.

3.1.l Cone-N-PlateViscometer The cone-n-plate viscometer is a widely used instrument for measurement of shear flow rheological properties of polymer melts [g-201. The principal features this viscometer are shown schematicallyin Fig. 3.1. The sample, whose rheological properties are to be measured, is trapped betweenthe circular conical disk at thebottomandthe circular horizontal plate atthetop. The cone is connected to the drive motor which rotates the disk at various constant speeds, whereas the plate is connected to the torque-measuring devicein order to evaluate the resistance of the sample to the motion. The cone is truncated at the top. The gap between the cone and plate is adjusted in such a way as to represent the distance that would have been available if the untruncated cone had just touched the plate. The angle of the cone surface is normally very small (6, or 0.0696 radians) as to maintain [4] cosec26, = 1. The cone angles are chosen such that for any point on the cone surface, the ratio of angular speed and distance to the plate is constant. This ensures that the shear rate is constant from the cone tip to the outer radius the conical disk. Similarly, the shear

-a-,

3.1 Schematic diagram showing the principal features of tional viscometer.

a cone-n-plate rota-

for

Rheometers

99

rate can be assumed to be constant for any point within the gap because of the predesigned method of gap adjustment as describedearlier. The flow curve for a sample held between the cone-n-plate is generated from measurements of the torque experienced by the plate when the cone is rotated unidirectionally at differentspeeds. The various parameters of relevance are determined as in the following subsections.

A. ShearRate For a constant speed of rotation of N rpm, the linear velocity = or)is 21~rNl cm/s where is the angular velocity (rad/s) and r is the radial position in centimeters. Then gap height at r is r tan eo,where is the cone angle. Hence, the shear rate in reciprocal seconds at r can be written as

p=

2~rN

60r tan

-

ITN 30 tan

ITN

=-

308,

Because the cone angleis always maintained to be verysmall, the approximation of tan = does hold well.

B. ShearStress The following expressiondefines the relationship between the measured torque and the shear stress: T = 2 1 ~ 7 ~ r2 ~ dr =

52 I -T R ~ T ~ ~

Thus

The shear stress is then obtained in dyn/cm2when T is expressed in dyn/cm and

E in cm. The ratio of Eq. (3.3) to Eq. (3.1) results in the apparent viscosity, expressed in poise.

C. NormalStressDifference The cone-n-plate configuration can be used for estimating the primary normal stress difference of the sample. If p is the pressure at a point on the plate in excess of that due to the atmosphere pressure, then it can be shown [ 4 ] that the total normal force NF on the plate is given by NF =

21~rpdr

Chapter

which on integration gives

(3.5) Thus,

Nl = W

F

Using Eqs. (3.1) and (3.6), a plot of primary normal stress versus shearrate can be generated. The shear stress and primary normal stress measurements can be made simultaneously on the sample when it is subjected to unidirectional rotation shear in the gap of a cone-n-plate viscometer.

D.

Shear

The cone-n-plate viscometer can be usedfor oscillatory shear measurements as well. In this case, the sample is deformed by an oscillating driver which may be mechanical or electromagnetic. The amplitude of the sinusoidal deformation is measured by a strain transducer. The force deforming the sample is measured by the small deformation of a relatively rigid spring or tension bar to which a stress transducer is attached. Because of the energy dissipated bythe viscoelastic polymer melt, a phase difference develops betweenthe stress and the strain. The complex viscosity behavior is determinedfrom the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions q', G',"', and others [21-281.

3.1.2 Parallel-DiskViscometer The parallel-disk viscometer used for measuring the shear stress and normal stress difference of molten thermoplastics is similar in principle to the cone-nplate viscometer except that the lower cone is replaced by a smooth circular disk. This type of viscometer was initially developed for measuring the rheological properties of rubber [29-331 and therefore made use of serrated disks placed in a pressurized cavity to prevent rubber slippage. When it was adapted for thermoplastic melts[15,34,35], measurements wereperformed using smooth disks and without pressure. The rheological properties in the parallel-disk viscometer are based on the shear rate at the outer radius of the disk. Thus, = oR1R

(3.7)

where o is the angular velocity (radls), R is the radius of the disk (cm), and is the gap between the two parallel disks (cm).

g

for

Rheometers

Shear stress and normal relationships:

differences are givenby

the following

?,==(l+--) 2nR3 31 dd In l n pT.

Oscillatory shear measurements can be done with the parallel-disk arrangement in a manner similar to the case of the cone-n-plate viscometer and, similarly, the material functions q', G', G", and others can be generated. However, a slightly different technique [36] is at times used wherein the polymer melt sample is deformed between two oscillating parallel eccentric disks as shown in Fig. 3.2. In this case, too, it has been shown that the fluid elements undergo a periodic sinusoidal deformation and the forces exerted on the disk are thus interpreted in terms of G' and G"

3.2

RHEOMETERS

Capillary rheometers of various types are used for determining the rheological properties of polymer melts as can be seen from Table 3.2. They are broadly

-

T

DISK

Figure3.2 Schematic diagram rotational viscometer.

the principal

of parallel eccentric disks

Chapter 3

categorized as (1) those operating at constant speed and (2) those operating at constant pressure. further categorization is possible based on the melt transport mechanism being of the plunger or the screw type and on the orifice shape, through which the melt is extruded, being of the circular or slit type. Each type of capillary rheometer is discussed in detail in the following subsections.

3.2.1

Constant Plunger Speed Circular Orifice Capillary Rheometer

Commercially available instruments such as the Monsanto AutomaticRheometer and the Instron Capillary Rheometer are examples of equipment which extrude the polymer melt through a capillary with a circular orifice using a plunger at constant speeds. The principle features of this rheometer are shown schematically in Fig. The major advantage of this type of capillary rheometer is that higher-shearrate levels than those attainable in rotational viscometers can be achieved. In fact, the achievable shear rates are within the realistic ranges that are actually observed in polymer processing operations, thus making the rheological data more meaningful for simulating processing behavior. Of course, the highest attainable shear rate data is limited due to the occurrence of flow instabilities resulting in extrudate distortion or melt fracture at die-wall shear-stress levels greater than lo6 dyn/cm2 The die-wall shear stress T~ can be easily calculated by taking a force balance across the capillary die as

or (3.11)

where RN and 4, are the radius and length of the capillary die,and is the pressure drop required to extrudethe polymer melt. Because the melt flows from a wider reservoir into a capillary die in a converging streamand then exits into open air or another widereservoir in a divergentstream, it is necessary to correct the shear-stress value for these entrance and end effects. It has been suggested thata long capillary with tNIR, = 200 be used in order to reduce the magnitude of these effects in comparison with the pressure drop value. A later study [41] showed that f?N& = 20-120 would also suffice in some cases to render these effects negligible. However, the customary method of incorporating end effects correction is through the use of an effective capillary length (tN + gR,> as suggested by Bagley The wall shear stress for fully developed

103

Rheometers for Polymer Melt

1

Flgure 3.3 Schematicdiagram rheometer.

flow over the length (4,

+ m,)

a constantplungerspeedcircular

or&% capillary

is then written as (3.12)

Theshearrateatthe die wall is expressed by the Rabinowitsch-Weissenberg [43] equation for steady laminar flow of a time-independent fluid as

Chapter 3

The term ln{4Q/rRi)/d In T~ is basically equal to lln, where n is the powerlaw index depicting the non-Newtonian character of the polymer melt. Thus, it is obvious from Eqs. and that a unique relationship exists as follows:

or

The above equation is a straight line when a plot of lN/RN versus m d i c is constructed at different constant values of as shown in Fig. This is done using dies of various ratios and the intercept on the ordinate at hp,, = determines the value of -5. There are possibilities of observing slight nonlinearity in the plots, as can be seen for data at and 10.8 S" in Fig. These are probably due the breakdown of the assumptions made

W

I (-

0

Flgure Plot for determination of the Bagley correction term during polymer melt flow through a capillary rheometer.

for

Rheometers

105

during the derivation of Eq. (3.15) of time independenceand no wall slip. True mechanical wall slip can occur duringpolymer melt flow whenthe shear stresses are large enough to overcome the static friction between the wall and the flowing material [M-501. Mechanical slip can occur aseither a steady-state phenomenon or as an unsteady phenomenon known as “~tick-slip~’[50-521. This wall slip may induce the slight nonlinearity in the plots shown in Fig. 3.4. From a linear regression of each of the plots in Fig. 3.4, the correction term 5 is determined. Using 5 in Eq. (3.12), the corrected shear-stress value at the wall is estimated. It should be notedthat because polymer melts are viscoelastic, the entrance effect needs an elastic-energy correction too. This is because when the melt converges into the capillary, elastic stresses develop and begin to relax inside the capillary. This effect is taken into account [53] by modifying (3.12) to include the recoverable shear term as follows:

m.

(3.16) The capillary rheometer be used for estimating the normal stress difference using the total ends pressure loss [53,54] and the exit pressure loss [55571, wherein the latter has a more rigorous theoretical basis. However, the assumption of fully developed flow existing up to the tube exit may not hold true, especially in slow flows [58], and the errors introduced by the velocity field distortions at the exit may prove significant.

3.2.2

Constant Plunger Speed Slit Orifice Capillary Rheometer

This rheometer is similar in all respects to that discussed in Section 3.2.1 except for the fact that it has a slit orifice cross section rather than a circular one. The major credit for the development of the concept and use of this rheometer goes to Han [55,57,58] although others [59] have also used it for polymer melt studies. The instrument makes use of a series of flush-mounted transducers located along the flow channel wall which measure the pressure gradients along the flow direction. These are then converted into wall shear stress values [55] as follows:

dP b-

Clx

where b is the half-thickness of the channel. The wall shear rate is determined from the following expression given in Refs. 4 and 55:

(3.18) where

is the half-width of the channel.

Chapter 3

In general, this instrument is capable providing data in the higher shearrate ranges comparable to those obtainable from the circular orifice capillary rheometer described in Section 3.2.1. Using exit pressure losses, this instrument can also be used for the determinationof normal stresses. However, theprobable velocity-profile distortions at the exit may introduce errors thatmaynot be negligible although experimental evidence based on limited data [14,57] suggests otherwise.

3.2.3

Constant Speed Screw-Extrusion-vpe Capillary Rheometers

These capillary rheometers are principally the same as those described in Sections 3.2.1 and 3.2.2 except for the melt transport system which is of the screw extrusion type rather than the plunger type discussed earlier. A schematic diagram an extrusion capillary rheometer is shown in Fig. 3.5. Commercially available extrusion capillary rheometers are the Haake Rheocord (Haake Buchler Instruments, Inc., Saddle Brook, NJ) and the Brabender Plasticorder (Brabender, Duisburg, Germany). The rheological property measurements canbe done using a circular or slit orifice, as these are separate attachments for the miniaturized single screw extruder. These type capillary rheometers are capable of generating rheological data from medium-to-high shear rates. The applicable equations for shear stress and shear rate are the same as those discussed in Sections 3.2.1 and 3.2.2. The data generated is automatically corrected for the Bagley correction and the

E!,] \ W / ! Dl MELT

Figure 3.5 Schematicdiagramshowingtheprincipalfeatures screw-extrusion-type capillary rheometer.

a constantspeed

Rheometers Melt for Polymer

107

Rabinowitch-Weissenberg correction throughacomputer softwareprogram 1601The screw-extrusion-type capillary rheometers have been usedfor rheological studies of polymer melts [61,62] but have not becomeas popularas the plungertype capillary rheometers becausethey need a much larger quantity of polymer feed. Care has to be taken that the polymer melt completely fills the extruder screw during melt transportation in order to avoid cavitation and erroneous results. Nevertheless, the utility of these types of instrument cannot be undermined. The single screw extrusion capillary rheometer is only one of the functions performed by the commercially available Haake Rheocord and Brabender Plasticorder. They come with a number of other accessories such as the miniaturized internal mixer and miniaturized twin screw extruder as well. In fact, the miniaturized internal mixer also has, at times, been used for assessing the rheological properties of polymer melts. The torque versus rpm data generated by internal mixer can be easily converted [63-651 to shear stress versus shear rate data. A more detailed understanding of torque rheometry and instrumentation can be obtained from the excellent article by Chung [60].

3.2.4

Constant Pressure Circular Orifice Capillary Rheometer (Melt Flow Indexer)

This rheometer is also similar to the one described in Section 3.2.1 except for differences. First, the capillary used is of very short length, and second, the polymer melt is extruded by the use of dead weights (i.e., constant pressure) rather than constant plunger speed. This instrument, commonly known as the Melt Flow Indexer, is very popular in the thermoplastics industry due toits ease of operation andlow cost, which more thancompensates for its lack of sophistication. In most monographs and texts on polymer rheology, the Melt Flow Indexer has been treated briefly because it has generally been considered an instrument meant only for quality control. However, it has been shown in the recent past [66] that the Melt Flow Indexer provides more than just a quality control rheological parameter. As the main accent of the present book is to show the multiple uses of the data from the Melt Flow Indexer, this particular instrument will be treated in the utmost detail. However, the discussion will not be done in this subsection butwill be carried forward to the next chapter wherethorough justice will be done.

3.3 The rotational viscometers and the capillary rheometers described in Sections 3.1 and 3.2 are those applicable for shear flows. However, there are polymer

Chapter

processing operations that involve extensional flows. These flows have to be treated differently for making measurements of extensional viscosity. The extensional viscosity of a material is a measure of its resistance to flow when stress is applied to extendit. In general, measurement of steady-state extensional viscosity has proven to be extremely difficult. In experiments to measure both the extensional rate and the stress must be constant. A steady extensional rate would be achieved by pulling the ends of the sample apart such that e = 4, exp(kt) or, in other words, at a rate that increases exponentially with time. Steady state is reached when the force is constant. However, often the sample breaks before steady state is achieved or the limits of the equipment are exceeded or, at the other extreme, the forces become too small for the transducer to differentiate between noise and response signal. Nevertheless, there have been various methods attemptedfor the measurement of extensional viscosity.

3.3.1

FilamentStretchingMethod

The most common method for measurement of extensional viscosity is to stretch the filament of material shown in Fig. 3.6 vertically as done by Ballman [67] or horizontally as done by Meissner [68]. The polymer melts must have a high enough melt viscosity of lo5 P or greater in order to be amenable for such extensional experiments. Hence, such data are restricted to high-viscosity polyolefins such as polyethylene and polypropylene rather than low-viscosity nylon and polyester. Further, the deformation rates are to be maintained at low values to prevent breakage of filament; hence, the deformation rates are limited to or less. In the method of Ballman [67], which has been used by others [69,70] vertical thermostated filament is clamped at both ends and stretched at the rate dt‘ldt such as to maintain a constant deformation rate. Thus. (3.19)

In the method of Meissner [68], a horizontal filament immersed in thermostated immiscible is held at both ends between pairs of toothed wheels rotating with a linear velocity of Vl2. Thus, the deformation rate is written as

(3.20) There are other variations of the filament stretching technique. For example, filaments are clamped at one end and taken up on a rotating roll [71,72]. This reduces the amount of filament stretching to a more uniform level and produces a more constant extensional rate. In fact, when the following filament is taken up on a cold roll 1721 a better constancy in the extensional rate is obtained.

Rheometers for Polymer Melt

t

(b) Figure 3.6 Schematic diagram showing the principalfeatures of filament stretching method for extensional viscosity measurements.

Extensional viscosity based on constant stress measurements have also been reported In one case the filament is extended vertically on top of a bath, whereas in the other case the vertical sample is immersed in the bath. The commercial equipment (Rheometrics Extensional Rheometer) available for the measurements of extensional viscosity from Rheometrics is based on the latter

Chapter 3

new universal extensional rheometer for polymer melts hasbeen described by Miinstedt [76]. It was specifically designed with the idea of making measurements on small samplespossible in research laboratories under a variety of physical conditions (e.g., at constant stress or constant stretching rate), as well as relaxation and recoil experiments. The rotary clamp consisting of a pair of gears is a basic construction element for the design of various types extensional rheometers for polymer melts described earlier. The fact that the design is amenable for use in uniaxial and biaxial extensional rheometry has been shown by Meissner et al. [77]. Other biaxial extensiometers for molten thermoplastics have also beendescribed [78,79] by other researchers, method for measurement of viscoelastic properties of polymer melts in the prestationary extensional flow has been investigated by Leitlands [80]. special experimental device using a vibrorheometer with automatic control has been suggested. Some other methods of experimental studies with regard to the extension of polymer melts have been discussed by Prokunin [81]. In terms of uniform extensional flow of molten polymers, a rather comprehensive review is that of Petrie and Dealy [82], which may be referred to for further information on the subject.

3.3.2 Extrusion typical example extensional flow is the flow at the entrance of a capillary die. Cogswell [83] has shown that the pressure losses through such dies can be used as ameasure of the extensional viscosity. This method has notgained popularity because of the skepticism in accepting the complex converging-flow patterns at the die entrance as representative of true extensional flow with constant extensional rate. Cogswell [84] did suggest later that the die should be lubricated to reduce the shear flow and the profile of the die wall should vary at all cross sections in such a way as to ensure constant extensional rate along the die axis. Such a rheometer has been known to be developed and used for extensional viscosity data of polystyrene melt [85]. The extrusion method using a lubricated die [85,86] allows the measurements of melt systems with viscosity levels as low as lo3 P. Thus, it can be used for extensional viscosity determinations in the case of nylon and polyester melts which are often spun to make synthetic fibers. Higher extensional rates even up to 200 S" are also achievable in this apparatus [85,86], thus making the information relevant for the polymer processing industries involved in fiber spinning.

1. Philippoff, W. and Gaskins, F. H., Viscosity measurements on molten polyethylene, J. Polym. Sci., 21, 205-22 (1956).

Rheometers Melt

Polymer

111

2. Cogswell, F. N., The influence of pressureon the viscosityof polymer melts,Plastics Polymer, 41, 39-43 (1973). 3. Van Wazer, J. R., Lyons, J. W., Kim, K. Y., and Colwell, R. E., ficosity and (1963). measurement:A LaboratoryHandbook ofRheolog, Interscience, New York 4. Walters, K, Rheometry, Chapman & Hall, London (1975); Rheometry: Industrial Applications, Research Studies Press, Chichester (1980). 5. Whorlow, R. W., Rheological Techniques, Ellis Horwood Ltd., Chichester (1980). 6. Schramm, G.,Introduction to Practical Viscometry, Gebruder Haake GmbH, Karlsruhe, Germany (1981). 7. Dealy, J. M., Rheometers for Molten Plastics, Society of Plastics Engineers and Van-Nostrand, Reinhold Inc., New York (1982). 8. Meissner,J.,Rheometry of polymer melts, Annu. Rev. Fluid Mech., 17, 45-64 (1985). 9. Weissenberg, K, Proc. Inst. Intl. Rheol. Cong. (1948). 10. Pollett, W. F. 0. and Cross, H.,J. Inst., 27, 209 (1950). 11. King, R. G., A rheological measurementof three polyethylene melts,Rheol. Acta, 5, 35 (1966). 12. Chapman, F. M.and Lee, T. S., Effect of talc filler on the melt rheologyof polypropylene, SPE J., 26, 37-40 (Jan. 1970). 13. Meissner, J., Modification of the Weissenberg rheogoniometer for measurement of transient rheological properties of molten polyethylene under shear, J. Appl. Polym. Sci., 16, 2877-2899 (1972). 14. Han,C.D., Kim, K U., Siskovic, N., and Huang, C. R., A comparison of measurements of the viscoelastic propertiesof polymer melts by means of the Han slit/ capillary rheometer and the Weissenberg rheogoniometer, J. Appl. Polym. Sci., 17, 95-104 (1973). Rheol., 18,467 (1974); Trans. Soc. Rheol., 15. Lee, B. L. and White, J. L., Trans. 19, 481 (1975). 16. Minagawa, N. and White, J. L., The influence of titanium dioxideon the rheological andextrusionproperties of polymermelts, J. Appl.Polym.Sci., 20,501-523 (1976). 17. Kataoka, T., Kitano, T., Sasahara, M., and Nishijima,K., Viscosity of particle filled polymer melts, Rheol. Acta, 17, 149-155 (1978). 18. Lobe, V. M. and White, J. L., An experimental study of the influence of carbon black on the rheological properties of a polystyrene melt, Polym. Eng. 19, 617-629 (1979). 19. Saini, D. R. and Shenoy, A. V., Viscoelastic properties of linear low density polyethylene melts, Eur. Pofytn. J., 19, 811-816 (1983). 20. Saini, D. R. and Shenoy, A. V., Dynamic and steady-state rheological propertiesof linear-low-density polyethylene melt, Polym. Eng. Sci., 1215-1218 (1984). 21. Andrews, R. D., Hofman-Bang, N., and Tobolsky, A. V., Elastoviscous properties of poly(isobuty1ene) relaxation of stress in whole polymer of different molecular weights at elevated temperature,J. Polym. Sci, 3, 669-692 (1948). 22. Adams, J. W.C., Janeschitz-Kriegl, H., Den Otter, J. L., and Wales, J. L. S., J. Polym. Sci., 871 (1968).

Chapter 3

112

Mills, N. J. and Nevin, A., Oscillatory shear measurements,J. Polym. Sci., Chen, I. J. and Bogue, D. C., Tie-dependent stress in polymer melts and review Rheol., of viscoelastic theory, Trans. Vinogradov, G. V., Malkin, A. Ya., Plotnikova, E. P., Sabsai, 0. Yu., and Nikolayeva, N. E.,Znt. J. Polym. Mater, Baily, E.D., Trans. RheoZ., Miinstedt, H., Proc. 7th Int. Rheol. Congr. p. Aoki, Y., J. Soc. Rheol. Japun, Mooney, M., shearing disk plastometer for unvulcanized rubber, Ind. Eng. Chem. Anal. Ed.,

Taylor, R., Fielding, J. H., and Mooney, M.,ASTM Symposium on Rubber Testing p. Mooney, M., The rheology of processing quality of raw rubber, J. Colloid Sci., 2, Mooney, M., Proc. Int. Rubber Conf. p. Mooney M., Some neglated problems in the rheology of high polymers, Chem. Technol.,

Rubber

XXVII-XXXI

Sakamoto, K, Ishida, N., and Fukusawa, Y., Normal stress effect of molten polyethylenes, J. Polym. Sci., Blyler, L. L., Normal stress behaviour of linear polyethylene melts in a rotational parallel plate rheometer, Trans. Soc. Rheol., Maxwell, B. and Chartoff, R. P., Studies of polymer melts in an orthogonal rheometer, Trans. Soc. Rheol., Spencer, R, S. and Dillon, R. E., The viscous flow of molten polystyrene II, J. Colloid.

4,

Tordella, J. P.,Fracture in the extrusionof amorphous polymers through capillaries, J. Appl. PhyS., White, J. L., Critique on flow patterns in polymer fluids at the entrance of a die and instabilities leading to extrudate distortion, Appl. PoZym. Symp., 20, McKelvey, J. M.,Gavis, J., and Smith, T.G.,SPE J., Mertz, E.H. and Colwell,R. E., higher shear rate capillary rheometer for polymer melts, ASTM Bull., Bagley, E. B., End corrections in the capillary flow of polyethylene, J. Appl. Phys., Eisenschitz, R., Rabinowitsch,B.,andWeissenberg, K, Mitt. Dtsch. Mat. Pruf Anst. Soderhefi, Wales, J. L. S., collaborative study of capillary flow of a highly lubricated unplasticized poly(viny1 chloride), J. Polym. Sci. Symp., hgerer, G. and Wolff, D., Rheol. Acta, Menning, G., Rheol. Acta, Uhland, E.,Rheol. Acta, Lin, Y. H., Explanation for slip melt fracture in terms of molecular dynamics in polymer melts, J. Rheol.,

for

Rheometers

113

Knappe, W. and Knunbock, E., Slip flow of nonplasticized PVC compounds, Rheol. Acta,

Ramamurthy, V.,Wall slip in viscous fluids and influence of materials of construction, J. Rheol., Lau, H. C. and Schowalter, W.R.,Amodelforadhesive failure of viscoelastic fluids during flow, J. Rheol., Leonov,A.I., linearmodelofthestick-slipphenomena in polymerflowin rheometer, Rheol. Acta, Philippoff, W. and Gaskin, F. H., The capillary experiment, Trans. Rheol., White, J. L. and Kondo, End pressure losses in extrusion of polymer melts through dies, J. Appl. Polym. Sci., Han, C. D., Rheology in Polymer Processing,Academic Press, New York Han, C. D.,Charles,M.,andPhilippoff,W.,Measurement of the axial pressure distribution. of molten polymers in flow through a circular tube, Trans. Rheol., Han, C. D., Trans. Rheol., Han, C. D., Measurement of the rheological properties of polymer melts with slit rheometer. I. Homopolymer Systems.11. Blend Systems.m. W OPhase SystemsHigh Impact Polystyrene and A B S Resins, J. Appl. Polym. Sci., (I), (II), (111) Wales, J. L. S., den Otter, J. L., and Janeschitz-Kriegl, H., Comparison between slit viscometry and cylindrical capillary viscometry,Rheol. Acta, Chung, J. T., Torque rheometer technology and instrumentation, in Encyclopedia of Fluid Mechanics (N. Cheremisinoff, ed.), Gulf Publishing Co., Houston, TX Vol. pp. Blake, W. T., Determining molecular weights of thermoplastics materials,Plastics Technol.,

Schmitz, A. O., How to use the torque rheometer to solve the pressing problems, Plastics Technol., Goodrich, J. E. and Porter, R. S., rheological interpretation of torque rheometer data, Polym. Eng. Sci., Blyler, L. L. and Daane, J. H., An analysis of torque rheometer data, Polym. Eng. Sci., Rogers, M. G., Rheological interpretation of Brabender Plasticorder (extruder head) data, Znd. Eng. Chem Process Des. Dm., Shenoy, V. and Saini, D. R., Melt flow index: More than just a quality control rheological parameter, Adv. Polymer Technol., (Part I), (Part II)

Ballman, R. L., Extensionalflowofpolystyrenemelt,

Rheol. Acta,

Meissner, J., rheometer for investigation of deformation-mechanical properties of plastic melts under defined extensional straining,Rheol. Acta, Viogradov, G. V., Radushkevich, B. V., and Fikham, V. D., Extension of elastic liquids; polyisobutylene, J. Polym. Sci.,

Chapter 3

114

Stevenson, J. F., Elongationalflowofpolymermelts,

AZChE J.,

Macosko, C. W. and Lorntsen, J. M., The rheology of two blow moulding polypp. ethylenes, SPEA n t a Tech. Papers Ide, Y. and White, J. L., Experimental study of elongational flow and failure of polymer melts, J. Appl. Polym. Sci., Dealy, J. M., Extensional rheometers for molten polymers: A review, J. Non-Newtonian Fluid Mech.,

Cogswell, F. N., The rheology

polymer melts under tension,Plastics Polymers,

Munstedt, H., Viscoelasticityofpolystyrenemeltsintensilecreepexperiments, Rheol. Acta, Miinstedt, H., New universal extensional rheometer for polymer melts measurements on a polystyrene sample, J. Rheol., Meissner, J., Raible, T., and Stephenson, S. E., Rotary clamp in uniaxial and biaxial extensional rheometry of polymer melts, J. Rheol., 2 5 , Denson, C. D. and Hylton, D. C., rheometer for measuring the viscoelastic responseofpolymermelts in arbitrary planar and biaxial extensional flow fields, Eng. Sci., PO~YIFZ. Rhi-Sausi, J. and Dealy, J. M., biaxial extensiometer for molten plastics,Polym. Eng. Sci., Leitlands, V. V., Investigations of unsteady elongational flow of polymer melts, Int. J. Polym. Mater, Prokunin, N., Some methods of experimental studies into the extension of polymeric liquids, Int. J. Polym. Matez, Petrie, C . J. S. and Dealy, J. M., Uniform elongational flow of molten polymers, Vol. pp. in Rheology (G. Astarita, G. Manucci, and L. Nicolais, eds.) Cogswell, F. N., Tensile deformations in molten polymers,

Rheol. Acta,

Cogswell, F. N.,Converging flow and stretching flow: A compilation, J. Non-Newtoniun Fluid Mech., Everage, A. E. and Ballman, R.L., The extensional flow capillary as a new method for extensional viscosity measurement,Nature, Winter,H. H., Macosko,C.W.,andBennett, K.E., Rheol. Acta,

From Melt

Index to Rheogram

Knowledge of the complete flow curve (rheogram) depicting the variation of the melt viscosity over the industrially relevant range of shear rates and temperature is essential in the design of polymer processing equipment, process optimization, and troubleshooting. The rheological data needed for constructing a rheogram are obtained on sophisticated scientific instruments, namely, rheogoniometers, capillary rheometers, mechanical spectrometers, and forth. These instruments are very expensive and require trained operators. Thus, the collection of the necessary flow data is not always possible due toconstraints of finance andthe limited technical capabilities of most polymer processors. The flow parameter that is readily accessible to most processers is the melt flow index (MFI). The MFI is either specified bythe thermoplastics raw material supplier or can be easily measured using a relatively inexpensive apparatus. It is a single-point viscosity measurement at a relatively low shear rate and temperature. Earlier, it was often said that MFI gives a dot when actually what is needed is a “plot” for the polymer processors. However, this is not true now because of a unique approach developed for estimating the rheogram merely from the knowledge of the MFI. This approach is discussed in detail in this chapter, and unified master rheograms for most polymers are presented.

4.1

MFI TEST

4.1 .l Origin The MFI test originated in the laboratories of Imperial Chemical Industries (ICI), in the early stages of development of polyethylene, and was mainly used in the

Chapter

past for characterization and specifications of polyethylenes. It was specified as a standard rheological quality control test in the ASTM, BS, DIN, ISO, and JIS (see Nomenclature list at back of the book for complete forms of these abbreviations). However, each of the standard tests had a number variants. For example, in Method 105C of BS 2782 (1970) [l] three variants are described corresponding to IS0 R292 (1967) [2]. These standards specifically referto only “The Determination of Melt Flow Index of Polyethylene and Polyethylene Compounds.” Later, its use was extended to other thermoplastics. The IS0 R1133 (1969) [3] and later BS Method 720A (1979) used the same principle of test but carried a broader variation of test procedures to cover a number of di€€erent thermoplastics. The ASTM D1238 (1979) [5] and DIN 53735 (1977) [6] differ from IS0 R1133 with respect to the number of different loading conditions that are allowed. When using the test method for polymers other than polyolefins, caution is necessary in selecting the appropriate conditions. IS0 R1133, ASTM D1238, DIN 53735, and JIS K7210 do not always agree on the suggested procedures. For example, IS0 R1133 and ASTM D1238 mention a temperature of 200°C and a piston loading of 5 kg for determining theMFI value of acrylonitrilebutadiene-styrene (ABS), whereas DIN 53735 offers alternatives of 200°C with a loading of 21 kg or 220°C with a loading of 10 kg; on the other hand, JIS K7210 offers three alternatives of 200°C with a loadingof 5 kg or 220°C with a loading of 10 kg or 230°C with a loading of 3.8 kg.

4.1.2BasicPrinciple The basic principle employed in the MFI test by any of the standards is that of determining the rate of flow of molten polymer through a closely defined extrusion plastometer whose important parts are shown in Fig. The cylinder is of hardened steel and is fitted with heaters, lagged, and controlled for operation at the required temperature with an accuracy of t0.5”C. The piston is made of steel and the diameter of its head is 0.075 2 0.015 mm less than that of the internal diameter of the cylinder, which is 9.5 m m . The die (or “jet”) has an internal diameter of 2.095 t 0.005 mm or 1.180 t mm (depending on the procedure used) and is made hardened steel. All surfaces of the apparatus that come into contact with the molten polymer are highly polished.

4.1.3 Definition The melt flow index (MFS) is basically defined as the weight of the polymer (in grams) extruded in 10 min through a capillary of specific diameter and length by pressure applied through dead weight under prescribed temperature conditions. ASTM D1238 specifies the details of the test conditions as summarized in Appendix A for commonly used polymers. The test conditions include temperatures between 125°C and 300°C and different applied dead loads from 0.325

From

to Rheogram

Figure Schematic diagram of the melt flow index apparatus showing tional view of the important parts.

117

a

to 21.6 kg giving pressures from 0.46 to 30.4 kgf/cm’. The specifications have been selected in such a way as to give MFI values between 0.15 and 25 for reliable results. D1238 gives the accuracy of the MFI value obtainable from a single measurement as carried outbydifferent operators at different locations to be in the range of 29% to 2 15%, depending on the magnitude of the MFI.

Sources of Variability in the MFI value and proper interpretation of test data have been the subject of discussion in the past Table 4.1, taken from Hanson lists the possible sources of errors resulting in the variability of MFI tests. Precise measurement of MFI calls for very strict control of all possible variables, and the test, although simple, must be performed with considerable care. The first and foremost care that should be taken is to ensure that the barrel, piston, and nozzle are scrupulously clean, by treatment with hot solvents and wiping with lint-free nonabrasive cloth. The barrel should be maintained at the test temperature for 15 min and then charged with the test sample weighingbetween 4 and g within a period not exceeding 1 min. The unloaded piston should be inserted

Table

Sources of Variability in MFI andtheMagnitude

Their Effects

Factor 1. Poor packing during charging (resulting in possible oxidative degradation) 2. Dirtybarrel 3. Partially blocked die 4. Piston height (a) In excess of standard 50 mm @) Within standard 50 mm 5.between charging Delays and packing (leading to possible oxidative degradation) 6. Extended time test (leading to degradation oxidative possible 7. Worn die

circpstances

+ 150% or higher + 100% (very variable) - 100%(very variable)

- 50% - 15%

+ 10%

+ 5%polymer the (more if stabilized) is poorly + 10%for 0.5 mm wear in diameter

load

piston 8. Worn temperature 9. Test 10. in Variations

+ 10%for 0.13 mm wear ? 1% for ? 1°C Pro rata

Source: Ref. 9.

and then the necessary weight is to be added only after When the molten polymer begins to extrude out, the rate of extrusion is to be measured by cutting extrudate at suitable intervals as specified in thestandards. Several such “cutoffs” should be taken up to 30 min maximum after insertion of the sample. Thefirst cutoff, as well asany containing bubbles, should be discarded; the remainder, at least three, are to be weighed to the nearest 0.001 g and the average mass is to be calculated. The maximum and minimum values of the individual weighings must be within specified values of the average; otherwise, the results are to be discarded, and a further test should be performed on the fresh portion of the sample. It is rather imperative that all the cutoffs are taken when the piston head is between 50 and 20 mm above the upper end of the die as marked by the scribed lines on the piston. The importance of this can be seen from Fig. which shows a significant rise in the pressure drops in the barrel, depending on the piston height abovethe die end for a shear sensitive polymer like polypropylene (PP). A number of workers [lo-151 have discussed the time dependence of polymer flow in a capillary viscometer witha very short die, such as in the melt flow indexer. The manifestations of this time dependence is the gradual rise on the output rate at a fixed temperature and applied pressure, from an initial to a

From

to Rheogram

119

9s

cm

Figure Effect of piston height on melt treatments. (From Ref. 9.)

higher, steady-state value. indexer [l31 shows that

index for sample with different pre-

simple calculation in the case of the melt flow

where subscripts B and N represent the barrel and nozzle and P , 4,and R are the pressure drop, length, and radius, respectively. Substituting the appropriate values showsthat the capillary pressure drop will increase from 78% to 100% of the applied pressure as the melt flow indexer goes from completely filled to empty. Thus, a 50% increase in the output rate is possible, as has been shownfor polyethylene (PE)by Skinner [13]. With PP, the position is shown to be worse by Charley [U], who performed extensive experiments for using the MFI test for molten PP. Despite this time-dependent deformation behavior, there is no correction recommended for entrance and exit abnormalities. This would also be difficult to specify, because the corrections would be expected to vary from polymer to polymer. Another sourceof error in the melt flow indexer is the load required to shear the film of polymer between the plunger and the barrel or reservoir, as was first

Chapter

noted by Marker et al. [16]. This could also tend to result in an increase in the effective applied stress and in the output rate as noted by Skinner [l31 as well as by Rudinand Schreiber [12]. Skinner [l31 emphasizes the importance of designing a proper barrel for use with short capillaries in order to minimize the error due to friction. If the reservoir is too small, then it can contribute significantly to the overall pressure drop. If it is large, the plunger friction would be considerable. Marshall and Riley [17], however, do not agree with Skinner [l31 with regard to the effect of large barrel diameters. They argue that at constant pressure and piston speed, the friction force increases in proportion to the circumference of the piston, whereas the pressure force increases with the area of the piston. Therefore, the friction force must decrease in importance, not only because of the ratio of circumference to area but also because required piston speeds are lowered. The smallerbarrel may cause significant piston friction?but it is still preferred due to shorter heat-up time. One possible method to reduce the frictional error is by use of a piston ring as suggestedby Marshall and Riley ~71. Care should be used when taking MFI data on moisture-sensitive polymers as well as filled polymeric systems. It might be advisable to modify the melt flow indexer suchthat dry nitrogen purgingis possible during measurements on nylons, polyesters, and polyester elastomers (Hytrel), which are known to degrade with moisture. For a highly filled system, there would be a tendency of yield stress behavior, and hence the proper choice of load condition would be essential for reliable results. Filled systems like polyvinyl chloride (PVC) would also create some difficulty due to migration of the fillers toward the capillary center, thereby giving erroneous results. There is a limitation on particle size and shape of the filler due to the standard diameter of the capillary and must be borne in mind when generating filled polymer system data. In such cases, too, a modified MFI apparatus may be used by changing the capillary diameter to suit the needs.

Utility Despite the fact that MFI is an empirically defined parameter with certain limitations as described above, it is still one of the most popular parameters in the plastics industry for distinguishing various grades of polymers. Polymer manufacturers have used it routinely to specify the most suitable end use of a particular grade of the polymer [see, for example, Tables 4.2 and 4.3 (from Krassig et al. [18])]. It is worth noting that the test conditions of MFI in each of the tables is different; hence, a similar value of MFI for two different polymers may not indicate the suitability of the material for the same typeof application. Krassig et al. [l81 have discussed the general experience in the choice of raw material for high density polyethylene (HDPE) and isotatic PP in the manufac-

From MFIto Rheogram Table

End Use Indication Through MFI for Various Grades of HDPE

MFI and kg load) Processing method preformed Profile,molding, Compression extrusion Extrusion extrusion film Blown molding blow Extrusion molding blow Extrusion

blocks

Qpical applications

Pipes, round bars Films Fuel oil tanks Hollow bodies (i.e., bottle)

household Toys, molding,blow Extrusion articles molding injection Screw molding Injection

caps,

beer cases article produced Mass

moldingInjection for household uses, nondeposit goods

Ref.

ture of tubes, films, and fibers, high"FI PP is preferred in the manufacture of flexible film products with a high fibrillation tendency. However, when fibrillation tendency is a disadvantage, as in the case of film tapes for sacks or fabrics for packaging purposes, it is preferable to use a PP grade with a lower MFI. In fact, MFI values greater than 2-3 &l0 min would tend to disturb the

Table

End Use Indication through MFI for Various Grades of PP

MFI application use and End load)kg Compression moldings, pipes Extrusion blow moldings Biaxially oriented films Film tapes Monofilaments General injection moldings High-speed injection moldings Flat films Staple fibers Spun-bonded fabrics Ref. 18.

Chapter

weaving performance due to fibrillation. In the case of strappings, for which maximum strength is required, the choice of grade should be an even lower MFI value, ranging from 0.3 to 0.5 g/10 min. For HDPE, due to its inherent low fibrillation tendency and lower intermolecular cohesion, it is most beneficial for film tape production and a lower MFI is the preferred choice. Although MFI is a good indicator of the most suitable end use for which the particular grade of a polymer can be used [19], it has not been considered as a fundamental polymer property. This is because the temperature and shear rate employed in the MFI test differ substantially from those encountered in actual polymer processing operations. This point has been well illustrated by Shida et al. [20] and Smith [21]. The latter has also shown the insensitivity of MFI to the effects of molecular-weight distribution. This is due to the fact that variation in molecular-weight distribution would normally affect the flow behavior at very low (lO"/s) and very high (104/s) shear rates, whereas MFI is measured at an intermediate shear rate. The effect of molecular-weight distribution on processibility and insensitivity of the MFI measurement to these effects have also been described by Borzenski [22]. Despite all these limitations, MFI still remains a simple, easily obtainable viscosity parameter from a relatively inexpensive apparatus within the technical and financial means of plastics processors. In fact, it has been shown [23] to be more than just a quality control rheological parameter.

RELNANT EQUATIONS FOR RHEOGRAM GENERATION

4.2

4.2.1

ShearStress/RateEquations

The apparatus for determining MFI is basically a circular orifice rheometer. By its very definition, MFI represents a point at specific shear rate and shear stress values on the viscosity versus shear rate curve. The expressions for shear stress T and shear rate -$ in the melt-flow apparatus are givenby the well-known conventional forms discussed in Sec. 3.2.1 as follows:

p = -4Q TR;

where piston radius R, = 0.4737 cm, nozzle radius RN = 0.105 cm, nozzle length e,., = 0.8 cm as per ASTM D1238 and 4, = 2.326 cm as per ASTM D3364, force F = testload L (kg) X 9.807 X lo5 dyn. The flow rate Q (cm3/s) is

From MFIto Rheogram

obtained from the definition of MFI as follows: MFI=lOX60Xw where t+ is the weight rate of flow (in W

@S)

and

=

where is the density (in &cm3). Combining Eqs. (4.4) and (4.5) gives

MFI Since the geometry of a melt flow indexer is fixed as given above,Eqs. (4.2) and (4.3) reduce to give 9.13 X 10%

(for all

3.1 X 104L (for PVC)

used polymers except PVC)

(4.7a) (4.7b)

m

= 1.83 P

MFI-Load Relationship As the MFI value is generated at a fixed temperature and a fixed load, a single point on the shear stress versus shear rate curve at that specific temperature can be obtained through Eqs. (4.7) and (4.8). This fact is useful for calculating the value of MFI from a known shear stress versus shear rate curve when the MFI is not reported. For example, for a load conditionof 2.16 kg, the constant shear stress in the melt-flow apparatus from Eq. (4.7a) is equal to 1.97 X lo5 dyn/ cm'. Thus, reading outthe value of shear rate for this constant shear stress at a specific temperature, a value of MFI could be calculated using Eq. (4.8). The propriety of this method for obtaining MFI from a shear stress versus shearrate curve has been tested for different polymers withreported MFIs. The calculated values and the reported values are always found to be in good agreement. This method has also been employed by Rideal and Padget [24]. They actually measured viscosity on a Weissenberg rheogoniometer at shear stress equivalent to that in the MFI apparatus and used it to estimate the MFIs of polymers, whose MF'Ivalues were too low to be measured accurately on the melt flow indexer. As noted earlier, MFI represents a point on the shear stress versus shear rate curve in the medium shear-rate region for most polymers. Thus, it falls within that area of the curve which can be modeled by the power-law equation discussed earlier, namely, T If two different load conditions are used for getting two different MFI values, then the following relationship would hold

Chapter

from Eqs. (4.7) and (4.8):

where MFIl and M FI, represent the MFI values at loads L1 and L,, respectively. This equation can thus be effectively used to estimate the MFI at an unknown load condition if its value is known at some standard condition.

4.2.3

MFI-Wscosity Relationships

Boenig [25] was the first to indicate that there existed an inverse relationship between MFI and zero shear viscosity. He showed that for PE the following relationship held: log MFI = const - log qo

(4.10)

Busse [26], on the other hand, gave the relationship between MFI and inherent viscosity, from which he derived the inverse correlation between MFI and zeroshear viscosity as follows: (4.11) Using the fact that q = T/*, the following relationship can be derived easily through Eqs. (4.7) and (4.8) as follows: [4.98

(for all

used polymers except

PVC) (4.12a)

(4.12b)

*

Taking note of the fact that the zero shear viscosity is only a special case of q for -,0, from Eq. (4.12) the inverse relationship between MFI and qo is evident. Dutta [27] has shown that for most thermoplastic melts (possessing a lowshear Newtonian viscosity plateau), a reasonably good estimate of zero-shear viscosity can be obtained from the knowledge of MFI using Eq. (4.12). Dutta [27] provided a comparison between experimentaland calculated zeroshear viscosities from Eq. (4.12) for 35 different polymer grades covering several polymeric species of widely varying melt flow indices. However, the given plot of qo(calculated) versus qo(measured) on log-log scales of 3 X 3 cycles masks a lot of error in the estimate. MFI is rather insensitive to changes in molecular-weight distribution (MWD) as can be seen from Fig. (from Hanson [9]). However, MWD has profound effects on low-shear viscosity behavior.

From

to Rheogram

125

mm-

A-

lo3

o5

10' (

/

CM2)

Figure 4.3 Effect of molecular-weight distribution ("D) on the shear sensitivity of melt viscosity showing how the MFI test load condition of 2.16 kg would give results leading to incorrect conclusions. (From Ref.9.)

Thus, MFI cannot truly predict zero-shear viscosity even within acceptable accuracy in such cases and, hence, the method of Dutta C271 could involve errors of even greater than 50%. Hence, the prediction of zero-shear viscosity from MFI using Eq. (4.12) must be done with utmost caution.

4.2.4

MFI-TemperatureRelationships

Saini and Shenoy [28] have provided the relationship between MFI and temperature. They suggesteda simple and standard procedure for obtaining E values as E, through a modified form of Eq. (2.69). In contrast to the conventionally followed method determining the activation energy of flow through viscosity measurements, especially from zero-shear viscosity, their work makes use of MFI for estimating E values. The advantage of the method lies in the fact that MFI is a much more easily determinable parameter in comparison to the zeroshear viscosity. Further, due to the MFI measurement being at a constant shear stress, it naturally results in a standard and meaningful value of E from the

Chapter

Arrhenius-type relation [Eq. (2.69)] which has been shown earlier to be truly valid only at constant shear stress. Porter and Johnson [29] have concluded that from somewhat above lo4 to somewhat around lo6 dyn/cm2of shear stress, the E values obtained at fixed shearing stress are justified to represent true viscous behavior. The loading conditions in " Ideterminations are such as to develop shear stress in the range of 3 X lo4 to 2 X lo6 dyn/cm*, as can be seen from Appendix thus falling naturally within the justifiable range of reliable results. For each polymeric system, the density is constant, and the load condition is fixed as per ASTM D1238 and ASTM D3364 in Eq. (4.12). Thus, q X MFI = constant and this fact can now be used to modify the Arrhenius-type Eq. (2.69) to give

m =B

exp( -

2)

(4.13)

E, can be calculated from the slope of the In MFI versus T" plot and would give a more meaningful and invariant result representing the activation energy of flow at a constant shear stress than those predicted through earlier analyses in the literature. Table 4.4 gives the values of E, calculated from the slope of In MFI versus T" plots for a number of common polymers [30-371. In a few cases, E, has been calculated for a number of different grades of each generic type, and the average value is reported along with the temperature range of validity for the E, values. Qpical processing temperature ranges for each polymer type are shown, and these can be seen to match rather well with the validity range of temperature of E,. With processing temperatures being far above TR+ 100 in the case of most polymers except for polystyrene and polycarbonate, use of the modified Arrhenius-type Eq.(4.13) is justifiable for reliable results. The reported values in Table 4.4 are consistent with low-shear Newtonian values of E (where E, = E?) given in the literature for most of the polymers over similar temperature ranges [28]. E, developed fromMFI values hasthe advantage of simplicity and easy availability in contrast to the zero-shear viscosity, which is difficult to obtain and amenable to questionable techniquesfor its determination. Note that although the activation energy should be denoted as E,, the general practice of using E without the subscript is followed in the future sections. The concept of obtaining activation energy from MFI be extended to copolymers as well. Shenoy and Saini [38] have shown that, in the case of copolymers, there exists an anomalous temperature dependence of melt viscosity leading to the existence of two distinct values of activation energies for each copolymer. Plots of MFI versus T" on semilogarithmic scale are shown in Figs. 4.44.6, which include three copolymer systems, namely, styrene-butadiene-styrene (SBS), ethylene-vinyl acetate (EVA), and liquid-crystalline copolyester, HBA-

From MFIto Rheogram

t-:

m

a

%

%

t-:

a

m

m

N

d-

?

\o

m

Polymer

Grade

con~tion EkgD

activation energy E (~cal/mole)

ern^. range

rcl)

range (“C)

Tg+ 100

eel

data (Ref.) 32

~athon

2.4 (180/2.16) 2.9 (200/2.16) 4.4 (22012.16)

6.78

LLDPE

Escorene

0.88 (17512.16) 1.00 (190/2.16) 1.11(20512.16)

3.20

3.20 (175-205)

185-205

--20

49

PP

Amoco 10-1046

3.7 (200/2.16) 6.3 (230/2.16) 10.0 (250/2.16)

9.87

9.76 (200-250)

210-240

90

33

hoco 10-6016

3.9 (200/2.16) 6.5 (230/2.16) 10.3 (25012.16)

9.64

PS

Ethyl cellulose

Amoco 5M

Ethocel 856

15.4 (210/5.0) 47.7 (230/5.0) 121.0 (250/5.0)

25.5

27.4 (210/5.0) 85.1 (230/5.0) 215.0 (250/5.~)

25.5

1.3 (170/2.16) 4.4 (190/2.16) 14.2 (210/2.16) 31.0 (23012.16)

23.3

33

25.5 (210-250)

185-225

200

33

33

23.3 (170- 230)

175-200

143

35

0.25 (190/2.16) 1.3 (210/2.16) 4.7 (23Q/2.16)

34.2

34.2 (190-230)

175-200

35

Cellulose acetate butyrate

Tenite 205 HZ

Cellulose propionate

Tenite 307H

3.7 (190/2.16) 19.0 (210/2.16) 68.8 (230/2.16)

34.2

Acrylic

Lucite 40

1.12 (220/3.8) 4.9 (240/3.8) 19.6 (260/3.8)

37.9

Plexiglass v-100

0.03 (170/3.8) 0.3 (190/3.8) 1.8 (21013.8) 7.9 (230/3.8)

41.0

35

Irnplex A

0.007 (19013.8) 0.036 (21013.8) 0.16 (230/3.8)

36.5

35

Nylon.

35

38.5 (170-260)

230-280

205

150

37

Nylon 6

8.2 (230/2.16) 16.7 (25012.16) 30.6 (27012.16)

18.6

Plaskon 8201

5.0 (23112.16) 13.7 (260/2.16) 29.5 (288/2.16)

17.6

35

Plaskon 8205

1.9 (260/2.16) 2.5 (268/2.16) 4.0 (288/2.16) 49 (28Q/2.16) 63 (29012.16) 80 (30Q/2.16)

16.3

35

16.2

35

Zytel 1QlNClO

17.2 (230-300)

180-220

34

[“Cllload condition Polymer PET

Grade

m7202A

PBT

VFR4716A

E W 1.22 (26512.16) 1.61 (27512.16) 2.23 (28512.16)

activation energy E (kc~mole)

(Temp. range Wl)

temp. range (“C)

2”’

ec)

Source of M H data (Ref.)

+ 100

20.1

20.1 (265-285)

250-280

170

32

31.8 (24012.16) 44.1 (25012.16) 68.2 (26012.16)

21.7

21.7 (240-260)

250-280

170

32

1.3 (250/1.2) 3.5 (27011.2) 6.1 (29011.2)

21.9

21.9 (250-290)

235-300

250

36

8.0

8.0 (190-290)

150-210

65

36

19.3 (240- 320)

240-320

308

70

PC

Lexan 140

PVDF

DYFXOR 2000

4.17 (1901125) 4.9 (210112.5) 6.8 (230112.5) 8.3 (2501’12.5) 10.8 (270112.5) 20.1 (290112.5)

PPO

Noyl-73 1

4.9 (26015) 12.3 (28015) 14.7 (30015) 30.4 (32015)

19.07

Noryl-N 110

5.9 (24015) 19.4 (26015) 39.3 (280/5) 59.9 ~30015~

23.01

Noryl-SE 100

PPS

RWON V-1

PES

Victrex 200P

PEEK

451GV

PEI

~-200-3

16.7 (250/5) 34.9 (280/5) 53.9 (300/5) 93.2 (32~/5) 6.87 (260/5) 13.7 (280/5) 25.5 ( 3 ~ / 5 ) 8.83 (240/5) 16.2 (2~0/5) 28.9 (280/5) 49.1 ( 3 ~ / 5 ) 9.81 (240/5) 20.2 (260/5) 44.1 (280/5) 63.7 (300/5)

18.00

20.03

17.04

18.60

195-250

71

4.6

4.6 (285-316)

300-360

4.4 (320/5) 19.6 (350/5) 49.5 (3~0/5)

36.0

36.0 (320-370)

340-400

333

72

10.8 (360/5) 22.6 (395/5)

12.0

12.0 (360-395)

350-420

244

74

8.34 (35515) 22.08 (3~5/5) 39.25 (395/5)

30.0

30.0 (355-395)

310- 400

317

75

3.53 (288/5) 8.3 (302/5) 17.6 (316/5) 35.3 (329/5)

35.0

35.0 (288-329)

250-380

802.2 (28515) 1000.0 (31615)

250-287

76 d d

Chapter 4

132

-2

".x

l

16~

Figure Semilogarithmic plot of MFI versus T" for SBS data from Ref. 39. (Reprinted from Ref. 38 withkindpermission from Elsevier Science Lausanne, Switzerland.)

PET, respectively. The data for these systems are given in Table along with the sources It can be seen that in each of the figures, the points are laid out in such a wayas to give two distinct straight lines.This indicates the existence of two separate values of activation energy accompanied by a sudden change at some characteristic temperature. It means that at this temperature a structural (or relaxation) transition occurs, leading to an alteration in the mechanism of flow. Copolymers have polymer chains comprised of more than one type of monomeric building block. The nature of comonomers and their placement in the chain have a major influence on the melt rheology. Except for random copolymers, all other types of block copolymers show microphase separation. Their melt viscosities are a manifestation of the existing two-phase structured system, probably a weaker version of the three-dimensional network which exists at lower temperatures. As the temperature is raised, only one of the domains of the two-phase system melts, but the system is able to flow as a whole due to the fluidity created by one of the domains, despite the fact that the two domains are not compatible. However, such a flow involves disruption of the melted domain and transfer of the segments througha thermodynamically incompatible unmelted second domain. This requires additional energy, giving rise to a high value of activation energy. As the temperature rises, a stage is reached when

From MFIto Rheogram

Figure Semilogarithmic plotof MFI versus T" for EVA using data from Refs. 40 and 41. (Reprinted from Ref. 38 with kind permission from Elsevier Science S.A., Lausame, Switzerland.)

both domains become fluidand the additional resistance to flow due to the presence the unmelted domain is removed. The temperature at which this occurs would then become the crossover point to the lower activation energy level as can be seen in Figs. 4.4-4.6. The values of the two activation energieson either side of the crossover point would be such that one would be tending to the activation energy of one phase, whereas the other would tend to the activation energy of the second phase. For example, in the case of SBS, between 110°C and lSO"C,the activation energyEl has a valueof 28.7kcal/mole which is not too differentfrom the activation energy for homopolystyrene (E = 25.5 kcdmole); whereasinthetemperaturerange between 150°C and 210°C,the activation energy tends to that of polybutadiene (E = 4.5-7.9 kcal/mole) and takes a lower value of E2 = 10.0 kcal/mole. It is essential to exercise caution when determiningthe activation energy for viscous flow in the case of copolymers. If the attempt at such determinationsis done through fewdata points, there is a likelihood of error because the crossover point would not come out distinctly and the plot of viscosity or MFI versus T-' may be mistaken for a Curve that cannot be approximated by a straight line. This was one of the reasons why the particular systems in Figs. 4.4-4.6 were specifically chosen, as they provide at least six data points. For other copolymers like

Chapter

134

I

I

I LIQUID-CRYSTALLINE POLYMER I

I

t

-

1 x le3 TEMP,K T 4.6 Semilogarithmic plot of MFI versus T" for HBA/PET copolyester using data from Refs. and meprinted from Ref. 38 withkindpermission from Elsevier Science S.A., Lausanne, Switzerland.)

ABS, vinyl chloride-vinyl acetate (VCVA), styrene acrylonitrile (SAN), and on, although data of MFI versus temperature is available as be seen in Table B2 of Appendix B, it is not sufficient to depict two distinct activation energies. When the activation energy E, for the polymer is known, be used for determining the MFI value at any unknown temperature from a known set of Mm and T. This is done using the rearranged formof as follows: Modified Arrhenius Equation [28]

-= exp[-(-0 MFI,

R

1 T2

$)l

(4.14)

~

opo1ymer

SBS

EVA

A

~ F I (g per 10 rnin)

T (“C)

Activation energy below crossover point (kca~mole)

0.05 0.34 1.7 2.7 4.8 7.8

110 130 150 170 190 210

28.7

0.02 0.12 0.18 0.34 0.50 2.37 4.8

60 65 70 75 80 100 125

54.9

0.1 0.44 6.8 19.1 63.7 122.5

275 285 295 305 315 330

109.8

Activation energy above crossover point (kcal/mole)

w.1 39

10.0

40

16.0

~ o ~ r cRef. e : 38. (~eprintedwith kind p e ~ i s s i o nfrom Elsevier Science S.A., Lausanne, Switzerland.)

40,41

42

31.0

2

3

Chapter

136

Using the inverse relationship between q and MFI, the WLF-equation can also be written in the reorganized form as follows: Modified WLF-Type Equation [30] log(%)

8.86(T, - T,) = 101.6 (T, - T,)

+

-

8.86(T1 - TJ 101.6 (TI - T,)

+

(4.15)

where TI is the ASTM-recommended test temperature (K), T, is the temperature at which MFI is required (K),T, is the standard reference temperature (=T, + 50) (K),Tg is the glass-transition temperature (K), R is the gas constant, and E is the activation energy for viscous flow (kcal/mole). The choice of equation for determining the temperature dependence of MFI is mainly governed by whether T < T, + 100 or T > Tg + At temperatures relatively closer to T,, free volume and its changes with temperatureplaya dominant role. Hence, the (Williams-Landel-Ferry) WLF-type Eq. (4.15) could provide better estimates. At temperaturesgreater than Tg + 100, the temperature dependence of MFI is decisively affected by overcoming of the forces of intermolecular interactions, in which case the Arrhenius-type Eq. (4.14) would give better predictions. Note that for copolymers, when MFI values at different temperatures are required, the modified WLF-type equation cannot be used because of the existence of two glass-transition temperatures in such systems.

Unification Technique The technique for unification of viscosity versus shear rate curves of various grades of polymers at different temperatures is based on Eqs. (4.8) and (4.12). These two equations are reorganized and written as follows: (4.16) X

lo4&

1.7 x lo4$

(for all

used polymers exceptPVC)

(for PVC)

(4.17a) (4.1%)

For a given polymer, the density and the testing load conditions are fixed, thus indicating that the MFI of the material is directly proportional to the apparent shear rate and inversely proportional to the apparent viscosity of the material under the conditions of temperature and pressure prescribed in the test. Although Eqs. (4.16) and (4.17) are valid only at the particular MFI test condition, in effect the validity of these equations overthe entire flow curve can beconstituted by a change of dead weight condition and hence the proportionality constant. It should therefore be possible to coalesce the q versus curves of various grades of polymer with different MFI by plotting -q X MFI versus on a log-

+

+/MFI

I

From

to Rheogram

137

log scale independent of temperature if the correct MFI value correspondingto the temperature of measurement of versus is used. The coalescence would be governed by the shape of the original versus curve. Similarly shaped curves would,undoubtedly, coalesce better. Shapes of the rheograms are known to vary with regard to molecular parameterslike long-chain branching and molecular-weight distribution. In arriving at the master curves, the viscosity and shear rate are normalized via the MFI. As MFI is itself insensitive to subtle changes in molecular parameters, this limitation would be expected to be present even in the master curves. The limitation of molecular-weight distribution would be more critical in the very low and very high shear-rate regimes. However, the working ranges for most polymer processing operations fall in the intermediate shear-rate region, and, therefore, the master curves would still be effective for use as a handy tool for the processors. Within the melt flow indexer die, capillary entrance effects are important, as LID is equal to 3.8 for ASTM D1238 and 11.1 for ASTM D3364. These have not been accounted for during the derivation of Eqs. (4.16) and (4.17). However, from the above discussion, it is clear that MFI would be usedmerely as a normalizing factor to obtainreduced viscosity versus shearrate curves. The MFI values used in generating the plots as well as those which would be used for obtaining the rheograms fromthe master plots would have the capillary entrance effects implicit in them and, hence, would annul each other. A further rationale for obtaining a master curve can be derived from Vinogradov and Malkin [43], who have shown that the viscosity data for a number of polymers like PE, PP, PS, and PIB [poly(isobutylene)] fall within an acceptable bandwidth when log (q/qo)versus log qoj is plotted. Taking note of the fact that qois only a special case of for 0, from Eq. the inverse relationship between MFI and qo is evident. Thus, a master curve should be possible by replacing qoby the reciprocal of MFI in the master curve function suggested by Vinogradov and Malkin [43]. Specifically for the viscosity of monomolecular melts of PVC, Jorgensen [41] developed master curves using plots of q/qoversus on log-log scale. The relaxation time X used was taken to be proportional to qdCgT (where C, is the polymer concentration and Tis the absolute temperature). Although the approach is good, it again relies entirely on the knowledge of zero-shear viscosity inorder to obtain the rheogram from the master curve. Menges et al. [44] have proposedamethod for converting the viscosity curves for filled systems into those for the unfilled base polymer by a shift factor under a constant shear stress. Although their viscosity function is temperature and concentration invariant, their approach requires a knowledge of the shift factor in the case of each system. Similarly, Menges et al. [45] have suggested a mathematical equation as a universal viscosity function based on the zero-shear viscosity and have shown

-

Chapter

138

that the function can be used to estimate the rheogram from a knowledge of zero-shear viscosity and glass-transition temperature. The zero-shear viscosity is a difficultparameter to obtain experimentally. The method discussed in this book uses MFI as a normalizing parameter. Therefore, the technique is more convenient for the processor because MFI canbe very easily measured.

In order to generate master rheograms, the authors went through an elaborate process of data collection and analysis. One, two, or all three of the following methods were employed for data collection in the case of each polymer.

Method Viscosity versus shear data was generated by the authors using the WeissenbergRheogoniometer R19 in the lower-shear-rateregion (10-l10z/s) and on the Instron CapillaryRheometerModel 3211 in the higher shear-rate region (lo-ldls). Method 2: Viscosity versusshear rate data was collected frompublished literature. Method Viscosity versus shear rate data was obtained directly from manufacturers of polymers. The MFI values needed for the unification process were obtained from one of the three following ways:

Method The MFI was actually measured using a Melt Flow Indexer Method B: The MFI was estimated fromthe available shear stress versus shear rate curve in the manner discussed in Sec. 4.2.2. Method C: The Mm values along with the detailed measurement conditions were obtained fromthe polymer manufacturers directly or throughtheir specified brochures.

AU this was done in order to consolidate the master rheogramsand eliminate any dependence that these might have on the measuring techniques, equipment, or operator. A summary of the systems analyzed is given in Appendix B. In order to exemplify the method of master curve generation, low-density polyethylene (LDPE) is chosen as a representative case and a step-by-step procedure is outlined, showing four graphs before the final master curve is given. In the case of all other polymers, the master rheograms are directly presented. 4.3.1

Low-DensityPolyethylene

Figure 4.7 shows a plot of viscosity versus shear rate for three grades of LDPE with MFIs of 0.2, 4, and 10 at a temperature of 190°C. The curves were gen-

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139

t

Figure Viscosity versus shear rate plots for three different grades of LDPE with different MFI at 190°C. (Reprinted from Ref. 30 with kind permission from Steinkopff Verlag Darmstadt.)

erated from data taken on the Weissenberg Rheogoniometer and Instron Capillary Rheometer. Unification of the three curves through the use of a plot of q X MFI versus is shown in Fig. 4.8. This curve is now grade independent but dependent on the MFI test conditions of 190°C and 2.16-kg load. Figure 4.9 shows a plot of viscosity versus shear rate at three different temperatures, 175"C, 190°C, and 205"C, for one grade of LDPE, namely, 24FS040 with a MFI of 4 (@ 190°C and 2.16 kg). In order to obtain a unified master curve of q X MFI versus it is essential to obtain MFI values at different temperatures but the same loading conditions, namely, 175°C and 2.16 kg as well as 205°Cand2.16kg. The equation discussed in Sec. 4.2.4 is used to obtain these effective MFI values at175°Cand205"C, knowing the MFI at 190°C. Using the appropriate MFI values with each of the curves in Fig. 4.9, a plot of q X MFI versus ?/Mm was generated as shownin Fig. 4.10.This unified curve is then temperature independent but dependent only on the MFI testing loadcondition of 2.16 kg. When aplot of q X MFI versus is tobe generated at a different load condition, Eq. (4.9) is used for obtaining the MFI at the required load condition. Under the fixed loading condition of 2.16 kg, curves in Fig. 4.8 and 4.10 canbe plotted together inFig.4.11 togive amaster curveindependent of polymer grade and temperature. The number ofdata points included in this curve

*/Mm

Chapter 4

140

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Figure Master plot for three different gradesof LDPE with different " Iat (Reprinted from Ref. with kind permission from Steinkopff Verlag Darmstadt.)

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Figure Viscosity versus shear rate plots for one grade of LDPEatthreedifferent temperatures.(ReprintedfromRef. withkindpermissionfromSteinkopffVerlag Darmstadt.)

From

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141

Figure4.10 Master plot for one grade of LDPE including three different temperatures. (Reprinted from Ref. 30 with kind permission from Steinkopff Verlag Darmstadt.)

and their sources are summarized in Table B1 of Appendix B.It should be noted that the data which were unified to form the LDPE master curve was a combination of those generatedby the author [30]and Mendelson [46]and included a total of 133 data points.

4.3.2

High-DensityPolyethylene

The master rheogram for HDPE is shown in Fig. 4.12.The data used in this curve involve three sources [30,31,46]and includes 146 data points. The viscosity versus shear rate curves were available for a shear-rate range from 0.01 to lOOO/s and at eight different temperatures from 170°C to 220°C (Table B1 of Appendix B).

4.3.3

Ultrahigh-Molecular-WeightPolyethylene

In the case of ultrahigh-molecular-weight polyethylene -E), the viscosity data was generated [47]at 230°C in the shear-rate range of 10-2-l/son aWeissenberg Rheogoniometer R19 for UHMWPE Fortiflex F50-06 from

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Master curve for HDPE at 2.16-kg test load condition for MFI using data and 46. (Reprintedfrom Ref. 30 with kind permission from Steinkopff Verlag Darmstadt.)

Figure from Refs.

From

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143

the Soltex Polymer Corporation. The MFI value as suppled by Soltex Polymer Corporation for this particular grade was 0.012 (@ 230°C and 21.6 kg). For data on a different grade of UHMWPE at a different temperature, the viscosity versus shear rate curve at 270°C in the shear-rate range of 0.3-3OO/s given by Plochocki [48] was used. The MFI value of this grade of UHMWPE is given as 0.02 (@ 270°C and 10 kg), which when converted to an estimated MFI at a load of 21.6 kg based on Eq. (4.9) gives a value of 2.5. Figure 4.13 shows a plot of q X MFI versus which is the master rheogram for UHMWPE that is grade and temperature invariant.

Figure 4.13 Master curve for UHMWPE at 10.0-kg test load condition for MFI using data from Refs. and

Chapter

4.3.4LinearLow-DensityPolyethylene A linear low-density polyethylene (LLDPE)sample obtained from Exxon Chemical Co., courtesy of Mr. Jack N. Shirrell of Plastics Technology Division, Baytown, Texas, was used for generating viscosity versus shear rate data [49]. The measurementsweredone on aWeissenbergRheogoniometer R19using the cone-plate geometry under six different shear rates and at three different temperatures. The data included in generating the master rheogram for LLDPE as shown in Fig. 4.14 involve only 21 points from just one source, in contrast to LDPE and HDPE where hundreds of points were usedfrom more than one source (Table B1 of AppendixB).Nevertheless, the mastercurve obtained would be reliable even with limited data based on the authenticity of the technique as proven by its applicability to almost all types of polymers. Because the MFI of the LLDPE grade used was 1 (@ 190°C and 2.16 kg), obtaining the viscosity versus shear rate curve at 190°C would have been sufficient because it would automatically establish the master curve by definition.

4.3.5 Polypropylene The master rheogram for PP shownin Fig. 4.15 involves data from three sources [30,33,50] and includes 180 data points (Table B1 of Appendix B). Similar to the case of LDPE, one set of data was generated on the Weissenberg Rheogoniometer and Instron Capillary Rheometer using three different grades of PP, each at three different temperatures. These data were complemented by viscosity versus shear data obtained on different grades of PP from two different manufacturers. One set was obtained courtesy of J.P. Whelan, Amoco Chemical Cor-

Figure Mastercurvefor data from Ref.

JLDPE at 2.16-kgtestloadconditionfor

m using

to Rheogram

From

145

T/MFI

Mastercurvefor PP at2.16-kgtestloadconditionfor MFI using data and 50. (Reprinted from Ref. 30 with kind permission from Steinkopff Verlag Darmstadt.)

Figure from Refs.

poration, Naperville, Illinois, and the second set from G.A. Vaughan, USS Novamont Inc.,WestVirginia. The MFI valuessupplied bythem at standard temperature and load (@ 230°C and 2.16 kg) was used and those required at other temperatures were estimated by the method discussed in Sec. 4.2.4. Considering the diversity from which data weremade available, the master rheogram was found to be quite well established.

Polystyrene The viscosity versus shear rate data for PS was obtained from three different manufacturers: (1) J.P. Whelan, Amoco Chemicals Corporation, Naperville, 11linois. (2) Dow Chemicals, Midland Michigan, (3) H.A. Biletech, LTDC, Polysar Inc.,Massachusetts. These data [33,51,52] along with those obtained froma research paper [53] were unified [30] to form the master rheogram as shown in Fig. 4.16. total of 112 data points were used over a very broad range of shear rate (Table B1 of Appendix B).

Cellulosics Chemical modification of naturally occurring polyether from wood pulp and cotton liners, namely, cellulose produce thermoplastics termed as cellulosics.

Chapter

146

t/MFI

Figure 6 Master curve for PS at 5.0-kg test load condition for MFI using data from Refs. 33 and 51-53. (Reprinted from Ref. 30 with kind permission from Steinkopff Verlag Darmstadt.)

They include (a) three organic esters, namely, cellulose acetate, cellulose acetate propionate, and cellulose acetate butyrate, @) one ether, namely, ethyl cellulose, and (c) one nitrate, namely, cellulose nitrate. A series of flow curves for different grades of cellulosics at various temperatureshavebeen compiled byWestover These data were used for the generation of the master curves [54]. A summary of the systems analyzed is given in Table B1 of Appendix B. The cellulose derivatives include cellulose acetate, cellulose propionate, cellulose acetate butyrate, and ethyl cellulose. In all cases, except ethyl cellulose, plots have been generated using at least two grades of the particular polymer at two or three different temperatures. The only available data on ethyl cellulose were for a single grade at four different temperatures. In the case of each type of cellulose derivative, it was found that a master curve could be generated by plotting q X MFI versus j/MFI on a log-log scale [54]. It is interesting to note that the three master curves for cellulose acetate, cellulose propionate, and cellulose acetate butyrate could be superimposed on one single curve given in Fig. 4.17a. However, the master curve for ethyl cellulose did not fit within the band of this curve, suggesting the existence of a

to Rheogram

47

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(b)

4.17 (a) Master curve for cellulose esterat 2.16-kg test load conditionfor MFI. @) Master curve for (ethyl cellulose) cellulose ether at 2.16-kg test load condition for MFI data from Refs. 35 and 55. (From Ref. 54.)

unique master curvefor cellulose esters (Fig. 4.17a) and a separate master curve for cellulose ethers (Fig. 4.1%).

The master rheogram for acrylic is shown in Fig. 4.18. total of 105 data points covering a shear-rate range from 1 to lO,OOO/s have been used The bulk of the data has been taken from Westover and complemented by 12 data

Chapter to'

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Master curve for acrylic at 3.8-kg test load condition for MFI using data (Reprinted from Ref. with kind permission from Steinkopff Verlag Dannstadt.) Refs. 35 and

points from another source[37]. The data cover a wide range of MFI and include a number of grades of three different commercially available acrylics ("able B1 of Appendix B).

4.3.9 Polyacetal For polyacetal (POM),viscosity versus shear rate data were obtained from sources [57,58]. One source was the renowned book on polyacetals by Barker and Price [57], whereas the other source was a research paper by Pritchard and Wissbrun [58]. Only 24 data points were analyzed and the master rheograms as shown in Fig. 4.19 were generated [56]. Even with limited data, two distinct master rheograms were revealed-one for linear polyacetal and the other for branched polyacetal. This is not an unexpected situation and is similar to that observed earlier for polethyleneswherein HDPE (linearPE)and LDPE (branched PE) have two distinct master rheograms.

4.3.1 0 Nylon Viscosity versus shearrate data from books [35,59] and research papers [34,60] were used to form [56] the master rheogram for nylon (PA) shown in Fig. 4.20. A total of 78 data points covering a shear-rate range from 1 to lO,OOO/s

149 From

to Rheogram

Ir

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I

Figure 4.19 Master curve for POM at 2.16-kg test load condition for Mm using data from Refs. 57 and 58. (Reprinted from Ref. 56 with kind permission from Steinkopff

Verlag Darmstadt.)

and temperatures from to have been included. It is seen that the Newtonian plateau in the case of nylon is more distinct than for the other polymers presented earlier.

4.3.1 1 Poly(Ethy1eneTerephthalate) The master rheogram for poly(ethy1ene terephthalate) (PET) is shown in Fig. Except for data points taken from a research paper by Wu the data points were obtained from a single source, namely, the manufacturer All types of grades of PET (i.e., fiber, molding, tire cord, bottle) have been unified on the master rheogram As in the case of nylon, the Newtonian plateau is very distinct and covers a wide range of shear rate. The deviation from the Newtonian behavior is also not too pronounced even at considerably higher shear rates.

4.3.1 2 Polycarbonate The viscosity versus shear rate data for polycarbonate covering a limited range of shear rate from to and temperatures from to were used in obtaining the master rheogram shown in Fig. Data on

Chapter

Master curve for nylon at 2.16-kg test load condition for Mm using data from Ref. 56withkindpermission from

from Refs. and58-60.(Reprinted Steinkopff Verlag Dmstadt.)

a particular grade of Makrolon was taken from Knutsson et al. [63], whereas the Lexan data was taken from the Product Technical Bulletin of GE [64] and a research paper [36].

4.3.13

Poly(Viny1ideneFluoride)

Viscosity versus shearrate data for several commercial poly(viny1idene fluoride) (PVDF) materials have been used [65] for the generation of the master rheogram given in Fig. 4.23. A set of 17 curves was made available by Dr. P. Gebauer [66]. For one grade of DYFLOR 2000 PVDF, viscosity versus shearrate curves were given at six different temperatures between190°C and 290°C. At particular fixed temperatures of 220°C and 250"C, curves included were for more than six different types of DYFLOR 2000, which had the high, medium, low, and very low viscosity grades. Similarinformationon 11 viscosity versusshear rate curves were made available by Puglia [67] for various grades of KYNAR, the trade name of PVDF manufactured by Pennwalt Corporation.All 28 curves were unified to the master rheogram shown in Fig. 4.23.

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151

r

Figure Master curve for PET at 2.16-kg test load condition for MFI using data from Refs. 61 and 62. (Reprinted from Ref. 56 withkindpermission from Steinkopff Verlag Darmstadt.)

4.3.1 4 Polyphenylene Oxide For polyphenylene oxide (PPO), viscosity versus shear rate data were collected from three sources [68-701 of which two were taken from published research papers [68,69] and one was obtainedby a request [70] made to General Electric, the sole manufacturer of this polymer. The GeneralElectric data on the standard grades of Noryl was taken on capillary rheometer Rheograph 2001 using a capillary0.120 cm in diameter anda eN/DN ratio of 30:l. The entire set of viscosity versus shear rate was available in the form of 37 curves which were unified [71] to form the master rheogram for P P 0 shown in Fig. 4.24. A total of 119 data points were used covering a range of shear rate from 3 to 9OOO/s and temperatures from 240°C to 320°C (Table B1 of Appendix B).

4.3.1

Polyphenylene Sulfide

For polyphenylene sulfide (PPS), viscosity versus shearrate data were generated at three different temperatures of 28OoC,285"C, and 316°C through in-house facilities [71]usingtheWeissenberg Rheogoniometer R19 for the low-shear region (10"-102/s) and the Instron Capillary Rheometer Model 3211 for the higher-shear region (l-103/s). Because the polymer would be exposed to high

152

Chapter

Figure 4.22 Mastercurvefor PC at 2.16-kg test load condition for MFI using data from Refs. 36,63, and (Reprinted from Ref.56 with kind permission from Steinkopff Verlag Darmstadt.)

lo'

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IO2

I o3

Figure 4.23 Master curve for PVDF at 12.5-kg test load condition for MFI using data fromRefs. 66 and 67. (ReprintedfromRef. 65 withkindpermissionfromAmerican Chemical Society, Washington, D.C.)

From MFI to Rheogram

Figure from Refs.

153

Master curve for PP0 at 5.0-kg test load condition for MFI usingdata (Reprinted from Ref. 71 with permission from Technomic Publishing

Co., Lancaster, PA.)

temperature for longer lengths of time during lower-shear data generation, nitrogen purging was done during the Weissenberg Rheogoniometer measurement that polymer curing was kept to a minimum. The capillary rheometer data were generated using a capillary of e,/& = 33.4 : 1 and corrected for end effects using conventional Bagley correction before plottingthe flow curves. The data covered a limited range of shear rate from 2 to 700/s but falls within the useful range of workability for this polymer. The three curves obtained were unified [71] to form the master rheogram for PPS shown in Fig. 4.25 using only 14 data points.

4.3.16

Polyether Sulfone and Polyaryl Sulfone

For polyether sulfone (PES), the existing data have been taken from available literature [72,73]. The variation of viscosity with shear stress at 350°C for injection-molding grades of polyether sulfone Victrex 200P and 300P is available in the IC1 brochure [72], whereas the effect of temperatures of 320°C and 370°C on the flow curve ofVictrex 200P is given by Cogswell [73]. For polyaryl sulfone the viscosity versus shear rate data were taken from a research paper by Bringer and Morneau [69]. The master rheogram shown in Fig. 4.26 is applicable to PES as well as PAS [71]. total of 22 data points covering a

Figure Master curve for PPS at 5.0-kg test load condition for Mm using data from Ref. 71.(Repnhted with permission from Technomic Publishing Lancaster, PA.)

shear rate range of 3 to 20,000/s and temperature range of 320-370°C have been used for PES. This is complemented by four data points in the shear-rate range of 10-1000/s at 403°C for PAS (Table B1 of Appendix B).

4.3.17PolyetherEtherKetone For polyether ether ketone (PEEK), a request was made to ICI, UK to supply the flow curves of their standard grades. Viscosity versus shear rate curves at temperatures of 360°C and 395°Cfor unfilled-grade PEEK * 451GV * STD and the 30% glass-filled-grade PEEK 4530GL were obtained courtesy of Dr. N.H. Taylor, Analytical and Polymer Science Group, Research and Technology Department of IC1 [74]. The high-shear-rate data was generatedby capillary rheometry, and Bagley corrected for “end effects” using an orifice die of zero length in conjunction witha long die of eJR, = 32. The low-shear-rate data was taken on a Weissenberg Rheogoniometer R18. The shear rate covered was from 0.1 to lOOO/s at three different temperatures, 360”C, 380”C, and 395°C (Table B1 of AppendixB). The entire set of data was unified[71] to form themaster rheogram for PEEK shown in Fig. 4.27. The master rheogram is seen to be independent of the type of the filler [CF (carbon fiber) or GF (glass fiber)] and is unique for filled as well asunfilled polymer, as discussed later in Sec. 4.3.34.

4.3.18 Polyetherimide For polyetherimide (PEI), viscosity versus shear rate data were taken from Ref. 75. These data were available for different levels of filler loadings in Ultem

Master curve forPAS and PES at 5.0-kg test load condition forMFI using data from Refs. 72 and 73. (Reprinted from Ref. 71 with permission from Technomic Publishing Co., Lancaster, PA.) 1000 at a single temperature. In order to encompass a broader range of data, viscosity versus shear rate curves at three different temperatures for the unfilled Ultem as supplied by R. Bourne of General Electric [70] was also used (Table B1 of Appendix B). A total of data points covering a shear-rate range of 5 to lO,OOO/s was used [71] for obtaining the master rheogram shown in Fig. It should benoted that the presence of filler does not affect the unification, and the master rheogram for filled and unfilled polymer is one and the same. This aspect is discussed in detail in Sec.

Polyarylate For polyarylate (PAr), all viscosity versus shearrate data was taken froma single source [76]. Nevertheless, the data covered a wide range which included four grades and four different temperatures. A total of data points covering a range of shear rates from 1to lO,OOO/s was used. The temperaturerange spanned from (Table B1 of Appendix B). The four grades covered a

Chapter 4

Flgure 4.27 MastercurveforPEEKat 5.0-kg test load condition for MFI using data from Ref. 74. (Reprinted from Ref. 71 with permission from Technomic PublishingCo., Lancaster, PA.)

Figure 4.28 Mastercurve for PE1 at 5.0-kg testloadconditionforMFIusingdata from Refs. 71 and 75. (Reprinted from Ref. 71 with permission from Technomic Publishing Co., Lancaster, PA.)

to Rheogram

From

157

wide range of MFI values. The master rheogram for PAr obtained [71] after unification of the entire set of data is shown in Fig. 4.29.

4.3.20StyreneAcrylonitrlte The master rheogram for S A N is shown in Fig. 4.30. A total of 80 data points have beenused [30], covering a limited shear-rate .range from 20 (Table B2 of Appendix B). The viscosity versus shear rate data was made available from themanufacturer[51]. This data was for two different grades of S A N 860B and Qril 867B) at four different temperatures (200"C,215"C, 230"C, and 250°C).

mril

4.3.21 Styrene-Butadiene-Styrene The master rheogram for SBS is given in Fig. 4.31. This unified curve has been produced [77] using limited data from a single source [39]. The 27 data points included have been taken from viscosity versus shear rate curve spanning 6 temperatures between 110°C and 210°C and covering a shear-rate range from 0.1 to lOOO/s (Table B2 of Appendix B).

4.3.22 Acrylonitrile-Butadiene-Styrene The master rheogram for A B S shown in Fig. 4.32 was formed [77] by using viscosity versus shear rate data from research papers [78,79]. In one case [78], the available data were for one grade of ABS, namely, Kralastic MH at three different temperatures of 18O"C, 200°C, and 220°C. In the other case [79], at a single temperature of21O"C, viscosity versusshear rate data for two

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Figure Mastercurvefor PAr at 5.0-kg testloadconditionfor Mm data from Ref. 76. (Reprinted from Ref. 71 with permission from Technomic PublishingCo., Lancaster, PA.)

158

Chapter 4

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Figure 4.30 Mastercurvefor S A N (random copolymer) at 3.8-kg test load condition for MFI using data from Ref. 51. (Reprinted from Ref. 30 with kind permission from Steinkopff Verlag Damstadt.)

Figure 4.31 Master curve for SBS (block copolymer) at 5.0-kg test load condition for MFI using data from Ref. 39.

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159

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Flgure Master curve for ABS MFI using data from Refs. 78 and 79.

I

copolymer) at 5.0-kg test load condition for

different grades of ABS were given. A set of 20 data points covering a shearrate rangefrom 0.01 to lOOO/s was used for unification [77] (Table B2 of Appendix B).

4.3.23VinylChloride-VinylAcetate The viscosity versus shear rate data for VCVA was obtained from a book [35] and a research paper [80]. The data were for a limited shear rate from 0.1 to 600/s and covered a limited temperature range from 140°C to 180°C (Table B2 of Appendix B). Nevertheless, this coverage was withinthe useful and relevant range for this polymer. A total of 32 data points were used [77] for unifying the curve to form the master rheogram for VCVA shown in Fig. 4.33.

4.3.24Ethylene-VinylAcetate The master rheogram for ethylene-vinyl acetate (EVA) is shown in Fig. 4.34. This unified curve was obtained [77] from limited data on just one grade of EVA, namely, ALATHON ENA 3185. The viscosity versus shearrate data were available [40,41] at seven temperatures between 60°C and 125"C, covering a shear-rate range from 0.01 to 500/s (Table B2 of Appendix B).

Chapter 4

160

l60.C

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Figure 4.33 Master curve for VCVA (random copolymer) at 2.16-kg test load condition for MFI data from Refs. 35 and 80.

4.3.25 .

PolyesterElastomer

The viscosity versusshear rate data for polyesterelastomer (Hytrel) wereobtained from its technical brochure on rheology and handling supplied by the manufacturer [81] (Table B2 of Appendix B). The data covereda shear-rate range of 10-3OOO/s for five different grades of Hytrel. For three of the grades, ,the data temperature was 220"C, whereas for the other two grades, they were 180°C and 240"C, respectively. A total of 20 data points were used [77] in the unification process to obtain the master rheogram for polyester elastomer (Hytrel) shown in Fig. 4.35.

4.3.26

Olefinic-TypeThermoplasticElastomer

The master rheogram for olefinic-type thermoplastic elastomer (TPE) obtained by theunification [77] of viscosity versus shear rate data from sources [82,83] is shown in Fig. 4.36. The data covers a number of different grades of W E over a range of shear rate from 1 to lOOO/s but is limited to the temperatures of 205°C and 230°C. A total of 44 data points have been used in the unification process (Table B2 of Appendix B).

From MFI to Rheogram

161

Figure Master curve for EVA (random copolymer) at 2.16-kg test load condition for MFI using data from Refs. 40 and 41.

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Figure 4.35 Mastercurveforpolyesterelastomer(blockcopolymer)at load condition for MFI using data from Ref. 81.

2.16-kg test

Chapter

Master curve for TPE (block copolymer) at 2.16-kg test load condition for MFI using data from Refs. 82 and 83.

4.3.27 Llquid-Crystalline Polymer: Hydroxy Benzoic Acid/Poly(Ethylene Terephthalate) Copolymer Viscosity versus shear rate curves for two different compositions of a hydroxy benzoic acid/poly(ethylene terephthalate) copolymer (HBA/PET) at various temperatures were used C841 in the formation of the master rheogram for the liquidcrystalline polymer shown in Fig. 4.37. The data for 80 mole% of HBA at six different temperatures ranging from 275°C to 330°C were taken from Ref. 42. In order to eliminate operator and interlaboratory errors, the data on the same composition at one representative temperature of 275°C was also used from another source[85]. The other composition of HBA/PET chosen was60 mole% and again from two different sources [42,86]. From one source [42], the visat three different temperatures between210°C cosity data for 60 mole% of and 300°C was used, where as from the other [86], three different temperatures between 260°C and 285°C was used. total of 51 data points covering a shearrate range from 2 to 8000/s have been included (Table B3 of Appendix B) in the unification process to form the master rheogram in Fig. 4.37.

4.3.28PP/HDPEBlend The master rheogram [87] for PP/HDPE is shown in Fig. 4.38. the viscosity versus shear rate data were available at one single temperature of 190°C but

MFI

Master curve for liquid-crystalline HBAiPET copolymer at 2.16-kg test load condition for MFI using data from Refs. and 86. (Reprinted from Ref. with kind permission from Gordon and Breach Publishers, Lausanne, Switzerland.)

Master curve for PP-HDPE blend at 2.16-kg test load condition for MFI using data from Ref. 88 (A-E) and Ref. (F-K). (Reprinted from Ref. with kind permission from Gordon and Breach Publishers, Lausanne, Switzerland.)

Chapter from sources [48,88]. The MFI value of PP in both casesis nearly the same, indicating the probable use of the same grade of polymer. However, the HDPE grades are distinctly different. In one case [88], the MFI value of HDPE is 9.3 at 190°C and 2.16 kg, whereas the other case [48], it is 0.51 under the same measuring conditions. A total of 67 data points for various composition ranges of PP/HDPE and covering a shear-rate range of 1-700/s was used (Table B3 of Appendix B) in establishing the master rheogram.

Blend The viscosity versus shearrate data for HDPE/P"A blend were obtained[87] from only a single source [89] and at a solitary temperature of 160°C (Table B3 of Appendix B). However, an exhaustive range of 12 compositions including the pure HDPE and pure PMMA polymer were covered. The data were available only in the low-shear-rate range from 0.01 to l/s. Using a set of 36 data points, the master rheogram shown in Fig. 4.39 was created.

Blend Only limited data on PS/PMIvfA blend have been used [87] in order to generate the master rheogram shown in Fig. 4.40. A total of three compositions (0.75/ 0.25,0.50/0.50,0.25/0.75)of P S / P " A besides the pure components were used. The viscosity versus shear rate data covering a shear-rate range of 20 to

Master curve for HDPE-PMMA blend at 2.16-kg test load condition for MFI using data from Ref. (Reprinted from Ref. with kind permission from Gordon and Breach Publishers, Lausanne, Switzerland.)

From MFI to Rheogram

165

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Figure Master curve for PS-P"A blend at 5.0-kg test load condition for " I using data from Ref. 90. (Reprinted from Ref. 87 with kind permission from Gordon and Breach Publishers, Lausanne, Switzerland.)

400/s at 220°C was taken froma single source [90] and included 20 data points

("able B3 of Appendix B).

4.3.31

PSPOM Blend

The master rheogram for PSPOM blend shown in Fig. 4.41 is based on limited data [87] taken from a single source C911 at a fixed temperature of 210°C. A total of 20 data points were taken from the viscosity data that spanned a shearrate range from 20 to 400/s for 3 compositions (0.9/0.1, 0.2/0.8) of PS/ POM, in addition to the pure components (Table B3 of Appendix B).

4.3.32

PMMAPOMBlend

Viscosity versus shearrate data for three compositions of P M W O M , apart from those for the pure components, were taken from a single source [91]. The data were available at a fixed temperature of 210°C and covered a shear-rate range of 10-400/s (Table B3 of Appendix B). A total of 20 data points were used [87] in establishing the master rheogram for P M W POM blends shown in Fig. 4.42.

4.3.33Poly(Viny1Chloride)Formulations The master rheogram established [92] for PVC formulations is shown in Fig. 4.43. This was formed using exhaustive viscosity versus shear-rate data from

Chapter

-

=

b PDM 0.2

L

I I IIII

loo

I IIII IOP

Master curve for PS-POM blend at 5.0-kg test load condition for MFI using data from Ref. 91. (Reprinted from Ref. 87 with kind permission from Gordon and Breach Publishers, Lausanne, Switzerland.)

2

?OM

" "

eo

0.a

-as

B

IO'

It

too

I I I11111

I O '

MastercurveforPMMA-POMblendat 3.8-kg test load condition for MFI using data from Ref.91. (Reprinted from Ref.87 with kind permission fromGordon and Breach Publishers, Lausanne, Switzerland.)

From

to Rheogram

167

Figure Master curve for PVC formulations at20.0-kg test load conditionfor MFI using data from Refs. 35,93, and 94. (Reprinted from Ref. 92 with kind permissionfrom Society of Plastics Engineers, Inc.)

three different sources [35,93,94]. Fourteen different, grades of PVC have been included and some with only stabilizers, others with only plasticizers, and some with both. In two cases, different types of lubricants have been used along with plasticizers to establish their effect. A total of 222 data points covering a shearrate range of 0.1-5OOO/s were used Fable B4 of Appendix B) in the unification to get the master rheogram for PVC shown in Fig.

4.3.34

FilledPolymers

The master rheograms for filled polymers have been givenby Shenoy et al. [95] on a limited number of systems. These are for filled systems of LDPE, HDPE, PP, PS, Nylon, PET, and PC using data from Refs. 59-61, 65,96-102. The various filled polymer systems analyzed by them are summarized in Table B5 of Appendix B. The viscosity data include the effect of seven filler types, namely, carbon black, titanium dioxide, quartz powder, calcium carbonate, talc, mica, and glass

Chapter 4

Figure Master curve for filled HDPE at 2.16-kg test load condition for MFI using data from Refs. 96 and 97. (Reprinted from Ref. 95 with kind permission from Society of Plastics Engineers, Inc.)

IO'

5 to'

IIIII~ I' I I ' I I I ~ ~ $0'

Figure Master curve for filled nylon at 2.16-kg test load condition for MFI using data from Refs. 59, 60, and 102. (Reprinted from Ref. 103 with permission from Chapman and Hall, Andover, UK.)

169 From

to Rheogram

Figure 4.46 Mastercurveforreprocessed LDPE at 2.16-kg testloadconditionfor MFI usingdata from Ref. (Reprinted from Ref. withkindpermission from Butterworth-Heinemann journals, Elsevier Science Ltd., Kidlington, U.K.)

fibers. These fillers represent a rangeof particle sizes. Carbon black particles are generally very fine, with the median particle size in the range of 0.02 to 0.08 p, depending on the method of preparation, whereas mica and quartz particles are relatively large, withthe median size ranging from to 80 p. A number of filler particle shapes are represented, including particulate, platelike, and fibrous. The various filler parameters influencing viscosity and their ranges, covered in the data analyzed are summarized in Table B6 of Appendix B. It was found that the master rheograms for each of filled systems was no different from that for the unfilled systems overa broad range of shear rates. Only in the very low-shear-rate region, the master curve is not unique due to the yield stress behavior, which is known to occur for filled systems as can be seen from one typical curve shown in Fig. This aspect was depicted only for HDPE, where a clear fork was shown to exist in the low-shear-rate region. Although in other curves the forks were not shown it does not imply their nonexistence. In fact, if sufficient data in the low-shear region on filled systems was available, then the fork would be present in all the curves. However,the matter of prime importance is that the mastercurve in the

Chapter 4

-

I NO.

t

$2 IS

3

I

l

I2

I5 L

Id

to'

Figure Master curve for reprocessed PP at 2.16-kg test load condition for MFI using data fromRef. 106. (Reprintedfrom Ref. withkindpermissionfrom Butteworth-Heinemann journals, Elsevier Science Ltd., Kidlington,U.K.)

higher-shear-rate region is the same irrespective of the filler type and amount, as well as the surface modifier type and amount. Further, in the case of nylon it has been shown [l031 that the presence of metal halides, too, does not alter the shape and position of the master curve, as can be seen from Fig. 4.45. Metal halides cannot be termed as fillers or diluents because their effects are much more pronounced than those expected of fillers or diluents. In fact, they could be better described as reactive additives because of their capability of interacting with the active amide-group sites along the nylon chain. Thus, it is obvious that any changes that occur in the rheological characteristics of the polymer due to the addition of reactive or inert fillers/ diluents are implicitly reflected in the MFI value as well. The master rheograms in Figs. 4.11-4.37 for all unfilled polymeric systems would, therefore, naturally hold even when fillers, reinforcing agents, coupling agents, or any reactive additives are present. This would be true more in the higher-shear-rate region, which is truly the region of relevance becausethe shear rates encountered in most processing operations, such as compounding, extrusion, and injection molding, are of the order of 1O/s or more.

From

to Rheogram

Mastercurveforreprocessed PS at 5.0-kg testloadconditionfor MFI using data fromRef. 107. (ReprintedfromRef. with kindpermissionfrom Butterworth-Heinemann journals, Elsevier Science Ltd., Kidlington,U.K.)

4.3.35

RecycledPolymers

The master rheograms. for certain recycled polymers have been given by Shenoy et al. [104]. Curves have been presented only for a limited number of polymers, namely, LDPE, PP, and PS. The systems analyzed by them are summarized in Table B7 of Appendix B. Polymer sampleswith different processing histories essentially represent polymer grades with different MFI values. Figure 4.46 shows the master curve formed by coalescing data on virgin and reprocessed LDPE fromRef. 105. The recycled sample wasprocessed in aBrabender mixer for 60 min at 190°C, which increased its MFI by about 60%. The master curve for reprocessed PP is shown in Fig. 4.47. The reprocessed materials were obtained from two grades of virgin PP by subjecting them to repetitive processing in an injection-molding machine [106]. Cuspor and Toth [l061 have provided a series of rheograms of the reprocessed materials representing 1, 6,9, 12, and 15 injection-molding cycles, andtheyhave also reported the MFI values of the reprocessed PP samples. It is interesting to note

Chapter

that all the curves of Cuspor and Toth [l061 coalesce into a single curve when plotted in terms of the modified viscosity and shear-rate functions. In Fig. 4.48 the unifying approach has been successfully demonstrated for reprocessed PS. Springer et al. [l071 used repetitive extrusion to impart varying shear history to the material. The virgin polymer, general purpose PS, was extruded at two different screw speeds of and 100 rpm with one to five passes in each case. Again, all the viscosity data of Springer et al. [l071 at fall on a single curve. Although only three different types of polymers are presented, it be seen conclusively that the master curves generated hold good for virgin as well as reprocessed material. Thus,the master curves reported for virgin polymers in Figs. 4.11-4.43 could be used reliably as master rheograms of reprocessed materials.

1. BS 2782 Method 105 C, “Melt Flow Index of Polyethene and Polyethene Compounds,” London (1970). 2. IS0 R292 (2nd ed.), “Plastics-Determination of the Melt Flow Index of Polyethylene and Polyethylene Compounds,” Geneva (1967). 3. IS0 R1133, “Plastics-Determinatioon of the Melt Flow Rate of Thermoplastics,” Geneva (1969). 4. BS 2782 Method 720A, “Determination of Melt Flow Rate of Themoplastics,” London (1979). 5. ASTM D1238, “Flow Rates of Thermoplastics by Extrusion Plastimeter,” Philadelphia (1979). 6. DIN 53735, “Determination of the Melt Flow Index of Thermoplastics” (1977). 7. Charley,R. V., Meltflowindexing of polypropylene, Br Plust., 34, 476-481 (1961). Plus8. Rudin, and Schreiber, H. P., Factors in melt indexingof polyolefins, tics Eng. J., 20, 533-539 (1964). 9. Hanson, D. E., The measurement of melt flow index of polypropylene, Plastics T’d~y, 46, 7-9 (NOV.1973). 10. Schreiber, H. P. and Rudm, A., Some elastic effects in the capillary extrusion of polyethylenes, J. Appl. Polym. 3, 122-124 (1960). 11. Schreiber, H. P., A study time dependence viscosity in capillary extrusion of polyethylene, J. Appl. Polym. 4, 38-44 (1960). 12. Rudin, and Schreiber,H, P., Time dependence viscosity in capillary extrusion of polyethylene, J. Polym. Sci., 44, 261-264 (1960). 13. Skinner, S. J., Polymer rheology, J. Appl. Polym. Sci., 5, 55 (1961). 14. Schreiber, H.P. and Rudin, A., Further remarks on time effects in capillary flow of polyethylene, J. Appl. Polym. Sci., 6, 545-546 (1962). 15. Charley, R. V.,Apparent increase in melt index during measurement, J. Appl. Polym 6, S19 (1962).

From

Rheogram

Marker, L., Early, R., and Aggarwal, S. L., Melt viscosity of polyethylene, shear dependence of viscosity, J. Polym. Sci., Marshall, D. I. and Riley, D. W., reply to Skinner’s polymer rheology,J. Appl. P~lym

Krassig, H., Lenz, J., and Mark, H. F., Fiber Technology, Marcel Dekker, New York Van Krevelan, D. W., Properties of Polymers, Elsevier Scientific Publishing, Amsterdam p. Shida, M., Shroff, R. N., and Cancio, L. V., Correlation of low density polyethPolym. ylene rheological measurements with optical and processing properties, Eng. Sci.,

Smith,D. J., The correlation melt index and extrusion coating resin performance, W P Z , Bonenski, F. J., An approach to the use of rheology in post reactor processing, Plastics Compounding, (SeptJOct. Shenoy, A. V. and Saini, D. R., Melt Flow Index: More than just a quality control (Part 11) rheological parameter,Adv. Polym. Technol, (part I), Rideal, G. R. and Padget, J. C., The thermal-mechanical degradation of high density polyethylene, J. Polym. Sci., Polym. Sympos. Ed., Boenig, H. V., Polyolefins, Elsevier, Amsterdam p. Busse, W. F.,Mechanical structures in polymer melts. I. Measurements of melt strength and elasticity, J. Polym. Sci., A2, Dutta, A., On viscosity-melt flow index relationship, Rheol. Acta, 23, Saini, D. R. andShenoy,A. V.,Anewmethod for the determination of flow activationenergy of polymermelts, J. Macromol. Sci.-Phys., Porter, R. S. and Johnson, J. F., Temperature dependence of polymer viscosity: The influence of shear rate and stress. The influence of polymer composition, J. Polym. Sci., C15, Shenoy, A. V., Chattopadhyay, S., and Nadkami, V. M., From melt flow index to rheogram, Rheol. Acta, Dutta, A., A theoretical analysis and experimental study of extrusion blow molding, Ph.D. Thesis, S U N Y , Buffalo, NY Boudreaux,E., Jr. andCuculo, J. A composition of theflowbehaviour of linear polyethylene, poly(buty1ene terephthalate) and poly(ethy1ene terephthalate), J. Appl. Polym. Whelan, J. P., Amoco Chemicals Corporation, private communication Bankar, V. G., Spruiell, J. E., and White, J. L., Melt spinning dynamics and rheological properties of nylon J. Appl. Polym. Sci, Westover, R. F., in Processing of Thermoplastic Materials (E. C. Bernhardt, ed.),

Van Nostrand, New York, pp. Yamada, M. andPorter, R. S., Compressionaleffectsinthecapillaryflow polycarbonate, J. Appl. Polym. Sci.,

of

Chapter

Lupton,J.M.,Flow

of polymermelts, Chem.Eng. Prog. Sympos. Series

Shenoy, V. and Saini, D. R., Effects of temperature on the flow of copolymer melts, Mater: Chem. Phys., Ghijsels, A. and Raadsen, J., collaborative studyon the melt rheology of styrenebutadiene-styrene block copolymer,Pure Appl. Chem., Lyngaae-Jorgensen, J. and Borring, A.-L., Melting of crystallites in polymers with II.Copolymers of ethylene low degreeof crystallinity under shear flow conditions. and vinyl acetate (EVA), in Proceedings of the VIIth International Congress on Rheology Lyngaae-Jorgensen, J., Capillary flowof poly(viny1 chloride) compounds. A comparison with viscometric flow data measured in continuous shear, J. Mucromol. Sci. Phys., wissbrun, K. F., Observations on the melt rheologyof thermotropic aromatic polyesters, Br: Polym J., Vinogradov, G. V. and Malkin, A. Ya., Rheological properties of polymer melts, J. Polym. Menges, G., Geisbusch, P., and Zigel, U.,Kunststoffe, Menges, G., Wortberg, J., and Michaeli, W., Kunststoffe,

Mendelson, R. A., Polyethylene melt viscosity shear rate temperature superposition, Trans. Soc. Rheol., Shenoy, V. and Saini, D. R., Compression moulding of ultra high molecular weight polyethylene,Plastics Rubber Process Applic.,5, Plochocki, Rheologicalproperties of polypropylene-polyethylenemolten blends, Trans. Soc. Rheol., Saini, D. R. and Shenoy A. V., Viscoelastic propertiesof linear low density polyethylene melts, Eur: Polym. J., Vaughan, G. USS Novamont Inc., private communication Dow Chemicals, private communication Biletech, H. LTDC, Polysar Inc., private communication Kataoka, T., Kitano, T., Sasahara, M., and Nishijima, K, Viscosity of particle filled polymer melts, Rheol. Acta, Shenoy, A. V., Saini, D. R., and Nadkami, V. M., Rheograms for cellulosic polymers from the melt flow index,J. Appl. Polym. Sci., Schulken,R.M.,Cox, R. H., andMinnick, L. Dynamicandsteadystate rheological measurements in polymer melts,J. Appl. Polym. Shenoy, V., Saini, D. R.,andNadkami, V. M.,Rheograms forengineering thermoplastics from melt flow index,Rheol. Acta, Barker, S. J. and Price, M. B., Polyacetab, Iliffe Books, London, p. Pritchard, J. H.and Wissbrun,K F., Reversible melt flow rate increase of branched acetal polymers, J. Appl. Polym. Sci., Kohan, M. I., Nylon Plastics, Wiley, New York Chap. Crowson, R. J. and Folkes, M. J., Rheology of short glass fiber reinforced thermoplastics andits application to injection molding11. The effectof material properties, Polym. Eng. Sci.,

Rheogram

Wu, S., Orderdisordertransitionintheextrusionoffiberfilledpoly(ethy1ene terephthalate) and blends, Polym. Eng. Sci, E. I. du Pont de Nemours and Co., personal communication Knutsson, B. A., White, J. L., and Abbas, K. A., Rheological and extrusion characteristics of glass fiber reinforced polycarbonate, J. Appl. Polym. Sci, Anon, L a a n for Extrusion, Lexan Products Dept., General Electric Co., Tech. Bulletin Saini, D. R. and Shenoy, A. V., Deformation behavior of poly(viny1idene fluoride), Znd. Eng. Chem. Prod Res. Dm.,

Gebauer, P., Dynamit Nobel AG, private communication Puglia, L., Pennwalt Corp., private communication Kramer, M., Noryl resins, Flexibility of basic technology in a broad family of products, Appl. Polym. Sympos., 15, Bringer, R. P. and Morneau, G. A., Polymer A new thermoplastic polysulfone for use at 50O0F,Appl. Polym. Sympos., Bourne, R., General Electric Plastics, private communication Saini,D.R.andShenoy,A.V.,Meltrheologyofsomespecialtypolymers, J. Elastomers Plastics,

Victrex-polyether sulfone brochure of Imperial Chemical Industries. Cogswell, F. N., Polymer Melt Rheology, George Godwin Ltd., London p. Taylor, N. H., IC1 Wilton, private communication Johnson, R. and Teutsch, E. O., Thermoplastic aromatic polyimide composites, Polym. Composites,

Ho, P. K. andWilliams,M.C.,Rheology of polyphosphazenemeltsand solutions-Some surprises, Polym. Eng. Sci., Shenoy, A. V. and Saini, D. R., Copolymer melt rheograms from melt flow index, Br. Polym. J.,

Cox, H.W. and Macosko, C. W., Viscous dissipation in die flows, AIChE J., Moroni, A. and Casale, A., ABS resins: The relation between composition and rheological behavior, in Proceedings of the VIIth International Congress on Rheology Hollister, E. H., Melt rheology ofvinyl chloride-vinyl acetate copolymers,J. vinyl Technol,

DuPont Company Booklet, Hytrel (Rheology and Handling), No. Walker, B. M. (ed.), Handbook of Thermoplastic Elastomers, Van Nostrand Reinhold, New York Han, C. D. and Rao, D. A., Measurement of the rheological properties of thermoplastic elastomers,J. Appl. Polym. S c i , Shenoy, A. V. and Saini, D. R., Melt flow behaviour of liquid crystalline polymer, Mol. Cryst. Liq. Cryst.,

Baird, D. G., Rheology of polymers with liquid crystalline order, in Rheology, Vol. 3: Applications (G.Astarita,G. Mamcci, and L. Nicolais,eds.),Plenum Press, New York pp.

Chapter

86. Jerman, R. E. and Baird, D. G., Rheological properties of copolyester liquid talline melts: Part I. Capillary rheometry, J. Rheof. 25, 275 (1981). 87. Shenoy, A. V.,Saini, D. R., and Nadkami,V. M., Melt rheology of polymer blends from melt flow index, Znt. J. Pofym. Muter., 10, 213-235 (1984). 88. Alle, N.andLyngaae-Jorgensen,J.,Polyethyleneandpolypropyleneblends: 1. Flow behaviour in capillaries, Rheol. Acta., 19, 94-103 (1980). 89. Martinez, C. B. and Williams, M. C., J. Rheof., 24, 421 (1980). 90. Kasajima,M.,Bull.Coll.Eng.HoseiUniv., 15,1(1979). 91. Carley, J. F. and Crossan, S. C., Viscosities of molten polymer melts, Pofyrn. Eng. Sci., 21, 249-258 (1981). 92. Shenoy, A. V., Saini, D. R., and Nadkami,V. M., Rheology of poly(viny1 chloride) formulations from melt flow index measurements, J. Vinyl Technol., 5, 192-197 (1983). 93. Singleton, C. J., Stephenson, T., Isner, J., Geil, P. H. and Collins, E. A., Processing

morphologypropertyrelationshipsofplasticizedpoly(viny1chloride),

94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106.

J. Mucromol. Sci. Phys., B14, 29-86 (1977). Sieglaff,C.L., SPETruns., 4, 1 (1964). Shenoy, A. V., Saini, D. R.,andNadkami, V. M., Rheogramsoffilledpolymer melts from melt flow index, Pofym. Compos., 4, 53-63 (1983). Chan, Yu., White, J. L., and Oyanagi, Y., Influence of glass fiber on the extrusion 1. Injection molding characteristics of polyethylene and polystyrene melts, Polym. Eng. Sci., 18, 268-272 (1978). Han, C. D., Sandford, C., and Yoo, H.J.,Effectsoftitanatecouplingagentson the rheological and mechanical properties of filled polyolefins, Pofym. Eng. 18, 849-854 (1978). Han, C . D., Rheologicalpropertiesofcalciumcarbonatefilledpolypropylene melts, J. Appf. Pofym.Sci, 18, 821-829 (1974). Monte, S. J. andSugerman, G., A newgenerationofageandwaterresistant reinforced plastics, Pofym. Plastics Technol. Eng., 13, 115-135 (1979). Boira, M. S. and Chaffey, C . E., Effects of coupling agents on the mechanical and Pofym. Eng. Sci., 17, rheologicalpropertiesofmicareinforcedpolypropylene, 715-718 (1977). Tanaka, M. and White, J. L., Experimental investigation of shear and elongational flow of polystyrene melt reinforced with calcium carbonate, titanium dioxide and carbon black, Pofym. Eng. Sci., 20, 949-956 (1980). Acierno, D., Amico, R. D., La Mantia, F. P.,andRusso, S., Rheological charac-

terization of poly-caprolactum anionically, synthesized in the presence of lithium chloride, Polym. Eng. Sci., 20, 783-786 (1980). Shenoy, A. V., Saini, D. R., and Nadkarni, V. M., Rheology of Nylon 6 containing metal halides, J. Muter. Sci., 18, 2149-2155 (1983). Shenoy, A. V., Saini, D. R., and Nadkarni, V. M., Estimation of the melt rheology of polymer waste from melt flow index, Polymer, 24, 722-728 (1983) Rokudai, M., Influence of shearing history on the rheological properties and processibility of branched polymers, J. Appf. Polym. Sci., 23, 463-471 (1979). Cuspor, I. and Toth, T., Study of the reprocessing of polypropylene, Znt. Pofym.

Sci. Technol., 7, T16-Tl9 (1980). 107. Springer, P.W., Brodkey, R. S. and Lynn, R. E., Pofym.Eng. Sci., 15,588 (1975).

Upgrade and Extension Unification Technique

the

The unified viscosity function curves given in the preceding chapter have an inherent limitation in the low-shear-rate region, namely, thatof being insensitive to molecular parameter variations,such as molecular-weight distributions. In this chapter, the use of a correction term to upgrade the viscosity-shear rate master rheograms in the low-shear region is demonstrated. Further, it is shown how the unification technique can be extended to other rheological material functions, such as normal stress difference, dynamic viscoelastic parameters, and extensional viscosity, to obtain coalesced curves which are grade and temperature invariant. Because of the limited amount of available data for all the cases to be treated in this chapter,the master curves are restricted to only afew candidate polymers, merely to demonstrate the wide applicability of the unification technique.

5.1

UPGRADINGTHE VISCOSITY MASTER RHEOGRAMIN THE LOW SHEAR REGION

During the generation of the unified curves (Figs. the coalescence was governed entirely by the shape of the original viscosity versus shear rate curves. Similarly shaped curves, undoubtedly, coalesced better. As the shape of the curves is known to vary with regard to molecular parameters like molecularweight distribution, the coalesced curves which are formed by normalizing viscosity and shear rate via the melt flow index (MFI) have this limitation present

ding

Chapter 5

178

in them. MFI is itself insensitive to subtle changes in molecular parameters as has been shown by Smith [l]and Borzenski [2] as well as Shida et al. [3]. The effects of the differences in molecular-weight distribution are seen in the lowshear-rate and very high-shear-rate regions as can be seen from Fig. 4.3. However, because the width of shear rates of interest for most polymer processing operations lie withinthe medium rangeas can be seen fromTable 5.1, the unified curves in Figs. 4.11-4.45 for the viscosity function are very handy tools for most processors. Nevertheless, from a fundamental viewpoint, it is essential to relieve the unification technique of this inherent limitation that even ambitious correlations of estimating zero-shear viscosity from MFI such as those attempted by Boenig [4], Busse [5], and Dutta [6] could becomemore meaningful. attempt at upgrading the unification technique in the low-shear region was done by Shenoy and Saini [7]. It has been suggested that a correction factor (MZ/Mw)’.’ should be used when coalescing low-shear viscosity data for polymer grades with widely different moelcular-weight distributions. Figures 5.1 and 5.2 show plots of X MFI versus on log-log scales for a number of polymer grades of polypropylene (PP) described in Table 5.2. The eight different polymer samples have been found to givetwo distinct unified curves in the low-shear-rate region, coalescing the data for four grades each. Molecular characteristics of the samples [g] show that B, C, E, and F form a distinct group withbroad and regular molecular-weightdistributions whereas D, G, and H another group with a narrow molecular-weight distribution. In the low-shear-rate region of 10-2-10/s, Figs. 5.1 and 5.2 do not superimpose as theystand.However, a plot of (q X MFI)/(Mz/M,,.)’.7 versus (Mz/MW)1.7(j/ MFI) yields a unique curve, taking care of the differences in the width of the molecular-weightdistribution. Figure 5.3 showssuch aplot. Thevalues of (Mz/Mw)1.7alone do not control the coalescence, as it is clear that the normalizing factor for unifying the viscosity versus the shear rate curves of widely different molecular-weight distribution is more truly (Mz/Mw)1.7/MFI. Now it is known that q varies directly as where takes the value of about for “

”

”

”

Table 5.1 Shear-Rate Ranges Encountered in Polymer

Processing Operations Typical shear rate hocess Compression Calendering Extrusion Injection

(S-’)

1-10

lo-ld lO”l0“ 103-104

”

Unification Extension Technique: and Upgrade

179

Figure Coalesced viscosity plot for broadandregularmolecular-weightdistribution PP at 2.16-kg test load condition for MFI using data from Ref. 8. (From Ref. 7.)

zero-shear viscosity and decreases slightly for viscosities at the higher shear rate. It has been shownearlier that MFI varies inversely as viscosity, and hence it can beassumed that MFI would vary inversely as a y as a first approximation. This suggests that the normalizing factor is controlled in the correct sense by the value of (az X Hence, all the grades of a particular generic type of polymer which have values of X MWcloser to each other would be expected to coalesce. The values of aZ X for the eight grades of PP are given in Table 5.2, and it is then clear why B, C, E, and F form a distinct group, whereas D, G,and H form another distinct group for the coalescence. This would explain why even two grades withsimilar MJM, but different MWcharacteristics and, consequently, different rheological characteristics would have a coalesced curve, because the sensitive X - _ parameter for unification is not MJM, but aw. In cases where MzIMw is the same, the viscosity versus shear rate curves would coalesce simply through MFI as the normalising factor because it would account for all changes in Figure shows a plot of q X MFI versus for four LDPE melts. In the low-shear-rate region of lO-*-lO/s, the coalescence is rather poor. plot of (q X MFI)I(M21Mw)'~7 versus (Mz/Mw)1.7(jMFI) for the same four LDPE melts is found to give a unified curve as shown in Fig. Figures and show

a,,,

"

"

a,.

"

"

a,

Extension Unification Technique: and Upgrade

181

( 92 /

iiJ7

1

Figure 5.3 Coalesced viscosity curve for broad, regular, and narrow molecular-weight distribution PP at 2.16-kg test load condition for MFI using data from Ref. 8. (From Ref. 7.) "

similar effects produced by the correction term of (MZ/Mw)'.' on HDPE lowshear data. Figures 5.3, 5.5, and 5.7 thus show plots which are independent of grade, temperature,andmolecular-weight distribution in the low-shear-rate region. The molecular-weight distribution is often expressed as one of the following ratios: MJM,,MzIMw, and on. An approximate interrelationship between the various expressions has been given in Chapter 1. Any of those expressions could be used in place of (Mz/Mw)'.7in the unified curve through proper conversion. The details of all the polymers [8-121 used for obtaining Figs. 5.1-5.7 have been given in Table C l of Appendix C. "

"

az+l/~n

"

5.2

EXTENDING THE UNIFICATION TECHNIQUE TO OTHER RHEOLOGICAL MATERIAL FUNCTIONS

The method of using MFI as a normalizing parameter to coalesce rheological parameter curves is not restricted to the shear viscosity function only. As a matter of fact, the unification technique be extended to obtain coalesced curves of normal stress difference, complex viscosity, storage modulus, and ex-

Chapter 5

Figure Coalesced viscosity plot for LDPE at 2.16-kg test load condition for MFI using data from Refs. 9 and 10. (From Ref. 7.)

E”,’”

Ffgure5.5 Coalesced viscosity curve taking into account the molecular-weight distribution for LDPE at 2.16-kg test load condition for MFI using data from Refs. 9 and 10. (From Ref. 7.)

Unification Extension Technique: and Upgrade

183

Figure Coalesced viscosity plot for HDPE at 2.16-kg test load condition for using data from Refs. 11 and 12. (From Ref. 7.)

Flgure Coalesced viscosity curve taking into account the molecular-weight distribution for HDPE at 2.16-kg test load condition for MFI using data from Refs. 11 and 12. (From Ref. 7.)

Chapter

tensional viscosity. In the following, the feasibility of the approach is shown for each of these material parameters under separate subheadings.

5.2.1

Normal StressDifferences

Equations and show that MFI is directly proportional to shear rate and inversely proportional to shear viscosity. Shear viscosity q is much less sensitive to structure than JI,,and for high-molecular-weight polymeric melts when qoa H:, it is known that a H? Thus, throughEq. it is not difficult to expect that

where C, is a constant incorporating the geometrical parametersof the melt flow indexer and the test load which is fixed for a particular polymeric system as per the standards. Now

+,

where each of X 0’ and is a constant depending on the fixed MFI testing conditions and fixed value of p for a particular polymeric system. Although this argument is valid at the particular MFI test condition, in effect the validity of (5.1) and over the entire normal stress difference curve can be constituted by a change of dead weight condition and, hence, the proportionality constant. It should, therefore, be possible to coalesce & versus p curves of polymer grades of different “ Iby plotting X (“I)2 versus (p/ on a log-log scale or, alternatively, by plotting NI versus on a log-log scale as per Eq. if the correct MFI value corresponding to the temperature measurement of versus p is used. The coalescence would be governed by the shape of the original versus p curve which is dependent on molecular parameters like the molecular-weight distribution. As MFI is itself insensitive to subtle changes in molecular parameters, this limitation would persist in the coalesced curves wherein MFI is used as a normalizing factor. The primary normal stress difference exhibits a strong dependence onmolecularweight distribution as would be predicted from the theory of second-order fluids Thus,

JI,

JI,

where the form for the steady-state compliance J, has been given by Refs. and Je

= KJ@

3.5

Extension 185 Unification Technique: and Upgrade ”

Equation(5.3) thus suggests thataplotof T~~ - T~~versus (Mz/M,,,)3.5X (j/MFI)’ on a log-log scale would yield a unique curve even in the case of polymer grades with variedmolecular-weight distributions [21]. Figures 5.8 and 5.9 show plots of NI versus on log-log scales for a number of polymer grades PP described in Table C2 of Appendix C. The eight polymer samples [8] have been found to give two distinct unified curves, coalescing data for four grades each. Molecular characteristics of the samples show that B, C, E, and F form a distinct group withbroad and regular molecularweight distributions, whereas A, D, G, and H form another group witha narrow molecular-weight distribution. Figure 5.10 also shows a plot of NI versus (j/MFI)*on log-log scales for PP [22] from an entirely different data source [12, 231 and a different temperature. It can be seen that Figs. 5.9 and 5.10 would superimpose on each other. However, Figs. 5.8 and 5.9 do not superimpose as they stand. Hence, in accordance with the arguments presented earlier, a plot of NI versus (MZ/M,J3.’X should yield a unique curve, taking care of the differences in the type molecular-weight distribution. Figure 5.11 shows such a plot and gives evidence of the strong dependence of the primary normal stress difference on molecular-weight distribution. Thus, for PP, Fig. 5.11 would form the unified plot independent of shear rate, temperature, and molecular-weight distribution. Figure 5.12 shows a plot of NI versus for four “

(+m’

Figure 5.8 Coalescednormalstressdifferenceplotforbroadandregularmolecularweight distribution PP at 2.16-kg test load condition for MFI using datafromRef. 8. (Reprinted from Ref. 21 with kind permission from Steinkopff Verlag Darmstadt.)

Chapter 5

186

Figure Coalesced normal stress difference plot for narrow molecular-weight distribution PP at 2.16-kg test load condition for MFI using data from Ref. (Reprinted from Ref. 21 with kind permission from Steinkopff Verlag Darmstadt.)

lo“

to’

lo4 l )*

Figure Coalesced normal stress difference plot forPP from different data source at 2.16 kg test load condition forMFI using data from Refs. 12 and (Reprinted from Ref. 21 with kind permission from Steinkopff Verlag Darmstadt.)

Extension Unification Technique: and Upgrade

/ww?’(1/MFl

Figure Coalesced normal stress difference curve for broad, regular, and narrow molecular-weight distributionPP at 2.16-kg test load condition for MFI using data from Ref. (Reprinted from Ref. 21 with kind permissionfrom Steinkopff Verlag Darmstadt.)

)2

Figure Coalesced normal stress difference plot for LDPE at2.16-kg test load condition for MFI using data from Refs. 9 and 10. (Reprinted from Ref. 21 with kind permission from Steinkopff Verlag Darmstadt.)

Chapter

LDPE melts. The scatter in the curve is due to the difference in the molecularweight distribution. plot of Nl versus (M,/M,J3.’ X for the same four LDPE melts was found to give a unified curve [21], as shown in Fig. 5.13. Figures 5.14-5.16 show the unified curves for high-density polyethylene (HDPE) [21], linear low-density polyethylene (LLDPE) [24], and Nylon [21], respectively. Wherever the molecular-weight distribution of the polymer samples of a generic type is not known, a plot of NI and (j/MFI)’ has been presented, as in the case of Nylon 112, 251. This would give a unified curve if the differences in the molecular-weight distribution of the various grades are not wide. In the case of Fig. 5.15 for LLDPE, the data are at various temperatures for a single grade of polymer [24]; hence, including the molecular-weight distribution correction term would not improve the coalescence any further. The method of unification described in the above paragraph is applicable to reprocessed polymers as well [26]. Figure 5.17 shows the normal stress difference versus shearrate data for virgin and reprocessed LDPE [27]. The recycled sample wasprocessed in a Brabender mixer for min at 190°C. The melt flow index of the virgin LDPE was increased from to about 1.3, thus indicating that the polymersamples withdifferent processing histories essentially represent polymer grades with different melt flow indices. When plotted in terms of the normal stress difference Nl versus the modified shear rate function (j/MFI)zon ”

(

/

nw

)*

5.13 normal stress difference curve taking into account themolecularweight distribution for LPDE at 2.16-kg test load condition for MFI using data from Refs. 9 and 10. (Reprinted from Ref. 21with kind permission from Steinkopff Verlag Darmstadt.)

Extension Uniflcatlon Technique: and Upgrade

189

Figure Coalesced normal stress difference curve taking into account the molecular-weight distributionfor HDPE at 2.16-kg test load conditionfor MFI using data from Refs. 11 and 12. (Reprinted from Ref. 21 with kind permission from Steinkopff Verlag Darmstadt.)

a log-log scale, the data in Fig. 5.17 give a single coalesced curve as shown in Fig. 5.18. Details of the data used for generating Figs. 5.17 and 5.18 are given in Table C3 (Appendix C). Only a single example is given to indicate the effectiveness of the coalescing approach to estimate normal stress difference of reprocessed polymeric material. However, because the unified curve for virgin material holds good in the case of reprocessed material too, the earlier curves in Figs. 5.8-5.10 giving unified normal stress difference versus shear rate plots for virgin materials could be used for reprocessed materials as well. It is known that polymer degradation during processing operations leads to changes inmolecular weight andmolecular-weight distribution inpolymers. MFI indicates the changes inmolecular weight but is rather insensitive to changes in molecular parameters such as molecular-weight distribution. In the case of reprocessed polymers, as long as the variations in molecularweight distribution are superimposed on the changes in molecular weight, a plot of Nl versus (“Q2 should suffice. However, when major changes in molecular-weight distribution occur, it is advisable to use the plots of Nl versus l i ; i w > 3 ~ ’ ( ~ / M F I ) 2as given earlier in order to obtain estimated normal stress difference versus shear rate curves for reprocessed polymers.

(az/

Chapter 5

Figure Coalesced normal stress difference curve for LLDPE at 2.16-kg test load condition for MFI using data from Ref. (Reprinted with permission from Elsevier Science Ltd., Kidlington, UK.)

Normal stress difference data are not as extensively available as the shear viscosity data. The plots, therefore, do not contain the abundance of data as in the unified curves for shear viscosity. The approach here was to establish the curve profile more than prove the validity of the unifying technique which, in the case of shear viscosity, has been conclusively shown to hold and the logic for the development of the unified curve in the case of the elastic material function runs on parallel lines. The unified curves provide the easiest way of getting an estimate of the elasticity of polymer melts simply through knowledge of their MFI.

5.2.2 ComplexViscosity and Storage Modulus The Cox-Mertz rule discussed in Chapter 2 (Sec. 2.3.3) can be written in the modified form for convenience as follows: at-=MFI

MFI

Figure Coalesced normal stress difference curve for nylon at 2.16-kg test load condition for MFI using data from Refs. 12 and 25. (Reprinted from Ref. 21 with kind permission from Steinkopff Verlag Darmstadt.)

It has been shownin Chapter that a unified curve, independent of temperature and grade for each generic type of polymer, can be generated by plotting q X MFI versus p m . Based on Eq. it is natural to expect that q* versus curves could also be coalesced by using MFI as a convenient shift factor. Thus, q* X MFI versus o/MFI should result in a master curve The same conclusion can be reached through first principles by noting that the superposition of dynamic functions is generally achieved through the use of reduced variables such as oh, or muT, where AT is the relaxation time and is an arbitrary shift factor. For a particular grade of a specific generic-type polymer, it is known that the dynamic functions corresponding to various temperatures are similar to one another in form and can be shifted along the frequency axis. The distance between the curves correspondingto different temperatures is generally taken to be the shift factor with respect to a reference temperature To. This enablesthe parallel shifting of the experimentallyobtained G’(w) and G”(w) curves through the use of the argument wuT. The choice of the reference temperature is arbitrary and is based on convenience. The physical significance of can be established by comparing the argument wuT with the dimensionless parameter whT. In practice, represents the ratio of the maximum relaxation times at different temperatures to the maximum relaxation time at the reference

192

Chapter 5

Figure Primarynormalstressdifferenceversusshearrateplotforvirginand reprocessed LDPE using data from Ref. (Reprinted from Ref. with kind permission from Elsevier Science Ltd., Kidlington, U.K.)

temperature To. Now hT can be represented in the following form:

where c' includes the quantities that remain constant when the temperature superposition is being accomplished. If c' is assumed constant, then aT may be given as

If the temperature density correlation factor is neglected in comparison with the variation viscosity, then

Extension Unification Technique: and Upgrade

193

Figure Coalesced primary normal stress difference plot for reprocessed LDPE at 2.16-kg test load condition for MFI using data from Ref. 27. (Reprinted from Ref. 26 with kind permission from Elsevier Science Ltd., Kidlington, U.K.)

The inverse relationship between melt viscosity and MFI used to give

may now be

-

= MFI(T0) MFI(T)

Because the choice of To is arbitrary, the reference MFI value may be chosen as identically equal to for convenience. Further, it is known that MFI changes from one grade of the polymer to another grade at the same temperature. Thus, using

1

=MFI it should be possible to coalesce curvesof different grades of a specific generic type of polymer. Because q* =

Jgy gy +

Chapter

it follows that

(5.12) Thus, q* X MFI versus u/MFI should result in agrade-independentand temperature-invariant master curve. The same conclusion wasreached earlierby a mere manipulation the Cox-Mertz rule along withthe expressions derived in Chapter for steady shear. Relating the steady-state normal stress difference T~~- T~ and the dynamic storage modulus G‘, both characterizing the elasticity of the material, has invariably been a more difficult task. Although the shapes the T~~- T~ versus and G‘ versus curves are generally similar, a relationship between the two has to be based on a suitable rheological model derived through an appropriate constitutive equation, such as was done by Pao [29, 301, Spriggs [31], Bogue [32], or Meister [33]. The results obtained by Pao [29, 301 for correlating the dynamic functions and steady-state stresses are limited to only low-frequency or low-shear data. The adjustable parameter among thematerial constants of the Spriggs [31], Bogue [32], and Meister [33] models make them more adaptable for wide ranges of shear rate or frequency data. The Bogue [32] and Meister [33] models representintegral-type models, each having adifferentphysical origin in its derivation. The Spriggs [31] model is of different type and involves material constants which are simple to determine and also have relevance to molecular parameters. In the present case, the Spriggs model has been chosen for correlating dynamic and steady-state elastic characteristics. The choice is somewhat arbitrary as each of the models is known to have almost the same capability for prediction [23] and by no means indicates the superiority of this model over that of Bogue [32] and Meister [33]. The dynamic functions that to the Spriggs model are expressed as follows:

(5.13)

whereas the steady-state functions are given as

(5.15) (5.16)

Extension unification Technique: and Upgrade

195

where qo,h, Z, and are model parameters and is an arbitrary adjustable constant expressed in terms of an independent parameter Z as

A comparison of the above equations yields the following: =

at

-

0

=

e

=

or q=-

a t o = e

W

Thus, it is obvious that the dynamic and steady-state characteristics ofa polymeric melt could be equivalent when appropriately shifted by an amount relative to each other. Note that only in the case of 2 = and hence = does the Spriggs model predict a correlation at = W. It has been shown in Sec. that the steady-state elastic response of polymer melts can be coalesced by plotting Nl versus Analogously, one can expect that coalesced versus (o/MFI)' could be obtained using the appropriate value of MFI. Figure shows the variation of the complex viscosity and storage modulus versus frequency for three different grades of HDPE each at a different temperature. Similarly, Fig. shows the variation of the complex viscosity and storage modulus versus frequency for two grades of PS at different temperatures. The details of the grade, the temperature of measurement, and the range of frequency are given in Table c4 of Appendix C. The melt flow index valueswere taken from the literature or calculated by the readout method from steady shear measurements which is discussed in Sec. Based on the foregone theoretical analysis, plots of q* X MFI versus o/MFI as well as versus (o/MFI)' have been made and these have resulted in the master curves for the viscoelastic data of HDPE and PS as shown in Figs. and respectively. It should be noted that whenever major changes in molecular-weight distribution exist, the shift factor MFI would need an additional (MJM,,,) term as suggested in the earlier sections of this chapter.

+

"

5.2.3 ExtensionalViscosity The fact that MFI is directly proportional to shear rate and inverselyproportional to shear viscosity as given by Eqs. and along with Eqs. and

Chapter 106

106

lo5

-105

l l l I IIIII

I 1 lo2

1

5.19 Variation of complex viscosity storage modulus with frequency for three different grades of HDPE, each at a different temperature. Grade

(2.66) giving the relationship between shear and extensional viscosities can be used to conclude the following: TEXMFIa-

2 tan%

(5.20)

tan M F 1 2

& -

Solving Eqs. (5.20) and (5.21) to eliminate tan

(5.21) gives

(5.22) The constant C, incorporates only the terms appearing in Eqs. (4.8) and (4.12) which are dependent only on the geometrical parameters the melt flow indexer

Unification Technique: Upgrade and Extension

Figure Variation of complex andstoragemoduluswithfrequency two grades of PS at two different temperatures.

Grade

170°C

678 U 686

B, D

-

for

C

-

and the test load which is constant for a particular system as per the standards. By using different loads, different. MFI values for the same system be obtained. Equation (5.22) is independent of which loading conditionis used and would hold for allgenerated MFI values, thus indicating thata plot of r ) E X MFI versus (M)’ could be expected to givea coalesced curve for polymer systems of the same generictype. The coalesced curve would be temperatureindependent if the MFI value used is obtained at the temperature of measurement of the versus C curve.

Chapter 5

198

Figure5.21 Master for the viscoelastic of HDPE using MFI as under a testing load condition of 2.16 kg. F see Fig. 5.19.

factor

' MFf Figure Master curves for the viscoelastic data PS using MFI as the under a testing load condition of 2.16 kg. to D see Fig. 5.20.

factor

Extension Unification Technique: and Upgrade

199

Figure 5.23 Coalesced extensional viscosity curve for HDPE at 2.16-kg test load condition for m using data from Refs. 23,38, and 39. (Reprinted from Ref. 37 with kind permission from Die Angewandte Makrornolekulare Chemie, Darmstadt.)

Figures 5.23-5.25 show the plotsof X MFI versus on logarithmic scales for HDPE, PP, and PS Each of the curves shows data [23, 38-40] for at least two different grades of the polymer at three different temperatures. The details of the data analysed are given in Table C5 of Appendix C for quick reference. Despitesuch diversity in the experimental data, the single curves obtained for each generic type of polymer show verylittle scatter. Dueto limited data, curves have not been obtained for a large number of polymeric systems.

Figure Coalesced extensional viscosity curve for PP at 2.16-kg test load condition for MFI using data from Refs.23 and 38. (Reprinted from Ref.37 with kind permission from Die Angewandte Makromolekulare Chemie, Darmstadt.)

Figure Coalesced extensional viscosity curve for PS at 5-kg test load condition for MFI using data from Refs. and (Reprinted from Ref. with kind permission from Die Angewandte Malaomolekulare Chemie, Darmstadt.)

Smith, D.J., The correlation of melt index and extrusion coating resin performance, WPZ,

Boaenski, F. J., An approach to the use of rheology in post reactor processing, Plastics Compounding, (Sept./Oct. Shida, M., Shroff, R.N., and Cancio,L. V., Correlation of low density polyethylene rheological measurements with optical and processing properties, Polym. Eng. Boenig, H. V., Polyolefns, Elsevier, Amsterdam Chap. p. Busse, W. F., Mechanical structures in polymer melts. I. Measurementsofmelt strength and elasticity, J. Polym. Sci., A-2, 5, Dutta, A., On viscosity-melt flow index relationship, Rheol. Acta, Shenoy, V. and Saini, D. R., Upgrading the melt flow index rheogram approach in the low shear rate region, J. Appl. Polym. Sci., Minoshima, W., White, J. L., and Spruiell, J. E., Experimental investigation of the influence of molecular weight distribution on the rheological properties of polypropylene melts, Polym. Eng. Sci., Ram, Viscoelastic parameters of characterized branched polyethylenes, Polym. Eng. Sci.,

Han, C. D., Kim, K. U., Siskovic, N., and Huang, C. R., An appraisal of rheological models as applied to polymer melt flow,Rheol. Acta, Macosko, C. W. and Lornston, J. M., The rheology of two blow molding polyethylenes, Soc. Plast. Eng. of ANTEC, Tech. Paper pp.

Extension Unification Technique: and Upgrade

201

12. H ~c.,D. and &e, S. M., Studies of melt spinningW I . The effects of molecular structureandcoolingconditionsonseverityofdrawresonance, J. Pobm. Sci., 24, 61-87 (1979). Adv.Appl.Mech., 20, 177-226 13. Becker,E.,Simplenon-Newtonianfluidflow, (1980). 14. Lodge, A. S., Rheology Research Center Report No. 60, University Of Wisconsin, Madison (1980). 15. Malm, A. Ya., Kulichikhin, V. G., Zabugina, M. P., and Vinogradov, G . V., The high elasticity of polyisobutylenes with different microstructure,PolYm. Sci. USSR, 12, 138-148 (1971). 16. Oda, K., White, J. L., and Clark, E.S., Correlation of normal Stresses in polystyrene melts and its implications, Polym. Eng. Sci., 18, 25-28 (1978). 17. Prest, W. M., Jr., Viscoelastic properties of blendsof entangled polymers,J. Polym. Sci., A-2, 8, 1897-1988 (1970). 18. Onogi, S., Masuda, T., and Kitagawa, K., Rheological properties of anionic polystyrene I. Dynamic viscoelasticity of narrow distributed polystyrene, 11. Dynamic Macromolecules, 3, viscoelasticityofblendsofnarrowdistributedpolystyrene, 109-116, 116-125 (1970). 19. Mills, N. J. and Nevin, A., Oscillatory shear measurementson polystyrene melts the terminal region, J. Polym. Sci., A-2, 9, 267-281 (1971). S., Steady state compliance of polymer 20. Masuda,T.,Takahashi,M.,andOnogi, blends, Appl. Polym. Sympos., 20,49-60 (1973). 21. Shenoy, A. V. and Saini, D. R., An approach to the estimation of polymer melt elasticity, Rheol. Acta, 23, 608-616 (1984). 22. Shenoy, V. and Saini, D. R., A simplified approach to the prediction of primary normal stress differences in polymer melts, Chem. Eng. Commun.,28,l-27 (1984). D., Rheology in Polymer Processing, Academic Press, New York (1976), 23. Han, C.. p. 52, 205. 24. Saini, D.R. and Shenoy, A. V., Viscoelastic properties of linear low density polyethylene melts, Eur. Polym. J., 19, 811-816 (1983). 25. Bankar, V.G., Spruiell, J. E., and White, J. L., Melt spinning dynamics and rheological properties of Nylon 6, J. Appl. Polym. Sci., 21, 2135-2155 (1977). 26. Shenoy, V. andSaini, D. R.,Estimationofmeltelasticityofdegradedpolymer from melt flow index, Polym. Degrad. Stability, 11, 297-307 (1985). 27. Rokudai, M., Influence of shearing history on the rheological properties and processibility of branched polymers, J. Appl. Polym. Sci., 23, 463-471 (1979). 28. Shenoy, V. and Saini, D. R., A new shift factor for coalescing dynamic viscoelastic data of polymer melts,Acta Polymerica,37, 504-507 (1986). 29. Pao, Y.-H., Hydrodynamic theoryfor the flow of a viscoelastic fluid, J. Appl. Phys., 28,591-596 (1957). 30. Pao, Y.-H., Theories for the flow of dilute solutions of polymers and non-diluted liquid polymers, J. Polym. 61, 413-448 (1962). 31. Spriggs, T.W., A four constant model for viscoelastic fluid, Chem. Eng. Sci., 931-940 (1965). 32. Bogue, D. C., Explicit constitutive equation based on an integrated strain history, Znd. Eng. Chem. Fundam., 253-259 (1966).

'

Chapter Meister, B. J., An integral constitutive equation based on molecular network theory, Trans. Soc. RheoL, Orbey, N.and Dealy, J. M., Isothennal swell of extrudate from annular dies. Effects of die geometry, flow rate, and resin characteristics,Polym. Eng. Sci., Nakajima,N., Shida, M.,and Wksbmn, K F., 26th Internat. Congr. Pure Appl. Chem. Lobe, V. M. and White, J. L., An experimental study of the influence of carbon black on the rheological properties of polymer melt, Polym. Eng. Sci., Shenoy, A. V. and Saini, D. R., Re-analysis of extensional flow data of polymer melts, Angew. Makrornol. Chemie, Shroff, R.N.,Cancio, L. V. and Shida, M.,Trans. Rheol., 21, 42 Au-Yeung, V. S. and Macosko, C. W., Biojluid Mech.,, Han, C. D. and Kim, Y. W., Studies on melt spinning Elongation viscosity and spinnability of two phase system, J. Appl. Polym. Sci., 18,

Rheological Models for Unified Curves

In Chapters and a number of unified curves have been presented for a variety of polymers of different generic types. These curvescan be readily used for generating specific material parameter versus shear rate curves at any required temperature of interest merely from the knowledge of the MFI at that temperature. As ASTM test conditions have tobe conferred to duringMFI test measurements, the MFI at the ASTM test temperature needs tobe converted to the MFI value at the required temperature of interest. This could be done throughtheuseofthemodified Arrhenius-type or the modified (Williams-Landel-Ferry) WLF-type described in Sec. Care must be taken to use the appropriate activation energy E value and the appropriate glass-transition temperature Tgvalue in the respective equations. The value of E must be determined within a narrow temperature band around the temperature of interest, that the obtained value has higher accuracy. In the case of copolymers [4], extra caution has to be exercised because of the presence of dual values that the appropriate value alone is chosen. The Tgvalues for various polymers are summarized in Table for ready reference. It is important to use the appropriate value of Tg in order to minimize errors in " Iestimation. The value of Tg could vary within a broad range from 10°C to 15"C, especially in the case of amorphous polymers. The values of the heat distortion temperatures of various gradesof polystyrene (PS) are reported in Table The heat distortion temperature, being a thermomechanical property of a polymer, is qualitatively related to the glass-transition 203

Rheological Models

Unified Curves

205

Table 6.2 Variation of T, with me of Plasticizer in PVC Formulation

Plasticizer mole fraction)

TB

ec>

Diethyl phthalate Di-n-butylphthalate Di-n-hexylphthalate

Di-n-decylphthalate Di-n-dodecylphthalate ~

Source: Ref. 7. (Reprinted withkind permission from American Chemical Society, Washington, D.C.)

temperature. Therefore, for these grades, the glass transition would also vary over about 8°C. The sensitivity of the effective M H value to the glass-transition temperature is illustrated in the last two columns of Table 6.1. In formulating the master curve for PS (Fig. 4.16), a single value of 100°C was used for the glass-transition temperature, whereas the glass-transition temperature of an impact polystyreneis expected to be lower than that the general-purpose crystal grade. The rheograms of the various grades at different temperatureswould coalesce in a narrower band, if the correct glass-transition temperatures for the grades were available. Even with a semicrystalline polymer like HDPE, where the density could vary from to the glass-transition temperature would be different for different grades. The range of this variation in semicrystalline polymers is generally narrower than that in amorphous polymers. In the case of poly(viny1 chloride) (PVC) formulations, extra care must be exercised, as the TBis known to vary [6] considerably depending on the type and amount of plasticizers [7, (see Tables 6.2 and and to a certain extent on the type and amount of

Table 6.3 Variation of TBwith Amount of the Plasticizer Dioctylphthalate POP) in PVC Formulation

CDOPIstabilizer Compound composition

TS

(parts by weight)

ec,

Viiol60d

Source: Ref. 8.

C

Chapter

other additives [g] (see, for example, Table Hence, wherever possible, it might be beneficial to use a measured value of T, for greater accuracy. In cases in which the MFI value is required at a load conditiondifferent from the determined one, Eq. from Sec. is to be used. The step-by-step procedure to obtain the material parameter versus shearrate curves is as follows. Initially, obtain the MFI value of the particular grade of the polymer under consideration using standard test load and temperature conditions as given in either from the polymer manufacturer or determined from the meltflow apparatus. If the loading condition of the obtained MFI does not correspond to that in the unified curve, then a new MFI value ought to be calculated using Eq. given in Sec. If the temperature at which the material parameter versus shearrate curve is desired is different from that at which MFT value is determined, then one of the equations and from Sec. ought to be used. The plot of material parameter versus shear rate under the required conditions then be readily obtained by substituting thecorrect value of MFI in the curve.

6.1 SUGGESTEDRHEOLOGICALMODELS In order simplify the final step in generating the required material parameter versus shear rate curves from MFI by the above procedure, appropriate rheological models have been suggestedfor fitting the unified curves.

Table 6.4 Variation T, with Amount of Filler in PVC Formulation

Percent filler Graphon

T, (“c)

Source: Ref.

Rheological Models

6.1

Unified Curves

Viscosity Versus Shear Rate

In the master rheograms of viscosity versus shear rate given in Chapter 4, the following four models have been suggested [10,11] covering a varied range of shear rates: Modified Carreau Model: qXMFI=q0XMFI

(Ay]-N

l + ( X x M m )*x

Modified Ellis Model: qXMFI=

qo X MFI

1

+

Modified Ostwald-de Waele Power-Law Model:

x MFI=K(&) General Rheological Model:

where qo X MFI is the modified zero-shear viscosity function, X MFI is the modified non-Newtonian viscosity function, is the modified shear-rate function, X X MFI is the modified time constant, K is the consistency index, N, a',P, and n are dimensionless parametersassociated with power-law behavior, is the shear stress given by the product X MFI and j/MFI, and T~~ is that special value of the shear stress for which X MFI = i(qo X MFI). The modifiedCarreaumodeland the modified Ellis model are limited to relatively low values of shear rates and shear stresses, respectively, whereas the modified Ostwald-de Waele power-law model is applicable to the higher-shearrate region where the data points fall in a straight line on the log-log plot of X MFI versus j/MFI. The General Rheological [l11 model,however, is applicable to the entire shear-rate range covering the region of validity of the modified Carreau model as well as the region of validity the modified Ostwald-de Waele power-law model.Hence, the General Rheological [l11 model is recommended for use when the entire master rheogram is to be fitted by a single best-fitting curve. The four constants of the General Rheological[l11 model can be systematically evaluated as follows:

Chapter

1. The plot of r\ X MFI versus on the log-log scale in the low-shearrate region yields the limiting values qoX MFI by mere readout as shown in Fig. 6.1. 2. The functional behavior at large shear rates onthe same log-logplotin Fig. 6.1 being linear defines K and n directly. The slope of the straight line defhes n - 1, whereas K is the value of X MFIwhen j/MFI = 1 (provided the point satisfies the power-law equation). The best values K and n - 1 can be easily computed by regressional analysis the data at high shear rates. The exponent P is most easily evaluated by determining the point of intersection the two limiting solutions which corresponds to qo X MFI). Thus, from Eq. (6.3), ?,,IMF1 can be calculated because qo

MFI =K(%) MFI

Denoting the ordinate on the actual rheogram by % X PJIFI (corresponding to an abscissa and substituting into Eq. gives

(.L( = 2(qo Thus, the exponent P can be readily determined from

Z e r o - shear limit

I I

I I

Slope * n ,ywer-law

-t /

\t

Figure 6.1 Evaluation GeneralRheologicalmodelconstantsforunifiedcurves viscosity versus shear rate.

of

Models Rheological

Curves for Unified

209

Table Rheological Parameters of the Modified Carreau Model [Eq.(6.1)] for Different Generic-vpe Thermoplastics Modified shear rate'

MFI [(Poise) (g per 10 min)] rlo

Thermoplastic

[(S")

per 10 min)]

Testb

X (g per load 10

(g

[(S)

N

min)"]

0%)

LDPE HDPE

3.0 X lo' 2.1 lo'

25.3 24.0

0.17 0.15

0.01-2 0.1-2

2.16' 2.16'

LLDPE PP PS C-Ester C-Ether

9.6 lo4 1.2 lo' 4.0 X lo6 8.5 X 104 8.5 X lo4

6.0 4.6 46.4 0.72 0.5

0.11 0.13 0.3 0.06 0.18

0.01-2 0.1-4 0.001-0.4 1-10 1-10

2.16' 2.16' 5.00' 2.16' 2.16

Acrylic Nylon PET PC PVDF PP0 PPS PASRES PEEK PE1 PAr

2.1 X 9.5 X 8.3 4.2 6.0 2.2 3.4 1.9 4.5 1.4 X 1.8

2.4 11.6 0.3 0.19 0.77 0.2 0.46 0.32 1.7 0.038 0.04

0.16 0.03 0.1 0.07 0.3 0.34 0.36 0.2 0.24 0.33 0.38

0.1-20 0.1-20 0.01-50 1-80 0.1-4 1-40 0.001-10 1-10 0.01-20 1-200 0.01-10

3.80' 2.16' 2.16' 1.2OC 12.50

SBS ABS

2.2 lo' 1.7 X lo6

1.0 12.8

0.12 0.31

0.1-20 0.01-1

5.00 5.00'

EVA Polyester Elastomer

2.5 9.0

0.01-1 0.1-7

2.16 2.16

PP-HDPE

1.5

PS-PMMA PS-POM PMMAPOM PVC Formulations

10'

lo4 lo4 lo4 lo'

lo5 lo' 16 lo'

lo5 lo5

5.00 5.00 5.00 5.00 5.00 5.00

lo'

lo4

7.7 0.88

0.23 0.17

lo5

0.66

0.34

0.1-10

2.16

3.4 lo' 2.7 X lo' 2.0 lo'

1.26 0.66 0.48

0.27 0.35 0.28

0.1-4 0.1-4 0.1-8

5.00 5.00 3.80

lo6

4.47

0.32

0.01-5

20.w

1.2

X

'Applicability range for j/MFI. bCondition for MFI used in the master rheogram. 'ASTM specified.

Chapter 6

210

Table 6.6 Rheological Parameters of the Modified Ellis Model [Eq. (6.2)] for Different Generic-Type Thermoplastics

Thermoplastic

qo X MFI [(Poise) X (g per 10 min)]

LDPE HDPE

3.0 X 10' 2.1 x 10'

LLDPE PP PS C-Ester C-Ether

9.6 1.2 4.0 8.5 8.5

Acrylic Nylon PET PC PVDF PPO PPS PAS/PES PEEK PEI PAT

2.1 9.5 8.3 4.2 6.0 2.2 3.4 1.9 4.5 1.4 1.8

SBS ABS

2.2 x 10' 1.7 X lo6

EVA Polyester

2.5 X 10' 9.0 x 104

x lo4 x 10' X

x x x x x x

lo6 lo4 lo4 10'

lo4 lo4 104

X 10'

x

10'

x los X

x

los

los

X 10'

x los

Shear stress" (dyn/cm2)

7112

Test loadb (kg)

(dyn/cm2)

a'

lo4

2.14 1.85

3

lo4

104-106

2.16" 2.16"

2 x 10' 1.54 X 10' 8 x lo4 1.27 X lo6 3.82 X los

2.00 2.89 2.53 2.13 2.63

10'-4 X 10' 6 X 103-106 104-106 10'- lo6 105-106

2.16" 2.16" 5.OW 2.16" 2.16

lo' lo6

2.23 2.24 2.28 1.82 3.00 2.70 2.70 2.10 2.10 2.56 1.50

2 x lo4-2 x

2.50 2.25

2 X 104-4 X 104-106

lo6

5.00 5.OW

1.85 2.10

3 X 10'-4 X 3 X 104-6 X

lo6 lo6

2.16 2.16

2.64 X 10'

3.10

2 x lo4-2

2.16

4.18 X l o s 4.4 x los 6.3 X 10'

2.17 3.33 2.33

6 X

5.00 5.00 3.80

1.92 X 10'

2.75

2 x104-107

6 X 3.67 x

4.09 1.18 3 2.73 1.4 9 7.6 2.2 2.8 3.6 2.3

X X

x lo6 X lo6 X

lo6

x los X 10'

x lo6 X

los

lo6 x lo7 2 x lo6 X

1 x 10'

5 x 1.6 X

lo4 lo6

x

103-106

lo6

104-107

1o3- lo7 5 x 104-107 2 X 1OS-5 X lo6 2 x 1oS-1o7 10'-2 x lo6 10'- lo7 5 x 104-107 2 x lo5-2 x 107 104-107

3.80" 2.16" 2.16" 1.20" 12.50 5.00 5.00 5 .OO 5.00 5.00 5.OO

Elastomer

los

PP-HDPE

1.5 X

PS-PMMA PS-POM PMMAPOM PVC Formulations

3.4 x 10' 2.7 X 10' 2.0 x 10' 1.2

x lo6

"Applicability range for T. bCondition for MFI used in the master rheogram. 'ASTM specified.

x lo6 105-3 X lo6 10'-1.5 X lo6 104-2.5 X lo6

20.00"

Rheological Models for Unified Curves

21 1

Table 6.7 Rheological Parameters of the Modified Ostwald-de Waale Power-Law Model [Eq. (6.3)] for Different Generic-Type Thermoplastics

Thermoplastic LDPE HDPE UHMWPE LLDPE PP PS C-Ester C-Ether Acrylic Nylon PET PC PVDF PPO PPS PASPES PEEK PEI PA SAN

SBS ABS VCVA EVA Polyester Elastomer TPE PP-HDPE HDPE-PMMA PS-PMMA PS-POM PMMA-POM PVC Formulations

K [(g/cm-s2-') X (g per 10 min).] 1.7 X 1.24 X 1.66 X 1.0 x 1.75 X 3.32 X 3.2 X 1.5 X 3.2 X 3.4 x 5.0 X 2.5 X 7.5 x 1.0 x 6.0 x 3.0 x 6.0 x 1.4 X 2.4 X 3.0 X 1.0 x 3.5 x 4.0 X 1.0 x 1.4 X

10' 10'

1.5 2.0 1.0 3.0 3.6 3.0 4.6

10'

X

lo6 105 10' 10' 10' 10' 10' 10' 10' 10'

lo5 lo6 105

ios 105

lo6 10' 10'

lo6 105 10'

lo5 10'

x 105 x 10' X X

10' 10'

x

lo5

X

10'

"Applicability range for j/MFI. bCondition for MFI used in the master rheogram. 'ASTM specified.

n

Modified shear rate" [(s-' X (g per 10 min)-'I

Test loadb (kg)

0.37 0.471 0.156 0.66 0.34 0.368 0.38 0.47 0.44 0.44 0.43 0.52 0.40 0.25 0.44 0.60 0.32 0.34 0.85 0.33 0.25 0.38 0.33 0.55 0.65

1- 1000 2-1000 10- 10,000 4- 100 4- 1000 0.4-200 20- 1000 2- 100 20- 1000 50- 1000 50- 1000 80- 1000 2-200 40- 1000 10- 100 10-1000 20- 1000 200- 1000 10-100 3 -1000 20-2000 1- 100 0.3 -7000 1- 1000 7-300

2.16" 2.16" 10.00" 2.16" 2.16" 5.00" 2.16" 2.16 3.80" 2.16" 2.16" 1.20" 12.50 5 .OO 5.00 5.00 5.00 5.00 5.00 3.80 5.00 5.00" 20.00 2.16 2.16

0.40 0.32 0.45 0.46 0.30 0.43 0.36

1- 1000 10- 1000 0.1-40 4- 100 4-100 8-100 5 -2000

2.16 2.16 2.16 5.00 5.00 3.80 20.00"

Table 6.8 Rheological Parameters of the General Rheological Model [Eq. (6.4)] for Different Generic-Type Thermoplastics ~

Thermoplastic

~

~~~

~~

TO X MFI [(Poise) X (g per 10 min)]

X

K [(g/cm-s2-") (g per 10 min>.]

~~

n

P

0.370 0.471 0.156 0.66 0.34 0.368 0.38 0.47 0.44 0.44 0.43 0.52 0.40 0.25 0.44 0.60

- 1.220 -0.934

~

Modified shear rate" [(s- '> x (g per 10 min)-'I

Test loadb (kg)

~

LDPE HDPE UHMWPE LLDPE PP PS C-Ester C-Ether Acrylic Nylon PET PC PVDF PPO PPS PASPES

3.0 x 2.1 x

lo5 lo5

9.6 x 1.2 x 4.0 X 8.5 X 8.5 X 2.1 x 9.5 x 8.3 x 4.2 x 6.0 X 2.2 x 3.4 x 1.9 X

lo4

-

lo5

lo6 lo4 lo4 105 104

lo4 lo4 10'

lo5 105 10'

1.70 x 1.24 x 1.66 X 1.0 x 1.75 x 3.32 x 3.20 x 1.5 x 3.2 x 3.4 x 5.0 x 2.5 x 7.5 x 1.0 x 6.0 x 3.0 x

lo5 lo5 lo6 105 105 105

lo5 105 105 105

lo5 105

lo5 lo6 lo5 lo5

-

-2.808 - 1.076 - 1.253 -2.58 -2.58 -1.336 - 1.624 -2.031 -2.282 -3.106 -1.318 - 1.306 -4.033

0.01- 1000 0.1-1000 10- 10,000 0.01-100 0.1-1000 0.001-200 1- 1000 1- 100 0.1-1000 0.1- 1000 0.01- 1000 1-1000 0.1-200 1- 1000 0.001 -100 1- 1000

2.16" 2.16" 10.00" 2.16" 2.16" 5.00" 2.16" 2.16 3.80" 2.16" 2.16" 1.20" 12.50 5.00 5.00 5.00

9 9,

2 Q)

PEEK PEI PAr SAN

SBS ABS VCVA EVA Polyester elastomer TPE PP-HDPE HDPE-PMMA PS-PMMA PS-POM PMMA-POM PVC (Formulations)

4.5 x 105 1.4x 105 1.8 x 105 3.3 x 105 2.2 x 105 1.7 X lo6 -

2.5 x 105

9.0 x 104 -

1.5 x 105 -

3.4 x 105 2.7 x 105 2.0 x 105 1.2 x

lo6

6.0 x 1.4 X 2.4 x 3.0 x 1.0 x 3.5 x 4.0 x 1.0 x

0.32 0.34 0.85 0.33 0.25 0.38 0.33 0.55

- 1.264 -2.060 -12.127 - 1.384 -1.318 - 1.090

-3.802

105 105 105 105

0.65 0.40 0.32 0.45 0.46 0.30 0.43

-2.00 -2.31 -4.265

4.6 x 105

0.36

- 1.286

105

lo6

105 105

lo6 105

lo5

105

1.4 x 105 1.5 x 105 2.0 x 105 1.0x 3.0 x 3.6 x 3.0 x

"Applicability range for j/MFI. bCondition for MFI used in the master rheogram. 'ASTM specified.

-

-2.110

0.01- 1000 1- 1000 0.001- 100 1-1000 0.1-1000 0.01-100 0.3-7000 0.01- 1000

5.00 5.00 5.00 3.80 5.00 5.OW 20.00 2.16

-k

4 -

-1.103 -

0.1-300 1- 1000 0.1- 1000 0.1-40 0.1-100 0.1-100 0.1-100

0.01-2000

2.16 2.16 2.16 2.16 5.00 5.00 3.80 20.00"

Chapter

Tables list the model constants and the range of applicability based on themodifiedCarreaumodel, the modified Ellis model,themodified Ostwald-de Waele power-law, model, and the General Rheological [l11 model, respectively, for the master rheograms of most of the polymers discussed in Chapter 4. The liquid-crystalline hydroxy benzoic acid/poly(ethyleneterephthalate) (HBA/PET) copolymer master rheogram given inFig. 4.37, however, is not amenable to a curve fit by any of the models discussed above. The reason is because the shape of the unified curve for this liquid-crystalline copolymer is radically different from those obtained for other thermoplastics. There are two shear-thinning regions separated by a short plateau as can be seen in Fig. 4.37. A new equation of the following form is thus suggested [12]:

where n1 is the slope of the linear portion in the low-shear-rate region 10°/s), n2 is the slope of the linear portionin the medium-shear-rate region (10'-102/s), K , is the viscosity function value at $MFI = 1.0 based on the initial linear portion of the curve,and KZ is the viscosity function value at */ME = 1.0 based on the second linear portion of the curve. From Fig. 4.37, it can be seen that in the medium to higher shear-rate regions, the slope of the curve is continuously changing. Hence, the determination of KZand n2 based on a linear correlation is not truly correct over the entire range. Thus, there has to be a restriction imposed on the upper limit of the shear rate. The parametric values of the suggested rheological model have been determined to be equal to the following [12]: = lo4,

n1 =

= 1.5 X

lo4,

= -0.35

The solid curves drawn on all other master rheograms in Chapter 4 represent the best-fitting curves based on the General Rheological [l11 model. In most cases, the fit is very good. It must be remarked that in regions where the fit is not good, the limitation is not due to the model but due to the inherent scatter inthe master rheogram. This scatter is attributed to the inaccuracies in the viscosity versus shear rate data obtained by various workers and techniques. Of course, any noticeable scatter in the low-shear-rate region is also attributed to the insensitivity of MFI to the molecular-weight distribution as discussed in Chapter 5. In such circumstances,the modified Carreau model has to be altered [l31 as follows: Altered Modified Carreau Model:

Unified Curves

Rheological Models

215

The model constants for the above equation which fit the curves given in Figs. 5.3, and 5.7 are tabulated in Table 6.9.

6.1.2

Normal Stress Difference Versus Shear Rate

The normal stress difference versus shear rate curves given in Chapter are for only five generic types of polymers as against the many more that are given in Chapter 4 for viscosity versus shear rate. This is basically due to the nonavailability of literature data and the general difficulty in generating this data using commercial rheometers. However, if a relationship between the unified viscosity curve and the unified normal stress difference curve is established, then estimated normal stress difference curves could be generated for each generic type of polymer. Wagner [14,15] has provided a method for the prediction of normal stress difference from shear viscosity using a strain-dependent single-integral constitutive equation of the perstein-Kearsley-Zapas) BKZ type. In an appropriately modified form, it can be written [l31 as follows: Modified Wagner'S Relationship: (6.10)

where m is an adjustable parameter. Normal stress difference data are generally collected on viscometers which have an effective upper limit of shear rate not greater than 1O/s. Thus, the relationship between the viscosity function and the normal stress difference function is sought in the low-shear-rate region [16]. The modified Carreau model as given by Eq. (6.1) is the obvious choice for selecting this relationship through its use in Eq. (6.10).

Table

Rheological Parameters the Altered Modified Carreau Model for Polyolefins Considered in Figs. 5.3, 5.5, and 5.7 qo

MFI

Polymer

(az/aw)1.7

PP LDPE HDPE

2.2

lo4

8.0 9.5

l@ l@

Source: Ref.

(az/aw)1,7

1.0 0.31 0.50

N

Applicability range (M,/Mw)1'7 (*m)

-_

0.22

0.1-1000

0.25

1-lo00

Chapter 6

216

From Eqs. (6.1)and(6.10), obtained as

the primary normal stress function

is readily

(6.11) l+(AXMJ?I)’

MFI

The plots of normal stress difference versus shearrate can then be obtained using (6.12) The adjustable parameter or damping constant used for fitting the unified normal stress difference curves inFigs.5.10,5.12,5.14,5.15,and5.16 are given in Table 6.10. It can be seen from Table 6.10 that the damping constant lies between 0.13 and 0.2, in line with the findings of Wagner [l51 for polymer melts. This fact can then be used to generate the unified normal stress difference curves for other polymers [16-181 whose parameter values for the modified Carreau model fit of unified viscosity data are given in Table 6.1. Figures 6.2-6.13 show the predicted normal stress difference curves for different polymer types. The solid line shows the plot generated for a median value of = 0.16 and the band

Table Values of the Adjustable Parameter of Eq. (6.11) for Obtaining Best-Fit Normal Stress Difference Versus Shear Rate Curves in the Case of the Five Polymers Mentioned in Chapter 5 Polymer

Damping constant (m)

LDPE HDPE LLDPE PP PA Source: Ref. 16. (Reprintedwithkind permissionfrom Gordon andBreachPublishers, Lausanne, Switzerland.)

6.2 Unifiednormalstressdifferencecurvepredicted from Eq. (6.11) using a median value of m = 0.16 for cellulose esters at 2.16-kg test load condition for MFI. (Reprinted from Ref. 16 with kind permissionfrom Gordon and Breach Publishers, Lausanne, Switzerland.)

218

Chapter 6

Figure Unifiednormalstressdifferencecurvepredicted from Eq. (6.11) using a median value of m = 0.16 for acrylic at 3.8-kg test load condition for MFI. (Reprinted from Ref.16withkindpermission from GordonandBreachPublishers,Lausanne, Switzerland.)

shows the upper and lower bounds defined by a damping constant m of 0.13 and respectively. A plot of Nl versus (j/MFI)’ on a log-log scale would suffice to obtain a unified curve when various gradesof polymers of a generic type all have broad and regular molecular-weight distributions or, alternatively, all have narrow molecular-weight distributions. If a unique curve is to be obtained which is independent of the width of molecular-weight distribution, then a correction term (Mz/Mw)3.5 is to be included that a plot .of N I versus (Mz/~w)3.’(j/MFI)2 has to be used to obtain the coalescence. The following relationship among N,, $I,, and j 2 can be written in the modified form as would fit the unique curves in Figs. 5.11, 5.13, and 5.14: ”

(6.13)

9

Rheologlcal Models for Unlfied Curves

Figure Unifiednormalstressdifferencecurvepredictedfrom Eq. (6.11) a median value of m = 0.16 for PET at 2.16-kg test load conditionfor h4FI. (Reprinted from Ref. 16 with kind permission from Gordon and Breach Publishers, Lausanne, Switzerland.)

Substituting

for

in EQ. (6.11) gives the following:

(6.14)

In fact, the modified Wagners relationship can be written in the altered as follows:

(6.15)

16 '

1

(t/MFI l2 Figure Unified normal stress difference curve predicted from (6.11) using a median value of m = 0.16 for PC at 1.2-kg test load condition for MFI. (Reprinted from Ref. 16 with kind permission from Gordon and Breach Publishers, Lausanne, Switzerland.)

G2

Id' (W/ MFI)'

Figure Un5ed normal stress difference curve predicted from Eq. (6.11) using a median value of m = 0.16 for PVDF at 12.5-kg test load condition for MFI. (Reprinted from Ref. 17 with kind permission from American Chemical Society, Washington, D.C.)

Rheological Models for Unified Curves

IO'

IC2

IO0

l o1

lo2

( Y I M F I 1'

from Eq. (6.11) using a Figure Unifiednormalstressdifferencecurvepredicted medianvalueof m = 0.16 for P P 0 at 5.0-kg testloadcondil [ionfor MFI. (R .eprinted from Ref. 18 with permission from Technomic Publishing Co., Lancaster, PA.)

Combining Eqs. (6.14) and (6.15), the altered modified Carreau model as given by Eq. (6.9) is easily derivable [13]. Normal stress difference data are not as extensively available nor easily determinable as shear viscosity data. Unified curves predicted through ws. (6.11) or (6.14) thus provide the easiest method of getting a reasonable estimate of the elasticity polymer melts simply through the knowledge of the MFX

6.1.3 ComplexViscosity Versus Frequency The rheological models for unified complex viscosity versus frequency curves can be easily written based on the modified Cox-Mertz rule discussed in Chapter 5. Thus, using &. (5.5), the modified .Carreau model for complex viscosity can be written from Eq. (6.1) as

222

Chapter

IO0

IO'

IC?

( t /MFI

6.9 Unifiednormalstressdifferencecurvepredicted from Eq. (6.11) using a median value ofm = 0.16 for PPS at 5.0-kg test load condition forMFI. (Reprinted from Ref. 18 with permission from Technomic Publishing Co., Lancaster, PA.) Modified Carreau Model:

[

-

(6.16)

Similarly, the other models may be written as follows: Modified Ostwald-de Waele Power-Law Model:

)MFI '(K=

MFI

Modified General Rheological Model:

[

(.

$

K

)'(&)"'"]"

MFI

(6.18)

Rheological

Unified

223

Figure 6.10 Unifiednormal differencecurvepredictedfrom Eq. (6.11) using a median value of m = 0.16 for PAS and PES at test load condition for MFI. (Reprinted from Ref. 18 permission from Technomic Publishing Lancaster, PA.)

6.1.4 Storage Modulus Versus Frequency The rheological model for storage modulus versus frequency curves is derivable from the equality given in Eq. (5.19), which can be written as NI

”

-

2G’

at -= c(&)

MFI

(6.19)

The altered modified the Wagners relationship in Eq. (6.10) on using Eq. (6.19) can be written as follows: Modified Wagners relationship:

I 00

IO'

MFI

Figure Unified normal stress difference curve predicted from , .11)usling a (6. median value of = 0.16 for PEEK at 5.0-kg test load condition for MFI. (Reprinted from Ref. 18 with permission from Technomic Publishing Co., Lancaster, PA.)

( % / M F I l2

Unified normal stress difference curve predicted from Eq. (6.11) using a median value ofm = 0.16 for PE1 at 5.0-kg test load condition forMm. (Reprinted from Ref. 18 with permission from Technomic PublishingCo., Lancaster, PA.) Figure

Rheologlcal Models

Unified Curves

Figure Unified normal stress difference curve predicted from Eq. (6.11) using a median value of m = 0.16 for PAr at 5.0-kg test load condition for MFI. (Reprinted from Ref. 18 with permission from Technomic Publishing Co., Lancaster, PA.)

In writing Eq.(6.20), c has been forced to take the value of 1 because while replacing q X MFI by q* X MFI the modified Cox-Mertz rule of Eq. is used, which holds only when p/MFI = It is, however, assumed that the new adjustable parameter m* will implicitly account for any variations in the model due to the relaxation of the assumption. Combining Eqs. (6.16) and (6.20) gives the equation for the storage modulus versus frequency as follows: (6.21)

1. ASTM test D1238 for all commonly used polymers except PVC formulations.

ASTM test D3364 for PVC formulations.

Chapter

3. Saini, D. R. and Shenoy,

V., new method for the determination of flow activation energy of polymer melts, J. Macromol. Sci.-Phys., B22, 437-449 (1983). 4. Shenoy, V. and Saini, D. R., Effect of temperature on the flow of copolymer melts, Mater: Chem. Phys., 19, 123-130 (1990). 5. Shenoy, V., Chattopadhyay, S., and Nadkarni, V. M., From melt flow index to rheogram, Rheol. Acta, 22, 90-101 (1983). 6. Shenoy, V., Saini, D. R., and Nadkami, V. M., Rheology of poly(viny1 chloride) formulations from melt flow index measurements, J. vinyl Technol., 5, 192-197

(1983). 7. Immergut, E.H. and Mark, H. F., Principles of plasticization, in Plasticization and Plasticizer Processes, (N. J. Platzer and R. F. Gould, eds.), Symposium Series American Chemical Society, Washington, DC, (1965), pp. 1-26. 8. Lyngaae-Jorgensen, J., The viscosity of monomolecular melts of poly(viny1 chloride), J. Appl. Polym Sci., 20, 2497-2509 (1976). 9. Howard,G. J. and Shanks, R. The influence of filler particles and polymer structure on the mobility of polymer molecules, J. Appl. Polym. Sci., 26, 30993102 (1981). 10. Shenoy, V. and Saini, D. R., Rheological models for unified curves for simplified design calculations in polymer processing,Rheol. Acta, 23, 368-377 (1984). 11. Shenoy, U.V., Bamane, S., and Shenoy, V., General Rheological model for polymer melts, 40th Canadian Chemical Engineering Conference (1990). 12. Shenoy, V. and Saini, D. R., Melt flow behaviour of liquid crystalline polymer, Mol. Cryst. Liq. Cryst., 135, 343-354 (1986). 13. Shenoy, A. V. and Saini, D. R., Upgrading the melt flow index to rheogram approach in the low shear rate region, J. Appl. Polym. Sci., 29, 1581-1593 (1984). 14. Wagner, M. H.,Analysis of time-dependent non-linear stress growth datafor shear and elongational flow of a low-density branched polyethylene melt, Rheol. Acta, 15, 136-142 (1976). 15. Wagner, M. H., Prediction of primary stress difference from shear viscosity data using a single integral constitutive equation, Rheol. Acta, 16, 43-50 (1977). 16. Shenoy, V. and Saini, D. R., A simplified approach to the prediction of primary normal stress differencesin polymer melts, Chem. Eng.Commun., 28,l-27 (1984). 17. Saini, D. R. and Shenoy, A. V., Deformation behavior of poly(viny1idene fluoride), Znd Eng. Chem Prod. Res. Dm.,25, 277-282 (1986). 18. Saini, D. R. and Shenoy, V., Melt rheology of some specialty polymers,J. Elastomers Plastics, 17, 189-217 (1985).

Spreadsheet Program Vmcosity Versus Shear Rate Curves from Master Rheograms

lWI123:

A spreadsheet program is often referred to as an electronic accountant’s pad. However, a closer look at an electronic spreadsheet reveals that it is much more than just a neat representation of rows and columns. Its facility for table lookup, its ability to do quick “what-if”calculations and to handle “if-then-else” logic functions, and its capability to provide graphic displays and define keyboard macros makes it a handy tool for performing both simple and complex engineering calculations [1-81. In this chapter, the various spreadsheet capabilities are used for developing a worksheet for determining the viscosity versus shear rate curves from the master rheograms given in Chapter The master rheograms have been obtained by plotting q X MFI versus and each has beenfittedbyappropriate rheological models as shown in Chapter The General Rheological model [g] given by Eq. is used in the spreadsheet program. The viscosity versus shear rate data for a particular polymer at the temperature of interest is determined simply by substituting the appropriate value of MFI in The known MFI value, normally determined under standard ASTM test conditions as given in Appendix A, is converted to the appropriate value through one of the following two equations:

For T2 < TG =~ ~

+ l , ~ I ~ ~ ~ ~ ) ~ h ~ ~ ~ ~ . ~ - ~ - ~ ~ ~ - ~ + ~ l . ~ ~ - t ~ . ~ - T G - S O ) / (

(7.1)

Chapter

where T1 is the ASTM recommended test temperature (K),T2 is the temperature at which MFI is desired (K), TG is the glass-transition temperature (K), R is the gas constant, and E is the activation energy for viscous flow. L1 is the ASTM recommended test load (kg), and L2 is the load (kg) specified in the master curves. Note that Eq. is combined with Eqs. and to form Eqs. and The symbols in these equations are slightly changed for convenience becausesubscripts would not be appropriate for use in the spreadsheet.

THE SPREADSHEET There are a number of commercial microcomputer-based spreadsheet packages. The general setup of each of these packages is the same; that is, there are predefinedrowsandcolumns. Numbers are used to designate the rows and letters are used to name the columns. Every combination of a column nameand row number locates a specific “cell.” In each cell, one can enter text, a number, or a mathematical formula. The information present in the entire set of cells forming the grid is stored in computer memory, and the resulting template can be altered as well as updated. In the present case, the spreadsheet is prepared by using Lotus No computer programming skills are required to prepare a spreadsheet templatelike the one shown in Fig. 7.1. The figure is a shortened version listing three of the polymers contained in the actual worksheet. A step-by-step procedure for preparing this spreadsheet demonstrates the ease with which it can be done. (For those completely unfamiliar with spreadsheet environments, itis recommended that the user’s reference manual be consulted.) When preparing a spreadsheet on Lotus, the fist thing to do is to make sure the worksheet is clean (as there is only a limited view of the worksheet on the computer screen, there is always a possibility of some remnantof previous entries). To ensure a fresh start: Command Sequence ( / m y = Slash - Worksheet - Erase - Yes) should be used. Before making any entries, it is a good idea to leave the first rows blank for future labeling. In the present case, begin by moving the cursor to cell and typing the word POLYMER as a label (Fig. 7.la). Labels are left aligned, right aligned, or centered within a cell using the prefix (apostrophe), ” (double-quote), or, * (caret), respectively. Because the names of the chosen polymers to be typed are considerably long, adjust the column width This is done by using

MFll23: Program A Spreadsheet

229

B

A

D

E

F

ASTM

ASTM

ASTM

no.

"C

5 5

190 190 190

C

G

H

Input

m

Desired Polymer load temp. cond. input temp. "C

&l0

1 1 1

190 LDPE 190 HDPE 190 UHMWPE a. User-input data

I

J

K

min.

L M

14

2.160 2.160

1o.Ooo

P

Q

K eta0

P

X

+ 05 + 05 + 05

-1.220 3.0E -0.934 2.1E -1.000

N

R

Known Parametric Model General Values Data

TG "K

Master load

E

n

1531.7E 0.37 2.16 7.25 153 0.471 2.16 1.2E 6.83 153 6.83 10.00 0.156 1.7E b. Known data and model parametric values. S

T

1 2 3 4 MFI Model 5 correct constant 1

U

V D C -

Model constant 2

7 1.00002.00E - 07 4.16E - 07 8 1.87E - 05 1.75E - 05 9 1.00000.20E + 001.66E + 06 c. Calculated data

Mm

+ 05 + 05 0.OE

+

W

X

YZ

Model constant 4

Model constant 5

DMA Model constant 3

1.000 7.69E - 01 -8.20E - 01 1.000 4.94E - 01 -1.07E + 1.000 -8.44E - 01 1.OOE 00

+

Figure Shortened versions of spreadsheet. Note: Numbersandlettersinbolddenote rows and columns in the spreadsheet.

Chapter

230

Command Sequence 2 (/WCS28=Slash-Worksheet-ColumnWidth-Set-28) with the cursor positioned in the columnofinterest.Alternatively,a global adjustment can be done by following Command Sequence (/WGC14 -Slash-Worksheet-Global-ColumnWidth-14) andthen using command sequence 2 to make specific changes in particular columns. In cells B4..D6, introduce appropriate labels (namely, desired temperature "C, MFI input in g per 10 min, and ASTM condition number). In cells B7..B25 as well as C7..C25, introduce values that represent user-input data. This is done by putting a value such as 1 in cell C7 and then using the Copy command (/C) to duplicate this into the remaining cells of the column by specifying the required range. Column D7..D25 is also for inputting data in terms of the ASTM condition number used in MFI measurement. This number corresponds to a set of temperature and load conditions as given in Table 7.1. Hence, Table 7.1 is prepared on the spreadsheet in the grid area space of A31..D50, with A31..A50 having numbers 1 to 20,B31..B50having appropriate temperature values, C31..C50 having appropriate load conditions, and D31..D50 having the corresponding alphabetic representation of the conditions as per ASTM D1238. In order to let the spreadsheet know that the data in these cells represent a closed set of information, assign the name TABLEl to the relevant group of cells by using the Command Sequence 4 (/RNC TABLEl = S l a s h

- R a n g e - Name - C r e a t e - TABLEl)

and specifying the range A31..D50 in which the data are stored. Values in column B31..D50 and C31..C50 are essentially to be selected in pairs, when the ASTM condition number is input in column D7..D25. This is done through the use of the LOOKUP function in columns E and F. For example, the formula stored in cell E7 is @VLOOKUP(D7,$TABLE1,1)

whereas that stored in F7 is @VLOOKUP(D7,$TABLE1,2)

The LOOKUP function instructs the program to look down columnsin TABLEl for the value corresponding to D7, and pick up the first offset for E7 and the second offset for F7. Note the use of the dollar sign in front of TABLE1. This is for convenience that the formula can be copiedinto all cells from E7..E25 through the Copy command. Further, the V in front of LOOKUP function signifies that TABLEl is in the vertical format. In case it was in the horizontal format, then @HLOOKUPfunction would be used instead to perform the same task, but with the difference of looking up rows.

231

MFI123: A Spreadsheet Program

Table

-

31 32 33 34 35 36 37 38 39 40 41 42 43

ASTM D1238 TestConditions

A No. 1 2 3 4 5 6 7 8 9 10 11 12 13

44

45 46 47 48 49 50

15 16 17 18 19 20

B Temp. (T)

C Load

125 125 150 190 190 190 200 230 230 265 275 230 190 190 300 190 235 235 235 250

0.325 2.160 2.160 0.325 2.160 21.600 5.000 1.200 3.800 12.500 0.325 2.160 1.050 10.000 1.200 5.000 1.000 2.160 5.000 2.160

D

Condition B

C D E F G

H I J

K L M

N P

R S

T

Note: Numbers and letters in bold denote and columns in the spreadsheet. Source: Reprinted from Ref. 8 with permission from DRVMcGraw-Hill, Milan, Italy.

Columns B to D complete the actual INPUT DATA from the end user and this label is now attested in Row 2 which was initially left blank at the start of the spreadsheet preparation. In order to make these columns obviousto the user as INPUT DATA columns, Bl..D25 are unprotected using Command Sequence 5 ( /RU = Slash - Range - Unprotect), which causes this range to appear in a different color or intensity on the video screen. Columns I to K are prepared by inputting known data on the polymers for TG,E, and L2, whereas Columns N to Q are used to store model parametric values for the General Rheological model given by Eq. (see Fig. 7.lb). Thus, columns N and 0 contain n and K values for the modified power-law model, whereas columns P and contain P and q,, X MFI. Because the values of K and qo X MFI are typically large, it is better to use scientific notation through

Chapter 7

232

Command Sequence 6 (/RFS = Slash - Range - Format - S c i e n t i f i c ) and then specify the number of decimal places desired, as well as the range to be formatted. Column S is allocated for calculating the corrected MFI value (Fig. 7 . 1 ~ ) from the known MFI value, the known ASTM test condition, and the appropriate use of Eqs. (7.1) or (7.2), whichever is applicable (depending on the value of the desired temperature inputted in column B). This requires the use of a logic function of the if-then-else type because, depending on whether the condition is satisfied or not, an appropriate choice of equation must be made. The logic function used by Lotus 1-2-3 is of the following form: IF (Condition, x,y), which may be translated as "If condition is true (nonzero), then x, or else y." In the present case put the condition T2 > TG + 100, followed by the two expressions separated by commas. Noting that the ASTM temperature and the desired temperature in columns E and B, respectively are in "C, the actual formula which is stored in cell U7 appears as follows: @IF(B7>(17 -173), +C7*(K7/F7)"(1/N7)*@EXP((lOOO*J7/ 1.9858)*((1/(E7 273)) - (1/B7 + 273)))), +C7*(K7/ F7)"(1/N7)*10A(8.86*(B7 - I7 - 50)/(B7 - I7 + 51.6) 8.86*(E7 - I7 - 50)/(E7 - I7 + 51.6))) where the variables are the cell locations. Only the final computed value (not the formula) appears in the cell; the formula itself resides at the upper lefthand corner of the screen when the cursor is in the cell. Columns T to X are calculation formulas derived from Eq. (6.4). Thus, use +Q7"P7 (incellT7), (07"P7)(incellU7), and (1/S7)"((N7-1)*P7) (in cell V7), (N7-1)"P7 (in cell W7) and 1/P7 (in cell X7). It must be noted that the Copy command ( /C) should be used to generate formulas in all the remaining rows of columns T to X (namely, T8..X25), because the generic formulas above are properly written for automatic modification by the spreadsheet program. Thus, the need to type in long, cumbersome formulas over and over again is avoided. In order to generate a graph, cells AA to AZ (not shown) are stored with the range for shear rate y. In AA7, we have @LOG(O.Ol); in AB7, we have +Mi7 + 0.2; in AC7 we have +AB7 + 0.2, and so on. These are simply generated in subsequent cells by using the Copy command. Columns BA and BZ (not shown) give the formulas for calculating the viscosity at corresponding shear rate values. For example, in BA7 we have @LOG (+(1/$S7)*($T7 + $U7*$V7*(10AMi7)"$W7)"$X7) Every stored formula in a cell is "relative" in representation to each of the other cells in the grid. However, each representation can be made "absolute"

+

MF1123: A Spreadsheet Program

233

by placing a dollar sign in front of the cell address in the formula. Introduction of the dollar sign in the proper position facilitates copying of the formula into different rows and columns. Rheograms are conventionally log-log plots, and most spreadsheet environments do not directly support log-log graphs; hence, the methodology adopted here to circumvent this limitation of spreadsheets is to store the logarithms of the values in the various cells. At this stage, it is appropriate to develop and view the rheogram (Fig. 7.2). Command Sequence 7 (/GTX= Slash - Graph - Type - XY) followed by the X-Range (e.g., AA7..AZ7 for shear rate values of LDPE) and the A-Range (e.g., BA7..BZ7 for viscosity values of LDPE) provides the basic setup. Next, one could use the Options command to do the following: Format A-Range (e.g., Lines), Title First (e.g., FIGURE 1:RHEOGRAM FOR LDPE), Title Second (e.g., viscosity versus shear rate), Title X-axis [e.g., log(shear rate in reciprocal sec)], Title Y-axis [e.g., log(viscosity in Poise)], and Grid [e.g., both]. Because we wish to define several different graphs (for various polymers) utilizing a single worksheet's data, we create a named graph by using NameCreate, followed by a graph name (e.g. LDPEGRAPH to store the LDPE graph settings). The graph is next viewed on the screen with the /GV command and, if satisfactory, saved (by the / GS command) for printing later using the PrintGraph

Log ( sheaf rate in

set')

Figure 7.2 Rheogram for LDPE: viscosity versus shear rate at 190°C as displayed by the spreadsheet program. (Reprinted from Ref. 8 with permission from DRI/McGrawHill, Milan, Italy.)

Chapter

program. The procedure must be repeated for the remaining polymersto create different groups of settings to which are assigned different names. For the sake of providing a quick reference for the user, TABLE2 is prepared giving the names of some common polymers along with the commonly used ASTM conditions (see Table 7.2). The user could refer to it by moving downward and sideways in the spreadsheet beforeinputting the value of the condition number in column D7..D25. This completes the preparation of the spreadsheet. The worksheet must be stored at regular intervals when inputting large sets of data and certainly before exiting Lotus 1-2-3 by using Command Sequence 8 (/FS= Slash - File - Save) followed by the file name (say, MFI123). The entire spreadsheet took only a few hours for preparation and checking. This is considerably less than what wouldhavebeenrequired for writing a normal algorithmic program to perform the same tasks, especially for a nonskilled programmer.

USING THE Command Sequence (/FR = Slash - File - Retrieve) along with the file name is issued to retrieve a stored spreadsheet. Table

TestingConditions for Some Common Polymers

J

K

L

"_""""""""" Testing conditions as per ASTM D1238 for some common polymers

_""""""""""

Polymer ACryliCS esters Cellulose Nylon Polyethylene Polycarbonate Polypropylene Polystyrene Polyterephthalate

H, D,E,F K, Q, R, S B, D, E, F,N 0 L G, H, I, P T

" " " " " " " " " " " " " " ~ " " " " "

Note: Numbers and letters in bold denote and columns in the spreadsheet. Source: Reprinted from Ref. 8 with permission from DRVMcGraw-Hill, Milan, Italy.

Spreadsheet Program

235

Three input values are needed (temperature at which rheogram is desired,

known MFI value, and known ASTM condition) before calculations can be performed. The spreadsheet is normally on its default setting of automatic recalculations that the entire worksheet gets calculated for every change in any cell value. To avoid time losses due to unnecessary recalculation, the iilanual option is preferred during data entry and the development of large programs. Therefore, the first step in the present case would be to hold the spreadsheet calculation facility on manual during the simultaneous data input of all three values. This is done through Command Sequence 10

(/WGRM = S l a s h - Worksheet - Global - Recalculation Manual ) After inputting the three values, the automatic recalculation mechanism is activated by using

Command Sequence 11 (/WGRA/WGRN=Slash-Worksheet-Global-Recalculation -Automatic-Slash-Worksheet-Global-Recalculation - Natural) This causes the spreadsheet to recalculate for the new input values. display of the viscosity versus shear rate can now be seen on the screen by using Command Sequence 12 ( / G N U = S l a s h - Graph

- Name - U s e )

followed by a graph name. Command sequences 10 to 12 require a good bit of keyboard manipulation. In such circumstances, it is advantageous to set up keyboard macros to do the same job. Keyboard macros are essentially programs stored and assigned to a key, and invoked by that key when desired. Spreadsheets support the use of such macros and these can be set up as follows. Any convenient vacant cell (which would normally be outside the common viewing area of the user) is chosen, for example, M52. The macro is entered as /WGRM and named as / I by using Command Sequence 13 (/RNC = S l a s h - Range - N a m e - Create) NOW,if Alt I is typed from anywherein the spreadsheet, the command sequence 10 is automatically executed. Set Alt R for command sequence 11, and Alt v for command sequence 12. Thus, beforeinputting the data, the user can initialhe by pressing Alt after inputting the data, he can run the program simply by pressing Alt R, finally, he can view the relevant graph by just pressing Alt V and selecting the appropriate graph name.

236

Chapter

special Calc function key, F9, is available in Lotus 1-2-3, which may be directly used for recalculation of the complete worksheet.Keyboard macros are particularly useful when no keys to perform certain specialized tasks are provided by the spreadsheet environmentitself. For example, although viewingof the current graph is possible through the F10 key (which is equivalent to the /GV command), no key exists for command sequence 12, which allows one to view a particular graph. The ease of recalculation afforded by spreadsheets allows the user to ask such questions as: What if the temperature at which the rheogram is desired is changed? If the recalculation is on Manual, entering the new temperature in column B, pressing F9 and then A t V provides the new rheogram within seconds. If the graph is saved, a hard copy of the rheogram can be obtained using a printer or plotter (see Figure 7.2) by selecting the PrintGraph option from the Lotus Access system and then choosing the appropriate graph file.

7.3 TESTINGTHE RESULTS The authors tested the efficacy of the spreadsheet for various cases. Using the Alt I key, desired changes were made in the columns B7..D25. Then the Alt R was used for recalculation and, next, theAlt V was used for viewing the graphs. The tests were performed to check howclose the predictions from the master curve are to the actual data of viscosity versus shear rate. For most polymers, the error in the predicted value was limited to a maximum of 520%. The imum deviation was less than as in the case of PET. For some polymers, however, the maximum error in the case of some data was within a broader band of The reason for this is that the master rheogram itself had a broad bandwidth. Because most of the data were coalesced from existing literature data, the errors in the original data naturally creep into the predictions from the master curve. It is known that, even with the most sophisticated equipments, errors to the extent of -150% can creep in if proper care is not taken when generating the data. Thus, the blame for the broader error band in the case of some polymers,that too only in the low-medium shear-rate ranges, does not rest entirely with the master curve. If the original data which are used for forming the master curve are error-free, then automatically the master curve predictions would be very accurate, as the unification technique itself is very reliable. The spreadsheet program developed by the authors was limited to only 19 homopolymers. Intentionally, copolymers, blends, and PVC formulations were omitted. This is because of the nonuniqueness in the glass-transition temperature and the activation energy in all these cases. Hence, columns I and J are difficult to fill for these systems and the formula in column S would need modification. Such developments are not impossible and the users are encouraged to try and develop the programs for copolymers, blends, and PVC formulations. Because

237 MFll23: Program Spreadsheet

the spreadsheet is limited to shear viscosity versus shear rate data, it would also be an interesting exercise to extend the idea to normal stress difference, complex viscosity, and dynamic storage modulus master curves as well, which the users attempt. can a

Selk, S., Spreadsheetsoftwaresolvesengineeringproblems, Chem. Eng., (June Schmidt, W.P. and Upadhye, R.S., Material balances on a spreadsheet, Chem. Eng., (Dec. Goldfarb, S.M., Spreadsheets for chemical engineers, Chem. Eng., (April Ferrall, J.F., Pappano, A.W., and Jennings, C.N., Process analysis on a spreadsheet, Chem. Eng., (March Sowa, C.J., Engineering calculations on a spreadsheet, Chem. Eng., (March Cheremisinoff, N.P., Statisticalregressionroutinesonspreadsheets, Chem. Eng., (Aug. Skaar, E.C. and G., Decisions: Can CERABULL spreadsheets help? Ceram. Bull.,

Shenoy, A.V. and Shenoy, U.V., Microcomputers speed up polymer engineering cal(July culations, Mod. Plastics Int., Shenoy,U.V.,Bamane, S., andShenoy,A.V.,AGeneralRheologicalmodel for polymer melts, Canadian Chemical Engineering Conference

Processing Parameters

One of the main objectives of thermoplastic melt rheology is to develop an u n d e r s t a n ~ ~ofg the responses of the polymeric materials to various types of deformations so that useful information is generated for the polymer processor. ~ormally,the processor resorts to simple rules of thumb based on prior experience in order to get answers to any processing problems. This is because the work of the theoretical rheologist, although accurate, is beyond the comprehension of the common processor, and the work of the experimental rheologist, although pragmatic, is often based on conditions that are too ideal to have actual importance. The master rheograms, developed and discussed in the earlier chapters, however, serve as a useful tool for practical applications of rheology to polymer processing. The route of obtaining i~ormationon various processing parameters from master rheograms provides an acceptable compromise between the accurate complex mathematical approach and the approximate simple ~le-of-thumbapproach. In the present chapter, this fact is demonstrated for a few representative cases selected from the most common processing operations such as injection molding, compression molding, calendering, extrusion, and compounding.

I Injection molding of a thermoplastic is, in essence, an operation which involves high-pressure squeezing of a polymeric melt through a very small hole (gate) 23

Master From Rheograms

to Processing Parameters

239

into a cold cavity and then packing of additional material into the cavity to allow for shrinkage as the material cools. Uniform pressure is desirable at the end the packing, as abnormal local pressure can lead to frozen-in stresses. Packing pressure is necessary to achieve relatively high part density and good gloss. It has a dramatic effect on orientation and hence directly relates to the tensile strength, yield, impact strength, crystallization, distortion and warpage of the final molded product. For example,decreasing the injection pressure gives an increase in the impact strength. Molddesign does affect thefilling, packing, and cooling of the molding process. The degree of packing depends primarily on the typeof polymeric material and the mold geometry. Overpacking may cause flashing. This occurs when the mold plates are slightly separated due to excesscavity pressure which may develop in a molded part if the process is not properly controlled after the cavity is filled. When the hot polymer melt flows into a cold mold cavity, it freezes rapidly to form a skin (a thermal boundary layer) from which solidification continues, gradually moving toward the center of the cross section. During solidification and cooling, the specific volume of the material decreases continuously as a function of temperature and pressure.Because the temperature and pressure vary from location to location in the mold and also with respect to time, the rate of shrinkage varies and,therefore, it is ratherdifficult to compensate for it. A method to minimize shrinkagein amorphous polymers during postfilling stages has been suggested by Hellmeyer and Menges [l]through the process control of pressure in the cavity. For crystalline materials, the situation is rather complicated and depends on the kinetics of the rate of crystallization. Simulation of flow behavior of the melts in mold cavities has beenthe subject of investigation for over three decades now. Spencer and Gilmore were the first to initiate work for understanding the flow phenomena in injection molding. This was followed by the extensive experimental work and detailed flow examination of Ballman et al. who predicted the proper behavior for flow in a simple cavity with a flow-controlling delivery system. Paulson investigated the pressure losses through the gate andmeasured the pressureprofiles along adelivery system andina singlecavity mold. Following the macroscopic analysis of the mold-filling process by the above-mentioned workers came the rather simple analysis of Barrie based on the spreading-disk theory of Cogswell and Lamb for isothermal based on the flow of a power-law fluid. The only empiricism in the work phenomenological or transport equation approach is the inclusion of arbitrary empirical constants to incorporate the nonisothermal effects. Harry and Parron computed filling rates and velocities for filling bynumerically solving a differential equation for flow of a power-law fluid. Kamal and Kenig [lo] used the Pearson [l11 approach to model the injection molding process for noniso-

Chapter

thermal flow of a power-law fluid theoretically. Wu et al. [l21 simulated the mold-filling process and concentrated on the pressure and force prediction during the filling of simple disk molds. Williams and Lord [13], on the other hand, concentrated on coupling the cavity-filling process with the flow in the sprue and runner system. Although, in most of the above simulation approaches, the point of cavity filling is predictable as a function of a number of variables like melt temperature, mold temperature,gate restriction, injection pressure, injection rate, and material properties, they are still unattractive to the processor, who might look for simpler ways of estimation rather than rigor. The plastics processor desires to operate injection molding machines at the minimum pressure for mold filling in order to keep the frozen-in stresses at the lowest level for maximum dimensional stability in the product and to maintain lower clamping forces to prevent mold opening during processing. The effect of a cooled mold on the pressure versus the flow rate curve must be known for calculating the minimum pressure requirements. However, this involves nontrivial equations solved through computer programs using numerical techniques. During the manufacturing process, the molten polymer is subjected to a shearing flow at shear rates of around 10-102/s in the runner and as high as 103-104/s in the mold gate of an injection molding unit. For most of the common molding materials, it is known [l41 that the molding temperatures lie in the range of 190-320°C. As polymermelt viscosity is dependent on both shear rateand temperature, the injection molder and the mold designer cannot ignorethe melt rheology of the molding material. knowledge of the complete flow curve or rheogram depicting the variation of the melt viscosity over relevant range of shear rates and temperatures (190-320OC) is essential for the design calculations. Generation of such data involves the use of highly sophisticated and expensive rheological instruments. With the large number of different types of polymers along with an almost equal number of molding gradesavailable in each generictype, the processor would have to go through a cumbersome and expensive procedureto obtain the minimum pressure requirementsfor mold filling. In such circumstances, when answers to processing problems are beyond their financial feasibility and technical capability, the processors usually resort to rules of the thumb based on their processing experience. However, what is required is a balance between accuracy at high costs and order-of-magnitude estimations through simplified procedures in order to upgrade the rule-of-thumb approach. Saini and Shenoy [l51 combine simplicity with an acceptable degree of accuracy in order to provide simplified calculations for mold filling during nonisothermal flow of polymer melts. The approach is based on the works of Barrie [6,7] who gave approximate relationships among material properties, machine design parameters, and mold geometry. The estimation of nonisothermal flow behavior inmolds is based on empirical observations in acenter-gated disk

Master From Rheograms

to Processing Parameters

241

molding that the frozen polymer layer is approximately uniform on the mold surfaces at the instant of mold filling; this means that at all times the melt flow occurs in acavity of approximatelyuniformthickness. Thus, if the original cavity has a thickness X, and a thickness AX of frozen material forms on each cold surface, then the effective remaining flow path is given by

x, = x, - 2Ax Experimental observation of the variation of mold suggests the following relationship:

Ax = Cf,"

(8.1)

AX with the time ff taken to fill the (8.2)

where the proportionality constant C is given [6] approximately by the empirical equation

where is the heat diffusion coefficientof the melt, 0, is the mold temperature, T is the melt temperature, and To is the freeze-off temperature. The cavity filling time tf for the circular disk cavity of radius R, is given by

TR;X

tf=

a

It is now assumed that isothermal flow equations can be applied using the reduced, effective cavity thickness X,, that the pressure drop from the center of a spreading disk flow can be written [6]

where MFI is the melt flow index, is the volume flow rate, R, is the instantaneous radius of the disk or the flow length, R, is the radius of the central hole, K is the consistency index, and n is the pseudoplasticity index of the polymer melt in the master rheogram. Equation (8.5) is derived from the basic definitions relating pressure drop to shear stress and using the modified Ostwald-de Waale power-law model for the master rheogram given by Eq. (6.3), such that

The above equation is found to fit best in the shear-rate range 10-103/s, which is of practical significance to the injection molder. Now assuming that the inlet hole is very small compared with the disk radius and that is approximately equal to unity, Eq. (8.5) can be written in the

Chapter

following simplified form using &S.

Po =

(8.1)-(8.4) as

=PR:-" (1 - n)(MFI)"

[l - ~C(ITR:/~)'"I''?"

Differentiating Eq. (8.7) with respect to Q and equating to zero yields the expression for the minimum pressure gradient Po, for center-gated disk mold cavity filling.

where

fln) =- 23n+1 (Il-n

(;;z ) ' + ~

Rearranging Eq. (8.8) gives

Po,

Z4" c -

"

R P

MFI"

(8.10)

where

-

C = Any.."

(8.11)

For each generic type of polymer, C is a constant and incorporates the values of K and n from Table 6.7. Table 8.1 lists the valuesof C and n for each generic typeof polymer. Knowing the geometric parameters of the mold, the temperature, and thermal properties of the melt, the minimum pressure requirement for filling a mold cavity can be estimated by inserting the values of C as well as of n from Table 8.1 in (8.10) and using the MFI value at the appropriate temperature. The MFI value is, however, governed by the test conditions of temperature and load as in D1238 and D3364. The K and n values in the case of polymers of each generic type have been determined from unified curves produced at test load conditions. Hence, the MFI value to be used in Eq. (8.10) must be at the same load condition as given in the master rheogram. Further, the MFI must be known at the melt temperature T during mold filling. The MFI determined at the ASTM test temperature would thus have to be converted to the MFI value at the required temperature. This could be done through either of the two equations given in Chapter 4, namely, Eqs. (4.14) or (4.15). The minimum clamping force F- can now be estimated using (8.12)

Master From Rheograms Table

to Processing Parameters

Values of

243

n Needed in Eq. (8.10)

-

Thermoplastics

Cload'

Test n

1.20 2.10 1.82 9.62 0.95 2.30 2.46 2.52 4.14 4.40 5.92 6.55 6.85 2.50 7.76 16.31 2.75 7.61 188.79 1.50 2.50 2.69 2.00 3.43 12.23 1.37 0.92 1.41 4.62 1.39 3.55 2.97

LDPE HDPE UHMWPE LLDPE PP PS C-ester C-ether Acrylic Nylon PET PC PVDF PP0 PPS PASPES PEEK PE1 PAr SAN SBS

ABS

VCVA EVA Polyester elastomer WE PP-HDPE HDPE-PMMA PS-PMMA PS-POM PMMA-POM PVC formulations ~

~~~~

'Condition MFI used in bASTM specified.

(8.10).

2.16b 2.16b

0.37 0.47 0.156 0.66 0.34 0.68 0.38 0.47

lo.oob 2.16b 2.16b 5.00b 2.16b 2.16 3.80b 2.16b 2.16b

0.44 0.44

0.43 0.52 0.40

1.20b

0.25

0.44 0.60 0.32 0.34 0.85 0.33 0.25 0.38 0.33 0.55 0.65 0.40 0.32 0.45 0.46 0.30 0.43 0.36

;

'

12.50 5 .00 5.00 5.00 5.00 5.00 5.00 3.80 5.00 5.00b 20.00 2.16 2.16 2.16 2.16 2.16 5.00 5.00 3.80 2omb

Chapter

The above equation is obtained from the general form of the equation for the mold-opening force and hence the clamping requirements as given by Barrie [6,7]. Here 2 is the effective projected area of the molding given as ITR:and B is a numerical function dependent on n as shown by Barrie [7] for circular, square, and rectangular moldings basedon a mathematical model of radial pressure distribution. of the expressions for B suggested by Barrie [g are given below: For a circular panel (center injection)

B=-n + 2

(8.13)

n

For a long, thin rectangular panel (center injection)

B=-n

+ l

(8.14)

n

The more centrally gated, flat, and symmetrical moldings would show higher values of B anda lower clamping force. For engineering design, one could choose Eq. (8.14) which would give the lower estimate of B. Thus, in the present B = 3.5 for n 0.44,and B = 3.0 for case, a value of B = 4.0 for n n 0.50 may be used as rough estimates. The simplistic approach given above is not a panacea but would certainly be a useful handy tool to the plastic processor to get quick order-of-magnitude estimates of the design parameters. Some practical results in support of the suggested design equations are, undoubtedly, desirable. However, for providing experimental evidence,relatively large machines and moldsare needed to enable the variables of injection speed and pressure to be separated. Laboratory setups do not provide suchflexibility and, therefore, such data are not available. Nevertheless, the design equationsdeveloped herein are in general accord with practice and, hence, would be of high pragmatic value despite the lack of experimental verification.

-

COMPRESSION MOLDING Compression molding is one of the most common methodsfor producing articles from thermosetting plastics. It is rarely used as a production process for thermoplastic materials due to the long cycle times required for cooling and reheating the press. However, there are some specialized materials like ultrahighmolecular-weightpolyethylene (UHMWPE) which are preferentiallycompression molded rather than injection molded to form the product. In fact, more than 50% of UHMWPE is processed via compression molding [l61 as can be seen in Table 8.2. The other major processing technique used for UHMWPE is

From Master Rheogmms to Processing Parameters

245

Table 8.2 m i d Processes used for UHMWPE Processing % of material processed

Technique Compression molding Ram extrusion Win-screw extrusion Injection molding

1 Source: Ref.

ram extrusion. Recently, Zachariades et al. [l71 have investigated the possibility of using solid-state extrusion of compacted powder for processing UHMWPE. Despite any new techniques or approaches suggestedfor processing this family of polyethylenes, compression molding will, undoubtedly, continue to have a major share in the processing of these ultrahigh-molecular-weightmaterials. Figure 8.1 shows a typical molding operation whereina preweighed charge, either as a powder or a preformed cake, is placed in the lower half of a heated mold and the upper half is then forced down, causing the material to be squeezed out to fill the mold cavity and take the mold shape. The viscoelastic material is preheated to reduce the temperature difference between itself and the mold. If the material is at a uniform temperature in the mold during the process, then the entire compression-molding operation is akin to a typical squeezing flow which has been analyzed in detail by a number of investigators [18-281. This particular flow can be schematically represented as in Fig. 8.2 giving the flow geometry and initial conditions. The moldingmaterial is contained between two flat horizontal plates, one being fixed while the other is free to move vertically.

-.""_.

Figure

Schematic diagram of a typical compression-molding operation.

Chapter 8

Figure (a) Schematic diagram the flow geometry; (b) schematic diagram initial conditions, during a typical squeezing operation.

the

It is assumed for simplicity that the mold plates are circular in the first instance and have a radius R& An initial separation h, is maintained for times t < 0, and at t = 0 a constant normal load FCdis applied to the upper plate. Often it is of interest to determinethe function h&) indicating the separation distance between the two plates at different times of t > 0. Such a flow can be analyzed after assuming a quasi-steady-state and lubrication approximation for a power-law non-Newtonian fluid as done by Scott [l91 to give the relationship among the force, the flow geometry, and the material flow parameter. Shenoy and Saini [29] used Eq. (8.6) to modify the derivations of Scott [l91 and present the results in the following modified form. For circular disks (8.15)

Substituting the appropriate values of K = 1.66 X lo6 (g per cm sec'"') (g per 10 min)" and n = 0.156 for UHMWPE and rearranging the terms gives (8.16)

Master From Rheograms

to Processing Parameters

247

For the case when the molding plates are flat strips instead of circular disks the equation suggested by Oliver [30] is used and written in the modified form [29] as For flat strips:

n+2 where W is the width and 4, the length of the strip. Again, substituting the appropriate values of K and n and rearranging the terms gives

(8.18)

In compression-molding operations, the typical squeezing rate and the initial separation would be such as to give values of (-h:”/ho) between 0.01 and 0.U S1 0. Figure 8.3 shows the variation of the left-hand sides of Eqs. (8.16) and (8.18) with the factor (-h:”/h,,) in the above range. It is now evident that knowing the geometrical factors of the mold, one can easily determine the compaction force merely from the knowledge of the MFI at that temperature and at a 10-kg load condition which corresponds to the load condition of the master rheogram. If the available MFI is at a temperature and load condition other than the desired one, then the method similar to that discussed in Sec. 8.1 is to be followed. Equations(8.16) and (8.18) may be used to decide the squeezing rate for a particular grade of the polymer knowing the capacity of the compressionmolding press and the MFI. UHMWPE processors would no longer have to resort to the common rule-of-thumb approach based on their experience, especially when they decide to change the polymer grade, the squeezing rate, or the molding press. Equations (8.16) and would certainly suffice in all circumstances. In compression molding, the shear rates are known to be of the order of In this region, the power-law model is likely to overestimate the viscosity in some cases, resulting in an underestimation of the rate at which the separation decreases, especially if the compression load is small. Nevertheless, Eqs. (8.16) and (8.18) would predict reasonably accurate estimates from a pragmatic viewpoint. Actually, Eqs. (8.16) and (8.18) are only valid for light compaction, as it is known [22] that viscoelastic effects begin to dominate at heavy compaction, resulting in a decrease in the separation, which is much slower than that predicted by the inelastic theory as given by (8.16) and (8.18). Brindley et al. [25] predicted that under some severe compactionconditions, even solidlike bouncing behavior is possible, rendering the results of Eqs. (8.16) and (8.18) meaningless.One way to check whether such conditions are likely to occur during the squeezing flow of UHMWPE is to estimate whether the Deborah

Chapter

numbers in such situations are low, medium, or high. For medium and high Deborah number squeezing flows, it has been shown [31] that elastic effects dominate and the viscosity effects alone cannot account for the observed slow rate of plate separation. The governing parameter in such cases is the Deborah number De which is the ratio of the material characteristic time X, (normally taken as the relaxation time h,) and a properly chosen process time(such as the time scale of deformation The relaxation time AT as defined below can be considered as a measure of elasticity:

AJ.

(8.19) Shenoy and Saini [32] have provided a simplified approach to the prediction of primary stress difference in polymer melts as discussed in Chapter 5. This is based on the relationship between the unified normal stress difference function and the unified viscosity function curves throughastrain-dependent single-

Master From Rheograms

Processlng to Parameters

249

integral constitutive equation of the Bernstein-Kearsley-Zappas type. Along similar lines, the unified normal stress difference function for UHMWPE can be estimated from the viscosity function shown in Fig. 4.13. plot of the predicted curve is shown in Fig. The value of the normal stress difference function would Iie within the bandwidth obtained for the damping constant m of 0.13 and 0.2, respectively, in the following equation: (8.20)

(6.3), the above equation can be written as

Using

(8.21)

Thus, the relaxation time can be estimated hT =

l - n

m9 Because n = 0.156, 0.13

(8.22)

m 0.2, and 1 p 10 for the present case, it be seenthat 0.4 hT 6.5. vpical squeezing rates in compression molding would be of the order of d m i n and the separation gap couldbe of the order of 0.5 cm. Thus, a characteristic process time of the order of 60 S may be

"

IO0

I

lo4

Figure Grade-andtemperature-invariantunifiednormalstressdifferencefunction estimated curve for UHMWPE.

250

Chapter

estimated. In such circumstances,the Deborah numbers would be less than 0.1, which is very low for the elastic effects to dominate. Thus, it may be concluded thatin the compression-molding operationof UHMWPE, one need not be concerned about the effects of elastic forces and, hence, it is possible to describe the entire mechanics through Eqs. (8.16) or (8.18), which are dependent on geometric parameters and a single-point rheological characteristic value of MFI.

8.3 The term “calendering’7as applied to thermoplasticmaterials refers to the shaping of the material into sheet or film by feeding it through a pair of corotating heated rolls. This processis commonly used for the manufacture of various PVC formulation products such as leather cloth, shrink films for packaging, resilient flooring tiles, and on. The “calendering” processfor shaping of thermoplastic materials into sheets and films has been the subject of investigation for many decades now. Gaskell was the first to analyze the process by developing a mathematical procedure in one dimension for Newtonianfluids,derived as a specific extension for apurely viscous (nonelastic and time independent) fluid.McKelvey extended the analysis to include non-Newtonianflow, assuming only that the viscosity depends onthe state of shear and making the corresponding modification in the momentum equation. He also showed the approximate cubic dependence of maximum pressure developed between rolls on the leavedistance, and specialized his non-Newtonian development for power-law fluids. Alston and Astill considered yet another special case-that of fluids whose shear-rate-dependent viscosity could be approximated by a hyperbolic tangent function. Tadmor and Gogos pointed out the limitations of an isothermal treatment and set up equations assuming the lubrication approximation for a Criminale-Ericksen-Filbey model that exhibited normal stresses in viscometric flows. Agassant andAvenas developeda thermopseudoplastic model using the approach of Brazinsky et al. to obtain force and torque as functions of plasticity index and process parameters, assuming conformity of thermodependence of the behavior with experimental data. It should be noted that the Agassant-Avenas model, which rightly takes into account the heating of the melt by the viscous dissipation and therefore attempts a simultaneous solution of the mechanical and energy equations, is cumbersome to use in calculations because itrequires iterations based onlinear relationships between inlet and outlet data. Vlachopoulos andHrymak suggested a hydrodynamic model for the calendering of rigid poly(viny1 chloride) (PVC), using both the isothermal as-

Master From Rheograms

to Processing Parameters

251

sumption and the lubrication approximation. Ray and Shenoy [40] extended the analysis [39] and presented a simplified technique based on the use of MFI (measured in accordance with test conditions D3364-73) for the prediction of pressure distribution on the rollers, the torque exerted by each roll, the power input into each roll, and the average temperature rise due to viscous dissipation. The work of Ray and Shenoy is recounted below.

8.3.1 The Hydrodynamic Theory WithoutSlip The present form of the hydrodynamic theory makes essentially the same assumption for the operating conditions as Gaskell [33] and Vlachopoulos and , Hrymak [39], namely, isothermal steady-state flow of an incompressiblepowerlaw liquid. The reconciliation of this assumption to the experimental data is also identical; namely, it is assumed that the calendering temperatureis equal to the average surface temperature plus the average temperature rise due to adiabatic viscous dissipation, as per Kiparissides and Vlachopoulos [41]. For the system shown in Fig. 8.5, the lubrication assumption, as applied to the mass flow balance and momentum conservation (details of which are available in Ref. 42) yield (8.23)

(8.24)

Y

Figure The nip area and notationfor a pair of corotating and equal-sizedrolls which have no translational motion of their (From Ref.

Chapter

At this stage, it is essential to substitute an expressionfor the shear stress based on an appropriate rheological model. During the manufacturingprocess,the molten PVC formulations are subjected to a shearing flow at shear rates around 10-102/swithin a temperature range 150-220°C. Within this region of shear rate and temperature, the Ostwald-de Waele power-law model best describes the flow behavior of the melt, and for the master rheogram given in Fig. the relationship between shear stress and deformation rate can be written

where K and n are temperature invariant and also independent of the type of formulation. In the case of PVC, the determined values of K and n are the following:

K=

n=

(gper

X

Substitution of in Eq. and assuming a no-slip condition, i.e., u(h) = U at the roll surface, the following can be written:

ap*

-

"

ax*

where x* = x(2R,H0)"",

a dimensionless distance

and a dimensionlessflow parameter

(8.29)

Assuming an infinite reservoir for the PVC melt feed, the boundary condition is given as

x*"ao3p*40 x* =

x*

Integration of

p* = 0 using Eq.

yields

From Master Rheograms to Processing Parameters

253

where

Note that C, is a constant for a given polymer system and C, for given process parameters. In the case of PVC, it follows from Eq. that

From the condition given in Eq. written as

an expression for determining h* is

A plot of h* as a function of n has been made by Vlachopoulos and Hrymak and this can be used for the specific case of n = given in Eq. to yield h* = Using this value of h*, Eq. can be numerically integrated for various valuesof to yield a unique plot of P*("' versus for given process parameters in the case of all PVC formulations as shown in Fig. In the present case, the numerical integration was carried out usingthe trapezoidal rule. Taking an appropriate value of MFI for the specific PVC formulation under consideration at the operating temperature, curvesof P* versus can be generated from this master plot.

8.3.2 The HydrodynamicTheory with Slip Any analysis of flow behavior in calendering is incomplete without a consideration of the corrections introduced .due to slip-which could arise, for instance, by sliding of high-molecular-weight material over a thin layer of low molecular weight comprised of oligomeric material and lubricants, and on, which tends to adhere to the wall. As per a power-law expression given by Chauffoureaux et al. approximating such slip, the following is written: 1

P Using this as a boundary condition at

au

-=0 ay

aty=O

= h along with the constraint that

Chapter

8.6 Plot of P*(Iv~FI)'.~~ versus dimensionless distance x*-master curve for the theory without slip. (From Ref.

integration

m.(8.25) yields (8.38)

Master From Rheograms Processing toParameters

255

Integration of the above expression to obtain the volumetric flow rate from the equation

yields

Introducing x* =

and

x*

+ x*')

where h =

- B@*)

ax* ap* where

-

-

=

(

I 1 ax* ap*

= (U -

B@*) =

-

from geometry, as shown by Middleman

I

+ C@*) ax*

gives

=

ap*

+

-

+ x*2)"(2RrH,)-"R

The equation only be solved by a tedious numerical technique which involves the calculation of aP*/ax* values for a number of x* positions by a bisection linear interpolation technique as given by McCormick and Salvadori These values are inserted into a fourth-order Runge-Kutta method to obtain the P* versus x* data. If simplifying assumption is made that the inaccuracies introduced by neglecting slip beyond a first-orderestimate are small, a master curvefor P* versus x* (taking MFI = can be determined by the numerical analysis suggested above and used to computeP* versus x* for other temperatures usingthe equation from the no-slip analysis, namely,

PT(MFI,y = P,*(MFITJ at constant

Chapter

It is found that this approximation introducesan error of the orderof -10% in P*, which is well within thelimits of the errors introduced by measurement uncertainties in other pressure prediction methods as, for example,in Ref. [39]. The master curve for the case involving the conditionsof slip is shown in Fig. 8.7. A comparison of the predictions of the present approach withthat suggested by Vlachopoulos and Hrymak [39] is carried out to demonstrate the propriety of the technique. The rheological data on the PVC formulations at various temperatures and the relevant process parameters have been taken from Ref. in order to provide a meaningful comparison. The MFI values at each temperature were calculated from the T versus p relationship provided by Vlachopoulos

DYNES I CM*

6

S

I

P"(

L

o

X*

os

8.7 Master curve as in Fig. 8.6 for the theory with the effect of slip taken into account. (From Ref. 40.)

Master From Rheograms

to Processing Parameters

257

and Hrymak [39] using the methodology discussed in Sec. 4.2.2. Table 8.3 lists the rheological parameters used; the process constants are given in Table 8.4. Using these values, the resulting pressure distribution as a function of was derived from the master plot given in Fig. 8.6 for various temperatures and compared with Vlachopoulos and Hrymak‘s predictions for the case without slip. The plots have been shownin Figs. 8.8a-8.8f. The predictions of maximum pressure fromthe present method are 10-25% lower than those of Vlachopoulos and Hrymak [39] for temperatures of 170-210°C. However, at 160”C,the values of P& are markedly different. It is observed that although Vlachopoulos and Hrymak C391 predict a P:= of 19.8 X lo’ dyn/cm’ at 160°C and a value of 9.7 X lo7 dyn/cm’ at 170”C, the present technique suggests corresponding values of 10.3 X lo7dyn/cm2 and 7.5 X lo7dyn/cm*, respectively. It is not very likely that such a phenomenal pressure change as derivable from Ref. 39 will occur over only a 10°C temperature change. The present approach thus predicts more realistic values. Further, it can be seen that the prenip slope of the pressure distribution changes more graduallyin the present analysis, which also is more realistic. A comparison of the two sets of predictions along withthe experimental data used in Ref. 39 is shown in Fig. 8.9. It can be seen that the present approach provides a closer fit to experimental data. A similar observation is made in the “with slip’’ case, as can be seen fromFig. 8.10. It is worth mentioningthat for this figure an exact analysis of (8.43) was carried out using the appropriate “ Ivalue to yield P:, = 5.2 X lo7 dyn/cm2. However, if the master curve (Fig. 8.7) is used, the predicted value of P:= is slightly lower and yields a value of P$, = 4.8 X lo7 dynlcm’ which compares favorably with the experimental value of 4.2 X 10’ dyn/cm2. It must be noted that the sensitivity of the dependence of P* on h*, which itself depends on n, is quite significant. Thus, the method suggested by Vlachopoulos and Hrymak [39] is prone to uncertainties for each measurement on n, i.e., each temperature of use. For the master rheogram, the valueof n is invariant, and the only temperature-dependent variable in the pressure prediction equation is the MFI, which be measured repeatedly for accuracy without resorting to curve-fitting techniques and data correction procedures,as in the case of Ref. 39. The sheet thickness has been shownby Middleman [42] to be approximately equal to

m.

H’ = Ho(l

+ h*’)

(8.48)

As h* is a constant for this technique, it is unable to predict sheet thickness. However, Vlachopoulos and Hrymak’s plots [39] show that sheet thickness dependence on n is such that H‘/Ho does not change appreciably in usual working ranges of PVC formulation. Proceeding along the lines of Vlachopoulos and Hrymak [39], other process parameters of interest can be estimated as follows. The torque exerted by each

7.81

Chapter 8

258

Table 8.3 RelevantRheologicalParameters

Vlachopoulos and -

Temp.

e")

160 170 180 182 190 200 210

X

5.26 5.06 4.61 2.82 X 1.38 X 0.583 0.59

Ray and Shenoy [40Ib

MFI

K

ec>

*T =

[39]'

lo' lo' lo' lo' lo' lo' 10'

(g per 10 min)

0.456 0.390 0.322 0.340 0.412 0.479

1.154

0.454 1.43 1.80 5.10 17.3 42.3

x

as given by Eq. (2.39). %quation for shear stress versus shear rate, T = K(j/MFI)",where K and n are given by Eq. (8.26). Source: Ref. 40.

roll is obtained by integration of the product of shear stress at the surface, roll radius, and contact area between the melt and roll, namely,

r

=

TJ,

a,

where

A, = W, ak = h(%) Making substitutions for

Table

and h in terms of

RelevantProcessParameters

Parameter

U

73.8

U0

0 cm/s

Ho R,

3 lo+ cm 12.5 cm 2.644 1O"/cm 2.28

S

P"

and H. gives

From Master Rheograms to Processing Parameters

Figure 8.8 Comparisons of predictionsbyVlachopoulosandHrymak'sanalysis (---)and Ray and Shenoy analysis (-), pressure profiles at temperatures varying from 160°C to 210°C at 10°C intervals. (From Ref. 40.)

Chapter

POLY (VINYLCHLORIDE) T=lBO'C

POLY (VINYLCHLORIDE) T=19O"C

a

(a Continued.

From Master Rheograms to Processing Parameters

-

I CM'

,".

(e)

I

-2

Q Figure 8.8 Continued.

261

262

Chapter 8

POLY (VINYL CHLORIDE) T=182'C

X"

Figure 8.9 Comparison predictions made in Vlachopoulous and those made in Ray and Shenoy (-), and an experimental curve theory without slip. (From Ref. 40.)

Applying

(---), (---) for the

(8.27), the above equation can be rearranged to

Once again, the expression yields a constant curve (as h* and n are constants) for versus x*. Similarly, the power input into each roll would be given

r(MFI)"

From Master Rheograms to Processlng Parameters

x

DYNES I CM?

-3

X‘

Figure Comparisons as in Fig. Fig. (From Ref. 40.)

forthetheorywith

dip. Same symbols as of

Again, by appropriate substitutions,

The average temperature rise as a consequence be estimated by dividing total power input to the

viscous dissipation can relevant rolls by the mass

Temp.

Chapter

flow rate and the specific heat of the melt, i.e., (8.56)

This can also be seen to be a function of MFI and C, alone for fixed process conditions, and because the variation in C, is small (1.84 J/g K at to 1.80 J/g K at the variation in MFI alone will adequately provide a reasonable estimate of (AT), at different process temperatures. Vlachopoulos and Hrymak [39] have used appropriate values of K and n at and in order to predict the torque exerted on rolls 3 and 4 of their experimental setup. Ray and Shenoy [40] simply use the MFI value estimated at the temperature consideration and derived thetorque value fromEq. (8.53). Similarly, the power requirements can befound using Eq. (8.55) and compared with values predictedthrough the Vlachopoulosand Hrymak [39] approach. Table 8.5 shows the comparison. It can be seen that the values predicted by Ray and Shenoy [40] are in reasonable agreement with the predictions of Vlachopoulos and [39], considering the simplicity of the approach [40] over that of the earlier work [39].

EXTRUSION With the growing importance of plastics, more and more industrial processes involve extrusion of polymer melts through dies of complex cross section to form the final product. The shape of these dies and the pressure losses through them are important factors in preventing flow defects and controlling the quality of the end product. Attempts at predictions of extrusion pressure losses in dies of complex cross-sectional geometries are those of Lenk [45-471, White and Huang [48], Tiu [49], and Shenoy and Saini [50].

Comparison Between Predicted Torque and Power Requirements by Vlachopoulos and Hrymak and Ray and Shenoy Table

Shenoy and Roll no.

Source: Ref. 40.

W>

Vlachopoulos and Ray Hrymak Torque

Power

Torque

Power

Master From Rheograms

to Processing Parameters

265

Lenk [45-471 has provided a rigorous treatment for a variety of case studies and derived expressions for pressure drops throughwide-slit dies with and without taper, elliptical channels, and regular polygonal channels. White and Huang [48] developed the pressure drop-flow rate relationship for the flow of polymer melts through rectangular and trapezoidal dies based on a one-dimensionalshear flow approximation [51]. Tiu [49], however, showed that one does not have to go through complicated derivations as done in Ref. 48 in order to develop such pressure drop-flow rate relationships in dies of complex section. straightforward approach based on a simple geometric parameter method proposed earlier [52] was used for the prediction of pressure losses through rectangular and trapezoidal dies. Each of the expressions developed in Refs. 45-49 for pressure losses was based on the power-law model for describing the rheological characteristicsof the polymer melt whichis being extruded. Hence, besides the geometric parameters, the developed expressions contain the flow parameters in terms of the consistency index and the power-law index. Shenoy and Saini [50], however, used the modified Ostwald-de Waele power-law model described by Eq. (8.6) in order to makepredictionsofpressure losses through dies of complex cross section only from the value of MFI. The choice for the basis of their theoretical development could have been either the work of Lenk [45-471 or of Tiu [49], both having provided simplistic expressions through straightforward approaches.The treatment of Lenk, [45-471 being both simple and rigorous, was chosen [50]. This is by nomeans an indication of the inferiority ofthe approach of Tiu [49]. In fact, the methodology presented by Shenoy and Saini [50] be extended to the expressions provided by Tiu [49] without difficulty. The method essentially involves the development of expressions for the pressure drop in terms of the channel geometry, the shear-viscous characteristics of the fluid, and the volume output rate. Four types of channel cross sections are considered (rectangular, cylindrical, elliptical, andregularpolygonal),andin each case, the flow in parallel-sided as well as in tapering geometries has been studied. In the case of an elliptical channel as well as in a regular polygonal channel, the assumption is made that the isovels resemble those of rectangular and circular channels, respectively. The validity of these assumptions increases when the aspect ratio of an ellipse approaches that of a wide slit in one case and the number of sides Np in a regular polygonal channel takes values far greater than 4 in the other case. The general expression proposedby Shenoy and Saini [50] is as follows:

where fi is the function of the geometric parametersbased on the shape of the die, fi is a function of the fluid properties, namely, K and n as given by Eq. (8.6), t d is the length of the die, and H,is a characteristic geometric parameter

Chapter

266

specific to each geometric shape. The expressions for and fi are given below for each specific die shape.

Case l : Rectangular Channel Without Taper H, = H

(8.58a) (8.58b)

+

h =K(+

(8.58~)

eWH

where is the shape factor given by the ratio of the width W to the height H of the channel. Thus, &,., + 1 represents a square, whereas SWH+ represents a wide slit. Case 2: Rectangular Channel Constant Width and a Vertical Taper

H, = H,

(8.59a) (8.59b)

fi

=

K

1"(

(8.59~)

where A, is the constant vertical taper factor given by the ratio of the height Hz at the end of the channel to the height H, at the start of the channel and SW is the shape factor given by the ratio of the constant width W to the entrance height Hl of the channel. Case 3: Rectangular Channel Constant Height and a Constant Lateral Taper

H, = H 1 1 - A:"' 1-A,)

(8.60a)

"c(

(8.60b)

fi = K

(8.60~)

l - n

where A2 is the constant lateral taper factor given by the ratio of the width W, at the end of the channel to the width W, at the start of the channel and &, is the shape factor given by the ratio of the entrance width W, to the constant height H of the channel. Case RectangularChannelwithaConstantVerticalTaperAlongwith a Constant Lateral Taper

H, = H,

(8.61a)

From Master Rheograms to Processing Parameters

- P)(l - h,) +-1 - 2n (h, (1 - hY)(l - h,)

K

4n+2

fz=z ( 7 y

(8.61b)

(8.61~)

where h, is the constant vertical taper factor given by the ratio of the height Hz at the end of the channel to the height H , at the start of the channel, h, is the constant lateral taper factor given by the ratio of the width at the end of the channel to the width at the start of the channel and 6 is the shape factorgiven by the ratio of the entrance width to theentrance height H , of the channel. Case 5: Rectangular Channel with a Taper Such That the Cross-sectional Shape Factor Is Constant

H, = H,

(8.62a) (8.62b) (8.62~)

where h, is the taper factor given by the ratio of the height H, at the end of the channel to the height I f l at the start of the channel and !&,is the shape factor which is constant, such that the tapering channel has a constant sectional rectangular shape. Case 6: Cylindrical Channel

H, = R,

(8.63a)

=1

(8.63b)

fl

&=K(+ 3n

+1

(8.63~)

where R, is the radius of the channel. Case 7: Ti-uncated Right Cone

H, = R ,

(8.64a) (8.64b)

Chapter 8

K 3n+1"

h=&)

(8.64~)

where A, is the constant taper factor given by the ratio of the exit radius R, to the entrance radius R,. Case 8: Untapered Polygonal Channel withNpSides

H, = R,

(8.65a) (8.65b) (8.65)

where R, is the radius of the circle whose cross-sectional area is equal to the 'cross-sectional area of the polygon. Case 9: Constant Taper Polygonal Channel with Np Sides

H, = R,

(8.66a) (8.66b) (8.66~)

where R, is the radius of the circle whose cross-sectional area is equal to the cross-sectional area of the entrance of the polygon, R, is the radius of the circle whose cross-sectional area is equal to the cross-sectional area of the exit of the polygon, and A, is equal to RJR,. The values of fi can be generated by using the appropriate values of K and n from Table 6.7. Knowing the geometry of the die, fi values in all cases considered canbe easily calculated.From &. (8.57), a relationship between Al'Hc/2ed and Q/Hz can thus be generated only through the MFI using an appropriate value of f, calculated from the geometry of the die, an appropriate value of f, based on the generic type of polymer, and the corresponding value of n from Table 6.7. Figures 8.11 and 8.12 showplots of hPH/2td versus Q/H3for polypropylene and polystyrene meltflow through a rectangular die. The data points correspond to the experimental valuesof White and Huang [48], whereas the solid lines are predictions of Shenoy and Saini For comparison, the dashed line is shown giving the predictions of Lenk [53] based on the rheological data as given by White and Huang [48]. It can be seen that the prediction merely through MFI

Master From Rheograms Processing toParameters

269

I

Figure 8.11 Pressuredropversus flow rateforpolypropylenemelt rectangular die. (Reprinted from Ref. 50.)

I o2

I

flow througha

is as effective as the other cumbersome approach throughthe knowledge of the entire flow curve.

WSCOUS HEAT In all polymer processes, the molten polymer is subjected to a wide range of shear rate and temperature, during which viscous heat is generated purely because of the friction between the viscous meltand various parts of the processing

Figure 8.12 Pressuredropversus flow rate for polystyrene melt flow througharectangular die. (Reprinted Ref. 50.)

Chapter

equipment with which it comes into contact. The viscous heat dissipated leads to a temperature rise, resulting in an offset of the setting of the extruder temperature profile with respect to throughput rate or screw speed. For all high output rate operations, it is essential to know the viscous heat generated inorder to appropriately design the conventional extruder screw as to minimize the temperature increase in the process of plastication and to optimize the extruder temperature profile with respect to throughput rate or screw speed. Viscous heat estimation is also critical when processingheat-sensitive polymers as to maintain the melt temperature well below degradationpoint. The viscous heat dissipation can be calculated using the following equation as given by Bird et al.

+

where is the viscous heat generated within the melt per volume. Shenoy and Saini [ S ] have shown that plots made in terms of versus p/MFI are unique with respect to each generic type ofpolymer. The generatedmaster curves for various polymers are shown in Figs. It has been shown that within the ranges of considered, each of the unique plots on log-log scale could be fitted by a straight line following the relationship as given by

+/MFI

The values of and PI for various polymers are given in Table The plot of versus p can readily be obtained by substituting the correct value of MFI in the unified curve. Alternatively, the need to read out values from the curves can be done away with by using Eq. Once the rate of generation of heat is known at the shear rate of relevance, then the maximum adiabatic temperature rise can be estimated through an analysis of the temperature profiles developed due to viscous heating by solving the energy equation. Although the solution of the energy equation for various geometriesis available in the literature, it is still a cumbersome method of calculating the amount of heat generated by viscous dissipation. Cox and Macosko have provided an elegant approach resulting in a simplified form for calculating the maximum adiabatic temperature rise as follows:

+

+

where is the density in g/cm', C, is the specific heat in caVg "C,and tr is the residence time in seconds. The magnitude of the temperature rise ATr can be represented and estimated, in general, through the dimensionless groupdefined

Master From Rheograms Processing toParameters

Figure 3 Unified viscous dissipation curve for LDPE at for MFI. (Reprinted from Ref. 55.)

271

test load condition

Chapter 8

.lo"

I

Id

IO1

I

MFl Flgure

Unified viscous dissipation curvefor HDPE at 2.16-kg test load condition

MFI. (Reprinted from Ref. 55.)

by the ratio the viscous dissipation to the heat conduction value, namely, the Brinkman number. The viscous heat dissipation estimate is extremely useful in determining the temperature-residence time relationship during the processing the polymer, thereby indicating the conditions that should be maintained in order to avoid excessive thermaldegradation. This is done through energy balance in the processing equipment whereby the relationship between specific power input, residence time, and power density the mixing system is obtained, as has been

From Master Rheograms to Processing Parameters

test load condi-

Chapter

Figure 8.16 Unified viscous dissipation curve for PP at 2.16-kg test load condition for MFI. (Reprinted from Ref. 55.)

From Master Rheograms to Processing Parameters

Figure

Uniiied viscous dissipation curve for PS at

test load condition for

MFI.(Reprinted from Ref. 55.)

done by Stade for the specificcaseof compounding on a continuous kneader. It is easy to determine the following relationship (for details, see Ref. 57) to calculate the specific energy inputforthermal loading of theproduct during compounding:

-=+ t,

(8.70)

Chapter 8

MFI Figure Unified viscous dissipation curve for cellulose ester/ether at load condition for MFI. (Reprinted from Ref. 55.)

2.16-kg test

where e, is the specific energy input in J/g. A number of specific cases such as those discussed above can be cited wherein the viscous dissipation data are absolutely essential during the compounding operation and processing of the polymers. The generationof this information must be quick and straightforward to make it useful. The approach followedin the present analysis provides a desk

From Master Rheograms to Processing Parameters

I oo

o3

I

Figure Unified viscous dissipation curve for acrylic at 3.8-kg test load condition for MH. (Reprinted from Ref. 55.)

Chapter

I

=26eoc -2oooc -2eIOc

-

- -- -W -- -- - - -- - -- --- -- -- --

- - - -

---

--- """ "" --

-

.f' 8.20 Unified viscous dissipation curve for Nylon at for MFI. (Reprinted from Ref. 55.)

test load condition

calculator route to the generation the data on viscous heat dissipation-an estimate of which is critical when dealing with polymer melts whose viscosity ranges are high enough (103-106P) to make theeffect viscous heat significant during processing.

From Master Rheograms to Processing Parameters

MFI Figure Unified viscous dissipation curve for PET and PBT at condition for (Reprinted from Ref. 55.)

test load

280

Chapter 8

8.22 Unified viscous dissipation curve for PC at 1.2-kg test load condi tion for MFI. (Reprinted from Ref. 55.)

From Master Rheograms Processing toParameters

Figure 8.23 Unified viscous dissipationcurve

281

PES at

test loadcondition

MFI. 8.6 BLENDING AND FILLING TO FORM MULTICOMPONENT POLYMERIC SYSTEMS During recent years, multicomponent polymeric systems have attained great importancedue to the possibility obtainingcompounds withnovel physical properties and optimumcost preformance benefits.From a macrostructural view-

Chapter 8

282

-

4530

* 451

*

%/MFI

Figure 8.24 Unified viscous dissipation curve

PEEK at

test load condition

"I.

point, polyblends and filled polymers would fall under the category of multicomponent systems consisting of a polymeric continuous phase or matrix and of a dispersed phase. The dispersed phase could either be another polymer (as in the case of polyblends, see, for example, Tables 1.7 and 1.8) or fillers such as calcium carbonate, talc, and on, or reinforcing agents such asglass fibers, mica, and on (as in the case of filled polymers, see, for example, Table 1.10). All PVC formulations would fall into the category of multicomponent systems due to the presence of a number of additives such as plasticizers, fillers, stabilizers, lubricants, and on, which go into suchformulations (see, for example, Table 1.9). The final properties of the products depend to a large extent on how well the dispersion level of one phase is into the other.

From Master Rheograms to Processing Parameters Table

Values of a,and PI Needed in

Thermoplastic LDPE HDPE UHMWPE LLDPE PP PS C-Ester C-Ether Acrylic Nylon PET PC PVDF PP0 PPS PASPES PEEK PE1 PAI SAN

SBS ABS VCVA EVA Polyester elastomer TPE PP-HDPE HDPE-PMMA PS-PMMA PS-POM PMMA-POM PVC formulations

s2-7 X (g

per 10 min)"]

m.(8.68) P*

Modified shear rate' [(S") X (g per 10 min)"]

1-1000 2-1000 10-10000 4-100 4-1000 0.4-200 20-1000 2-100 20-1000 50-1000 50-1000 80-1000 2-200

lo' 104 lo6 l6 lo' l6 lo' l6 105 lo' 5.0 X 1 6 2.5 X l 6 7.5 1 6 1.0 lo6 6.0 X 16 3.0 X lo' 6.0 X lo' 1.4 X lo6 2.4 lo5 3.0 X 1 6 1.0 lo6 3.5 16 6 4.0 X l 1.0 1 6 1.4 X lo'

1.37 1.471 1.156 1.66 1.34 1.368 1.38 1.47 1.44 1.44 1.43 1.52 1.40 1.25 1.44 1.60 1.32 1.34 1.85 1.33 1.25 1.38 1.33 1.55 1.65

10-100 10-1000 20-1000 200-1000 10-100 3-1000 20-2000 1-100 0.3-7000 1-1000 7-300

1.5 2.0 1.0 3.0 3.6 3.0 4.6

1.40 1.32 1.45 1.46 1.30 1.43 1.36

1-1000 10-1000 0.1-40 4-100 4-100 8-100 5 -2000

1.7 X 1.24X 1.66 X 1.0 1.75 X 3.32X 3.2 X 1.5 X 3.2 X 3.4

X

X X X X

lo' lo' 16 16

l6 l6 lo'

'Applicability range ?M. bCondition MFI used in Eq. (8.68). specified.

40-1000

Test loadb (kg)

2.16' 2.16' 10.00' 2.16' 2.16' 5.00'

2.16" 2.16 3.80' 2.16" 2.16' 1.20' 12.50 5.00 5.00 5.00 5.00 5.00

3.80 20.00 2.16 2.16 2.16 2.16 2.16 5.00 3.80 20.00"

Chapter 8

284

8.6.1

Determination of Optimum Conditions During Polyblending

In the case of polyblends, the critical steps involved are (a) selection of the component polymers, their appropriate grades, and their composition based on property requirements, and (b) the proper choice of compounding method and conditions. Most often, the component grade selection and choice of compounding process parameters are arbitrary. It is the general practice to finalize the appropriate grades of the components and the compounding conditions based on the end property evaluations. This becomes more a trial-and-error procedure. Single-phase blends would generally provide synergistic property advantages, and this cannot be achieved even in thermodynamically miscible polymers if they are mechanically incompatible due to large differences in the melt viscosities at conditions of compounding. Shenoy [58] has suggested a simplistic approach of getting a quantitative estimation of compounding conditions or grade selection of blend components through MFI. A methodology is developed by specifying the temperature and shear-rate conditions for melt blending two component polymers whose grades have already been selected based on other considerations. Alternatively, it is shown that the method could be used for grade selection if the compounding process parameters are fixed a priori.

A.

Compounding Temperature Determination

In order to establish the most suitable compounding temperature, the modified Arrhenius-type equation [Eq. (4.13)] proposed by Saini and Shenoy [59] is used. Let us assume that polymers PI and P, are to be blended and their MFI values are known. It can be seen from ASTM D1238 (Appendix A) that there are different load and temperature conditions under which MFI values for various polymers are determined. Presently, let us assume the load conditions under which the MFI values for the two polymer components PI and P, are the same. In cases where the test load conditions are different, Eq. (4.9) is to be used for estimating the MFI value at the load of interest. Let the temperature of MFI measurement for each component be TpI and Tp2,respectively. Thus, from Eq. (4.13) the following can be written: (8.71a)

(8.71b) Now let us assume that the compounding is done at a temperature T,. Then the MFI value at the compounding temperature for each of the components can be

From Master Rheograms to Processing Parameters

205

written as (8.72a) (8.72b) In order to achieve mechanical compatibility, it is essential to maintain the viscosity levels of the two components at the same value. In other words, the optimum blending temperature would be that which gives (8.73) Equations (8.72a) and (8.72b) can be solved for the condition given by Eq. (8.73) to yield the temperature of compounding of the blends in order to achieve maximum mechanical compatibility. Thus, (8.74) The compounding temperature represents graphically the point of intersection of the two curves shown in Fig. 8.25(a) for polymers P, and P2. If the grade selection of the two component polymers has been done a priori, then the compounding temperature as determined from the intersection of the two curves is fixed. The value of T, as determined through Eq. (8.74) may not always be relevant for the selected grades. For example, the cases shown in Figs. 8.25b8.25d clearly indicate that the blending temperature as determined by the point of intersection is, respectively, too low (even below the melting temperature), too high (even above the degradation temperature), or does not exist if the activation energy of the two components are equal. Such values may not be relevant for use but are certainly indicative of the incompatibility of the component polymers. In such circumstances it might be advantageous to fix the blending temperature at two or three different relevant values and determine the ratio of MFIp2,Tp2 to MFIpl,Tpl which would satisfy the following condition obtained by rearrangement of Eq. (8.74): (8.75) Having fixed the grade of one component polymer, we can then determine the correct grade of the other component for achieving mechanical compatibility. After determining the correct compounding temperature T,, it is possible to obtain the value of (8.76)

Chapter 8

Figure Four possible ways in which curves MFI versus 1/T on a semilogarithmic plot would exist for two given component polymers. (Reprinted from Ref. 58.)

through (8.72). Note that M F I , Tc denotes the melt flow index value each of the components at the compounding temperatureT,. This value is useful for determining the shear-rate conditions for the compounding operations as will be shown in the subsequent analysis.

B. CompoundingShear-RateDetermination In order to establish the most suitable shear rate for compounding

polymer blends, the modified Ostwald-de Waele equation in the form given by Eq. (8.6) is used. For thetwo component polymers, P, and P*, at the temperature of blending and melt flow index of MHB,T,, we can write (8.77a)

From Master Rheograms to Processing Parameters

287

Figure 8.26 Four possibleways in whichcurves of T versuson a log-log plot would exist for two given component polymers. (Reprinted from Ref. 58.)

(8.77b)

In order to achieve mechanical compatibility conditions, both of the component polymers must see the same level of shear stress 7, and this ought to be achieved at the same shear-rate level p because the compounding equipment does not distinguish between the two components. Graphically, thisis akin to the intersection point on the two curves defmed by Eqs. (8.77a) and (8.77b) as shown in Fig. 8.26. Thus, = = rc and pp, = jp2 = pcin Eqs. (8.77a) and (8.77b) yield (8.78)

Chapter

289

From Master Rheograms to Processing Parameters Table 8.8a

VariousLLDPEandLDPE

Grades Used

m (19OoC/2.16 code Sample kg) LLDPE LLDPE LDPE C LDPE LDPE

2.0

Source: Ref. 60.

Table 8.8b Relevant Parameters for LLDPE and LDPE

Applicability range of E

Applicability range Load temp.

Polymer (kcal/mol) LLDPE LDPE

VC)

3.2 7.25

175-205 175-205

per

qo X

(kg)

MFI

2.16 9.6 2.16 3.0

[(S")

X

N

min)"]

25.3

0.11 0.17

0.01-2 0.1-1

X 10' X 16

Source: Reprinted from Ref. 61 with permission from Elsevier Science Ltd., Kidlington, UK.

Table 8.9 Compounding Temperature for Various

LLDPELDPE Blends Compounding temperature calculated from Eq. (8.74) 2l/Component Component LLDPE A/LDPE C LLDPE A/LDPE D LLDPE ALDPE G LLDPE B/LDPE C LLDPE BLDPE D LLDPE BLDPE G Source: Ref. 58.

(g

A X MFI

127 110

312 190 168 457

10

290

Chapter 8

Table 8.10 Required MFI ValueRatiofor LLDPELDPE Blends at Various Compounding Temperatures

~LLD,rlMFIL,per

2.0 1.5 1.16 1.0

Compounding temperature T, ("C) 125 150 175 190

Source: Ref. 58.

I

1

HDPE-PP

/

J Figure 8.27 Melt flow index variation with blend composition for high-density polyethylene/polypropylene blend at 190°C and 2.16-kg test load condition for MFI using datafromRef.83(validityover18O"C-21O0C).(ReprintedfromRef.78withkind permission from Steinkopff Verlag Darmstadt.)

From Master Rheograms to Processing Parameters I

HOPE

-

-

3-

P 2-

I

I

12

2

P2

Figure Melt flow index variation with blend composition for high-density polyethylene/polymethyl methacrylate blend at 160°C and 2.16-kg testloadconditionfor MFI using data from Ref. (validity over (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

Note that when np, = npzthe curves do not intersect and hence no shear-rate condition can be specified for achieving mechanicalcompatibility. When the K and n values for the component polymers are the same (as in the case of PET and PBT wherein K = X lo“ (&cm S””) (g10 min)” and n = for both polymers), then the blending shear rate truly does not matter, as the two components are undoubtedly compatible, provided they have the same MFI value at the blending temperature. A few case studies are presented below to illustrate the above postulated method of determining the compounding conditions (temperature and shear rate) and, alternatively, to select the appropriate grades of the component polymers if the compounding process parametersare fixed a priori.

C. Case Studies Case I : To determine the temperature andshear-rate conditions for SAN/acrylic blend, weconsiderthesamegrade of S A N with three different grades of PMMA.

292

Chapter

8.29 Melt flow indexvariationwithblendcompositionforpolystyrene/polymethyl methacrylate blend at 220°C and 5.0-kg test load condition for MFI using data from Ref. 85 (validity at all temperatures). (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

(a)Component 1: 860B [MFI = 9.5 (230°C/3.8 kg)]; Component 2: Lucite 140 [MFI = 5.0 (230°C/3.8 kg)]. The relevant parameters needed for the calculations are shown in Table 8.7. The loading and temperature conditions of MFI values for the two components are identical. Thus, TI = Tz = 503 K Substituting the appropriate values in Eq. (8.74) gives T, = 548 K = 275°C. @) Component 1: 860B [MFI = 9.5 (23OoC/3.8 kg)];Component 2: Plexiglas W 1 0 0 [MFI = 7.9 (230°C/3.8 kg)]. Here again, TI = Tz = 503 K and hence, from Eq. (8.74), T,= 515 K = 242°C. (c) Component 1: 860B [MFI = 9.5 (23OoC/3.8 kg)];Component 2: Plexiglas VS100 [MFI = 11.2 (210°C/3.8 kg)]. Here, TI = 483 K and Tz = 503 K and thus, from Eq. (8.74), T, = 410 K = 137°C.

From Master Rheograms to Processing Parameters

PS

293

-

Flgure Melt flow index variation with blend composition for polystyrene/polyamtal blend at 210°C and 5.0-kgtest load condition for MFI using data from Ref. (validity over 180-250°C). (Reprinted from Ref. 78 with kind permission fiom SteinkopffVerlag Darmstadt.)

It can be seen that in (a) and (c) the blending temperatures are a little high and too low, respectively. The better acrylic grade to blend with "yril 860B is thus Plexiglas V M O l O at a blending temperatureT, = 242°C. The MFI value of these two polymers at 242°C can be calculated from Eq. (8.72). In the present case blending, Qril 860B with Plexiglas VM100, we get MFI,, Tc = (23OoC/3.8 kg). Using Eq. (8.78), the shear rate for blending can be calculated as equal to 10.8/s. Case 2: Speed [60] showed that better blown films can be designed by using blends linear low-density polyethylene (LLDPE) and low-density polyethylene (LDPE). The conditions of blending have not been given, but we illustrate in the following how easily this could be done by the above technique. The various grades LLDPE and LDPE used by Speed [60] are given in Table 8.8a. The other parameters useful for performing the necessary calculations are given

Chapter 8

-

I

I

I

I i

8.31 Melt flow indexvariation with blendcompositionforpolyacetal/polymethyl methacrylate blend at 210°C and 3.8-kg test load condition for MFI using data from Ref. 86 (validity over 180-250°C). (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

in Table 8.8b Various combinations of LLDPE and LDPE grades have been considered, and in each case, the blending temperature has been determined as above. The results are given in Table It can be seen that LDPE G requires a very high temperature of blending if it has be mechanically compatible, thus showing that it is probably not the right grade choice for synergistic properties. The other component combinations would be mechanically compatible at different temperatures;hence,a blendingtemperatureforallcombinations of LLDPE and LDPE grades would not result in optimum property products. In order to illustrate the method for determining the shear-rate conditions for blending, we choose LLDPE A and LDPE C. Instead of the power-law model used in Eq. (8.77), we choose the modified Carreau model as given in Ref. 61 whose parametric valuesare tabulated in Table 8.8b. Thus, comparingthe shear

From Master Rheograms to Processing Parameters

2

295

S 4 -

162

Figure 8.32 Melt flow index variation with blend composition for high-density polyethylenellow-densitypolyethyleneblendat 190°C and 2.16-kgtestloadcondition for MFI using data from Ref. 87 (validity at all temperatures). (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

Chapter I

HDPE

-

- LDPE

.-

6

I

C l

2

8.33 Melt index variation with blend composition for high-density polyethylenenow-density polyethylene blendat and 190°C and test load conditionfor MFI usingdatafromRef. 88 (validityoverentiretemperaturerange).(Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

Master From Rheograms Table

to Processing Parameters

297

Temperature Sensitivity of the Reciprocal log [MFI(T, O)FIFI(T,&)l Temperature

ec>

lflog[Mm(T, O)/MFI(T, l)]

HDPEPP

180 190 200 210

1.00 1.07 1.20 1.25

HDPEPMMA

150 160 170 180

0.41 0.61 1.00 2.19

PSPMMA

200 210 220 230

0.51 0.52

200 210 220

1.05 1.16 1.26 1.37

components Blend

PARS

230

200 210 220 230

0.53

0.53

1.05 1.20 '1.35 1.51

Source: Ref. 78. (Reprinted with kind permission from Steinkopff Verlag Darmstadt.)

(+mB,

the value can be determined, which in the present case is found to be equal to 37.6. NOW M F I B , Tc can be calculated using (8.72) to give a value equal to 0.578, and hence the shear rate for blending is 21.7/s. Case 3: Suppose we decide to make blends of LLDPE and LDPE. Then the choice of grades can be made as follows, based on the predetermined blending temperature. Using (8.75), the ratio of MH,, to MFI,, can be calculated for different blend temperatures. Thus, for LLDPE and LDPE blends, the results obtained are as shown in Table 8.10. It can be seen that based on the choice of the blending temperature and the grade of one of the components, selection of the grade of the other component can be made. number of such cases can be cited wherein the above method can be effectively used for selecting the grades of polymer components for the blends or,alternatively, the blending conditions. This technique is the simplest and

Chapter 8 2

-

/ 'c

QI* 8.34 Melt index variation with plastickm compositionfor poly(viny1 chloride) formulation at 193°C and Wkg test load conditionfor MFI (ASTMD3364) using from Ref. 89. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

quickest route for determining compounding conditions compatiblity.

8.6.2

for mechanical

Determinationof of Blends and Filled Systems at Any Composition

There have been a number of models for predicting the viscosity of blends [62-661 based on the viscosities and amounts the respective components. Most of the models have been empirically proposed and are based on the additivity law inparallel, series, or logarithmic.Hanand Yu have repeatedly demonstrated the invalidity of any simple law of additivity in terms of blending ratios and the rheological properties of the individual components to predict the viscosities of the polyblends. In the case of filled polymers, someof the empirical formulas proposed for describing the dependence of viscosity on concentration fit experimental data rather well; however, there is no uniqueness and no general physical significance the numerous available formulas. Details about the complexity in the rheological behavior of polyblends and filled polymers has not been included here, as these are readily available

From Master Rheograms to Processing Parameters I

I

PVC-450 /

I

I

/

-462 205.C

6 -

2 -

0

I

I

I

I

2

4

6

8

12

Figure8.35 Melt flow index variation with plasticizer composition for poly(viny1 chloride) formulation containing stabilizer and lubricant at 205°C and 20-kg test load condition for MFI (ASTM D3364) using data from Ref. 90. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

through a number of comprehensive reviews[72-771. There is, nevertheless, a need for obtaining a theoretical generality to describe the concentration dependence of the dispersed phase on the rheological behavior of the continuous phase that the viscous behaviorof multicomponent systems can beelucidated under a unified framework. This need has been satisfied by Shenoy and Saini [78] who have developed a method of predicting the MFI of the blend or filled system at any composition based on the altered free-volume model [79]. The following relationship between viscosity of a polymer melt and its free volume can be written on the lines of Doolittle’s equation [80] as (8.80)

where q(T,&) represents the shear viscosity of the system at temperature T and containing & weight fraction of the dispersed phase. In the case of nonNewtonian systems, q(T,&) is taken asthe zero-shear viscosity.A’ is a constant

Chapter 8

300 I

i! -

LL

I J

$2

8.36 Melt flow index variation with plasticizer composition for poly(viny1 chloride) formulation containing stabilizer and lubricant at and 20-kg test load condition for MFI (AsTh4 D3364) using data from Ref. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Dannstadt.)

dependent on the nature of the continuous phase. B' is a constant which can be set equal to unity based on the arguments provided by Fujita and Kishimoto [81]. f(T,&) represents the free volume of the system at temperature T and containing & weight fraction of the dispersed phase. systems are known tobe nonMelts of polyblends andfilledpolymer Newtonian in nature and their viscosities are shear dependent. Hence, q(T, in Eq. would have to be logically chosen to represent the zero-shear viscosity. However, in the case of filled polymer systems, it is known (see, for example, Ref. 82) that the melt viscosity in the very low-shear-rate ranges does not show a plateau as in the case of homopolymers but shows a continuous increase representing ayieldpoint. This viscosity buildup atverylowshear

Master From Rheograms

Processing to Parameters

301

I

LOPE

I

- QTZ

2

& Figure8.37 Melt flow index variation with filler composition for low-density polyethylene/quartz powder system at 220°C and 2.16-kg test load condition forMFI using from Ref. 91. ( R e t e d from Ref. 78 kind permission from Steinkopff Verlag Darmstadt.)

rates + 0) is dependent on the physical nature, chemical type, and amount of the filler presentinthepolymer matrix. Ifa generality in the theoretical development is being seeked for multicomponent systems which includes polyblends as well as filled polymers, it is essential to use a viscosity parameter other than the zero-shear viscosity. MFI becomes an ideal choice because it is determined in the medium shear-rate range far from the very low shear-rate range where filled systems would show a yield point. Using the inverse relationship between MFI and q as given by Eq.(4.12) of Sec. Eq. (8.80) can be rewritten in. the following form: 1

In MFI(T, &) = 1nA" - f(T, 0 2 )

(8.81)

where MFI(T, &) is the melt flow index of the system at temperature T and containing & weight fraction of dispersed phaseA" is a constant. At this stage of the theoretical development, the altered free-volume model concepts can be used. It is assumed that the addition of another polymer or a filler alters the state of the reference medium which could bea pure homopolymer, or a polyblend containing a known amount of the dispersed phase, or a filled polymer containing a known amount of filler. The relevant parameter

302

Chapter 8 I

I

5

I J

2

I

12

2

Id2

8.38 Melt flow index variation with Mer composition for polypropylene/calcium carbonate system at 200°C and 2.16-kg test load condition for MFI data from Ref. 92. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

which characterizes the altered state is chosen to be the "free volume" of the system. If the polymer is chosen to be the continuous medium, then its free volume can be specified by AT7 0). Its melt flow index be written as MFI(T, 0) and the following relationship derived from Eq. would hold: In MFI(T, 0) =

-f(T7 0)

The altered free-volume state the multicomponent system (polyblend or filled polymer) could be denoted by f(T, &)' where & is the volume fraction the dispersed polymer phase in case polyblend and the dispersed filler phase in the case of filled polymer. The melt flow index of the multicomponent system is denoted MFI(T, &) and would show the relationship between melt flow index and the altered free volume.

From Master Rheograms to Processing Parameters I

l

PS

303

-

5 -

-

2-

I

I

io

2

1

12

82

Figure Melt flow index variation with filler composition for polystyrene/carbon black system at 180"Cand 5.0-kg test load condition for MFI using data from Ref. 93. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

As a first approximation, the free volume in the altered free-volume state resulting from the addition of a dispersedpolymer phase or a fdler can be considered to reducethe free volume of the reference medium and to be a linear function of the volume fraction +z of the added entity similar to that given by Fujita and Kishimoto [81]: where p(r) represents the difference between the free volumes of the polymer and the dispersed phase. Combining Eqs. (8.81) to (8.83) and rearranging the terms gives 1 - - 2.303jTT, log a m

"

1 + 2.303f2(T, 0) -

Pm

+2

(8.84)

Chapter

I

Q

I

2

8.40 Melt index variation with filler composition for polystyrene/titanium dioxide system at 180°C and 5.0-kg test load condition for MFI using data from Ref. 93. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Dannstadt.)

where

(8.85) Based on the assumptionIS made during th.e derivation of Eq. (8.24), it is imperative to choose*I(T, 0) > MFI(T7 &). In the case of filled systems, this condition is naturally satisfied when the polymeric matrix is taken as the reference medium. For polyblends, a deliberate choice of the reference medium would have to be made that the MFI value of the reference medium would be greater than that of the multiphase system. Eq. (8.84)predicts that a plot of l/log am versus l/&should be linear, and the propriety of this model has been examined quantitatively in the light of the reported experimental data. Existing viscosity data in the literature available for all multicomponent systems is in the form of viscosity versus shear rate or shear stress versus shear rate curves. In each case, the data are transformed into specific MFI values using Eqs. (4.7) and (4.8)and the specified load condition for each systembased on D1238.

From Master Rheograms to Processing Parameters

305

I

-

2

g2

Melt flow index variation with filler composition for polycarbonate/glass fiber system at 290°C and 1.2-kg test load condition for MFI using data from Ref. 94. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

Figures 8.27 to 8.31 show plots of l/log am versus l/&for different blend systems [83-861. The multicomponent systems studied are such as to include blends of homopolymers from different classes such as olefinics, styrenics, and engineering thermoplastics. The amount of free volume available for motion of the individual molecules depends to a large extent on the generic type of the individual polymers involved in the blend. Most addition polymers have relatively long extended chain lengths of the order of 5000 A, whereas condensation polymers represent short polymer chains extending to lengths of the order of 1000 A. Interactions between two addition polymers and those between an addition and a condensation polymer would be expected to be different and hence the effect on the relative free volumes of the blends wouldbe different depending on the constituents. However, despite the obvious diversities and complexities in the rheological behavior of the polyblends with speciestype and amount, they all seem to fit within the framework of the model proposed through Eq. Figures 8.32 and 8.33 also include systems which would fall in the class of polyblends [87,88]. However, the difference is that HDPE-LDPE blends specif-

306

Chapter 8 I

I

PET

I

I

I

-

82

Figure Melt index variation with filler composition for poly(ethy1ene terephthalate)/glass fiber system at 275°C and 2.16-kg test load condition forMFI using data from Ref. 95. (Reprinted from Ref. 78 with kind permission from Steinkopff Verlag Darmstadt.)

ically form systems which bring outthe effect of branching on the melt viscosity. By adding different amounts of LDPE (branched PE) to HDPE (linear PE) the effect of branching on the rheological properties can be elucidated. The proposed model through Eq. (8.84) is seen to work excellently in this case, too..In Fig. 8.33 the linear plot contains data points at and and,hence, is temperature invariant. This is because the glass-transition temperatures of HDPE and LDPE are almost the same. In order to bring out the effect temperature on the linear plots shown in Figs. 8.27-8.31, the MFI values of the individual components in each blend were calculated at different temperatures.This was done using the modified (Williams-Landel-Ferry) WLF-type equation (4.15).

Master From Rheograms Processing toParameters

307

comparison of the values of reciprocal log[MFI(T, O)/MFI(T,&)l at $z = 1 has been shown inTable 8.11. It is evident thatwhen the Tg values of the individual components a blend are not too different, the values of reciprocal log[MFI(T, O)/MFI(T, l)]canbetaken to be approximately constant. Thus, HDPE/P"A forms the worst case in Table 8.11 due to the largest TBvalue differences between the individual components withinthe blend systemsstudied. By the same token, P S / P " A shows an approximately constant value of reciprocal log[MFI(T, O)/MFI(T, l)]. In the linear plots of reciprocal log[MFI(T, 0)/ MFI(T, 4 4 3 versus reciprocal &,the maximum deviations due to temperature wouldbe expected at +z = 1. Thus, if reciprocallog[MFI(T, O)/MFI(T, l)] remains approximately constant within a temperature range, the linear plots shown in Figs. 8.27-8.31 can be assumed to be temperature invariant within that temperature range and the entire range of &. Figures 8.34-8.36 show plots of lflog versus l/+z for a few typical PVC formulations [89,90]. In Fig. 8.34, the effect of the amount of plasticizer, namely, dioctyl phthalate on the MFI, can be readily calculated from the linear plot. The presence of the plasticizer increases the free volume and reduces the Tgof PVC. The effect of this on theflowbehaviorof PVC can be readily understood through the use of the suggested model given by (8.84). The effect the plasticizer amount on MFI in the presence of stabilizers and lubricants has been shown in Figs. 8.35 and 8.36. Figures 8.37-8.42 show plots of lflog am versus l/& for different filled polymer systems. The systemsare chosen as to include different generic types of polymers as the matrix and to include fillers with different shapes and types [91-951. In all cases, despite the apparent diversity, there is a uniqueness in the altered free-volume state model. The proposed model through Eq. (8.84) can thus be used in all generality for all types of multicomponent systems.

Hellmeyer, H. 0. and Menges, G., Application process computersfor the control andoptimization injectionmouldingprocess,Proc.AnnualTech. Soc. Plastics Engrs. p. Spencer, R. S. and Gilmore, B.D., Some phenomena in the injection moulding of polystyrene, J. Colloid Sci., Ballman, R.L., Shusman, L., and Toor, H. L., Injection moulding: a molten polymer into a cold cavity, Ind. Eng. Chem., Ballman, R.L., Shusman, L., andToor,H. L., Injection moulding: A rheological interpretation, Mod. Plastics, (Sept. Paulson, D. C., Pressure loss in the injection mould, SPEANTEC Tech. Paper, No. p. Barrie,1.T., An application of rheologytotheinjectionmoulding of large-area articles, Plastics Polym.,

308

Chapter

7. Barrie, I. T., An analysis of large-area moulding technology, Plastics Polym., 37, 463 (1969). 8. Cogswell, F. N. and Lamb, P., Polymer properties relevantin melt processing,Plast i c ~Polym., 38, 331-342 (1970). 9. Harry, D. H. and Parrott, R. G., Numerical simulation of injection mould filling, Polym. Eng. Sci., 10, 209 (1970). 10. Kamal, M. R. and Kenig, S., The injection moulding of thermoplastics, PartI: Theoretical model, Polym. Eng. Sci., 12, 294 (1972). 11. Pearson, J. R.A., Mechanical Principles of Polymer Melt Processing, Pergamon Press, Oxford (1966). 12. Wu, P. C., Huang, C. F., and Gogos, C. G., Simulation of the mould filling process, Polym. Eng. Sci., 14, 223 (1974). 13. Williams, G. and Lord, H. Mould filling studies for the injection moulding of thermoplastics, Polym. Eng. Sci., 15, 533 (1975). 14. Le&, R. S., Polymer Rheology, Applied Science Publishers, London(1978), Chap. 13, p. 141. 15. Saini, D. R. and Shenoy, A. V., Simplified calculations for mould filling during nonisothermal flow of polymer melts, Plastics Rubber Process Applic.,3,175-180 (1983). 16. Birnkraut, W. H.,Braun, G., and Falbe,J., Ultra high molecular weight polyethylene processing and properties, Appl. Polym. Sympos.,36, 79-88 (1981). 17. Zachariades, A. E., Watts, M. P. C., Kamamoto, T., and Porter, R. S., Solid state

extrusion of polymer powders illustrated with ultra high molecular weight polyethylene, J. Polym. Sci., Polym. Lett. Ed., 17, 485-488 (1979). 18. Stefan, J. and Sitzgber, K., Experiments on apparent adhesion, Akad. Wws.Math. Wein, ZZ, 69, 713 (1874). 19. Scott, J. R., Trans. Inst. Rubber Znd., 7, 169 (1931). 20. Diennes, G. J.’ and Klemn, H. F., Theory and application of parallel plate plastometer, J. Appl. Phys., 17, 458 (1946). 21. Tanner, R.I., Some illustrative problems in the flow of viscoelastic non-Newtonian lubricants, ASLE Trans., 8, 179 (1965). 22. Metzner, A. B., The significant rheological characteristics of lubrication technology, Trans. ASME, 9OF, 531 (1968). 23. hider, P. J. and Bird, R. B., Squeezing flow between parallel disks: 1. Theoretical analysis, Ind. Eng. Chem. Fundam., 13, 336-341 (1974). 24. hider, P. J., Squeezing flow between parallel d i s k 11. Experimental result, Znd. Eng. Chem. Fundam., 13, 342-346 (1974). 25. Brindley, G., Davies,J. M., and Walters, K., Elastico-viscous squeeze films,J. NonNewtonianFluid Mech., 1, 19 (1976). 26. Binding, D. M., Davies, J. M., and Walters, K., Elastico-viscous squeeze films III. The torsional balance rheometer,J. Non-Newtonian Fluid Mech., 1, 277 (1976). 27. Zahorski, S., Viscoelastic propertiesin plane squeeze-film flows,J. Non-Newtonian Fluid Mech., 4, 217 (1978). 28. Tichy, J. A. and Modest, M. F., A simple low Deborah number model for unsteady hydrodynamic lubrication, including fluid inertia,J. Rheol., 24, 829 (1980).

Master From Rheograms Processing toParameters

309

29. Shenoy,A. V. and Saini, D. R., Compression moulding of ultra high molecular weight polyethylene, Plastics Rubber Process. Applic., 5, 313-317 (1985). 30. Oliver, D. R., The influence of fluid inertia, viscosity and extra stress in the load bearing capacity of a squeeze film of oil, Appl. Sci. Res., 1 (1979). 31. Binding, D. M., Avita, F., Maldenado, A., and Sen, M., Medium and large Deborah number squeezing flows, in Proc. VIIIth Int. Congr. Rheol. (1980), Vol. 2,p. 111. 32. Shenoy, A. V. and Saini, D. R., A simplified approach to the predictionof primary normal stress differences in polymer melts,Chem. Eng. Commun.,28,l-27 (1984). 33. Gaskell, R. E., J. Appl. Mech., 17, 334 (1950). 34. McKelvey, J. M., Polymer Processing, Wiley, New York (1962). 35. Alston, W.W. and Astill, K N., An analysis for the calendering of non-Newtonian fluids, J. Appl. Polym. Sci., 17, 3157-3174 (1973). 36. Tadmor, Z. and Gogos, C. G., Principles of Polymer Processing, Wiley, New York (1979). 37. Agassant, J. F. and Avenas, P., Calendering of PVC: Prediction of stress and torque, J. Macromol. Sci., Phys., 14, 345-365 (1977). 38. Brazinsky, I., Cosway, H. F., Valle, Jr., C. F., Jones, R. K, and Story, V., A theo-

39. 40. 41. 42.

retical study of liquid film spread heights in the calendering ofNewtonianand power law fluids, J. Appl. Polym. Sci., 14, 2771-2784 (1970). Vlachopoulos, J. and Hrymak, A. N., Calendering poly(viny1 chloride): Theory and experiment, Polym. Eng. Sci., 20, 725-731 (1980). Ray, A. and Shenoy, A. V., PVC calendering: A simplified prediction technique, J. Appl. Polym. Sci, 30, 1-18 (1985). Kiparissides, C. and Vlachopoulos, J., A study of viscous dissipation in the calendering of power law fluids, PoZym Eng. Sci., 18, 210-214 (1978). Middleman, S., Fundamentals of Polymer Processing, McGraw-HiJl, New York

(1977). 43. Chauffoureaux,J.C.,Debennan,C.,and (1979).

vanRijckevorsel,J.,

J. Rheol., 21, 1

44. McCormick, J. M. and Salvadori, M. G., Numerical Methods in FORTRAN, Prentice-Hall, Englewood Cliffs, NJ (1964). 45. Lenk, R. S. and Frenkel, R. A., Pressure drop through tapered wide-slit dies-A revised version, J. Appl. Polym. Sci., 26, 2801-2804 (1981). 46. Lenk, R. S., Flowin ellipticalchannels, J. Appl. Polym. Sci., 26, 3171-3173 (1981). 47. Frenkel, R.A. and Lenk, R. S., Flow through regular polygonal channels, J. Appl. Polym. Sci., 26, 3939-3994 (1981). 48. White, J. L. and Huang, D., Extrudate swell and extrusion pressure loss of polymer meltsflowingthroughrectangularandtrapezoidaldies, Polym. Eng. Sci., 21, 1101-1107 (1981). 49. mu, C., Prediction of extrusion pressure loss of polymer melts flowing through noncircular dies, Polym. Eng. Sci., 22, 1049 (1982). 50. Shenoy, A. V. and Saini, D.R., Prediction of pressure losses through typical die shapesbasedonasimple,novelapproach, Polym. Plastics Technol. Eng., 23, 169-183 (1984).

Chapter

0

51. Ishida, M., Kikuchi, N., and Ito,

K, Broken section method for analyzing flow of polymer melts in dies having cross-sections of varying heights, Appl. Polym.Sym-

POS.,20, 99-108 (1973). in ducts of arbitrary 52. Kozichi, W., Chou, C. H., and Tiu, C., Non-Newtonian flow cross-sectional shape, Chem. Eng. Sci., 21, 665-679 (1966). 53. Lenk, R. S., Pressure drop through tapered dies, J. Appl. Polym. Sci, 22, 17751779 (1978). 54. Bird, R. B., Stewart, W. E., and Lightfoot, E. N.,TransportPhenomena, Wiley, New York (1960). 55. Shenoy, A. V. and Saini, D. R., A simplistic route to viscous heat estimations in polymer processing, Polym. Plastics Technol. Eng., 23, 37-68 (1984). 56. Cox, H. W. and Macosko, C. W., Viscous dissipation in die flows, AIChE J., 20, 785-795 (1974). 57. Stade, K H., The production of glass fiber-reinforced poly(buty1ene terephthalate) on a continuous kneader, Polym. Eng. Sci., 18, 107-113 (1978). 58. Shenoy,A. V., Estimation of compounding conditions and grade selections in the preparation of thermoplastic melt blends,Polym Plastics Technol. Eng., 24,27-41 (1985). 59. Saini, D. R. and Shenoy, A. V., A new method for the determination of flow activation energy for polymer melts, J. Macromol. Sci. “Phys., B22,437-449 (1983). 60. Speed, C. S., Formulating blendsof LLDPE and LDPE to design better film, Plustics Eng. (July 1982), p. 39. 61. Saini, D. R. and Shenoy, A. V., Viscoelastic properties of linear low density polyethylene melts, Eur. Polym. J., 19, 811-816 (1983). 62. Hayashida, K., Takahashi, J., and Matsui, M., Proc. Fifth Int. Congress on Rheology (1970), Vol. 4, p. 525. 63. Heitmiller, R.F.,Naar, R. Z., and Zabusky, H. H., Effect of homogeneity on viscosity in capillary extrusion of polyethylene,J. Appl. Polym. Sci., 8,873-880 (1964). 6 4 . Nielson, L. E.,Polymer Rheology,Marcel Dekker, New York (1977), p. 90. 65. McAllister, R.A., The viscosity of liquid mixtures,AIChE J., 6, 427-431 (1960). 66. Takayanagi, M., Kobunshi, 10, 285 (1961). 67. Han, C. D, and Yu, T. C., Rheological propertiesof molten polymers II.W Ophase systems, J. Appl. Polym. Sci., 15, 1163-1180 (1971). 68. Han, C. D., Measurement of the rheological properties of polymer melts with slit rheometer II. Blend systems, J. Appl. Polym. Sci., 15, 2579-2589 (1971). 12, 81 (1972). 69. Han, C.D. and Yu, T. C., Polym. Eng. 70. Thomas, D. G.,Transient characteristics of suspensionVIII. A noteon the viscosity of Newtonian suspension of uniform spherical particles, J. Colloid Sci., 20,267277 (1965). 71. Frankel, N. A. and Acrivos, A., On the viscosity of a concentrated suspension of solid spheres, Chem. Eng. Sci., 22, 847-853 (1967). 72. Van Oene, H., in Polymer B l d , Rheology Polymer Blends and Dispersion (D.R Paul and S. Newman, eds.), Academic Press,New York (1978) Vol. 1, pp. 296-352. 73. Plochocki,A. P., Polyolehs blendsrheology,meltmixingandapplications,in Polymer Blends (D.R. Paul and S. Newman, eds.), Academic Press, New York (1978), Vol. 2, pp. 319-368.

From Master Rheograms Processing to Parameters

311

Brenner, H., SuspensionRheology,in Progress inHeat Mass Transfer (W.R. Schowalter, ed.), Pergamon Press, Oxford Vol. Batchelor, G.K, Transport propertiesof two-phase materials with random structure, Annu. Rev. Fluid Mech., Jinescu, V. V., The rheology of suspension, Int. Chem. Eng., Jeffrey, D. J. and Acrivos, A., The rheological properties of suspension of rigid particles, AZChE J., Shenoy, A. V. and Saini, D.R., Interpretation of flow datafor multicomponent polymeric systems, Colloid Polym. Sci., Kulkami, M. G. and Mashelkar, R. A., A unified approach to transport phenomena Chem. Eng. in polymeric media,I. Diffusion in polymeric solutions, gels and melts, Sci., 11. Diffusion in solid structured polymers, Chem. Eng. Sci., Doolittle, A. K, Studies in Newtonian flow, 11. The dependence of viscosity of liquids on free space, J. Appl. Phys., III. The dependence of the viscosity of liquids on molecular weight and free space (in homogenous series), J. Appl. Phys., Fujita, H. and Kishimoto, J., Interpretation of viscosity data for concentrated polymer solutions, J. Chem. Phys., Menges, G., Wortbert, J., and Michaeli, W., Kunststofie, Alle, N. and Lyngaae-Jorgensen, J., Polypropylene and polyethylene blends, I. Flow behaviour in capillaries,Rheol. Acta, Martinez, C. B. and Williams, M. C., J. Rheol., 24, Kasajima, M., Bull. Coll. Eng. Hosei Univ., Carley, J. F. and Crossan, S. C., Viscosities of molten polymer blends,Polym. Eng. Sci., Bersted, B. H., Slee, J. D., and Richter, C. A., Prediction of rheological study of long branching in polyethylene by blending, J. Appl. Polym. Sci., Jacovic, M. S., Pollock, D., and Porter, R. S., A rheological studyof long branching in polyethylene by blending, J. Appl. Polym. Sci., Sieglaff, C. L., SPE Trans., Shah, P. L.,Encyclopedia of PVC (L. I. Nass, ed.), Marcel Dekker, New York Vol. Menges, G., Geisbusch, P., and Zingel, U.,Kunztstoffe, 485 Han, C. D., Rheological propertiesof calcium carbonate-filled polypropylene melts, J. Appl. Polym. Sci., Tanaka, H. and White, J. L., Experimental investigationsof shear and elongational flow properties of polystyrene melts reinforced with calcium carbonate, titanium dioxide and carbon black, Polym. Eng. Sci., Knutsson, B. A., White, J. L., and Abbas, K A., Rheological and extrusion characteristics of glass-fiber-reinforced polycarbonate, J. Appl. Polym. Sei., Wu, S., Order-disorder transitions in theextrusion of fiber-filledpoly(ethy1ene terephthalate) and blends, Polym. Eng. Sci.,

MFI Correlations with Other Parameters

Over the years, it has been found that MFI correlates well with a number of other useful parameters. For example, duringpolymer manufacture, it has been observed that the reaction temperature, the catalyst activation temperature, and molecular weight buildup can all be correlated with MFX Similarly, MFI relates well with certain rheological parameters such as die swell, melt strength, and breakage stretch ratio besides all the other rheological parameters discussed in Chapters and Further, it has been observed that in polymer processing operations, melt flow index (MFI) correlates well with mold-filling behavior during injection. molding, curing, and cross-linking behavior as well as degradation and stability of thepolymer.Similarly, the physical,mechanical,and thermal as well as certain optical properties of the finished products can be related well with MFI of the raw material. With all these enhanced abilities to predict a variety of properties, MFI has certainly been upgraded in value from its original belief that it is a mere quality control rheological parameter. The objective of this chapter is to put forth the possible existing correlations between MFI and various other important parameters encountered in polymer manufacture, polymer product fabrication, and polymer product property evaluation that have not been discussed in the preceding chapters.

9.1

MFI CORRELATIONS IN POLYMERMANUFACTURE

The possibility of manufacturing polymersdiffering from one anotherin density and MFI enables the manufacturers to prepare gradessuitable for specific fields 31

arameters Other Correlations with MFi

313

of application. The different grades are a result of controlled manipulations in the reaction conditions and reactants during the polymer synthesis stage leading to polymers with different molecular weights,different molecular-weight distribution, and different levels of branching. It will be shown in the following how MFI can becorrelated directly or indirectly with eachof the relevant parameters that it can act as a rather beneficial control parameter.

9.1.l Effect of Reaction Conditions onMFI During the polymer synthesis stage, the polymer manufacturer can optimize his reaction conditions to get a particular grade of choice through the use of MFI. For example, the reaction temperature as well asthe catalyst activation temperature can be seen to vary- significantly with MFI as shown in Figs. 9.1 and 9.2 taken from Hoganet al. [l]for polyethylene (PE). It can be seen that seemingly small temperature gainsare quite significant in terms of achieving higher polymer MFI. Similarly, increase in activation temperature sharplyincreases the MF'I increase potential to the point where the catalyst undergoes a sharp decline in pore volume becauseof sintering.

POLYETHYLENE

I

I

Figure 9.1 Effect of reaction temperature on polymer melt flow index. (From Ref. l;)

Chapter

314

Effect

Figure Ref. 1.)

9.1.2

catalystactivationtemperature

on melt flow index. (From

Effect of Weight-Average Molecular Weight and Branching on MFI

relationship between MFI and weight-average molecular weight of linear and branched polyethylene was exploredby Dark It was foundthat two families of curves resulted one for branched and other for the linear PE. Thus, the following relationships were given by Dark Branched P E log "I, =,

-

Linear PE: -log M&

=

lo-'@,

-

m,,

Even though the above expressionsrepresent the best fit of as given in Fig. the data for branched PE cannot be in good order, as it indicates a higher MFI for ahigher M,, which is incorrect. The data for thebranched PE as collected by Dark covered a broad range of branching, reactor configuration, and telogen level. eliminate the effect of these variables, data were taken that, but from a single autoclave configuration, where AT was held constant for H,,all other variables were nearly constant. Only the pressure drop across

31

MFI Correlationswith Other Parameters

-

I

POLY ETHYLENE

I

Figure 9.3 Variation of weight-average molecular weightof linear and branched polyethylene versus logarithmof melt flow index. (From Ref. 2.)

the autoclave wasvaried, and the effect the changes in pressure wasreflected in MFI and values as shown in Fig. It can thus be concluded that, with branched PE, consideration must be given to long-chain branching, nonrandom termination, and reactor profile, whereas for linear PE a simple expression like that given in Eq. (9.2) would suffice. Rokudai and Okada [3] made a detailed study of a number of low-density polyethylenes in order to develop relationships among MFI, density, and molecular parameters. Figure shows the variation weight-average molecular weight with MFI. It is seen that the curves are different for the dif€erent series. The A, B, and C series are different in density,but they are all from the autoclave reactor, whereas the T series, which has a density intermediate between the B and C series is exclusively from the tubular reactor. Thus, the T series differs radically in the level of branching from the A, B, and C series. It was found that in order to get a relationship between MFI and weight-average molecular weight, Rokudai and Okada [3] had to use the parameter g, which is defined as the ratio of root-mean-square radii the branched and linear polymers withthe

a,,,

Chapter

\ 9.4 Variationofweight-averagemolecularweightofbranchedpolyethylene versus logarithm of melt flow index when changes in the autoclave pressure are made during the polymerization. (From Ref. 2.)

same molecular weight. Thus, therelationshipsfor the autoclave andtubular grades of PE as obtained by Rokudai and Okada [3] are as follows and shown in Fig. 9.6.

31

MFl Correlations with Other Parameters

(grn/lOmin)

Figure Variation of weight-average molecular weight of LDPE versus melt flow index for autoclave and tubular grades of resin: (0)series; (n) B series, (A) C series, and T series. (From Ref.

x,

Rokudai and Okada also defined a parameter which represented the longchain branching frequency and attempted to get the correlation between it and MFI as shown in Fig. 9.7. The curves obtained had a scatter for the polymer from the tubular reactor which was no better than that in Fig. 9.5 for aw versus MFI. Thus, they developed a summation series to predict MFI from weightaverage molecular weight and long-chain branching frequency as follows: log MFI =

c

aij(log Bwy(logr;>i

ij=O

where the values of the coefficient are given in Table 9.1. A comparison of the predictions of Eq. (9.5) with actual determined values of MFI shows excellent agreement, as can be seen from Fig. 9.8. The relationship between MFI and branching parameters was studied by Pilati et al. Figure 9.9 shows a plot of MFI versus ?i?, on log-log scales. It is seen that, only for unbranched samples, an equation of the following form can be fitted: log

MFI, = 16.9 - 3.5 log

(9.6)

318

Chapter

”

Q

Figure Relationship between melt flow index and weight average molecularweight Using a correction factor, g: (0, 0,A) autoclave grades and tubular grades. (From Ref. 3.)

where M F I e is the melt flow index for linear samples in grams per 10 minutes obtained under a load of 5 kg at 250°C. For branched samples with different trifunctional unit content, it can be seen that the MFI is different for the same molecular weight and the relationship between MFI and also changes. Using weight-average degree of branching defined as B,,,, Pilati et al. [4] modified the equation suggested by Bates [5], namely,

in terms of MFI, knowing the inverse relationship between viscosity and melt flow index as given by Eq. (4.12). They thus made a plot of M F I e M F I , , versus 1 0.25Bwand obtained a straight line for branched samples of poly(buty1ene terephthalate) (PBT) as shown in Fig. 9.10.

+

31

MFI Correlationswith Other Parameters

Figure Relationship between long-chain branching frequency and melt flow index (symbols same as in Figs 9.5 and 9.6). (From Ref. 3.)

The relationship between MFI and weight-average molecular weight can be effectively used to check the structural arrangement of teleblock polymers such as butadiene-styrene as shown in Fig. 9.11 taken from Ref. By evaluating MFI and the correspondingmolecularweight, it would be easy to adjudge whether the polymerized species is linear, trichain, or tetrachain. At equal moTable 9.1 Coefficients in Eq. (9.5) for Polymers from 'hbular Reactor i = l nu j = O j=l j=2 Source: Ref.

i=O -0.3757 X 10' -0.1731 X 10' -0.1933 X 10'

-0.3643 X 10' -0.4464 10' -0.1579 X 10'

-0.5287 X 10" -0.2370 X 10' -0.1982 X 10"

Chapter 9

320

M Figure Comparison of the observed melt flow index with that predicted from EQ. (9.5) (symbols same as in Figs. 9.5 and 9.6). (From Ref. 3.)

lecular weight, viscosity is lower for tetrachain (or trichain) radial polymers than for linear polymers; at equal viscosity level, one can use a radial tetrachain polymer of higher molecular weight.

9.1.3

Effect of Number-Average Molecular Weight on MFI

Basically, MFI is known to relate better with weight-average molecularweight, however,Mortimer et al. [7] found excellent correlations betweennumberaverage molecular weight %, and MFI, as canbe seen from Fig. 9.12. The curve in Fig. 9.12 is described by the following equation:

-

M,,= 37,000 - 8610 log MFI

MFI Correlations with Other Parameters

"l

mo I TMT I DMT

0

c

\;

I 4.5

I

\, Log

I

l

5.5

Rw

Figure Experimental valuesof melt flow index at 250°C with a5-kgload condition versusweight-averagemolecularweight for unbranchedsamples PBT (0) and branched samples with different trifunctional unit content (A, 0, 0, A). (From Ref. 4.)

(

mol D M f

A

-

I

Log (l+0.25

Figure

Plots of modified Bate's Eq. (9.7): log(MFI,/MFIb)versus log(1 + 0.25 9.9). (From Ref. 4.)

E,,,) for branched samples of PBT (symbols same as in Fig.

Chapter 9

Figure 9.11 Variation of melt flow index with weight-average molecular weight for teleblock polymers butadienehtyrene. (From Ref. 6.)

a,,

which gives the with a standard deviation of +2200. The standarddeviation of points about this line is close to the standard deviation of osmotic measurement. The particular advantage in determininga,, by the above equation is that it is faster than the conventional method. In fact, Mortimer et al. [7] used for narrowit ona routine basis for over 2 yearsin orderto estimate distribution batch-type polymers. During this period, there was no change detected in the correlation. The line given by Mortimer et al. [7] differs from that of Sperati et al. [8] especially at higher molecular weight. The reason for this apparent discrepancy is not very clear. Although it is known that MFI is related rather than M,,[9-111, the above correlation given by Eq. (9.8) holds to M,,, because MJM,, was constant for the samples of Mortimer et al. [7]. Forbranched PE, using the following relationship between and MFI, Tomis [l21 has shownthat the value of is not toodifferent from that obtained

a,,

”

a,,

a,,

23 arameters Other MFI Correlations with

Figure 9.12 Variation of melt flow index with number-average molecular weight for narrow distribution high-pressure polyethylenes. (FromRef. 7.)

by Gel Permeation Chromatography(GPC) measurements, as can be seen from Table (M,)1n=

-

log MFI

n,,

It is seen that, in particular for PE grade FB the values of by GPC and by MFI agree very well. For other pairs of values, the M,, calculated from is slightly lower. Grades FA and RB have almost identical -Eq. _ MJM, values and contain some very high-molecular-weight fractions of the order of This particular fraction is missing in FB 1-28 and hence behaves differently in the rheological sense. If Eq. had closely predicted M,, for FA as well as RB it would have failed to predict the same for FB This is because any relationship between MFI and M,, would hold only when the molecular-weight distribution is not different between the grades of the particular polymer.

Chapter 9

324

Table 9.2 Comparison between 5 .Obtained Through MFI and GPC

Bralen Bralen Bralen FA7-15 FB1-28 Polyethylene

G,,by GPC

M. by MEI

30,000 26,440

RB 03-23 35,000 35,340

50,000 40,670

Source: Ref.

9.1.4 Effect of Molecular-Weight Distribution on MFI Shekhtmeister et al. [l31 have suggesteda method of determining the molecular) MFI using the values of MFI evaluated under weight distribution ( W from not less than two different load conditions. The selection of right loads is of prime importance that the determined MFIs are different to a maximum extent and yet should be within the measurable limits for a reliable correlation with MWD. Shekhtmeister et al. [13], based on extensive data analysis, recommend the following combinationsof loads: region I, 5.0/1.2; region 11,21.6/0.325;and region 111, 21.6/0.325. Region I corresponds practically to all injection-molding grades and some extrusion grades of high-density polyethylene (HDPE). They expressed the interrelationship between MFI and MJMn as follows: "

(9.10)

where M F 1 5 . 0 and respectively.

9.1.5 Effect

MFI1.*

represent the MFI values at 5.0- and 1.2-kg loads,

of Solution viscosity on MFI

There is a close relationship between solution viscosity and MFI, as has been pointed out by a number of workers [7,14-161. Garcia-Borras [l41 states that whenever solution viscosity was required, an initial calibration plot was prepared between solution viscosity and MFI with results within 5% error, as shown in good that plotting only Fig. 9.13. Accuracy of such a plot was found to be three or four data points spaced by a few centipoise each hasbeen recommended ~41.

Figure 9.14 shows yet another correlation between [q]and MFI and is given for polyethylenes by the equation [q] = 1.064 - 0.183 log MFI

(9.11)

It can be seen that the above correlation suggested by Mortimer et al. [7] is consistent with the data of Moore [15]. The samples of Moore [l51 did not

MFI Correlations with Other Parameters

325

A

2.0 2.5

MFI(9 m / 1 0 m i n ) Figure 9.13 Plot of solution viscosity versus melt flow index at results within 5% better. prom Ref. 14.)

load with

contain a high-molecular-weight tail. It was observed by Mortimeret al. that variations in short-chain branching had no observable effect on this correlation. However, experimental points for samples containing long-chain branching lie to the left of the MFI-[q] line for a linear polymer in Fig.9.14. The relationship for long-chain branched samples were fitted by a straight line given as [q] = 0.966

- 0.189 log Mm

(9.12)

With a view to investigate the effect of branching parameters on solution viscosity, Pilati et al. [l61 extended their melt viscosity work [4] and studied the effect of branching parameters on intrinsic viscosity. Interestingly, the plot made by them of log[q] versus log Mm shows a nearly straight line, can be seen from Fig. 9.15, which is the same for branched and linear samples. Pilati et al. [l61 suggest that this behavior is probably because the effect of branching on melt viscosity gets partly balanced by a decrease in intrinsic viscosity. Combs and Nation [l71 studied 52 polyesters having variousmolecular structures anddeveloped relationships among Mm, glass-transition temperatures, melt temperatures, and the inherent viscosities. regression analysis was used for determining the constants of the following models investigated: lnMm=A

- 273) + B(T, (T - 273) + c l n t l )

(9.13)

Chapter

326

9.14 Variation of melt flow index with intrinsic tion high-pressure polyethylenes. (From Ref. 7.)

hMFI=A

+ -BT, + T

Ch{q}

B C hMFI=A+-+-+Dh{q} T T*

viscosity fornarrow

(9.14) (9.15) (9.16)

The values the constants for each the above equations are given in Ref. 17. Equations (9.13) and (9.14) arethe simplest models thatadequatelyfitted

MFI Correlations with Other Parameters

mol

,

A '

BS"

-

-

A

aA

-

I

1

-0.2

Log

'11

Figure Experimental values of melt flow index versus intrinsic viscosity for unbranched (0)and branched samples of PBT (A, 0, A). (From Ref.

the data. They differ only in that T, and T are expressed in degrees Celsius for (9.13), whereas they are in degrees Kelvin for (9.14). Equation (9.15) has a more theoretical backing because it is of the Arrhenius form and hence would be preferred. Equation (9.15) is the analytical form implied by the graphical procedures to relate corrected MFI with T, by Gray et al. [18]. Equation (9.16) is a generalized form of the Williams, Landel, and Ferry (WLV equation, which hasbeen successfully used to describe the rheological properties of amorphous polymers within 100°Cof their TBvalues. Any of the equations could be used to predict the MFI an unknown polyester if its T, and {q} are known. The measurements T, and {q} require <0.3 g of material, whereas an MFI measurement requires at least 5g. Combs andNation [l71 suggest thatthe equations given above could be used for predicting the MFI values of other polymers. However, this needs to bechecked. Nevertheless, the method is useful for oxygen-sensitive and moisture-sensitive polymers whoseMFI determination might involve a certain degree of error.

Chapter

328

9.2 MFI CORRELATIONS IN POLYMERPRODUCT

FABRICATION During polymer product fabrication, the only flow parameter to which the processor has ready access is the MFI.It has been shownin the preceding chapters how various rheological and processing parameters canbe obtained through knowledge of the MFI. In this section, correlations between MFI and some more rheological and processing parameters are given.

9.2.1

Die Swell with MFI

Nakajima [l91 has shownhow die swell could be predicted through knowledge ofthe MFI. The salient features of his approach are follows. He useda material function, namely, the time constant A defined as

where is the shear rate in the capillary rheometer and related to die swell as follows:

is the strain which is

(9.18) where S, is the die swell defined as the ratio of the extrudate diameter to the die diameter after the extrudate completes recovery. If the time constant A is known, then S, can be calculated. For the linear PE samples analyzed with MFI values of 0.4-9, Nakajima [l91 foundthat the time constant varied inversely as the power of the shear rate: A = k;P'

(9.19)

For convenience, in order to eliminate the proportionality factor, Eq. (9.19) was written as (9.20) where X,, is the value of A at 9 = 501s. Further, A,, and p' were expressed as A,, = A,(MFI)-''

(9.21)

p' = B,(MFI)-"

(9.22)

The values of A,, q', B,, and r' were determined from the time constant versus shear rate curves [19]. Using Eqs. (9.17)-(9.22), the die swell could be calculated for a given sample of knownMFI.The valuesof die swell thus calculated for over 100 observed

Parameters Other MFI Correlations with

329

data showed a variation from the actual within 23.5% at confidence level. However the above cannot be taken as a general method to correlate MFI with die swell. It is known that die swell is sensitive to molecular parameters such as molecular-weight distribution, branching, and forth, whereas MFI is not. Hence, the present method needs to be extended to incorporate these parameters on the lines done [20] for upgrading the melt flow index to rheogram approach in the low-shear region. The introduction of plasticizers, naphthenic and paraffm oil as well as paraffin wax into polystyrene (PS) and high-impact polystyrene (HIPS) is known to reduce itseffective viscosity while improvingits elastic properties as assessed from the degree of swelling, Actually, the reduction in viscosity that occurs upon plasticization would lead to the easier movements of conglomerates of chains and, consequently, to an acceleration of the process of transition of PS macromolecules and the PS matrix of HIPS to a forced high elastic state [21]. The display of higher elasticity by the polymer according to the rate of movement of the macromolecules is confirmed by the existence of a correlation between and MFI which is invariant in relation to the plasticizers despite different plasticizing capacity as shown in Fig.9.16. Duringmechanochemical mixing of HIPS containing 5% and 9% polybutadiene rubber with a thermoplastic rubber taken in a quantity of 5%, lo%, and 15%, a similar relationship between and MFI is observed, as can be seen from Fig. 9.17, for reasons akin to those mentioned above. Melt flow index recovery (MFIR) is the term given to a percentage increase in the diameter of the extrudate over the diameter of the orifice, that is, loo(&D,)/D,, where D, is the orifice diameter. In order to understand the effect of MWD and branching on the rheological parameters of PE melts, Guillet et al. [22] correlated MFIR with the ratio [q]9s/[q]so, as shown in Fig. 9.18, where [qIg5 and [q]50are the intrinsic viscosities at Z wt% of 95 and from the distribution curves. It is clear that the MFIR decreases asthe distribution broadens, because the ratio [q]95/[q]50 is truly proportional to the breadth of the molecular-weight distribution and is particularly sensitive to the high end of the distribution. Guillet et al. [22] also found that there was a strong relation between critical shear rate at which roughness on the extruded sample first occurs and melt recovery as shown in Fig. 9.19. It shows the regression line obtained by analysis of critical-shear-rate measurements on 38 samples of PE. Romanini et al. [23] also studied the dependence of the swelling ratio p. (after annealing of PE samples) to the MFI (taken under identical conditions of load and temperature) with changes in the molecular weight as shown in Fig. 9.20. The numbers in Fig. 9.20 corresponds to the sample numbers given in Table which outlines the main chemicophysical characteristics of the polymers used; samples 1-5 were obtained in a tubular reactor under specific reaction conditions, whereas samples6-14 were produced in an autoclavedivided

Chapter

330

---POLYSTYRENE

2.1

HIGH

POLYSTYRENE

2.' ( gm

mln)

9.16 Correlation between the degree of swelling of the melt and the MFI for PS (- - -) and HIPS (-) containing different plasticizers:(0) butyl ester, naphthanic compressor (A) mixture of paraffins; (U)medicinal paraffin )(. initial polymer. (From Ref. 21.)

into conveniently arrangedmultiple-reactionchambers. The mainaimof the study of Romanini et al. was to get a correlation between the rheological parameters and the molecular structure of PE. The ratio P,/MFI in Fig. 9.20 may be associated with the balance between the elastic and viscous component of the material. The molecular weight being the same, long-chain branching causes a shift of this ratio toward lower values, corresponding to a decrease in the recoverable elastic deformation and/or to an increase in the material flowability.

9.2.2

Melt Strength with MFI

When a polymer is extruded from the of the melt flowindexer, the diameter of the extrudate changes due to the stress applied to the melt by the weight of the material that has preceded it. Bymeasuring the change in diameter of the extruded cord over the first 1.27 cm of length, a melt viscosity at extremely low shear rates is calculated andtermed as the melt strength because it indicates how well the melt can support its weight. Guillet et al. [22] suggested the following equation to calculate the melt strength based on

MFI Correlations with Other Parameters

Correlation between the degree of swelling of the melt and MFIfor HIPS Figure containing 5% (-) and (- - -) rubberand 5% (O), 18% and 15% (A) thermoplastic.

their experience:

MS =

3.584 X 105(At)*r;

MFI

(9.23)

where A t is the length of the cord (cm) for a 50% decrease in diameter and r,, is the radius (cm)of the cord as it first emerges from the die (measured by extrapolation of measurements at 0.16, 0.64, and 1.27 cm). The above equation suggested by Guillet et al. [22] was later corrected by Combs et al. [24], as a mistake was noted in the conversion of the units used in the equation. The actual data reported by Guillet et al. [22] were correct, but the equation was different by a factor 41.62 or (2.54)4. The correct equation, therefore, relating MS and MFI is as follows:

MS =

1.492 X lO’(At)*r;

MFI

(9.24)

It was assumed that the melt strength expressed above was equivalent to the melt viscosity at low shear rates. A number of other assumptions were made

Chapter 9

332 eo POLYETHYLENE

+

MFIR

-

-z E L

I

/cnlso Figure Correlationbetweenthemelt indexrecoveryandratio of intrinsic viscosities of of 95 and 50 from the distribution curves. (From Ref. 22.)

9k

I I l IIIII

I lI

I I I to‘

loz CSR,SEC?

Figure Correlation between the critical shear rate at which roughnesson extruded sample first occurs versus the melt index recovery. (From Ref. 22.)

MFi Correlations with Other Parameters

Figure Dependence of the swelling ratio/melt flow index parameter on the molecularweightfordifferentannealed PE samples;symbolsareas given in Table 9.3, (From Ref. 23.)

during the derivation of Eq. (9.24) which are not truly correct and accurate [24]. It was found that the above equation showed a 37-fold difference in the melt viscosities of polyethylenes having similar melt indexes. The reason for this is simply because " Iis insensitive to molecular-weight distribution and hence a normalizing factor of the kind used [20] would be required for more accurate predictions.

Chapter

334 Table 9.3 Speciscations

the Polyethylenes Density

Synthesis atSample process no.

23°C

(g per IOmin)

'hbular Tubular Tubular Modified tubular Modified tubular Vessel Vessel Vessel Vessel Modified vessel Modified vessel Modified vessel Modified vessel Modified vessel Linear polyethylene Linear polyethylene Linear polyethylene Linear polyethylene

1 2 3 4 5 6 7

8 9 10

11 12 13 14

15 16 17 18 ~

m

0.9176 0.9172 0.9176 0.9205 0.9234 0.9128 0.9144 0.9156 0.9176 0.9198 0.9204 0.9211 0.9207 0.9217

2.14 1.57 0.24 3.12 2.13 84.0 22.4 6.58 1.19 1.75 1.69 0.25 0.24 0.24

0.9780

15.1

qo

lo+

at 180°C (Poise) 13 18 130 8.5 12 0.16 0.95 4 18 10 16

110 130 80 0.78

G,,,X

"

10-~

280 360 550 120 135 210 340 420 690 140 185 275 330 600

5.4 5.7 5.7 4.9 4.1 6.6 8.8 9.6 9.4 4.5 5.5 4.7 5.8 6.6

52

3.0

0.9500

0.98

22

110

5.3

0.9445

0.21

160

140

8.0

0.9440

0.04

1200

220

9.1

~~

Source: Ref.

Figure 9.21 shows a plot of melt strength determined at 180°C versus MFI for a number of PE samples described in Table 9.3 as studied by Romanini et al. [23]. It can be seen that long-chain branching causes an increase in melt strength.

9.2.3 Breakage Stretch Ratio with The breakage stretch ratio (BSR) gives a measure of the stretchability of the polymer. The behavior of the polymer in extension is important in certain processing operations like fiber spinning and film blowing. Basically, breakage stretch ratio is the ability of the polymer extruded from a die to withstand breakage when stretched at a particular rate on drawing roll system. Figure 9.22 shows a plot of breakage stretch ratio determined at 19OOC

with orrelations MFI

335

too

l$

op

MFl(gm/lOmin)

Figure 9.21 Correlation between melt strength and melt flow index for different annealed PE samples; symbols are as given in Table (From Ref. 23.)

versus MFI for the polyethylenes specified in Table 9.3 as studied by Romanini et al. [23]. It can be seen that long-chain branching causes a decrease in BSR for the same MFL. The same behavior was observed earlier by Busse during the study of melt strength and elasticity and their relationship to the mechanical structures of the polymers.

9.2.4

Spiral Flow Length with MFI

Cipriani and Trischman [26] have shown how the injection-molding behavior of a polymer can be assessed via MFI during the product development and improvement stages. In their efforts to develop a new class polypropylenes, they compared the behavior of a standard polymer with that the developed Polymer A with a broader weight distribution. It was observed that Polymer A had greater melt fluidity than the standard polymer at shear rates found in a commercial injection-molding operation asis brought out bythe spiral melt flow data given in Figs. 9.23-9.25. Spiral flow length is a good indicator of the ease

336

Chapter 9

Figure 9.22 Correlationbetweenbreakagestretchratioandmelt indexfor different annealed PE samples; symbols are as given in Table 9.3. (From Ref. 23.)

of mold filling, and it was devised as a quick and simple test by IC1 to assess the mold-filling properties of materials [27]. From Fig. 9.23 it is evident that under the same melt processing conditions, Polymer flows easier than the standard polymer. For example, a 10 MFI polymer A processes similarly to a 14 MFI standard polymer, other conditions being equal. It would, therefore, be possible to mold Polymer A at about 15-20°C lower temperature (Fig. 9.24) or, for the same temperature under corresponding pressure(Fig.9.25),than the standard material of the same MFI. The broader molecular-weight distribution of Polymer truly accounted for the difference in the rheological behavior of the two groups of polypropylene PP).

MFI Correlations with Other Parameters

337

Figure 9.23 Effect of melt flow index on the flow distance in spiral cavity mold for Polymer A and standard polymer at molding temperature 205°C and pressure of 10’ dyn/cm2. (From Ref. 26.)

= l2

338

Chapter

=

MOLDING

PRESSURE(x107 wNES/CM2)

Effect of molding pressure on the flow distance in spiral cavity mold for different melt flow index Polymer and standard polymer at molding temperature of (From Ref.

9.2.5

Mooney Valuewith MFI

Kandyrin et al. [28] found an interesting correlation between Mooney value and MFI of rubber mixes which would help in monitoring injection-moldingoperations. The MFI was determined at 130°C under 21.6-kg load condition with a capillary of diameter equal to 0.2 cm and length equal to 1 cm.The values MFI obtained by them were in the range 0.08-75 g per 10 min. It was observed that the relationship was a power-law type such that Mooney value = K(1/ MFI)”, where the value n corresponded to the normal Ostwald-de Waele power-law index in the shear stress-shear rate relation. Thus, a plot of Mooney value ML versus log(l/MFI) resulted in a straight line as shown in Fig. 9.26 within limits 3 decades of MFI values. It follows that the MFI characterizes the viscosity of rubbers and mixes with greater accuracy than the Mooney values. Thus, the MFI value may be used for estimating the processing behavior mixes and the viscosity properties mixes in working out formulations, and also to effect operational monitoring the injection-molding properties of the mixes under production conditions long as the shear stresses do not exceed 2 X lo6 dyn/cm2. Beyond this value shear stress, the value n alters considerably and, hence, it might become necessary to determine the MFI of the mixes under different load conditions.

MFi Correlations with Other Parameters

ro XES

339

/

Figure 9.26 Relationship between the Mooney value and melt flow index of rubber mixes. (From Ref. 28.)

9.2.6

Energy for Injection into Mold with MFI

In injection molding, controlled quality molding is dependent on process conditions, like temperatures, pressures, and cycle times as well as on the rheological properties ofthe molding compound.Higher MFI polymers are easily molded with a conventional ram-type injection-molding machine. As the MFI decreases, the molding of the material becomes moredifficult. Some of the difficulties can be overcomeby injection pressure and enlarging the diameter of the gates. However, this would often lead to degradation of the polymers, sink marks, and distortions of the product. With a view to look for a good control parameter for maintaining the quality of molding, Knappe [29] studied the relationship between the energy for injection into mold W, with the melt flow index for PP. From Fig. 9.27 it is evident that with a higher Mm, the injection energy decreases and the pressure in the filled mold increases because the pressure drop in the nozzle and gates becomes lower. It was found by Knappe [29] that a plot of W, versus the reciprocal of MFI gave a linear correlation with some deviation for the lowest MFI. In contrast to MFI, which is measured under constant shear-stress conditions, W, is caused by a viscosity at constant shear gradient; yet the linear relationship between W, and MFI” is surprisingly good.

Chapter

1s BESO

Pmloeo

]

230/50

9.27 Variation

injection energy with melt flow index for PP.(From Ref. 29.)

9.2.7 CuringandCross-Linking Certain polymers have the ability to undergo “change” upon heating or irradiation resulting in enhanced properties. l h o of the common polymers, wherein this “change” phenomenon often referred to as curingor cross-linking has been drawn to advantage, are polyphenylene sulfide (PPS) and ,PE. Brady conveniently monitored the progress of the curing reaction of PPS by following changes in the MFI. As curing proceeds, the molecular weight increases and does the melt viscosity, evidenced by a decrease in MFI from an initial high value as shown in Fig. MFI truly acts like a fingerprint of the degree of cure because the higher the degree of cure, the lower is the value of MFI. The effect of electron irradiation on MFI of PE was investigated by Harper It was observed that for a number of different types of polyethylenes, the MFI changed uniformly with radiation dose, as shown in Fig. and that this change could be represented by the following equation:

-

D=-

+ b x- - - bx -x ’ + “ z o g ~ I

k

-x

x

where B is the radiation dose in megaradians E, and b are constants, and = (MFI, - MFI,)/MFI,represents the degree of change in the melt flow index. MFI, is the initial melt flow index and MFI, is the melt flow index at dose 0.

MFI Correlations wlth Other Parameters POLYPHENYLENESULFIDE

20 TLME

(H)

Flgure 9.28 Curing rate of PPS asmonitored Ref. 30.)

through melt

index. (From

-.

LOW DENSITY POLYETHYLENE

DOSE

Figure 9.29 mica1 curveofmelt pressure polyethylene. (From Ref. 31.)

indexversusdose

of irradiationfor high-

Chapter

Although only a single equation as defined above fitted all types of polyethylene samples, the shape of these curves were affected by the properties of the PE such that the initial slope of the curve of (MFI,. - MFI,)/MFI, versus dose as shown in Fig.9.30 is proportional to either the molecularweight or the crystallinity of the high-pressure PE. Due to the approximate inverse relationship of the molecular weight and crystallinity, a firm conclusion with regard to the controlling factor was not reached.

9.2.8

DegradationandStability

During processingof polymers throughextrusion, injection molding, blow molding,and other conversion processes, the material is exposed to thermal and oxidative as well as shear degradation, causing molecular scission and hence a reduction in molecular weight. The decrease in molecular weight is generally characterized through changes in MFI of the polymer[33-361. Figure 9.31 shows the changes in MFI that occur due to repeated extrusions of the material. Processing conditions for the virgin and recycled material would be different in accordance with changes in the melt rheology. Curves like those in Fig. 9.31 are, undoubtedly,indicative of the changes in molecularstructure; however, what is required is the entire viscous and elastic response with shear rate and temperature in order to assess the processibility. This has been provided by Shenoy

A SOOM-+

A

E-

LOW DENSITY POLYETHYLENE

2 DOSE

(h4FI,

IN

9.30 Variation of - MFI,)/(MFI, with irradiation dose for nine types high-pressure polyethylenes. (From Ref. 31.)

of

sr Correlations with MFI

343

It

l

I

I

POLYPROPYLENE gm

I

UNITS -

Lb

3-

I

I

I

I3

3

NUMBER OF PROCESSING CYCLES

et al.

and discussed in the preceding chapters (see Secs.

and

It is known that PP has a great tendency to decompose during melting and at elevated temperature. Hence, all commercially available PP grades contain thermostabilizers. Even then it is advisable to maintain the processing temperature in the metering zone of extruders as low as possible in order to minimize losses in the mechanical properties of the end product due to thermal decomposition of the polymer. The extent of degradation canbe easily checked by the amount of rise in MFI, which increases at more elevated extrusion temperatures. Examples of the increase of MFI in correlation with the extrusion temperature for various isotactic PP grades stabilized to a different degree are given in Fig. taken from Ref. Bagheri et al. studied the melt stabilization of PP by galvinoxyl at three different temperatures and for various periods of mixing under three sets of conditions: (a) open to the atmosphere, (b) mixer closed, and (c) mixer closed after purging with nitrogen. In each case, the effect of galvinoxyl on the melt stability of PP was tracked by observing the changes in MFI with processing time as shown in Fig. for the open mixer. It was observed

Chapter

POLYPROPYLENE

9 -

-

7 -

.

0

2:o

220 2;o

2230 b0

2 : o

240 21io

2;o

280 250 2A0 2bo

Figure 9.32 Effect extrusiontemperature on thefilmmelt different melt stability. (From Ref. 39.)

290 260

index for PP

that, in the presence of oxygen, the additive was less effective. There was no real induction period and the additive was less effective at than at Whereas, in the open mixer; oxidation was effectively inhibited for up to 10 min, it was observed that, ina closed mixer for the same concentration of galvinoxyl and the same temperature, the PP was stabilized for 20 min of processing time. The end of the MFI induction period actually corresponds to the commencement of an irreversible removal of hydrogalvinoxyl from the system. Zavadsky et al. studied the melt viscosity of PP and its dependence on temperature, shear, time of shear,andmolecularweight. A master curve of viscosity versus shearrate in the of q/aTas a function of aTqwas attempted

MFi Correlations with Other Parameters

/"

345

I

l5

5

TIME

Figure 9.33 Effect ofgalvinoxylonthemelt flow index of PP in an openmixer; numbers on the curves are concentration (lo4 mol per g) with temperature in parentheses. (From Ref.

for all temperatures with a temperature with shift factor aT defined as

where E is the activation energy, R the universal gas constant, To the absolute reference temperature, and T the absolute temperature of measurement. It was however, found that the data for temperatures lower than lie on a single curve, whereas the data for temperatures and higher did not. This was due to the degradation of the polymer. Nevertheless, a master curve incorporating even the degradedpolymer data could be obtained by using a modified shift factor defined as the ratio of the temperature shift factor and The dependence of the melt the melt flow index MFI as shown in Fig. index on the temperature t,,, and the residence time t, as shown in Figs. and was determined in the form MFI =

+

+

The relationship was found to fit experimental data rather well, as can be seen from Fig. Most of the flow curves were fitted by a power-law model of the following type by Zavadsky et al. as to be suitable for calculations of transport

Chapter 9

346

,A

%+o

,-

A

I

\“

*

A t

L

-

;*A

x *

P

A

ffA

Oll

l

S

Lmlfa,/MFI)

Figure 9.34 Shear viscosity ratefor PP plottedas MFI). (From Ref. 41.)

as a function of ln(ja,/

4

.-

E

z. G

I

I

I

I

L (“C)

Figure 9.35 Effect of temperatureandresidencetimeduringextrusion melt index. (From Ref. 41.)

of PP onthe

Correlations with MFi

347

POLYPROPYLENE

ro -

-

= 230°C

-

l+

I

OO

tc (SEC)

Figure 9.36 Effect of temperature and residence time during the extrusion of the melt flow index; (x) experimental data. (From Ref.

PP on

problems of polymer melt in pipes and extruders: q=

exp(-k't&""

The variation of k', n, m, and q with " Ifor various levels of shear rate are given in Figs. The influence of the shear rate and the melt flow index on the parameters k and n is distinct but diminishes withincreasing values of both quantities. There is a strong influence of melt flow index on the parameter m,,but the effect of shear rate on it is rather negligible. The hydrolicstability of glass-fiber-reinforced PBT, PET and PC was studied by Kelleher et al. following the changes in MFI during the process of hydrolytic degradation. Their entire approach was similar to that used earlier for reinforced PBT and PC Because MFI is related to the degree of polymerization, the following relationship was developed to study the rate of hydrolysis:

M :F IM 1: F I

kt=ii(---) 1

348

Chapter 9

1SEC'I

-

-

IOS'."

sec"

*

v*3OOSE&

a

qSo%

I

20

30 min 1

Figure9.37 Effect of shear rate and melt flow index on the parameter k' of the powerlaw model given in Eq. (9.28). (From Ref. 41.)

POLYPROPYLENE :

SEC-'

= l 0 SEC-'

p

=

Figure 9.38 Effect of shear rate and melt flow index on the parameter n of the powerlaw model given in Eq. (9.28). (From Ref. 41.)

MFI Correlations with Other Parameters

POLYPROPYLENE

"8

2

-

zi

loqooo

-

3 t=

P

SEC1

IO

MFI

Figure9.39 Effect of shear rate and melt flow indexon the parameter m, of the powerlaw model given in Eq. (9.28). (From Ref.

where k is the rate in terms of percent scissions per unit time t, MFI, is the MFI at time t, MMo is the initial value, is a constant which includes the monomer molecular weight, the polydispersity of the polymer, and the value of the intercept at the curve, and m is the slope of axis of the log MFI versus log this curve. A plot of versus hydrolysis time was used to calculate the reaction rates as shown in Fig. for glass-reinforced PBT, whereas a plot of M F I " 2 3 versus hydrolysis time as shown in Fig. was used for glassreinforced PET. The value of the exponent is the value of the slope of the curve in accordance with the mathematical treatment given by Pryde et al. It can be seen that the reaction rates decrease with decreasing temperature in both cases, whereas PET hydrolyzedat a faster rate than PBT. possible explanation for this behavior is that, for a given PET has more ester groups than PBT. Although MFI was foundto be considerably sensitive to hydrolytic degradation, the same was not the case for tensile properties. It was found that, for unfilled PBT, there was a fundamental correlation between MFI and elongation that the final MFI could be equated to a value of elongation which was considered a failure point. also it was found that the tensile strength of unfilled PET almost changes linearly with MFI. However, no such adequate correlation was seen to exist for the reinforced systems. This is because MFI is not sensitive to the polymer-filler bond and cannot predict strength properties due to the uncertainties at the interface. Consequently, MFI cannot be used to define a failure criterion for filled systems; instead, to predict serviceability of filled polymers, one must rely primarily on actual mechanical property data. Generally, rheological experiments require a stability test, which makessure that

MF'I'=

Chapter

350

POLYPROPYLENE

1woo

-

-E

-

the rheological properties do not change during a rheological test. The duration ofmost rheological tests is much longer thanthe average residence timein processing equipmentsand, hence, thermalstability tests do assume considerable importance. Winter [45] has presented the results of the thermal stability tests on low-density polyethylene (LDPE) samples at by measuring the MFI kg) at different residence times. Thethermal stability at seemed to be good for the two LDPE samples up to 200 min, whereasfor larger residence times, the MFI decreased rapidly, as can be seen from Fig. 9.43. Thus, MFI can be considered to be asufficiently sensitive material function when monitoring material changes with time. Prolonged storage of polymeric materials may lead to deterioration in properties as a result of aging. This point was investigated for polyethylenes by Neverov and Vasil'eva [46]. It was observed bythem that mechanical properties such as tensile strength at yield, tensile strength and elongation at break, the

MFI Correlations with Other Parameters

351

Figure Variationofmelt flow index with hydrolysistime for 30% glass-fiberreinforced PBT pellets at relative humidity. prom Ref.

dielectric constant, and the electrical strength changed very little over a storage periodof years ofunstabilizedand stabilized LDPE. However,themost marked change during storage of the polymer occurred in the MFI of the material. It can be seen from Fig. 9.44 that for the unstabilized material there was a monotonic, practically linear change in Mm with time even up to a storage period of 17 years. For the stabilized material, only a slight reduction in MFI is observed during the early stages of storage, followed by a slight increase and then a decrease after years. Changes of this type in MFI can be attributed to the process of oxidative cross-linking.

9.3

MFI CORRELATIONS IN POLYMERPRODUCT PROPERTY EVALUATION

Despite the fact that MFI is generally considered as a measure of the rheological behavior and processability of the polymers, it has been shown over the years to correlate with final product properties. A general outline of the effect of MFI

Chapter 9

xlO"

/H

Figure Variation of melt flow index with hydrolysis time for reinforced PET at relative humidity. (From Ref.

30% glass-fiber-

increase can be seen fromTable which enlists most the common physical properties of the end products In the following, a detailed discussion the response product properties upon changes in MFI will be done.

9.3.1 TensileStrength The tensile strength at yield as well as the ultimate tensile strength at break have a reasonably strong dependence on MFI. The changes in the tensile yield strength with MFI are shown in Fig. It can be seen that, at lower MFI values, the changes are sharper, but eventually the increase proceeds in a linear manner. When plotted against temperature, the tensile yield strength of polymers with two MFI values (which are an order magnitude different

MFI Correlations with Other Parameters

353

Figure Stability test: Variation of the melt flow index at and 2.16-kg test load condition with the residence time of the polymer in the apparatus. (From Ref. 45.)

II

LOW DENSITY POLYETHYLENE I

I

TIME ( Y E A R S )

Figure Relative variation in the melt flow index for nonstabilized LDPE (curve 1) and stabilized polymer (curve 2) with time of storage. (From Ref. 46.)

Chapter

354 Table 9.4 Product Properties Affected by an Increase Melt Index

Degreeproperty Physical

and

Tensile Yield Rigiditylstiffness Toughness Modulus of elasticity Creep Hardness Impact Thermal to Resistance temperatures Solubility Permeability Resistance environmental to stress cracking Gloss Transparency Note:Pronouncedincrease = + +, pronounceddecrease = increase = +, slight decrease = -, no change = x. Source: Refs. 47 and 48.

in the of effect

-

X "

+ + -_ ++ X

- -, slight

from each other) are practically the same at lower temperature and gradually merge as the temperature increases, as shown in Fig. The simultaneous effect of MFI and crystallinity on the yield stress can be seen fromFig. taken from Crespiand Ranalli who studied the behavior of PP with different isotactic indices. It is seen that, with the same isotactic index, that is the equivalent crystallizable part, lower MFI samples are less crystalline and hence show a lower yield stress, but a small variation of the isotactic index leads to a remarkable increase in the yield stress from samples with the same flowability or, in other words, similar MR. The tensile strength at break is almost entirely dependent on molecular weight and hence is sharply sensitive to changes in MFI. This is demonstrated by the plots in Figs. and For linear low-density polyethylene (LLDPE) films, the decrease in the ultimate tensile strengths in cross-direction and machine directions is approximately by the same amount for increasing values of MFI;however, for high densities while the cross-direction ultimate tensile strength increases, the machine direction strength decreases for the same MFI (Fig. In the case of ethylene-vinyl acetate (EVA) copolymers, too, the decrease in the tensile strength at break is also very sharp as can be seen from Fig.

MFI Correlations with Other Parameters I

I

l

355 I

l

I

-

z

,x

-

.

E U

U)

z

> I

t;

-

5

-

Y =! 30.3(I)

z W

I

I

I

I

I

I

3

1

Figure Variation oftensileyieldstrengthwithmeltflowindexas D412-61T, &min. (From Ref. 49.)

per ASTM

Figure Variation of tensile yield strength with temperature for different melt flow index samples as per ASTM D412-61T,5 cm/min. (From Ref. 49.)

Chapter

-h z

c

4

Variation of tensile yield stress with melt flow index and isotactic index of PP. (From Ref.

9.3.2

UltimateElongation

The effect of MFI on elongation at break is not as radical as in the case of tensile strength at break, as is evident from Fig. The elongation at break is seen to remain fairly constant with changing MFI at constant vinyl acetate content. At least, the ultimate elongation may increase slightly with increasing MFI, as can be seen from Fig. from Ref. for PP film tape stretching. In fact, ultimate elongation is more dependent on density than MFI. Thus, in polymers that are near or above the glass-transition temperature, the ultimate elongation generally decreases as crystallinity increases this is due to the decreasing mobility of the system. Whereas polymers of medium crystallinity (20-60%) will cold draw beyond the yield point, polymers of high crystallinity (70-90%) become brittle and break The yield point normally increases linearly with density in the case of both high and low MFI. As a general rule, the lower the MFI, the higher the stretch limit at the same density.

9.3.3 Tenacity The effect of MFI on tenacity of film tapers can be seen from Fig. taken from Ref. 53, which shows that increasing the MFI decreases the tenacity of

MFi Correlations with Other Parameters

357

!S

MFI (gm I

min)

Figure Variation of the ultimate tensile strength with melt films. (From Ref. 51.)

index for LLDPE

the film. During film processing, in order to achieve high tenacity in the film, the extrusion conditions must be adjusted that the compression and metering zones are maintained at low temperatures to avoid molecular orientation while the die is maintained at a high temperature to reducepreorientation of the film in the melt form as it would prevent maximum orientation during the stretching operation. The filmmeltflow index indicates that for achieving the highest tenacities, the MFI value has to be as low asis permissible, as can be seenfrom Figs. and taken from Refs. and respectively. The effect of draw ratio as well as extrusion temperature becomes evident fromthese figures.

358

Chapter 9

9.49 Variation of the tensile strength at break with melt flow index for EVA copolymers at constant vinyl acetate content. (From Ref. 52.)

9.3.4 Elastic Modulus and Flexural Stress The elastic modulus and flexural stress both serve as criteria for the rigidity of the material. In general, the elastic modulus is higher for lower MFI The flexural stress at maximum deflection also follows the same trend as the elastic moduli, although the values of flexural stress at maximum deflection are of the order of 10 times less than those for elastic moduli Flexural modulus or stiffness increases rapidly with increasing density but not with MFI. In fact, a polymer with a MFI of 2 will hardly be perceptibly stiffer than the one with a MFI of 20 if the densities are the same The curve of stiffness in flexure in Fig. is almosta straight line.Whenplottedagainsttemperature, the stiffness of PP with two MFI values (which are an order of magnitude different from each other) is almost the same, and, in fact, identical at higher temperatures, as can be seen from Fig. taken from Ref.

9.3.5ImpactStrength Impact strength could be regarded as the determination of flexural stress at a rapid rate of load increase. The impact strength is known to depend on both density as well as MFI It is highest for the lowest MFI material, but drops sharply as MFI increases and then fairly evens out at higher values as

MFI Correlations with Other Parameters

359

2

(gm

Figure Variation of ultimatepercentageelongation(at130°C) index (at 230"C, 2.16 kg) for PP. (From Ref.

with melt flow

shown in Fig. 9.56. Decreasing density also results in higher impact strength as can be seen from Fig. 9.57. Higher density implies higher crystallinity and expectedly lower impact strength because crystallinity reduces the mobility of the segments of the adjacent amorphous polymer phase. Thus, in order to avoid brittleness, the higher-density polymers must have a lower MFI [57]. Temperature changes affect lower-MFI materials more, as can be seen from Fig. 9.58 wherein the differences in the impact strength behavior with temperature between materials of MFI of 10 and 30 are minor. Figure 9.59 is a plot of melt flow index (under D1238 condition G: 5 kg load) versus impact strength (on injection-molded 3-mm notched bars at 23°C). The product is a crystalline PS containing varying amounts of styrene butadiene rubber(SBR). This plot has been suggested for use as a quality control check for polystyrene incomingraw material or those reextruded two or more times The estimated results from this plot were found by Garcia-

Chapter 9

360

j l l \ t-

e

M F I (gm

IO

mirr

Figure Variation of tenacity (at 130°C) with melt flow index (at for PP. (From Ref. 53.)

2.16 kg)

c D

2

(qm/lO

Figure Variation of tenacity with melt flow index at different draw ratios for PP film tapes. (From Ref. 53.)

MFI Correlations with Other Parameters

I

I

361

I

S

Figure 9.53 Variation of tenacity with melt flow index at different draw ratios and on the effect of extrusion temperaturefor PP film tapes. (From Ref. 39.)

Borras [l41 to be accurate within 10% whencompared with the test results obtained by two independent laboratories. It is not unrealistic to expect similar plots when other melt flow index conditions and different impact bar sizes are used. Thus, MFI can beused as aneffective quality control parameter for impact strength. The effect of applying load very rapidly permits measurementof the strength property related to impulses such as impact strength as discussed above. The work performed in effecting fracture has proved in practice to be a goodmeasure of the impact resistance of a material and is generally represented as the area under the stress-strain curve for a rapidly applied load. The amountof work that has to be performed is considerably affected by density and MFI [48].It decreases as the density and MFI increases, as can be seen from Fig. 9.60.

Chapter

362

Variation

stifhess with melt flow index as per ASTM D747-58T.(From

Ref. 49.)

9.3.6

BrittleTemperature

typical catastrophic mechanical failure is brittle breakage under high-speed impact. When a tough rigid polymer is cooled to lowerand lower temperatures, its impact strength tends to drop, and this decrease is often most marked near the glass-transition temperature. When a flexible or a rubbery polymeris cooled, it reaches a temperature at which it becomes inflexible and often brittle. It is that the brittle temperature is considerably higher than theglass-transition temperature because Tg actually gives the mobility of much smaller molecular segments than those involvedinembrittlement.With increasing molecular weight, the brittle temperature decreases because its increasing length of the molecules providesgreater mechanical strength. This effect can be seen by plotting brittle temperature versus MFI, as given in Fig. 9.61 for high-density polyethylene (HDPE) The brittle temperature depends to a great extent on the crystallinity. Crystallinity reduces the mobility of the segments and increases the temperature at which they have enough mobility to produce a glass transition. This results in

MFI Correlations with Other Parameters

363

Figure Variation of stiffness with temperature for two melt flow index samples as per ASTM D747-58T. (From Ref. 49.)

an increase in the brittle temperature (Fig. 9.62), whereas it is higher for a higher MFI [60].

9.3.7 TearStrength In film mechanical strength properties, the tear strength is of immense importance and is related to MFI and density as shown in Fig. 9.63. Elmendorf tear is inversely related to density as well as MFI. Cross-direction tear strength is higher due to low cross-direction orientation. In fact, orientation is more difficult to achieve in either direction with LLDPE than with conventional LDPEbecause of the absence of long-chain branching. Kendalland Sherliker [61,62] have shown that milling carbon black into LDPE results in a dramatic decrease in the tear strength of the system despite the fact that the polymer molecules are bound to the filler in a layer 2-3 nm

Chapter

364

MFI (gm I

min)

Figure Variation of Izodimpactstrength D256. (From Ref.

with melt flow index as per ASTM

thick However, the catastrophic failure is due to the weak secondinterface between the bound surface layer and the remainder of the polymer matrix. Kendall and Sherliker investigated the effectof molecular weight of the polymer in alleviating the brittleness of the filledsystem. They observedthatlowmolecular-weight polymers deteriorated badly in the presence of 10% volume fraction of the filler, whereas the high-molecular-weight materials remained tough even up to a volume fraction level. The transition in behavior due to the molecular weightof the polymer can be shown more obviously byplotting the tear strength at a particular volume fraction against MFI, as shown in Fig. The tear strengths on the graph have been normalized by the values of the unfilled polymer to remove the effect of molecular weight on polymer toughness and thereby isolate the influence molecular weight filler behavior. In Fig. it can be seen that the polymer becomes embrittled when filled with 10% filleraround MFI equal to 1. The nature of the curve and the transition is independent of the chemical typeof the filler as shownby the results for carbon black as well assilica of similar size. Figure contrasts the behavior of high-

365

MFI Correlations with Other Parameters

*C

Figure

Variation of Dart impact strength with melt flow index and density. (From

Ref. 51.)

and low-density polyethylene as wellas ethylene-vinyl acetate copolymer filled with 10% silica. Above MFI equal to both HDPE and LDPE show similar embrittlement. However, whereas filled LDPE is not toughened at high molecular weights, filled EVA shows slight reinforcement, filled HDPE shows considerable reinforcement. In the case of EVA, the transition to poorer properties occurs at such high melt flow indices that all production grades are more often than not likely to show good toughness when filled. The manner in which small amounts of colloidal filler destroy the cohesion of low-molecular-weight polyethylenes was thought by Kendall and Sherliker to be similar to the phenomenon of environmental stress cracking, in which minor amounts of detergents or alcohol produce inordinate embrittlement of polymers, especially of low molecular weight In order to check their hypothesis, a drop of isopropanol was placed at the tip of the tear, and the fall in force at constant speed was measured. The results were plotted as a function of

Chapter 9

RUBBER

\

A

- 20%

RUBBER

I

kJ

2

z 6'

l

IOmin 1

Figure 9.59 Variation of impact strength (0.3 cm notched bar at 23°C) with melt flow index (at 200"C, 5 kg) for crystalline PS. (From Ref. 14.)

MFI Correlations with Other Parameters

Figure Variationofworkdone density of LDPE. (From Ref.

Figure

in effectingfailurewithmelt

indexand

Variation of brittle temperature with melt flow index of HDPE. (From Ref.

Chapter 9

368

- 30

-I"

In

-l

m

(gmlcel

Figure 9.62 Variation of brittletemperaturewithdensity Ref. 60.)

and melt

index. (From

MFI as shown in Fig. 9.66 wherein the transition in the cracking behavior was found to closely resemble the colloidal filler effect.

9.3.8

EnvironmentalStressCracking

Environmental stress cracking ( E X ) is the name given to a phenomenon by which a polymer under high stresses may crack in contact with certain active environments such as detergents, fats, and silicone fluids. There are a number chemical and physicochemical effects involved in any given ESC phenomenon [64]. The susceptibility of a polymer to ESC decreasesrapidly as the MFI is decreased and, hence, can be tracked quite sensitively through the MFI test, as shown in Fig. 9.67 taken from Ref. 65 which studies the cracking of PE in weak organic detergents. Pelagatti and Baretta [65] concluded that water solu-

Correlations with MFi

LINEAR LOW DENSITYPOLYETHYLENE

'\

CROSS \,/DIRECTION

'

\

\

DENSITY

Figure 9.63 Ref. 51.)

Variation of Elmendorftear with densityandmelt

flow index.(From

Chapter

)

Flgure 9.64 Variation of tearstrengthwithmeltflowindexshowingthetransition between rough and embrittled behavior of LDPE. (From Ref.

l

1.2

.o

l0

l

1

9.65 Variation of tear strength with melt flow index showing the tough-brittle transition for LDPE, HDPE, and EVA copolymers at 10%volume fraction of silica. (From Ref. 61.)

371

MFI CorrelationswRh Other Parameters

Figure 9.66 Variation of tear strength with melt flow index showing the link between the effects of colloidal silicaand isopropanol on cracking of LDPE. (From Ref. 61.)

-

I

-

U

=

roo

l

oo

lo2 TO

I

l8

I

Figure 9.67 Effect of anhydrousdetergent and its watersolutionson cracking of PE of various melt flow indices. (From Ref. 65.)

Chapter

tions of detergents are more effective for highermolecularweight, whereas anhydrous detergents and small molecule organic liquids would attack preferentially lower molecular weight polymers. The effect of molecular weight in terms of long-term stress crack resistance is shown in Fig. 9.68 for LDPE [66] Experimental studies indicate that the decrease in the resistance to cracking with MFI is primarily due to extraction or leaching of the low-molecular-weight fraction [66]. The effect of crystallinity and MFI on environmental stress cracking is shown in Fig. 9.69 and found to vary inversely with these two parameters ~31. In practice, it is important not to use polymers of high MFI for applications in which they are severely stressed, especially in contact with active environments. Figure 9.70 shows an application in which a severe external stress is applied in service in contact with an active environment. In this case, it can be seen that a low MFI is essential and the use of the best possible molding conditions will not prevent a high-MFI polymer from cracking. Figure 9.71 shows

9.68 Effect Ref. 66.)

melt

index upon the

resistance

LDPE.(From

MFI Correlations with Other Parameters

373

l00

Figure Effect of crystallinity and melt cracking of LDPE. (From Ref. 58.)

index upon time of failure in stress

the same two polymers, again subjected to an active environment, but in an unstressed application. Both polymers were molded at a melt temperature of which was suitable for the polymer of high MFI but not for the low MFI. Consequently, the molding of low MFI failed, whereas that made from high MFI remained perfectly satisfactory This was because the material withlower Mm hadahigher level of frozen-in strain,thereby leadingto warpage. The resistance to chemicals of any polymer can also be determined by the swelling test. Swelling is related to a decrease in strength and deterioration of the properties. This can be easily checked through MFI determination using a plot of the type shown in Fig. 9.72 taken from Ref.

374

Chapter 9

4i

l

MFI

2

MFI

20

Figure 9.70 Illustration of the effectof severe external stress applied in serviceduring contact with an active environment. (From Ref. 57.)

MFI

2

MFI

20

Figure 9.71 Illustration of the effect of an active environment in the absence of stress during service. (From Ref. 57.)

MFI Correiatidns with Other Parameters

0 (gm/lOrnin 1

Figure (From Ref.

Swelling of PE of various melt flow indices in an active environment.

9.3.9 Thermal Effects Stress cracking as discussed above need not only be due to an active environmental. Often a thermal stress can lead to cracking. Heat shock failure is also seen to relate rather linearly with Mm, as given in Fig. taken from Ref. The effect of temperature is also important during film and fiber processing, for example, in determining thermoshrinkage. Low MFI enhances thermoshrinkage and there exists an optimum MFI wherein the percentage of shrinkage is minimal, as can be seen from Fig. (from Ref. for PP films. Polybutylene polymers’ slowcrystallization rates are helpful in forming hotmelt adhesives with long open times [67]. Besides the hot-melt viscosity, lap shear strength, T-peel,and open time, the shear adhesion failure temperature (SAFT) of the bonded substrate is an important property for evaluating the effectiveness of the adhesive. The polymer MFI plays an important part in determining the surface temperature at which shear adhesion failure could occur, as can be seen from Fig. The top curve represents the SAFT test results with a 0.5-kg load, whereas the bottom curve shows test results with a l-kg

376

Chapter 9

Figure 9.73 Effect of density and melt flow index upon thermal stress cracking of PE wire insulation at 21°C.(From Ref.

-.

I

POLYPROPYLENE

I

M

(

min

Figure 9.74 Variation ofpercentageshrinkage(at 230°C, 2.16 kg) for PP film tapes. (From Ref. 53.)

130°C) withmeltflowindex(at

K9

POLYBUTYLENE

K9

t

0

40 80 l - B - C o E 1l:O m e l t Index (9WIOmin D-1238Cond"E)

9.75 Relationship between surface temperature and melt flow index for a polymer blend in hot melt adhesive formulation. (From Ref. 67.)

9.76 Effect of molding temperature and melt flow index on the gloss of moldings. (From Ref. 57.)

377

Chapter 9

378

load. Some drop in service temperature is found as polymer MFI is increased. At the higher load, the magnitude of the drop is about over the 10-100 MFI range. At a heavier load (1 kg),theeffect is morepronounced-about 11°C drop over the same MFI range. Although the service temperature doesnot vary strongly as a function of MFI, Fig. 9.75 indicates that high-MFI polybutylene polymers would be preferable for preparing useful adhesive compositions.

Gloss and Clarity The gloss of a molded article is often assessed both visually and by measuring the light reflected from the surface of the moldings under standard conditions. The effect of MFI on gloss is very significant, as be seen from Fig. 9.76. The higher the MFI, the lower is the temperature at which high gloss moldings can be produced [57]. In the case of extruded films, too, gloss and clarity are directly related to MFI. Figure 9.77 shows the effect of MFI upon the gloss of extruded HDPE film. Thus, it be seenthat low molecular weightbasically favors high clarity and gloss [68]. It is also known that narrow molecular-weightdistribution favors

l

IS

Effect

melt

index upon the extruded HDPE film.(From Ref. 68.)

Parameters Other MFI Correlations with

379

high clarity,whereas a broaddistributionfavors high gloss. Thus, byproper balance of the molecular-weight distribution and MFI, the desired level of clarity and gloss can be achieved.

1. Hogan, J. P., Norwood, D. D., and Ayres, C. A., Phillips Petroleum Company Loop Reactor Polyethylene Technology,Appl. Polym. Sympos., 36, 49-60 (1981). 2. Dark, W. Determination of polyethylene melt index from GPC data, Liquid Chromatography of Polymers and Related Materials III (J. ed.), Marcel Dekker, New York (1981) 3. Rokudai, M. and Okada, T., Characterization of low-density polyethylenes and re-

lationships among melt index, density and molecular structural parameters. J. Rheol. Japan, 8, 154-160 (1980).

4. Pilati, F., Munari,

and Manareshi, P., Randomly branched poly@utylene terephthalate): Correlation between melt flow index and branching parameters, Matex

Chem., 7, 661-674 (1982). 5. Bates, T. W., The melt viscosity of branched polydisperse polymers, Eul: Polym. J., 8, 19-34 (1972). 6. Haws, J. R., Radial Block Thermoplastic Rubbers (R. D. Deanin, ed.), American Chemical Society, Washington, DC (1972), pp. 1-14. 7. Mortimer, G. Daues, G. W., and Hammer, W. F., Relationships between molec-

ular weight, solution viscosity and melt index for narrow distribution, high pressure polyethylenewholepolymers,fastdeterminations, J. Appl. Polym. Sci., 8, 839-847 (1964). 8. Sperati, C. Franta, W. A., and Starkweather, H. W., The molecular structure of polyethylene V. The effect of chain branching and molecular weight on physical properties, J. Am. Chem. 75, 6127-6133 (1953). 9. Busse, W. F.and Longworth, R., Effect of molecular weight distribution and branching on the viscosity of polyethylene melts, J. Polym. Sci., 58, 49-69 (1962). io. Schrieber, H. P. and Bagley, E. B., The Newtonian melt viscosity of polyethylene: An index of long-chain branching, J. Polym. Sci., 58, 29-48 (1962). 11. Peticolas, W. T.and Watkins, J. M., The Molecular structure polyethylene W. The melt viscosity and the effect of molecular weight and branching. J. Am.. Chem. SOC., 79,5083-5088 (1957). 12. Tomis, F., Problems of rheology in plastics processing, Int. Polym Sci. Technol., 7, T90-T92 (1980). 13. Shekhtmeister, I. E., Mnatsakanov, S. S., Nesterov, V. V., andBelen’kii, B. G.,

Determining the MMD and mass-average MM linear PE from the melt flow index, Int. Polym Sci. Technol., 5, T47-T48 (1978). 14. Garcia-Borras, T., Follow these steps for better PS quality control, Plastics Technol., 89-95 (June 1977). 15. Moore, L.D., Relations among melt viscosity, solution viscosity, molecular weight and long-chain branching in polyethylene, J. Polym. ScL, 36, 155-172 (1959).

380

Chapter

16. Pilati, F., Munari, A., and Manaresi, P., Viscosity of dilute solutions of randomly branched poly@utylene terephthalate),Mufez Chem., 7,649-660 (1982). 17. Combs, R. L. and Nation, R. G., Relationships among melt flow, glass transition temperature and inherent viscosity of thermoplastic polyesters, J. Polym. Sci., 30, 407-414 (1970). 18. Gray, T. F., Combs, R. L., Slonaker, D. F., and Wooten, V. C., Jr., Tech. Papers, in SPE Meeting (1967), Vol. 13, pp. 370-372. 19. Nakajima, N., Can die swell be predicted?Rheol. Acta, 13, 538-541 (1974). 20. Shenoy, A. V. and Saini, D. R., Upgrading the melt flow index to rheogram approach in the low shear rate region, J. Appl. Polym. Sci., 29, 1581-1593 (1984). 21. Nikiton, Yu. V., Belova, E. A., and Chudinov, P. B., Influence. of plasticisers and fillers on the viscoelastic behaviourof polymer melts, Int. Polym. Sci. Technol., 10, T59-T60 (1983). 22. Guillet, J. E., Combs, R. L., Slonaker, D. F., Weemes, D. E., and Coover, H. W., in rheological parameters Jr., Effect of molecular weight distribution and branching ofpolyethylenemelts,Part I. Unfractionatedpolymers, J. Appl. Polym. Sci., 8, 757-765 (1965). 23. Romanini, D., Savadori, A., and Gianotti, G., Long chain branching in low density polyethylene. 2. Rheological behaviour of the polymers, Polymer, 21, 1092-1101 (1980). 24. Combs, R. L., Slonaker, D. F., and Coover, H. W., Jr., Derivation and correction of polyethylene melt strength equation,J. Appl. Polym. Sci., 11, 747-750 (1967). 25. Busse, W. F., Mechanicalstructuresinpolymermelts I. Measurements of melt strength and elasticity, J. Polym. Sci., A-2, 5 , 1249-1259 (1967). 26. Cipriani, C. and Trishman, C. A., Jr., Polypropylene process and product improvements with new high yield catalyst, Appl. Polym. Sympos., 36, 101-112 (1981). 27. Griffiths, L., Spiral-flow moulding, compare the flow behaviour of thermoplastic materials under actual injection molding conditions, Mod. Plastics, 34, 111 (Aug. 12, 1957). 28. Kandyrin, L. B., Al’tzitser, V. S., Anfimov, B. N., and Kuleznev, V. N., Correlation of rubber between the apparent viscosity, Mooney viscosity and melt flow index Technol., 4, T53-T55 (1977). mixes, Int. Polym. on the injection 29. Knappe, W., Rheological measurements with technical polymers moulding machine, Rheol. Acta, 21, 478-480 (1982). 30. Brady, D. G., Poly(pheny1ene sulfide)-How, when, why, where and where now, Appl. Polym. Sympos., 36, 231-239 (1981). 31. Harper, B. G., The effect of radiation on the melt flow index of polyethylene. II. Effect of initial polyethylene properties, J. Appl. Polym. Sci., 5, 601-605 (1961). 32. Harper, B. G.,Theeffect of radiation on themeltflowindex of polyethylene, J. Appl. Polym. Sci., 2, 363-366 (1959). 33. Mitterhofer, F., Processing stability of polyolefins, Polym. Eng. Sci., 20, 692-695 (1980). 34. Rokadai, M., Mihara, S., and Fujiki, T., Influence of shearing history on the rhe-

ological properties and processability of branched polymers. 11. Optical properties of low-densitypolyethyleneblownfilms, J. Appl. Polym. Sci., 23,3289-3294 (1976).

Parameters Other MFI Correlations with

381

35. Rideal, G. R. and Padget, J. C., The thermal-mechanical degradation of high density polyethylene, J. Polym. Sci. Sympos. 57, 1-15 (1976). 36. Cuspor, I. and Toth, T., Study of the reprocessing of polypropylene, Int. Polytn. Sci. Technol., 7, T16-Tl9 (1980). 37. Shenoy, A. V., Saini, D. R., and Nadkami, V. M., Estimation of the melt rheology of polymer waste from melt flow index,Polymer, 24, 722-728 (1983). 38. Shenoy, A. V. and Saini, D. R., Estimation of melt elasticity of degraded polymer from melt flow index, Polym. Degrad. Stabil., 11, 297-307 (1985). 39. Evans, M. E., The influence of ,additives on processing and properties of polypropylene film filbers, inConf. Textiles from Film I1 (1971). 40. Bagheri, R., Chakraborty, K. B., and Scott, G., Mechanisms of antioxidant action: Evidence for a regenerative cycle during the melt stabilization of galvinoxyl, Polym. Degrad. Stabil., 5, 145-160 (1983). 41. Zavadsky, E., Karnis, J., and Pechoc, V., The time temperature and shear depen-

dence of the viscosity of polypropylene and its influence upon the extrusion process, Rheol. Acta, 21, 470-474 (1982).

42. Kelleher, P.G.,Wentz,R.P.,Hellman,M.Y.,andGilbert,E.

H., The hydrolytic stability of glass fiber reinforced poly@utylene terephthalate), poly(ethy1ene terephthalate) and polycarbonate, Polym. Eng. Sci., 23, 537-542 (1983). 43. Kelleher, P.G.,Wentz,R.P.,andFalcone,D.R.,Hydrolysisofpoly@utylene terephthalate), Polym. Eng. Sci., 22, 260-264 (1982). 44. Pryde, C. A., Kelleher, P.G.,Hellman,M.Y.,andWentz,R. P., The hydrolytic stabilityofsomecommerciallyavailablepolycarbonates, Polym. Eng. Sci., 22, 370-375 (1982). 45. Winter, H. H., A collaborative study on the relation between film blowing perfor46. 47.

49. 50. 51. 52.

53.

54.

mance and rheological properties of two low-density and two high-density polyethylene samples, Pure Appl. Chem., 55, 943-976 (1983). Neverov,A.N.andVasil’eva,N.P.,Thechangeinproperties of LDPE during storage, Znt. Polym. Sci. Technol., 11, T85-T86 (1984). Dubois, J. H. and John, F. W., Plastics, Van Nostrand Reinhold, New York (1967), Chap. 1. BASF Technical Brochure, Lupolen Volume 1: Properties. Anonymous, Melt flow and properties of polypropylene, Plastiw Technol., 31-32 (July 1962). Crespi, G. and Ranalli,F., Polypropylene, Trans. PlasticsInst., 55-73 (April 1959). Dow Chemical, Technical Bulletin, Dowlex Polyethylene Resins. Gilby,G.W.,Ethylene-vinylacetatecopolymers, in Developments in rubber technology-3 (A. Whelan and K. S. Lee, eds.), Appl. Sci. Pub., Essex, England (1982), pp. 101-144. Krassig, H., Lem, J., and Mark, H. F., Fiber Technology, Marcel Dekker Inc., New York (1984). Nielsen, L. E., Mechanical Properties of Polymers, Reinhold, New York (1962), p.

252. 55. Billmeyer, F. W., Jr., (1984).

Textbook of Polymer Science, Wdey-Interscience, New York

Chapter

382

Krassig, H., Film to fiber technology, J. Polym. Sci. Macromol. Rev., Dunkley, C. D., The injection moldingof polyethylene, Collected Papersfrom ZCZ, Wild, L., Woldering, J. F., and Guliana, R. T., SPE Annual Technical Conference Vol. pp. Phillips Chemical Company, Technical Bulletin, Marlex Dupont, Technical Bulletin, Alathon Polyethyene. Kendall, K. and Sherliker, F. R., Colloidal reinforcement: The influence of bound polymer, BE Polym J., Kendall, K. and Sherliker, F. R., Effect of polymer molecular weight on colloidal reinforcement, BE Polym. J., Howard, J. B., Engineering Design for Plastics (E. Baer, ed.), Reinhold, London Chap. Kambour, R.P., Environmental stress crackingof thermoplastics, in Corrosion Fatigue (0. F. Devereaux, J.McEvilyandR.W.Staehle,eds.), NACE-2, Nat. pp. Assoc. Corrosion Eng. Publishers, Houston,TX Pelagatti, U. and Baretta, G., Stress cracking of polyethylene, Mod. Plastics, (June Baer, E., Engineering Design for Plastics, Reinhold, New York pp. Korcz, W. H., Polybutylene polymers for hot melt adhesives, Adhesive Age, (November Celanese Polymer Company, Technical Bulletin, Fortiflex Polyethylene Properties.

l0 Concluding Remarks

This is the last chapter of this book; it has been included in order to provide a forum for the authors to make some remarksof significance. It gives an opportunity to retrospect the ideas presented in the book and to draw conclusions in a systematic manner. Theremarks are categorized underthree subheadings, namely, comments on the presented work, suggestions for future work, and a few words of caution.

10.1 COMMENTS ON THE PRESENTED WORK The first three chapters were basically meant to refresh the fundamental ideas relating to conventional thermoplastic melt rheology and processing. Chapter briefly discussed polymers and their classifications along with the role of rhe.ology in polymer processing. Most of the fundamental rheological parameters have been defined and described in Chapter 2; the methods of rheological measurements havebeen presented in Chapter There is nothing much to comment on these three chapters, as their contents are not too different from those discussedin many other books on rheologyand processing available in the market. The chapters thatneed comments are those from Chapter onward. For example, in Chapter a different type of unification approach is described and used for obtaining master rheograms for a number of polymers which include 383

384

Chapter

olefinics, styrenics, engineering thermoplastics, specialty polymers, copolymers, vinyl polymers, blends, and filled and recycled polymers. The master rheograms are unique for each generic type of polymer. The advantages of such master rheograms are numerous. First, because they are polymer grade-invariant and temperature-invariant, they eliminate the need for generating viscosity versus shear rate curves at temperatures of interest using expensive rheological equipments. Second, any two generic types of polymer could be compared with each other, and quick conclusionscould be drawn. For instance, a look at Fig. 10.1, which gives the master rheograms for LDPE, HDPE, and LLDPE on the same plot, helps to draw some quick conclusions like those given below. If the MFI value of each of these three types of PE is the same, then LDPE will have the highest viscosity value, while LLDPE the lowest viscosity, and HDPE a medium viscosity. This statement will be true, at least within the low-shear-rate range as shown in Fig. 10.1. Similarly, if the MFI value of each of these three types of PE is the same, the viscosity value of HDPE will be much closer to LDPE, whereas the LLDPE will be significantly different from LDPE. This fact will achieve im-

10.1 Mastercurvesofviscosityversusshearrateforthree types ofpolyethylenes in the low-shear-rate region. (Reprintedfrom Ref. 49 of Chapter

Concluding Remarks

385

portance when the processor decides to replace LDPE with LLDPE of the same MFI on an existing product line. also it can be concludedthat the Newtonian plateau in the case of LLDPE extends to a far higher shear rate than that in the case of HDPE or LDPE. The polymer processor normally has ready access to the MFI value of the raw material that he intends to use. The master rheograms provide the quickest and simplest method for obtaining viscosity versus shear rate data at the temperature of interest simply through the use of the appropriate MFI value. The master rheogramsare unique for each generic typeof polymer over a wide range of shear rates, excepting the very low and very high shear rates wherein the effect of molecular-weight distribution is felt. Chapter introduces the method of upgradation of the master rheogram in term the very low-shear-rate region through the use of an additional (MzIMw)1.7 in the normalizing factor. The master curves plotted in this fashion are certainly unique in the very low-shear-rate region, especially, at the Newtonian plateau. Thus, if an estimation of zero-shear viscosity is to be obtained, it is best to use the constant value (qoX MFI)/(Mz/M,,,)1~7 rather than qo MFI values given in Tables and Chapter also provides extensions of the unification technique to include master curves for normal stress difference, complex viscosity, storage modulus, and extensional viscosity. The advantages of such master curves are the same as thosediscussed earlier. For example,when the master curvesof normal stress difference for three polyethylenes are put on the same plot (Fig. 10.2), they give a quick idea of the elastic response of the three systems and conclusions be drawn as done before from Fig. 10.1. The uniqueness of the master curves for the various rheological parameters mentioned above is as far as the medium to high shear rate region is concerned. In the very low-shear-rate region, as in the case of the viscosity curves, there is a need to use the MJM, term with the appropriate power in order to establish a uniqueness. For example, the term MZ/Mw3.'is needed to unify the data on NI versus plot. In the case of extensional viscosity, unified curves have been presented for a limited number of cases and the data used for coalescence are also.limited. Because of the difficulties in measurement of extensional viscosity, thereliability of the data is often questionable. Because the original data cannot be as trustworthy as in the case of shear viscosity or complex viscosity data, the master curves of extensional data should be looked at in the same light. In the case of shear viscosity, the master rheograms for the filled polymers have been shown to be the same as those for the unfilled system. This is, of course, true in the medium to high shear-rate region. In the low-shear-rate region, however, the effects of yield stress would dominate and the uniqueness of the curve will be "

"

"

"

386

Chapter

." Figure Master curves of normal stress difference versus shear rate for threetypes of polyethylenes in the low-shear-rate region. (Reprintedfrom Ref. of Chapter

disrupted at lower and lower shear rates. The same is the case with the master curves for extensional viscosity of filled systems. At lower strain rate, the yield stress will result in a forklike region and the curve would no longer be unique, as can be seen from Fig. 10.3. The utility of the master curves presented in Chapters and is obvious in the sense that the individual curves for any grade of polymer at any temperature of interest can be merely readout and replotted byknowing the appropriate MFI value. However, the easier method is to use equations which fit the master curve and regenerate data by mere substitution of the MFI value. In order to do this, rheological models have been suggested in Chapter Four different models have been given covering different ranges of shear rates. Some of the model parameters are useful for drawing quick conclusions regarding the nature of particular generic type of polymer in comparison to another. For example, comparing the A X MFI values of the various polymers would give an idea of which polymer has an intrinsically higher relaxation time than the other. In other words, A X MFI = for LLDPE when compared with A X MFI = for LDPE shows that polymerchain disentanglement for LLDPE begins earlier than for LDPE. Similarly, by comparing the power-law index n, the intrinsic shear-thinning characteristics of each generictype of poly-

r

.-

'U

'U

388

Chapter

mer are comparable with the other. It can be seen that, in the case of condensation polymers like polyesters and nylons, the Newtonian plateau is pronounced and covers a much greater shear-rate range than for addition polymers like PE and PP. Further, the shear-thinning is far lower for polyesters and nylons than for PE and PP. Chapter 7 presents the command sequences in order to build a spreadsheet program. The program makes use of the GeneralRheological model which covers the entire range of shear rate. By appropriately inserting the value of MFI and temperature at which the rheogram is required, the viscosity versus shear rate curve can be viewed on the screen andevenahardcopy can be obtained. The program can be developed by the reader by following the stepby-step procedure as given in Chapter 7. One of the authors actually prepared a diskette in this manner and found that it takes only a few hours for the complete development and testing. The master rheograms presented in Chapter have a high utility value in polymer processing. Various processing parameters can be calculated using the rheological models givenin Chapter for the master rheograms.Chapter 8 gives examples of how the equations for the master rheograms could beused in order to establish simple expressionsfor various processing parameters. In the developed expressions, only the appropriate MFI value is needed, besides the geometric parameters, in order to evaluate the process parameters. The meltflow index is more thanjust a single-point quality control parameter and this fact has beenreemphasized in Chapter 9. It outlines various correlations among parameters involved in polymer manufacture, product fabrication, and property evaluation with MR. During polymer manufacture, the quality of the material can be monitored by the reaction temperature, the catalyst activation temperature, the reaction pressure, and forth. These important parameters hold a definite relationship with the MFI of the resulting polymer and, hence, can be adjusted quite easily to obtain the polymer grade of interest through MFI measurement. The specification of the polymer grade includes fundamental structural properties such as molecular weight,molecular-weight distribution, branching,glass-transition temperature, zero-shear viscosity, and on, all of which have also been shown to relate effectively with MFI. The processes involvedin the fabrication of polymer products includeinjection molding, compression molding, extrusion, calendering, and forth. In all these processes, the polymer melt is subjected to shearingand extensional flow. A knowledge of the entire rheological characteristics is a must for process optimization and product quality control. MFI has been shown to relate to shear viscosity, normal stress difference, extensional viscosity, die swell, melt strength, and on, and, hence, attains great importance during polymerprocessing. Preprocessing and postprocessing operations like cross-linking and curing can be

monitored through MFI measurements, besides degradation, stabilization, and aging of the polymer. Important product properties like tensile, flexural, tear, and impact strength as well as tenacity and ultimate elongation have been shown to relate to MFI. It has also been shown that with a proper balance of MFI and processing temperature, a desired level of clarity and gloss can be achieved. Further, the environmental stress cracking susceptibility can also be adjudged through MFI values. In summary, it has been shown that MFI is a parameter useful at all stages right from polymer synthesis to polymer processing, and final product performance. other known parameter has such a large gamut of utility. It is important that the potential ofMFI is realized byraw material manufacturers, polymer processors, and product development groups, as it truly is more than just a quality control rheological parameter.

10.2 SUGGESTIONS FOR FUTURE WORK Although the book appears quite exhaustive, there are a number of areas where more research work is needed to elucidate the subject matter further. Some of the possible areas, as perceived by the authors, are presented below. Chapter gives master rheograms for a long list of polymers but, by no means, includes all known thermoplastics. It is, therefore, beneficial to establish the master rheograms for the thermoplastics not considered in the book along with any new polymers, time and again, when they get developed in the future. The upgraded master rheograms have been given in Chapter for a handful of polymers. systematic study can beinitiated that master rheogramsof (q X MFI)/(a2/a,).'7 versus (372/%w)'.79/MFI can be established for all polymers listed in Chapter The data generation needs molecular-weight distribution along with MFI, and viscosity shear-rate data in the low-shear region. The advantage of such data generation is thata constant value of (qo X MFI)/ (Mz/M,,,)'.7can be established for each generic typeof polymer from which zeroshear viscosity evaluation can be done in the future through mere arithmatic calculations. Master curves for other rheological parameters such as normal stress difference, complex viscosity, storage modulus, and extensional viscosity for all polymers listed in Chapter could be attempted throughsystematic study.It is essential to reiterate the generality of the unification approach for these rheological parameters. In the case of normal stress difference, curves have been presented in Chapter which are estimates based on rheological models. These theoretical master curves need experimental verifications. In fact, the theoretical master curveshave been drawn using an average value for the damping constant "

390

Chapter

It is necessary to establish the correct value of m for each generic type of polymer through experimentaldata. The presented command sequences in Chapter 7 for developing the spreadsheet program are given only for viscosity versus shear rate curves. The stepby-step procedure is very simple and needs to be extended for other rheological parameters once master curvesare established for a long list of polymers. The method of obtaining process parameters from master rheogramsbeen has given only for a limited number of cases in Chapter 8. The idea was only to demonstrate the procedure and hence it was limited to only a few processing techniques and that too only to selected processing parameters within the treated processing technique. Extensions of these ideas to other processing parameters would be more than welcome as it would increase the utility value of the presented unification approach. Correlations such as those presented in Chapter can also be established probably for a number of other parameters. Although such work can certainly be undertaken in future, a better effort can be directed toward the establishment of possible master curves when correlating various process, product, and property parameters withMFI. Such master curvesare likely to exist, but a consorted effort is often needed to identify the correct normalizingparameter when coalescing sets of curves.

m of

10.3 A FEW WORDS OFCAUTION Care has to be exercised when regenerating plots from master rheograms. It is absolutely essential to make sure that the MFI value corresponds to the load condition of the master curve. If not, it is necessary to convert the MFI value to that at the appropriate load condition by the use of also, it is absolutely essential that the MFI used for obtaining curves from the master rheograms must be at the temperature at which the curve is desirable. If this is not the case, then Eqs. or ought to be used depending on whether the temperature of interest has a value greater than TB + 100 K or less than Tg K Only when using Eq. care has to be taken that the correct value of E is used, especially when dealing with copolymers which show two different values of E in different temperature ranges. Having established in this book that the MFI is sensitive to a numberof fundamental parametersin polymer synthesis, processing, and final product properties, it is absolutely essential that the measurement itself should be performed with extreme care that the possible errors are minimized. In order to emphasize this, a few caution points are givenbelow,whichmust beborne inmindwhen performing the MFI measurement.

+

The cylinder, piston, and capillary of the melt flow indexer should be meticulously cleaned before measurement.

Concluding Remarks

391

The measurement should be done strictly under the conditions specified in any of the standards ASTM, BS, DIN, ISO, or JIS. During charging of the sample, the packing should be uniform and without air gaps. Delays between charging and packing must be kept to a minimum. The piston height during all measurements should be keptbetween 50 and 20 mm. For polymers sensitive to oxygen and moisture, the melt flow indexer should be appropriately modified that dry-nitrogen purging can be done. For highly filled polymeric systems, higher loads must be employed and the capillary diameter modified if necessary, depending on the size and shape of the fillers. In case two polymers of the same generic type are found to give identical MFI values but show large differences in other properties, then the load condition must be altered until the MFI values are significantly different and sensitive. Any errors in MFI measurements are likely to depict amplified derogatory results when correlating with other fundamental properties. Hence, it is worth using microprocessor-controlled melt flow indexers which are now available in the market, that the highest order of accuracy and reliability can be maintained. The names and addresses of some of the leading manufacturers/suppliers of melt flow indexers from all over the world are given in Appendix D.

Glossary

is a chemical reaction in which simple molecules (monomers) are added to eachother to form long-chain molecules (polymers) without the formation of by-products.

Addition polymerization

Amorphous polymer is one that has no crystalline component and there is no

order or pattern to the distribution of the molecules.

is the ratio of shear stress by shear rate which has not been corrected for entrance length effects in a capillary rheometer.

Apparent viscosity

Barus effect or die swell or extrudate swell is the increase in diameter of the

polymeric melt extrudate upon emergence from the die. Branched polymer is one in which the main chain in the molecular structure

is attached with side chains, that is in contrast to a linear polymer. Breakage stretch ratio (BSR) is the ability of the polymer extruded from a

die to withstand breakage when stretched at a particular rate on a draw roll system. , is the temperature at which a flexible or a rubbery polymer when cooled becomes inflexible or brittle. Note that the brittle temperature is often considerably higher than the glass-transition temperature.

Brittle temperature

is a measure of the ability of a material to withstand internal hydrostatic or gas-dynamic pressure without rupture.

Burst strength 392

393

Glossary

Clamping force is the force required to prevent mold opening during injection molding. Complexmodulus consists of the real and imaginary parts of the modulus. The real part is called the storage modulus and the imaginary part is called the loss modulus. Compliance is the reciprocal of modulus. Compressive strength is the ability of a material to resist forces that tend to crush or compress it. Consistency is a rheological property representing the viscous behavior of a non-Newtonian material. Constitutive equation is an equation relating stress, strain, time,and sometimes other variables such as temperature or pressure. Couette flow is the shear flow in an annular gap betweentwo coaxial cylinders in relative rotation. Deborah number is defined as the ratio of characteristic time (in other words, the relaxationtime) of the material to the scale of deformationthat it is subjected (i.e., the duration of observation). Die swell or extrudate swell or Barus effect is the increase in diameter of the polymeric melt extrudate upon emergence from the die. Distortion of an iqjection molded product is the deformation resulting from differential shrinkageoccurringdue to therelaxation of residual stresses formed because of non-uniform solidification. Dynamic viscosity is the ratio of the stress in-phase to the rate of strain under sinusoidal conditions. Elasticity represents a reversible stress-strain behavior. Environmental stress cracking @SC) is the name given to aphenomenon by which a plastic product under high stresses may crack when in contact with certain active environments such as, detergents, fats and silicone fluids. Equation of state or constitutiveequation is anequation relating stress, strain, time, and sometimes other variables such as temperature or pressure. Extensional strain

is the relative deformation in strain due to stretching.

Extensional viscosity is the ratio of tensile stress to the extensional rate. Extra stress tensor is the difference between the stress tensor and the isotropic pressure contribution.

Glossary

394

Extrudate swell or die swell or Barus effect is the increase in diameter of the polymeric melt extrudate upon emergence from the die. Fatiguefailure is the failure ofa stresses below its elastic limit.

material whensubjected repeatedly to

Flashing is the thin excess webof material that is forced into crevices between mating mold surfaces. Flexural strength is the ability of a material to resist forces that tend to bend it. Flow activation energy is the energy required to activate the viscous flow. Flow curve or rheogram is a curve relating shear

or viscosity to shear rate.

Freeze-off temperature is the temperature at whichthe polymeric melt solidifies in the mold. Glass-transition temperature is the temperature at which an increase molecular mobility results in significant change in properties. Heat shock failure is the mechanical failure of a material due to sudden exposure to high heat. Hysterisis is a material characteristic which results in different values of the responses for the same values of corresponding stress or rate of strain when applied in increasing and decreasing order. Impact strength is the ability of a material to resist forces that tend to break it when dropped or struck by a sharp blow. Incompressible fluid is one that does not undergo a volume change, i.e., it is density-preserving. Intrinsic viscosity is the value of reduced viscosity as the concentration approaches the limiting value of zero. Izod impact is a test for shock loading in which a notched specimen held one end is broken by striking and the energy absorbed is measured.

at

Loss modulus is the imaginary part of the complex modulus. Melt Flow Index (MFI) is the weight of the polymer in grams extruded in minutes through a capillary of specific diameter and length by pressure applied through dead weight under prescribed temperature conditions as per set international standards. Melt flow indexer

is the apparatus used for measuring MFI.

Melt Flow Index Recovery ("IR) is the percentage increase in the diameter of the extrudate over the diameter of the orifice.

Glossary

Melt fracture is the irregular distortion of a polymeric melt extrudate upon passing through a die due to improper melt or processcharacteristics. Melt strength is the ability of the melt to support its weight during die extrusion and it is related to the melt viscosity calculated by the change in the diameter of the extruded polymer cord over the first 1.25 cm (0.5 in.) of length. Model is an idealized relationship of behavior expressible in mathematical terms. Molecular weight is a measure of the chain length of the molecules that make up the polymer. Mooney value is the resistance of an elastomeric material subjected to shearing action in double cone rotor cup arrangement. Normal stress coefficient is the ratio of the normal stress by the square of the shear rate. Normal

difference is the difference between the normalstress components.

No-slip condition at a solid boundary implies that the molecules in the thin fluid layer adjacent to the solid surface move at the same viscosity as that of the surface. Paraffin wax is a chemical substance obtained as a residue from the distillation of petroleum and is made up of higher homologs of alkanes with a melting range of Plasticizer is a material generally of low molecular weight that is incorporated into a thermoplastic melt to improve its workability during processing and flexibility in the hished product. Power-lawmodel is behavior characterized byapower tween shear stress and shear rate.

(n) relationship be-

Pragmatic means dealing with matters according to their practical significance. Reduced viscosity is the specific viscosity per unit of polymer concentration in solution. Relaxation time is the time takenfor the stress to decrease to an exponentially inverse of its initial value under constant strain. Rheogram or flow curve is a curve relating shear stress or viscosity to shear rate. Rheometry is an instrumental technique for measuring rheological properties.

Glossary Secondary flow is the result of nonzero components of flow velocity in aplane orthogonal to the main direction of flow. Shear strength is the ability of a material to resist forces that tend to shear one portion of it from a stationary section. Sink-mark is a shallow depression or dimple on the surface of an injectionmolded part due to the collapse of the surface following local internal age after the sealing of the gate. Steady flow is the flow in which the velocity at every point is the same. Storage modulus

is the real part of the complex modulus.

Stress cracking is the appearance of minute cracks when a material is stressed, thereby leaving a site which is particularly vulnerable to physical and chemical attack. Stresscrazing is aprelude to stress cracking whereinveryminute appear on the application of stress.

cracks

Tear strength is the ability of a material to withstand a force that would rip a crescent-shaped nick under standard specified conditions. Tensile strength is the ability of a material to withstand forces tending to pull it apart. Tensile strength at yield is the strength corresponding to the transition from elastic to plastic deformation. Ultimate elongation is the amount of stretch that a material fore taking a permanent set or failing otherwise.

undergo be-

Ultimate tensile strength is the maximum attainable load acting on a specimen in a tensile test divided by the original cross-sectional area of the specimen. Viscous heat is the energy dissipated in the form of heat due to the friction between the highly viscous polymeric meltand the various parts of the processing equipment that it contacts during processing. Vortices are intense spiral motions in a limited region of a flowing fluid. Weissenberg effect is an effect exhibited by certain non-Newtonian fluids and involves the climbing of the fluid up a rod rotating in it. Weld lines are flaws on molded objects that are caused by the meeting of two flow fronts. Yield stress is the stress corresponding to the transition from elastic to viscous deformation of the flow curve.

Appendix Specifications

Table

stress Condition *A *B *C *D *E *F *G *H *I *J *K *L *M

*N *O *P

*R *S

*T

Conditions and MFI

Standard Testing Conditions of Temperature and Load

Temp. Shear

Load piston weight

+

Approximate pressure

(T)

(kg)

(kg/cm2)

(psi)

125 125 150 190 190 190 200 230 230 265 275 230 190 190 300 190 235 235 235 250 175

0.325 2.160 2.160 0.325 2.160 21.600 5.000 1.200 3.800 12.500 0.325 2.160 1.050 10.000 1.200 5.000 1.000 2.160 5.000 2.160 20.000

0.46 3.04 3.04 0.46 3.04 30.40 7.03 1.69 5.34 17.58 0.46 3.04 1.48 14.06 1.69 7.03 1.41 3.04 7.03 3.04 28.12

6.50 43.25 43.25 6.50 43.25 432.50 100.00 24.00 76.00 250.00 6.50 43.25 21.00 200.00 24.00 100.00 20.05 43.25 100.00 43.25 400.00

Note: An asterisk (*) denotes ASTM D1238 and a dagger

lo' dydcni?

0.3 1.97 1.97 0.3 1.97 19.7 4.6 1.1 3.5 11.4 0.3 1.97 0.96 9.13 1.1 4.6 0.91 1.97 4.6 1.97 18.4

denotes ASTM D3364.

A2 TestingConditionsforCommonly Used Polymers Polymer

Condition

*Acetals *Acrylics *Acrylonitrile-butadiene-styrene *Cellulose esters *Nylon *Polychlorotrifluoroethylene *Polyethylene *Polyterephthalate *Polycarbonate *Polypropylene *Polystyrene tPoly(viny1 chloride) *vinyl acetal Note: An asterisk (*) denotes ASTM D1238 and a dagger notes ASTM D3364.

TestTemperature Summary Test temperature (“c)

Condition

* 125 *l50 ?l75

*

Note: An asterisk (*) denotes ASTM D1238 and a dagger (7) denotes ASTM D3364.

de-

399 TestLoad Summary Load Condition

(kg)

*5.000

Nore: An asterisk (*) denotes dagger (7) denotes D3364.

D1238 and a

ASTM Specifications for Pistonand Die Dimensions

Piston

Die

Diameter ? ?

2 O.OOO2 in. = 2 mm

in. = mm

Length 2

in. =

in. =

?

mm

mm

in. =

2 ?

Note:

asterisk (*) denotes

D3364.

Dl238 and

mm a dagger

(7) denotes

Append= B: Data Details for Master Rheograms

Table

Details

Data Used for Master Rheograms

Homopolymers in Figs. 4.11-4.29

Temperature which at Source: MFI points data are data (temp. "Cfload generated Chap. (shear rate

ition GradePolymer Indothene LDPE 22FA002 Indothene 22FA002 Indothene 22FA002 Indothene 24MA040 Indothene 24MA040 Indothene 24MA040 Indothene 24FS040 Indothene 24FS040 Indothene 24FS040 Indothene 26MA200 Indothene 26MA200 Indothene 26MA200 LDPE-B LDPE-C LDPE-D

0.16' (175/2.16) 0.2b (190L2.16) 0.25' (20W2.16) 3.0' (17Y2.16) 4.0b (190/2.16) 5.0' (205/2.16) 3.0' (175L2.16) 4.0' (190/2.16) 5.0' (206/2.16) 16' (17W2.16) 20b (190/2.16) 25" (205/2.16) 1.2b (190/2.16) 2.1b (190/2.16) 6.gb (190/2.16)

Sources

175 190 205 175 190 205 175 190 205 175 190 205 190 190 190

No. of 4

range, S-')

Ref.

(0.01-looo) 9 (0.01-1000) 9 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 10 (0.01-1000) 4 (0.01-1000) 4 (0.01-1000) 4 (0.01-1000)

30 30 30 46 46 46

B

Table B1 Continued Temperature at

No.

pointsdata are data MFI (temp. "CDoad generated Chap. (shear rate

nGradePolymer

kg)

4 ("c)

range, S-')

Ref.

6 (2-700) 6 (2-700) 6 (2-700) 6 (2-700) 6 (2-700) 6 (2-700) 18 (0.01-1OOO) 18 (0.01-1000) 18 (0.01-1000) 17 (0.01-500) 18 (0.01-1000) 18 (0.01-1000) 5 (0.01-1000)

30 30 30 30 30 30 31 31 31 31 31 31 46

HDPE

GD 6260 GD 6260 GD 6260 GF 5740 GF 5740 GF 5740 Marlex EHM-606 Marlex EHM-606 Marlex EHM-606 Marlex EH"606 Marlex EHM-606 Marlex EH"606 HDPE 4

2.34' (175D.16) 3.6b (190D.16) 3.17 (205/2.16) 0.35' (175/2.16) 0.45b (190D.16) (20Y2.16) 0.54' (170/2.16) (180E.16) 0.7Sb (190/2.16) 0.83' (200/2.16) 1.0' (210/2.16) 1.2' (220/2.16) 0.8b (190/2.16)

175 190 205 175 190 205 170 180 190 200 210 220 190

UHMWPE

Fortiflex

0.012b (230K21.6) 0.02' (270/10.0)

230 270

8 (0.01-1) 7 (0.3-300)

47 48

0.88' (17Y2.16) l.OOb (190/2.16) 1.11' (205D.16)

175 190 205

7 (0.02-100) 7 (0.02-100) 7 (0.02-100)

49 49 49

0.3b (200D.16) 0.5' (215D.16) 0.7b (230D.16) 0.75' (20012.16) 1.2' (215/2.16) 1.7b (230/2.16) 1.3' (200/2.16) 2.0' (2W2.16) 3.0b (230/2.16) lSb(230D.16) 4.0b (230/2.16) 12.0b(230K2.16) 3.T (210/2.16) 6.3b (230/2.16) 10.0' (250/2.16) 3.9' (210/2.16) 6Sb (230/2.16) 10.3' (250/2.16)

200 215 230 200 215 230 200 215 230 230 230 230 210 230 250 210 230 250

17 (0.005-700) 16 (0.005-700) 16 (0.005-700) 12 (0.03-700) 13 (0.05-700) 13 (0.05-700) 16 (0.03-700) 13 (0.1-700) 13 (0.1-700) 4 (20-1000) 7 (20-1000) 4 (20-1000) 6 (10-500) 6 (10-500) 6 (10-500) 6 (10-500) 6 (10-500) 6 (10-500)

30 30 30 30 30 30 30 30 30 50 50 50 33 33 33 33 33 33

200 220 240

10 (5-5000) 10 10 (5-5000)

51 51 51

Escorene LLDPE

Koylene PP

0730EB

Koylene EB 0730 Koylene EB 0730 Koylene 1730 Koylene 1730 Koylene 1730 Koylene 3030 Koylene 3030 Koylene 3030 Moplen 015 Moplen Moplen 120 PP 10-1045 PP 10-1045 PP 10-1045 PP 10-6016 PP 10-6016 PP 10-6016

PS

Styrene 666 U Styrene 666 U Styrene 666 U

7Sb (200/5) 37.0' (220/5) 130.0' (240/5)

Appendix B Table

Continued

MFI

(temp. "Cfload condition kg)

Polymer

XP 6065.00 XP 6065.00 XP 6065.00 Styrene 666 Polysar 201 Polysar 201 Polysar 205 Polysar 205 Polysar 205 Polysar E 520 Polysar E 520 Polysar M 520 Polysar M 520 Polysar M 520 Polysar H 5M Polysar H 5M Polysar H 5M Polysar G 2 Polysar G 2 Polysar G 2 Cellulose acetate

Tenite 036-H2 Tenite 036-MS

Tenite 307-H Cellulose propionate Tenite 307 H5 Cellulose acetate butyrate

Tenite 205-H2 Tenite 205 MS CAB

Ethyl cellulose

Ethocel856

Temperature at which dataare generated

eo

No. of data points (shear rate range, S")

Source: Chap. 4 Ref.

210 230 250

10 (5-5000) 10 (5-5000) 10 (5-5000) 7 (0.01-0.55) 1 (100) 1(100) 1 (100) 1 (100) 1 (100) 1 (100) 1 (100) 1 (100) 1 (100) 1(100) 6 (10-500) 5 (20-500) 6 (10-500) 6 (10-500) 6 (10-500) 6 (10-500)

51 51 51 53 52 52 52 52 52 52 52 52 52 52 33 33 33 33 33 33

0.T (19OD.16) 3.6b (210D.16) 4.8" (190n.16) 24Sb (210/2.16)

190 210 190 210

3 (10-100) 5 (10-5000) 3 (100-5000) 3 (100-5000)

35 35 35 35

3.T (190D.16) 19.0'' (210/2.16) 68.8b (230/2.16) 1.3' (210/2.16) 4.T (230n.16)

190 210 230 210 230

3 (300-5000) 3 (300-5000) 3 (300-5000) 3 (100-1000) 3 (100-1000)

35 35 35 35 35

0.25' (190/2.16) 1.3b (210D.16) 4.T (230/2.16) 2.5' (190D.16) 13.0b (210/2.16) 5.W (230/2.16)

190 210 230 190 210 230

3 (30-500) 4 (30-500) 4 (30-500) 3 (100-5000) 3 (100-5000) 4 (1-1000)

35 35 35 35 35 55

1.3' (170/2.16) 4.4" (190/2.16) 14.2' (210D.16) 31.0' (230D.16)

170 190 210 230

3 (10-100) 3 (10-100) 3 (20-1000) 3 (50-900)

35 35 35 35

8.0b (200/5) 42.0' (220/5) 139.0' (240/5) 9.4b (200/5) lSb(200/5) 7.4' (220/5) 0.9' (180/5) 6.8b (200/5) 33.5' (220/5) 2.4b (200/5) 12.0' (220/5) 0.7' (180/5) 5.4b (200/5) 26.5' (220/5) 15.4' (210/5) 47.T 121.0' 27.4' (210/5) 85.1' (230/5) 215.0' (250/5)

200 220 240

200 200 220 180 200 220 200 220 180 200 220 210 230 250

Appendix B

403

Table 61 Continued ~~

Polymer Acrylic

~

~

Grade Lucite 40 Lucite 40 Lucite 40 Lucite 129 Lucite 129 Lucite 130 Lucite 130 Lucite 130 Lucite 140 Lucite 140 Lucite 140 Plexiglas V-100 Plexiglas V-100 Plexiglas V-100 Plexiglas V-100 Plexiglas VM 100 Plexiglas VM 100 Plexiglas VM 100 Plexiglas VM 100 Plexiglas VS 100 Plexiglas VS 100 Plexiglas VS 100 Plexiglas VS 100 Implex A Implex A Implex A

POM

Type 1 Type 1 Type 2 SV-249 SV-284 SV-310

Nylon

Plaskon 8201 Plaskon 8201 Plaskon 8201 Plaskon 8205 Plaskon 8205 Plaskon 8205 Nylon 6

Temperature at which data are generated ("C)

No. of data points (shear rate range, s-l)

Source: Chap. 4 Ref.

19.6" (260/3.8) 0.9' (200/3.8) 27" (250/3.8) 0.002" (150/3.8) 1.28" (200/3.8) 38" (250/3.8) 5.0' (230/3.8) 25.2" (260/3.8) 58.8" (280/3.8) 0.03" (170/3.8) 0.3" (190/3.8) 1.8" (210/3.8) 7.9b(230/3.8) 0.003" (150/3.8) 0.08" (170/3.8) 0.9' (190/3.8) 4.7" (210/3.8) -0.007" (150/3.8) 0.2" (170/3.8) 2.2' (190/3.8) 11.2 (210/3.8) 0.007" (190/3.8) 0.036" (210/3.8) 0.16" (230/3.8)

220 240 260 200 250 150 200 250 230 260 280 170 190 210 230 150 170 190 210 150 170 190 210 190 210 230

4 (1-1000) 4 (1-1000) 4 (1-1000) 5 (1-10,000) 5 (1-10,000) 3 (1-100) 4 (10- 10,000) 4 (100-10,000) 5 (1-10,000) 5 (1-10,000) 4 (1-10,000) 3 (3-30) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 4 (3-60) 3 (3-20) 3 (3-20) 3 (3-20)

37 37 37 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35 35

1.1' (180/2.16) 5.1b (200/2.16) 4b (180/2.16) 2.8" (190/2.16) 2.5' (190/2.16) 1.1' (190/2.16)

180 200 180 190 190 190

4 (30-3000) 4 (30-3000) 4 (30-3000) 4 (6-200) 4 (6-200) 4 (6-200)

57 57 57 58 58 58

5.0" (23U2.16) 13.7" (260/2.16) 29.5" (28W2.16) 1.9' (260/2.16) 2.5" (26W2.16) 4.0"(28W2.16) 8.2' (230/2.16)

231 260 288 260 268 288 230

4 (10-4000) 4 (10-1000) 4 (10-10,000) 4 (10-2000) 4 (10-2000) 4 (10-2000) 3 (1-100)

35 35 35 35 35 35 34

MFI (temp. "C/load condition kg) 1.12'(220/3.8)

4.9'(240/3.8)

Continued

MFI (temp. "Cfload condition kg)

Polymer Nylon 6 Nylon 6 Nylon 6 Zytel42NClO Zytel42NC10 Zytel lOlNClO Zytel lOlNClO Zytel lOlNClO Nylon 66 Nylon 66 Marmy1 A-l00 Nylon 610 Nylon copolymer

PET

Grade Fiber

Temperature at which data are generated

No. of data points (shear rate range, S-')

Source: Chap. 4 Ref.

16.T (25OD.16) 30.6' (270/2.16) 33' (230/2.16) 4' (280/2.16) 6.3' (295/2.16) 49' (280D.16) 63' (290/2.16) 80' (300/2.16) 45' (288/2.16) 49' (29U2.16) 113' (280D.16) 235' (280/2.16) 79' (280/2.16)

250 270 230 280 295 280 290 300 288 291 280 280 280

2 (10-100) 2 (10-100) 4 (10-4000) 5 (1-10,000) 5 (l-10,000) 4 (10-10,000) 4 (10-10,000) 4 (10-10,000) 4 (10-10,000) 4 (10-10,000) 5 (10-10,000) 4 (10-10,000) 4 (10-10,000)

34 34 59 35 35 35 35 35 59 59 60 58 58

54' (275D.16)

275

9 (1-5000)

62

9 (1-5000)

62

IV = 0.57 Fiber Grade IV = 0.57 Fiber Grade IV = Fiber Grade IV = 0.57 Molding Grade IV = 0.722 Molding Grade IV = 0.722 Molding Grade IV = 0.722 Molding Grade IV = 0.722 T i e Cord. Grade IV = 0.887 Tire Cord. Grade IV = 0.887 T i e Cord. Grade IV = 0.887 line Cord. Grade IV = 0.887 Bottle Grade I IV = 1.004

(285D.16) 86' (295D.16)

295

9 (1-5000)

62

103' (305L2.16)

305

9 (1-5000)

62

275

9 (1-5000)

62

9 (1-5000)

62

15.7 (27512.16) 19.6' (285D.16) 24.5' (295D.16)

295

9 (1-5000)

62

28.5' (30W2.16)

305

9 (1-5000)

62

4.T (27W2.16)

275

8 (1-1000)

62

5.6' (285D.16)

285

8 (1-1000)

62

6.6' (295D.16)

295

8 (1-1000)

62

7.4' (305D.16)

305

8 (1-1000)

62

1.5' (275D.16)

275

8 (1-1000)

62

B

405

Table B1 Continued

MFI (temp. "C/load generated

Temperature at are data

ade Polymer

("c)

Bottle Grade I Iv = 1.004 Bottle Grade I Iv = 1.004 Bottle Grade I

N = 1.004 Bottle Grade II Iv = 1.102 Bottle Grade II Iv = 1.102 Bottle Grade II Iv = 1.102 Bottle Grade II Iv = 1.102 PC

PVDF

Makrolon 2805 Lexan 121 Lexan 141 Lexan 141 Lexan 141 Lexan 151 Lexan 151 D W R ME 2000 DYFLOR 2000 ME DYFLOR 2000 ME DYFLOR 2000 ME DYFLOR 2000 ME DYFLOR 2000 ME DYFLOR 2000-H DYFLOR 2000-M DYFLOR 2000-ME DYFLOR 2000-E DYFLOR 2000-LE DYFLOR 2000-L DYFLOR 2000-H DYFLOR 2000-M DYFLOR 2000-ME DYFLOR 2000-E DYFLOR 2000-LE

No. of data points (shear rate range, S-')

Source: Chap. 4 Ref.

1.6' (28572.16)

285

8 (1-1000)

62

1.7T (295D.16)

295

8 (1-1OOO)

62

1.96' (305D.16)

305

8 (1-1000)

62

0.88' (275D.16)

275

7 (1-500)

62

8 (1-1000)

62

1.0' (285/2.16) 1.1' (295D.16)

295

8 (1-1000)

62

1.13' (305Ll.16)

305

8 (1-1000)

62

4.9' (27Y2.16)

275

4 (10-5000)

61

6.13' (290D.2) 7.36' (288/1.2) 1.3' (250D.2) 3.5' (270/1.2) 6.1' (290D.2) 0.86' (288/1.2) 1.0' (290D.2)

290 288 250 270 290 288 290

3 (10-1000) 3 (20-2000) 4 (20-300) 4 (20-300) 4 (20-300) 3 (20-2000) 3 (20-2000)

63

4.1T (190/12.5) 4 9 (210/12.5) 6.8' (230h2.5) 8.3' (250/12.5) 10.8' (270/12.5) 20.1' (290D2.5) 5.15' (250/12.5) 18.1' (250/12.5) 24.5' (250D2.5) 34.3' (250/12.5) 53.9' (250/12.5) 107.9' (250D2.5) (220/12.5) 8.3' (220/12.5) 12.3' (220D2.5) 17.6' (220/12.5) 23.5' (220/12.5)

190 210 230 250 270 290

3 (5-2000) 3 (5-2000) 3 (5-2000) 3 (5-2000) 3 (5-2000) 3 (5-2000) 3 (30-300) 3 (30-300) 3 (30-300) 3 (30-300) 3 (30-300) 3 (30-300) 3 (15-200) 3 (15-200) 3 (15-200) 3 (15-200) (15-200)

66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66 66

250

250 250 250

250 250 220 220 220 220 220

36 36 36 64

Table

Continued Temperature at which No. of data are points data

MFI

(temp. "Cfload generated Chap. (shear rate

onGradePolymer

PP0

kg)

("c)

range, S")

Source: 4 Ref.

DYFLOR 2000-L DYFJAR 2000-MFI 300 KYNAR-460 KYNAR-461 KYNAR-720 KYNAR-730 KYNAR-740 KYNAR-760 KYNAR-821 KYNAR-881 KYNAR-901 KYNAR-931 KYNAR-961

46.6' (220/12.5)

220

3 (15-200)

66

73.6" (220D2.5) 0.84' (232L2.5) 0.64' (23U12.5) 44.1' (232/12.5) 9.8' (232D2.5) 4.4' (232/12.5) 1.8' (232h2.5) 3.2' (12.5) 4.6" (232/12.5) 8.8' (232/12.5) 24.5' (232L2.5) 30.4' (232D2.5)

220 232 232 232 232 232 232 232 232 232 232 232

3 (15-200) 4 (1-1000) 4 (1-1000) 3 (10-1000) 3 (10-1000) 3 (10-1000) 4 (1-1000) 4 (1-1000) 3 (10-1000) 3 (10-1000) 3 (10-1000) 3 (10-1000)

67 67 67 67 67 67 67 67 67 67 67 67

NORYL-731 NORYL-731 NORYL-731 NORYLGFN3 NORYGGFN3 UNKNOWN NORYL-731 NORYL-731 NORYL-731 NORYL-731 NORYGN 110 NORYL-N 110 NORYL-N 110 NORYL-N 110 NORYL GFN2 NORYL GFN2 NORYL GFN2 NORYL GFN2 NORY"SE90 NORYL-SE90 NORYGSE90 NORYL-SE90 NORYL-SE100 NORYL-SE100 NORYLSElOO NORYL-SE100

1.03' (262/5) 3.92' (264/5) 9.8V (306/5) 4.90' (318/5) (340/5) 2.94' (322/5) 4.9' (260/5) 12.3' (280/5) 14;T (300/5) 30.4' (320/5) (240/5) 19.4' (260/5) 39.3' (280/5) 5 9 9 (300/5) 10.8' (260/5) 16.T (280/5) 20.6' (300/5) 39.8' (320/5) 21.1' (240/5) 49.1' (260/5) 78.5' (280/5) 99.1' (300/5) 16.T (260/5) 3 4 9 (280/5) 53.9' (300/5) 93.2 (320/5)

262 264 306 318 340 322 260

4 (7-7000) 5 (4-7000) (4-7000) 4 (40-9000) 4 (40-9000) 4 (10-700) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-500) 3 (3-500) 3 (3-500) 3 (3-500) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400)

68 68 68 68 68 69 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70 70

280

300 320 240 260 280 300 260 280 300 320 240 260 280 300 260 280 300 320

Appendix B Table

407

Continued Temperature at which MFI pointsdata are data (temp. "Cfload generated Chap. (shear 4rate

on GradePolymer

("c)

NORYL-SE1 NORYL-SE1 NORYL-SE1 NORYLPX1112 NORYL-PX1112 NORYL-PX1112 NORYL-PX1112 NORYL-PX1180 NORYGPX1180 NORYL-PX1180 NORYL-PX1180

No. of Source:

range, S-')

Ref.

6.87 (260/5) 13.T 25.5' (300/5) 8.83' (240/5) 16.2' (260/5) 28.9' (280/5) 49.1' (300/5) 9.81' (240/5) 20.2' (260/5) 44.1' (280/5) 63.7 (300/5)

260 280 300 240 260 280 300 240 260 280 300

3 (3-300) 3 (3-300) 3 (3-300) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400) 3 (3-400)

70 70 70 70 70 70 70 70 70 70 70

26.2' 802.2' (285/5) 100O.V (316/5)

280 285 316

6 (2-700) 3 (2-40) 5 (2-40)

71 71 71

PPS

RYTON R-l RYTON V-l RYTON V-l

PES

VIcTRE!x 200P VIcrREx 200P VICTREX 200P VIcrREX 300P

4.41' (320/5) 19.6' (350/5) 49.5' (370/5) (350/5)

320 350 370 350

5 (30-20,000) 6 (10-10,000) 5 (3-4000) 6 (6-2000)

72 72 72 72

PAS

ASTREL 360P

1.5' (403/5)

403

4 (10-1000)

73

PEEK

PEEK 45G PEEK 4530CA (30 wt% CF) PEEK 4530GL (30 wt% GF) PEEK 4530 GL (30 wt% GF) PEEK 451GV PEEK 451GV

1.9' (380/5) 16.T (380/5)

380 380

5 (0.1-1000) 5 (0.01-1000)

74 74

2.96' (360/5)

360

7 (15-1000)

74

4.2' (395/5)

395

7 (15-1000)

74

10.8' (360/5) 22.6' (3995)

360 395

4 (15-1000) 4 (4-1000)

74 74

ULTEM 1000 ULTEM 1000

19.81' (360/5) 5.9' (360/5)

360 360

5 (5-5000)

5 (5-5000)

75 75

ULTEM 1000

3.96' (360/5)

360

5 (5-5000)

75

(20 wt% GF) ULTEM 1000 (30 wt% GF)

2.45' (36015)'

360

(5-5000)

75

8.34' (355/5) 22.08' (375/5) 39.25' (395/5)

355 375 395

PE1

(10 wt% GF)

ULTEM ULTEM ULTEM

3 (200-7000) 3 (200- 10,000) 3 (200- 10,000)

70 70 70

Table

Continued Temperature which at MFI Source: points data are data (temp. "Cfload generated Chap. 4(shear rate

ion GradePolymer

88

No.of range, S")

(288/5) PAI 3.53' E-200-3 E-200-3 E-200-3 E-200-3 E-100-1 E-100-2 E-100-3

76 (1-1000)4 302 8.3' (302/5) 76 (1-1000)4 316 17.6' (316/5) 76 (2-1000)4 329 35.3' (329/5) 376 316 2502.7 (316/5) (110-10,000) 316 1128.6' (316/5) (100-10,000) 376 316 466.2' (316/5)

Ref.

3 (100-10,000) 76

M 'F I value calculated from Eqs. (4.14) or (4.15) knowing the MFI value as per footnote b. "MFI value given by manufacturer or measured under ASTM test conditions. 'MFI value read out T versus curve using Eqs. (4.7) and (4.8) by the method discussed in Sec. 4.2.2. Source: Refs. 47, 49, 54,56, 65, and 71 of Chapter 4.

Table

Details of Data Used for Master Rheograms of Copolymers in Figs. 4.30-4.36 Temperature which at MFI Source: points dataaredata (temp. "Cfload generated (shear Chap. rate 4

Polymer S A N (random)

kg) condition Grade "yril860B W860B Tyril860B Tyril860B Tyril867B Tyril867B Tyril867B Tyril867B

0.05' (110/5.0) 0.34' (130/5.0) 1.7(150/5.0) 2.7 (170/5.0) 6.8' (190/5.0) 7.8' (210/5.0)

SBS (block)

ABS (graft)

1.0' (200/3.8) 3.4' (215/3.8) 9.5b (230L3.8) 29.5' (250/3.8) 0.5' (200/3.8) 1.6' (215L3.8) 4.5b (230L3.8) 14.0' (250/3.8)

Kralastic MH Kralastic MH Kralastic MH ABS1 ABS2

0.Y (180/5.0) 4.2' (200/5.0) 12.3' (220/5.0) 12.2' (210/5.0) 4.0' (210/5.0)

No. of

cc>

range, S")

Ref.

200 215 230 250 200 215 230

10 (20-600) 10 (20-600) 10 (20-600) 10 (20-600) 10 (20-600) 10 (20-600) 10 (20-600) 10 (20-600)

51 51 51 51 51 51 51 51

250

110 130 150 170 190 210

5 (0.1-1000) 5 (0.1-1000) 5 (1-1000) 4 (1-1000) 4 (1-1000) 4 (1-1000)

39 39 39 39 39 39

180 200 220 210 210

5 (0.01-100) 5 (0.1-1000) 4 (1-1000) 3 (10-800) 3 (1-100)

78 78 78 79 79

Table

Continued

kg) condition Grade

Polymer VCVA (random)

(175/20) 0.02' (175/20) 0.38' EVA (random)

Polyester elastomer (block)

Temperature at which m Source: points dataaredata (temp. "CAoad generated (shear Chap. rate ("c)

VYNS VYNWl VYNW2 vYNw3 VYNW4 A1

35(0.1-6) 3 140 0.36' (140L20) 160 4.3' (160/20) 35 (5-100)3 170 4.T (170/20) 35 4 170 1.1"(170/20)(0.3-100) 35 (3-100)4 180 1.8' (180/20)

A2

(20-600) 80 3 175 0.03' (20-600) 80 3 175 (175/20) 80 (20-600) (20-600) 80 3 175 7.8'

A3 A4 A5

No.of 4 range, S")

Ref.

335 (3-100)

80 3 (20-600) (175/20)

(175/20)

ALATHON(60L2.16) E/VA 0.02' 3185 60 3185 E/VA ALATHON(70/2.16) 0.18' E/VA 3185 70 ALATHON (7Y2.16) 0.34' 3185 E/VA 75 ALNTION ENA 3185 ALATHON E/VA 3185 ALATJ3ON (12W2.16) 4.8' 3185 E/VA 125 HYTREL 4056 HYTREL 5556 HYTREL 6346 HYTREL 7246 HYTREL 5526

TR-400 Olefinic thermoplastic TR-402 elastomer TR-100 TR-101 @lock) 205 (205L2.16) 3.68'TR-301 TR-405 TR-302 TPR-2800 TPR-1600 TPR-1800 ET-H-3100 ET-G3100 ET-H-1100

(0.01-1) 40 3 3 (0.01-1) 3 (0.01-1) 3 (0.01-1) 40

O.OT (65/2.16) 65

0.50'(0.01-500) (80/2.16) 7 80 1.3T (100L2.16) 100

4.41' (20W2.16) 8.8' 205 (205L2.16) 0.lg (205D.16) (20W2.16) 0.5' (20Y2.16) 0.15'(230/2.16) 0.22' (230/2.16) 0.03' (230/2.16)

40 40

40,41 (2-500) 414 (2-500) 414

(10-3000) 4 180 3.T (18OL2.16) 220 8.8' (220D.16) 220 9.2' (220/2.16) 240 14.6' (240/2.16) 220 20.4' (220/2.16) 0.53' (205D.16) 0.83' (20W2.16) 2.06' (20Y2.16) 3.18" (205L2.16)

3

(10-3000) 4 (10-3000) 4 (10-3000) 81 4 (10-3000) 4

3 (10-1000) 205 205 (10-1000) 82 3 205 (10-1000) 3 205 (10-1000) 82 3 82(10-1000)3 205 3 (10-1000) 82 82(10-1000) 3 205 (1-1000) 4 205 (1-1000) 83 4 205 (1-1000) 83 4 230 (1-1000) 82 4 230 (1-1000) 82 4 230 382 (1-100)

81 81 81

81 82 82

83

"FI value calculated from Eqs. (4.14) or (4.15) knowing the h4FI value as per footnote b.

"MHvalue given

manufacturer or measured under ASTM test conditions. value read out from T Versus P curye using Eqs. (4.7) and (4.8) the method discussed in Sec. 4.2.2. Source: Refs. and 77 of Chapter 4.

Appendix B

410

Table B3 Details of Data Used for Master Rheograms of Liquid-Crystalline Copolymer and Polymer Blends in Figs.

m' (temp."CAoad

rade Polymer Liquid-crystalline copolymer

Temperahre at which data are generated

kg)

HBAPET

4

HBAPET

5 240

PP/HDPE blend PP/HDPE

HDPWF"MA HDPE/I"MA blend

No. of data points (shear rate range, S-')

5

Chap. Ref.

Appendix B Table B3 Continued

Temperature which at MFr pointsdata are data (temp. "Cfload generated Chap. (shear rate

Source: 4

cc,

range, S-')

Ref.

112.8 (220/5) 75.8 (220/5) 41.7 (220/5) 9.8 (220/5) 1.47 (220/5)

220 220 220 220 220

5 (20-400) 5 (20-400) 5 (20-400) 5 (20-400) 5 (20-400)

90 90 90 90 90

5.4 (210/5) 12.3 (210/5) 22.5 (210/5) 31.4 (210/5) 39.3 (210/5)

210 210 210 210 210

5 (20-400) 5 (20-400) 5 (20-400) 5 (20-400) 5 (20-400)

91 91 91 91 91

3.6 (2lOn.8) 7.6 (210/3.8) 13.2 (210/3.8) 34.3 (210/3.8)

210 210 210 210

4 (10-400) 4 (10-400) 4 (10-400) 4 (10-400)

91 91 91 91

onGradePolymer ~~~~~

No. of

~~

PS/P"A blend

PS/PMMA 1.00/0.00 0.75/0.25 0.2W0.75 0.00/1.00

PS/POM blend PS/POM 1.0/0.0 0.9/0.1 0.2/0.8 0.0/1.0

PM"OM blend

PMMAPOM 1.0/0.0 0.8/0.2 O.YO.5

0.0/1.0

~~

'MFI value read out from vs. curve Source: Refs. 84 and 87 of Chapter 4.

Eqs. (4.7) and (4.8) by the method discussed

Section 4.2.2.

I I I I I

nnnn

I I I I I

d

d

d

e m

e m

h

h

h

h

m

g

8

I

I

2. 2.

8 m I

8 m I

8 I

W

G;' G;' 2. N W

m

m

m

G;' 2.

h

I

gm

0

N

dn dn

m

m

S

8 m

m

m

m

0

0 N W

0

0

m

m

m

h

h

m

l

G;' G;' 2. 2. N m

W

N

I

m

"! t-

I?

I

v)

0

I

m

2. 2.

d

e m

d

h

h

h

13

13

13

8 I

8 I

8 I

W

W

W

S

S m

S 0

v

W

2 2

N

2 I

I

I

I

I

d S

S

*

W

I

W

I

*

h

0 0

8

0

0

S

x

0 0

I

I

I

I

8

8

I I I I I

B

4m000004400

O w w - t m m m m m b

oooo8S3

8 3 3 N N N N N N N

B

B

Parameters of the Filled Polymer Systems Covered in the Data Analyzed for Master Rheomams of Filled Polymers

Table ~~

~~~

~

pe oupling gentphr Amount typeFiller Matrix LDPE

HDPE

PP

PS

Nylon PC PET

Quartz powder I Quartz powder 11 Calcium carbonate I Calcium carbonate II Fiber glass Talc Talc Calcium carbonate Calcium carbonate Mica Mica Carbon black Titanium dioxide Glass fibers Calcium carbonate Calcium carbonate Zytel (minerals) Glass fibers Glass fibers Glass fibers

Source: Ref.

of Chapter

100 11

25

11 25

-

-

-

0.5-1.0 phf titanate

-

phf silane 11 11 25 43 25

15 10

-

Treated Untreated

-

Particulate Particulate Prismatic Prismatic Fibrous Platelet Platelet Prismatic Prismatic Platelet Platelet Particulate Particulate Fibrous Prismatic Prismatic

-

Fibrous Fibrous

Appendix B Table

419

Details of Data Used for Master Rheograms of Recycled Polymers in Figs. 4.46-4.48

No. of data MFI (temp., "C/ Polymer Grade

PP

Profax

PS

Styron 686

Styron 686

points (shear rate range, S")

Source: Chap. 4 Ref.

Shearing history

load condition, kg)

Data temp. CC)

Viigin Brabender worked sample for 60 mm 1 cycle through an injectionmolding machine 3 cycles 6 cycles 9 cycles 12 cycles 15 cycles 1 cycle 3 cycles 6 cycles 9 cycles 12 cycles 15 cycles 1 pass through 6.35-~m extruder at 50 rpm screw speed 2 passes 3 passes 4 passes 5 passes 1 pass through 6.35-~m extruder at 100 rpm screw speed 2 passes 3 passes 4 passes 5 passes

0.8' (190/2.16) 1.3' (190/2.16)

190 190

7 (0.03-6) 7 (0.03-6)

105 105

1.3b (190/2.16)

190

4 (10-300)

106

1.6b (190/2.16) 2.2b (190/2.16) 3.2b (190/2.16) 4 . p (190/2.16) 6.7b (190/2.16) 1.B(190/2.16) 2.2b (190/2.16) 3.6b (190/2.16) 5.1b (190/2.16) 6.5b (190/2.16) 9.2b (190/2.16) 17.1' (220/5)

190 190 190 190 190 190 190 190 190 190 190 220

4 (10-300) 4 (10-300) 4 (10-300) 4 (10-300) (10-300) 5 (20-500) 5 (20-500) 5 (20-500) 5 (20-500) 5 (20-500) 5 (20-500) 3 (20-700)

106 106 106 106 106 106 106 106 106 106 106 107

18.1' (220/5) 21.5' (220/5) 31.8' (220/5) 41.T (220/5) 24.5' (220/5)

220 220 220 220 220

3 (20-700) 3 (20-700) 3 (20-700) 3 (20-700) 3 (20-500)

107 107 107 107 107

26.4' (220/5) 32.3' (220/5) 39.2' (220/5) 44.1" (220/5)

220 220 220 220

3 (20-500) 3 (20-500) 3 (20-500) 3 (20-500)

107 107 107 107

"FIvalue read out from versus 9 curves F+ (4.7) and (4.8). bMFI value given under testing conditions. Source: Ref. 104 of Chapter 4.

Appendix C: Data Details and Sources Upgrade and Extension Curves

Table Cl Details of the Polymers Used in Coalesced Shear Viscositv Curves in Figs. 5.1-5.7

M= GradePolymer

)2.59 PP-H-N PP PP-H-R-B PP-H-B-R PP-M-N PP-M-R PP-M-B PP-L-N PP-L-R-N LDPE (14W2.16) 0.12b B C PEP 21 1 M-l HDPE (180/2.16) 0.42b7.78 M-2 HIZEX 5000 HIZEX 5200

MFI (temp., "C/ loadcondition, kg)

Data temp.

cc>

No. of data points (shear rate range, S - ' )

180 (0.01-400) 6 3.57 180(0.01-1000) 6 1.12* (180/2.16) 180(0.01-1000) 6 3.54 0.85' (180/2.16) 2.81 180(0.13-3000) 7 2.67' (180/2.16) 4.82 (0.01-10) 4 180 2.85' (180/2.16) 4.46 (0.01-1000) 6 180 2.53' (180/2.16) 2.47 (0.01-7500) 7 180 5.75" (180/2.16) (0.2-6500) 7 180 3.18 5.2g8 (180/2.16) 6.63 9(0.1-100) 148 4 4.12 148 9(0.1-100)4 0.32b (148/2.16) 3.37 148 9(0.1-100)4 0.94b (14W2.16) 9.46 (0.1-100) 10 4 200 4.10b (200/2.16) 6.90 (0.3-100) O4S l b (180/2.16) 180 (0.3-100)4 180 (0.01-10) 4 240 5.60 2.16b (240/2.16) (0.01-10) 4 240 9.30 0.70b (240/2.16)

'MFI values calculated through equations in Sec.

Source: Chap. 5 Ref.

8 8 8 8 8 8

11 11 12 12

knowing the MFI values under standard conditions as given in Table values calculated from the shear stress versus shear rate curves in accordance with the readout method (Ref. of Chapter 4) described in Sec. Source: Ref. 7 of Chapter

Appendix Table 5.8-5.16

Details of the Polymers Used in Coalesced Normal Stress Difference Curves in Figs.

Source: points Data -

M -

GradePolymer

P-H-N PP

m "C/ (temp., load condition, temp. (shear rate Chap. kg)

No. of data 5 ("C)

range, S")

2.59 (180/2.16) 0.97" 180 3 (0.05-0.5) (0.01-0.5) 8 4 180 3.57 1.12' (180/2.16) 180(0.01-0.5) 4 3.54 0.85" (180/2.16) (0.05-1.0) 8 4 180 2.81 2.67" (180/2.16) 180 5 (0.02-0.8) 4.82 2.85" (180/2.16) (0.03-0.5) 8 4 180 4.46 2.53' (180/2.16) (0.20-1.5) 8 4 180 2.47 5.75" (180/2.16) 180(0.05-1.0) 8 4 3.18 5.29' (180/2.16) (0.3-10) 6 200 6.40b (200/2.16) (0.1-10) 7 200 2.20b (200/2.16) 23 (1-100)7 200 2.20b (200/2.16) LDPE A 6.63 (148/2.16) 148 0.12b 5 9(0.04-0.2) 148 5 (0.04-1.0) 4.12 B 0.32b (148/2.16) 148 (0.1-0.2) 9 4 C 3.37 0.94b (148/2.16) (0.4-200) 7 200 PEP 211 9.46 4.10b (200/2.16) HDPE(0.1-4)M-l 4 (180/2.16) 180 0.51b 6.90 (180/2.16) 0.42b 7.78 M-2 (0.1-4) 180 4 HIZEX (0.1-10) (240/2.16) 2.16b 3 5.60 240 HIZEX9.30 5200 O.7Ob (0.1-10) (240/2.16) 4 240 LLDPE 24 (0.1-3) - 4 175(175/2.16) l.OOb (100/2.16) 190(0.1-3) 4 l.OOb (0.1-3) (205/2.16) 4 205 Nylon NYLON 6 25(5-20) - 4 (230/2.16) 230 9.8b NYLON 6 25(5-20) - 4 (250/2.16) 16.6b 250 NYLON 6 24Sb 4 (270/2.16) 270 NYLON 6 -(100-200) 245.0b (280/2.16) 2 280

PP-H-R-B PP-H-B-R PP-M-N PP-M-R PP-M-B PP-L-N PP-L-R-N AMOCO 6014 EXXON E115 ENJAY E115

Ref. 8

8 8

12 12

9 10 11 11 12 12 24 24

25 12

"FIvalues calculated through equations in Sec. 4.2.4 knowing MFl values understandard condition given in Table 5.2. values calculated from the shear stress versus shear rate curves in accordance with the readout method (Ref. 22 of Chapter 4) described in 4.2.2. Source: Refs. 21 and 24 of Chapter

C

C3 Data on the Reprocessed Polymer Used in Coalesced Normal Stress Difference Curves in Fig. 5.17 and 5.18

ma Polymer LDPE LDPE

source: pointsData "C/(temp., Shearing condition, load Chap. temp. (shear rate history kg) Virgin Brabender worked sample for 60 min

data No. of 5

range, ('C) 190 190

0.8 (190n.16)

1.3 (190/2.16)

S")

7 (0.03-6) 7 (0.03-6)

Ref. 27 27

"FI valueobtained from shearstressversusshearratecurvesinaccordancewiththereadout method (Ref. 22 of Chapter 4) described in Sec. 4.2.2. Source: Ref. 26 of Chapter

Details of the Polymers Used in Coalesced Dynamic Shear Curves in Figs. 5.19-5.22 ~

~

~~

~

No. of

Temp. at MFP points which (temp., "C/ data are load condition, taken Ref. kg)radls) ('C)

Grade Polymer HDPE PS

DMDJ 5140 DMDJ 4309 SHOLEX 60098 STYRON 678U STYRON (0.1-100) (170/2.16) 36 1.5 3 686 170

0.51 (170/2.16) 3 170 200 0.42 (200/2.16) 0.90 (190/2.16) 190 (0.1-100) 2.5 23(200/2.16) 3 200

data (frequency Source: range, Chap. (0,l-10) 3 (0.1-10)

4 (0.1-100)

"FI valueobtainedfromshear stress versusshearratecurvesinaccordancewiththereadout method (Ref. 22 of Chapter 4) described in Sec. 4.2.2. Source: Ref. 28 of Chapter

5 34 23 35

C5 Details of the Polymers Used in Coalesced Extensional Viscosity Curves in Figs. 5.23-5.25

m (temp., "C/ load condition, Grade Polymer

kg)

MX6002 HDPE MX6002 MX6002 DMDJ 4309 DMDJ 5140

unknown Unknown

PP

PS

HDPE-1 HDPE-2 HDPE-3 E115 E115 E115

Shell Styron 678 Styron 686 Styron 686 Styron 686 Styron 686

0.14 (170/2.16) 0.20 (190/2.16) 0.27 (210/2.16) 0.23 (200/2.16) 4.55 (200/2.16) 0.23 (200/2.16) 1.00 (200/2.16) 0.32 (170/2.16) 0.18 (170/2.16) 0.13 (170/2.16) 1.40 (180/2.16) 2.76 (200/2.16) 4.78 (220/2.16) 1.0 (200/2.16) 12.0 (200/5.00) 2.9 (190/5.00) 2.5 2.5 (200/5.00) 12.3 (200/5.00)

Temp. at which No. of data data are points generated (extension ( Trange, ) rate S-') 170 190 210 200 200 200 200 170 170 170 180 200 220 200 200 190 200 200 220

"FI valueobtainedfromshearstressversusshearratecurves method (Ref. 22 of Chapter 4) described in Sec. 4.2.2. Source: Ref. 37 of Chapter 5.

4 (0.5-20) 4 (0.6-20) 4 (0.5-20) 5 (0.4-30) 3 (0.2-4) 3 (0.1-0.4) 3 (0.1-0.4) 4 (0.08-0.5) 4 (0.04-0.4) 4 (0.02-0.3) 3 (1-2) 3 (1-2) 3 (1-2) 5 (0.5-20) 3 (0.2-4) 5 (0.5-50) 3 (1-20) 4 (0.1-1) 3 (1-100)

Source: Chap. 5 Ref. 38 38 38 38 23 23 23 39 39 39 23 23 23 38 40 38 38 40 38

in accordancewiththereadout

Appendii D: ManufacturerdSuppliers MFI Equipment

Names and Addresses of Some Manufacturers/ Suppliers of MeltFlow Indexers Automatik Machinery Corporation 9724-A Southern Pine Boulevard Charlotte, NC 28210, USA Tel: (704) 523-7921 Telex: 387056 Brabender Ohg Duisburg Kulturstrasse Postfach 350162 D-4100 Duisburg 1 Germany Tel: 0203-73801-0 Telex: 855603 Carl G. Brimmekamp & Company, Inc. 102 Hamilton Avenue Stanford, CT 08902, Tel: (203) 325-4101 Telex: WUD 965955

D

Ceast S.p.A. Via Asinari di Bernezzo, 70 10146 Torino Italy Tel: (11) 790909-791092/93/94 Telex: 220147 Fax: (11) 799041 Custom Scientific Instruments, Inc. 13, Wing Drive Cedar Knolls, NJ 07927, USA Tel: (201) 538-8500 Telex: 25-4328 Daventest Ltd. Tewin Road Welwyn Garden City Hertfordshire AL7 1AQ England Tel: (0707) 327571 Telex: 23729 DAW F.F. Slocomb Corporation 1400 Poplar Street P.O. Box 1591 Wilmington, DE 19899, USA Tel: (302) 654-8863 Giittfert Werkstoff, Priifmaschinen GmbH Postfach 1220 Siemensstrasse 2 D-6967 Buchen/Odenwald Germany Tel: (06281)691 Telex: 04-66415 Karl Frank GmbH Postfach 1320 D-6940 Weinheim Germany Tel: 06201/84-1 Telex: 465514

425

Kayeness Inc. P.O. Box 709 Morgantown, PA 19543 Tel: (610) 286-7555 Fax: (610) 286-9396 Maruto Testing Machine Company No. 15-4, 2-Chome Shirakawa, Koto-ku Tokyo 135-91 Japan Tel: (03) 643-2111 Pacific Scientific Company 1100 East-West Highway Silver Spring, MD 20910, Tel: (301) 589-4747 Ray-Ran Engineering Kelsey Close Attleborough Fields Industrial Estate Nuneaton CV11 6RS Warwickshire England Tel: (0203) 342002 Telex: 312242 MID= G

Testing Machines Inc. 400 Bayview Avenue Amityville, NY 11701, USA Tel: (516) 842-5400 Telex: 96-1302

Toyo Seiki Seisaku-sho Ltd. 15-4 Takinogawa 5 Chome Kita-ku Tokyo 114 Japan Tel: (03) 916 8181 Telex: 272-2097 TOSEl J Fax: (03) 916-8173

Nomenclature

Symbol

-

A" A

Units

(9.25) (2.53)-(2.56), (5.13)-(5.16) (8.3) (9.5)

Constant Model parameter Heat diffusion coefficient of melt Coefficient whose values are given in Table 9.1 MFI ratio Shift factor Surface area of plate in Fig. 2.1 Function Constant dependent on the nature of the continuous phase Constant dependent on the nature of the continuous phase Coefficient Effective projected areaof molding Frequency term depending on the entropy of activation for flow Constant Coefficient Contact area between melt and roll during calendering

Equation

(8.85) (9.26)

(8.81) (9.13)-(9.16) (8.12) poise

(2.69) (2.73) (9.21) (8.50)

n

Nomenclature

Symbol Constant Constant Function Constant which is normally set equal to unity Constant Coefficient Coefficient Numerical function dependent

(9-25) (4.13) (8.45) (8.80) (2.73) (9.22) (9.13)-(9.16) (8.13), (8.14)

on n

Weight average degree of branching Coefficients Arbitrary adjustable parameter Polymer concentration Constant Constants Proportionality constant Function Coefficient Proportionality constant whose values are given in Table8.1 Proportionality constant Constant Function Proportionality constant Constant Function Specific heat of polymer melt Coefficient Radiation dose Extrudate diameter Nozzle diameter Orifice diameter Draw ratio Critical value of draw ratio Symmetric part of the gradient or' deformation tensor Exponential (where = 2.71828) Independent parameter Specific energy input for thermal loading of a product during compounding

(9.7) (8.71) (2.571, ( 5 . m (6.18)

-

(5.6) (2.63) (8.2), (8.3) (8.46) (9.13)-(9.16) (8.10), (8.11) (5.1) (2.68) (8.33) (5.22) (2.68) (8.34) (8.56), (8.69) (9.15)-(9.16) (9.25)

-

(2.3), (2.4) (5.17) (8.70)

Nomenclature

Symbol Activation energy for viscous flow of polymer melt Value of E detennined under constant shear-rate conditions Value of E determined under constant shear-stress conditions Value of E for polymerP , in the preparation of polyblends Value of E for polymerP , in the preparation of polyblends Force in Figs. 2.1-2.3 Function of Free volume of the continuous phase without the presence of any other dispersed phase Free volume of the multicomponent system at temperature T containing $2 weight fraction of the dispersed phase Function of the geometric parameters based on the shape of the extrusion die Function of the fluid properties, namely, K and which take different forms for different shaped dies Force exerted by the test load L on the polymer in the melt flow indexer Compaction force for a circular disk Compaction force for a flat strip Minimum clamping force Ratio of root-mean-square radii of branched and linear polymers Dynamic storage modulus Dynamic loss modulus Complex modulus Distance from center plane to roll periphery in calendering

kcaVmole kcal/mole kcaVmole kcal/mole kdmole dYn

-

cm’

(8.72a), (8.74), (8.75) (8.72b), (8.74), (8.75) (2.2) (8.9) (8.82), (8.83), (8.84) (8.81), (8.83)

-

dYn dyn

(8.15)

dyn dYn

-

dyn/cm2 dyn/cm* dyn/cm2

(2.15), (5.13), (6.20) (2.16), (5.14) (2.21) (8.23)

Nomenclature

Symbol Initial separation distance cm between two plates during compression molding Separation distance between cm plates at different times during compression molding Height of channel for rectangular cm die One-half of minimum gap width cm in calendering Entrance height of channel for cm rectangular extrusion die Exit height of channel for cm rectangular extrusion die Characteristic geometric cm parameter specific to each geometric shape Gap between two parallel disks cm of a viscometer cm Half-width of gap at exit equal to half-thickness of sheet produced in calendering cm'ldyn Steady-state compliance S" Rate in terms of percent scissions per unit time Constant Coefficient Constant S*-" Consistency index (dcm S'"') X Consistency index whose values (g per 10 min)" are tabulated in Tables 6.7 and 6.8. S'-") Vicosity function value at (g per 10 min)" 9/Mn = 1 for low-shear-rate region line. (&cm S'-") X Viscosity function value at (g per 10 min)" ?/Mm= 1 for high-shear-rate region line. Consistency index for Polymer 1 (dcm S'-") X (g per 10 min)" in the blend Consistency index for Polymer 2 (dcm S*-") X (g per 10 min)" in the blend cm Variable length of cylindrical rod in Fig. 2.3

(8.15)-(8.18)

(8.15)-(8.18)

(8.58a), (8.60a) (8.27)-(8.29), (8.41), (8.42) (8.59a), (8.61a), (8.62a) (8.59b), (8.61b), (8.62b) (8.58a)-(8.66a) (3.7) (8.48)

(2.50) (9.29) (2.74), (2.75) (9.19) (9.25) (2.39), (2.40)

-

(6.8) (6.8) (8.77a), (8.78) (8.77b), (8.78) (2.25), (2.26)

Nomenclature

431 ~

Symbol

Equation Length of barrel Length of extrusion die Length of nozzle Initial length of cylindrical rod Length of flat strip in compression molding Length of extruded thread at decrease in diameter Test load, i.e., dead weight + piston weight Test load Test load Damping constant

Adjustable parameter Adjustable parameter Damping constant Mass flow rate Model parameter Molecular mass of the polymer defined as the sum of the atomic masses of the elements Power index Molecular weight of polymer Number-average molecular weight Viscosity-average molecular weight Weight-average molecular weight Critical. weight-average molecular weight z-Average molecular weight z + l-Average molecular weight Melt flow index Melt flow index at time = 0 Melt flow index determined under test load Melt flow index determined at load condition Melt flow index determined under test load

Units

Nomenclature

432

Symbol

Description

Units

Equation

Melt flow index determined at 5.0-kg load condition Melt flow index for autoclave reactor polymer Melt flow index of branched polyethylene Melt flow index after radiation dosage Melt flow index before start of radiation dosage Melt flow index of linear polyethylene Melt flow index at time t Melt flow index for tubular reactor polymer Melt flow index at first temperature in calendering Melt flow index at second temperature in calendering Melt flow index of the continuous phase at temperature T but without the presence of any other dispersed phase Melt flow index of the multicomponent system at temperature T and containing $2 weight fraction of the dispersed phase Melt flow index of polymer PI at temperature Tp, in blends Melt flow index of polymer P2 at temperature Tp2in blends Melt flow index of blend at compounding temperature T, Melt flow index of polymer 1 at compounding temperature T, Melt flow index of polymer 2 at compounding temperature T, Melt flow index recovery defined as the percentage increase in diameter of extrudate over that of the orifice Melt strength

g per 10 min

(9.10)

g per 10 min

(9-3)

g per 10 min

(94

g per 10 min

(9.25)

g per 10 min

(9.25)

g per 10 rnin

(94

g per 10 min g per 10 min

(9.29) (9.4)

g per 10 min

(8.47)

g per 10 min

(8.47)

g per 10 min

(8.85)

g per 10 min

(8.85)

g per 10 min

(8.71a)

g per 10 min

(8.71b)

g per 10 min

(8.72a)

g per 10 min

(8:72a), (8.73)

g per 10 min

(8.72b), (8.73)

-

(9.23), (9.24)

Nomenclature

433

~~~

Symbol

Po,min

P*

Description Molecular-weight distribution Power-law index whose values are given in Tables 6.7 and 6.8 Term in the slope n1 - 1 of the linear portion in viscosity versus low shear-rate curve Term in the slope n, - 1 of the linear portion in viscosity versus high shear-rate curve Power-law index for polymer PI in the blend Power-law index for polymer Pz in the blend Power index in the Carreau model Primary normal stress difference Secondary normal stress difference Number of molecules of polymers having molecular weights MI, M2, M3, Mi Number of sides of polygonal channel die Power index for polymer PI in the blend Power index for polymer P2 in the blend Pressure Power index Pressure hole error Measured pressure Degree of polymerization Power-index in the General Rheological model whose values are given in Table 6.8 Pressure drop in barrel Pressure drop in nozzle Pressure drop from the center of a “spreading disk” flow in the mold cavity during injection molding Minimum value of Po Pressure between rolls in calendering

Units

Equation

(8.77a), (8.78) (8.77b), (8.78)

(8.65b), (8.66b) (8.79) (8.79)

(9.19), (9.22)

dyn/cm2 dyn/cm2 dyn/cm2

dyn/cm2 dyn/cm2

(8.8), (8.10) (8.27), (8.32), (8.43)

Nomenclature

434 Symbol Pressure between rolls in calendering at first temperature Pressure between rolls in calendering at second temperature Pressure drop in extrusion die Power index Volumetric flow rate Polydispersity index = =,,./R,, Power index Radius of the extruded cord emerging from the die Gas constant =

dynlcm’ dynlcm2

,

dyn/cmz

-

cm’ts

cm cal/mol K

Radius of disk or cone of a viscometer Radius of barrel Radius of nozzle Radius of piston Radius of cylindrical channel extrusion die Instantaneous radiusof the disk or flow length in the mold during injection molding Radius of circular die during compression molding Roll radius in calendering Entrance radius of extrusion die for truncated right cone and constant polygonal channel Exit radius of extrusion die for truncated right cone and constant polygonal channel Elastic strain recovery Die swell ratio of extrudate diameter to die diameter Time Residence time

cm

Cavity filling time during injection molding Melt temperature Trace of the deformation tensor

S

cm cm

cm cm

cm cm

(8.64a)

(8.64b)

S S

“C

-

-

Nomenclature Symbol

Loss tangent = G"/G' Polymer melt temperature Compounding temperature Degradation temperature Glass-transition temperature of polymers Melting temperatureof polymers Highest processing temperature Standard reference temperature equal to T, + 50 Freeze-off temperature in the mold during injection molding ASTM recommended test temperature Temperature at which MFI is required Temperature of M F I measurement for polymerP , Temperature of MFI measurement for polymer P2 Symmetric Cauchy stress tensor Velocity of fluid element during calendering Velocity at the roll surface during calendering Parameter in slip boundary wndition during calendering Velocity components alongxl, x, axes, respectively Weight flow rate Width of channel for extrusion die Width of channel at start of extrusion die Width of channel at end of extrusion die Width of flat strip in compression molding Sheet width in calendering Coordinate in the direction of flow for calendering Dimensionless distance in calendering

-

(2.17)

"C "C "C

-

K

(4.15)

"C "C

-

K

(4.15)

"C

(8.3)

K

(4.14), (4.15)

K

(4.14), (4.15)

"C

(8.71a)) (8.72a)

"C

(8.71b), (8.72b)

dyn/cm2

(2.3) (8.23), (8.36)

W

S

-

(8.27), (8.40) (8.36)

cm

(4.41, (4.5) (8.58b), (8.59b)

cm

(8.60b), (8.61b)

cm

(8.60b), (8.61b)

cm

(8.17), (8.18)

cm cm

(8.53)

-

(8.28)

-

Nomenclature Symbol Effective remaining flow pathin the mold cavity during injection molding Thickness of the frozen material formed in the mold cavity during injection molding Thickness of the original cavity of the mold in injection molding Distances along thexl, x;l, x, axis Positional vector Degree of change in the melt flow index = - MFI)/MFIo Model parameter

-

( M F ,I

Greek Symbols Parameter in the slip boundary condition in calendering Ellis model parameter Coefficient in the viscous heat dissipation equation Proportionality constant Parameter in the slip boundition in calendering Degree of swelling in Figs. and Swelling ratio after annealing in Fig. Power index in the viscous heat dissipation equation Strain related to die swell Shear rate Compounding shear rate Amplitude of the sinusoidal variation of shear rate Shear rate for polymer P , Shear rate for polymer P2 Torque in calendering Phase angle Uniaxial extensional rate Biaxial extensional rate Planar extensional rate

-

(2.28)

Nomenclature

Symbol

units

Equation

Steady shear viscosity Steady shear viscosity function Steady shear viscosity of multicomponent system at temperature T and containing weight fraction of dispersed phase Zero-shear viscosity

poise poise poise

(4.12) (2.9) (8.80)

poise

Viscosity of branched polymer Viscosity of linear polymer Viscosity value read from intersecting asymptotes Uniaxial extensional viscosity Biaxial extensional viscosity Planar extensional viscosity Volume viscosity Dynamic viscosity Imaginary part of complex viscosity Complex viscosity Inherent viscosity

poise poise poise

(2.34), (2.36), (4.10), (4.11) (9.7) (9.7) (2.45), (6.6)

poise posie poise poise poise poise

(2.31) (2.32) (2.33) (2.3) (2.18) (2.19)

Intrinsic viscosity

poise

Half-angle of convergence of stream line at the entrance of die Mold temperature T i e constant and model parameter Value of time constant at shear rate of 50 S” Dimensionless flow parameter Constant vertical taper given by the ratio of the height H , at the end of the channel to the height H , at the start of the channel Constant lateral taper given by the ratio of the width W, at the end of the channel to the width W, at the start of the channel

poise poise

radians “C S

S

(2.20) (4.11), (9.13)(9.16) (4.11), (9.11), (9.12) (2.65)-(2.67)

(8.3)

(2.43), (2.52)(2.561, W ) , (9.17) (9.20), (9.21)

-

(8.29), (8.42) (8.59b), (8.61b)

-

(8.60b), (8.61b)

Nomenclature

Symbol Characteristic time Constant taper factor given by the ratio of the exit radius R2 to the entrance radiusR1 T i e scale of deformation Relaxation time Long-chain branching frequency Shape factors of the extrusion dies Polymer melt density Average extensional stress Shear stress Shear stressfor polymer P, Shear stressfor polymer P2 Special value of shear stress when steady shear viscosity is half the zero-shear viscosity Amplitude of the sinusoidal variation of shear stress Shear stress components of the stress tensor Shear stress component at the wall Normal stress components of the stress tensor Normal stress difference at the wall Extra stress tensor Viscous heat dissipation Power input to calender rolls A and B Weight fraction of dispersed phase Primary and secondary normal stress coefficients Primary normal stress coefficient function Secondary normal stress coefficient function Frequency of oscillations or angular frequency

S

-

(8.64b)

S S

(5.6), (8.19)

-

(8.58b)-(8.62b)

dyn/cm2 dyn/cm2 dyn/cm2 dyn/cm2 dyn/cm2

(2.65) (2.36) (8.77a) (8.7%) (6.2)

dyn/cm2

(2.14)

-

dyn/cm2 dyn/cm2 dyn/cm2

(2.6)-(2.8)

dyn/cm2 dyn/cm2 Joule

(2.3, (2.6) (8.67)-(8.69) (8.56)

-

(8.80)-(8.84)

w/cm3

dcm dcm

(2.10)

dcm

(2.11)

radls

(2.12)-(2.21), (6.15)-(6.20)

Nomenclature

BS DIN IS0

JIS

American Society for Testing and Materials Standards British Deutsches Institut f i r Norrnung International Standards Organization Japanese Industrial Standards

Index

Abbas, K A., Abdel-Khalik, S. I., Abe, D. A., Aciemo, D., Acrivos, A., Adams, J. W. C., Agassant, J. F.,250 Agganval, S. L., May, G., Alle, N., Allen, P. W., Allport, D. C., Alston, W. W., Althouse, L. M., Al’tzitser, V. S., Amico, R. D., Andrews, R. D., Anfimov, B. N., Angerer, G., Aoki, Y., Apte, S. M., Annstrong, R. C.,

440

Ashare, E., Astarita, G., Astill, K N., Autran, M., Au-Yeung, V. S., Aventas, P., Avita, F., Ayres, C. A., Baer, E., Bagheri, R., Bagley, E.B., Baily, E. D., Baird, D. G., Ballenger, T. F., Ballman, R. L., Balmer, T., Bamane, S., Bandyopadhyay, G., Bankar, V. G., Baraam, Z., 23 Bares, J., Baretta, G.,

441

Index Barker, S. J., Barlow, J. W., Barnes, H. Barrie, I. T., Bartenev, G.M., Bastida, S., Batchelor, G. K, Bates, T. W., Baumann, D. K., Baumann, G.F., Becker, E., Belcher, H. V., Belen’kii, B. G., Belova, E. Benbow, J. J., Bennett, K.E., Bernhardt, E. C., Berry, G.C., Bersted, B. H., Berstein, B., Bestul, B., Bhardwaj, I. S., Bhattacharya, S. K, Bigg, D. M., Biletech, H. A., Billmeyer, Jr., F. W., Binding, D. M., Bird, R. B., Birks, A. M., Birnkraut, W. H., Blake, W. T., Bludell, D. J., Blumstein, Blyler, L. L., Boenig, H. V., Bogue, D.C., Boira, M. S., Borring, L., Borzenski, F. J., Boudreaux, Jr., E., Bourne, R., Bouvart, D., Brady, D. G., Brandrup, J.,

Braun, G., Brazinsky, I., Brenner, H., Bretas, R. E. S., Bright, P. F., Brindley, G., Bringer, R. P., Brodkey, R. 171,419 Brydson, J. Burke, J. J., Busse, W. F., Cancio, L. V., Carley, J. F., Carreau, P. J., Casale, Chaffey, C. E., Chakraborty, K B., Chan, C. F., Chan, Y., Chan, Yu., Chapman, F. M., Chard, E. D., Charles, M., Charley, R. V., Chartoff, R. P., Chattopadhyay, S.,

Chauffoureaux, J. C., Chen, I-J., Cheremisinoff, N. P., Chhabra, R. P., Chipalkatti, M. H., Chou, C. H., Chudinov, P. B., Chung, J. T., Churchill, R. W., Churchill, S. W., Ciferri, Cipriani, C., Clark, E. S., Clegg, D. W., Clegg, P. L.,

Index

442 Cogswell, F. N., Coleman, B. D., Collins, E. A., Collyer, A. A., Colwell, R. E., Combs, R. L., Conde, A., Cooper, S. L., Coover, Jr., H. W., Cosway, H. F., 250 Cox, H. W., Cox, R. H., Cox, W.P., Crawley, R. L., Crespi, G., Cross, H., Cross, M. M., Crossan, S. C., Crowder, J. W., Crowson, R. J., Cuculo, J. A., Cuspor, I., Czamecki, L., Daane, J. H., Danckwerts, P. V., Darby, R., Dark, W. A., Daroux, M., Daues, G. W., Davies, J. M., Dealy, J. M., Debennan, C., Denn, M. M., Den Otter, J. L., Denson, C. D., Devia, N., De Kee, D., De Waele, A., Dhimmar, I. H., Diennes, G. J.,

Dillon, R. E., Dobrescu, V., Doolittle, A. K., Dow, J. H., Dubois, J. H., Duffey, H. J., Dunk, R., Dunkley, C. D., DuPre, D. B., Dutta, A., Early, R., Edirisinghe, M. J., Eise, K., Eisenschitz, R., Eldon, R. A., Equiazabal, J. I., Ester, G. M., Evans, J. R. G., Evans, M. E., Everage, A. E., Eyring, H., Falbe, J., Falcone, D. R., Fenner, R. T., Ferguson, J., Femandez, A. M., Ferrall, J. F., Ferry, J. D., Fielding, J. H., Fikham, V. D., Fisa, B., Fisher, E. G., Flory, P. J., Folkes, M. J., Fox, T. G., Frados, J., Frankel, N. A., Franta, W. A., Frechette, F. J., Fredrickson, A. G., French, K W.,

Index Frenkel, R. Fujiki, T., Fujita, H., Fukusawa, Y., Galgoci, E. C., Garcia-Borras, T., Gamer, F.H., Gaskell, R. E., Gaskins, F.H., Gavis, J., Gebauer, P., Geil, P. H., Geisbusch, P., German, R. M., Ghijsels, Gianotti, G., Gilbert, E. H., Gilbert, R. D., Gilby, G. W., Gilmore, G. D., Glanvill, B., Glasscock, S. D., Goedhar, D.J., 5 Goel, D. C., Gogos, C. G., Goldfarb, S. M., Goodrich, J. E., Graessley, W.W., Grateh, S., Gray, T. F., Griffiths, L., Gruver, J. L., 88 Guillet, J. E., Guliana, R. T., Gupta, R. K., Hagler, G. E., 80 Hamieton, C. W., Hammer, W. F., Han, C. D.,

443 Hancock, M., Hanson, D.E., Harper, B. G., Harris, J., Harry, D.H., Hassager, O., Haw, J. R., Haward, R. N., Hayashida, K., Heitmiller, R. F., Hellman, M. Y., Hellmeyer, H. O., Higashitani, K., Hill, C. T., Hill, J. W., Hinkelmann, B., Hlavacek, B., Ho, P. K, Hofman-Bang, N., Hoftijzer, P. J., 5 Hogan, J. P., Hold, P., Holdsworth, P. J., Hollister, E. H., Holmes, L. 80 Holmes-Walker, W. Hori, Howard, G. J., Howard, J. B., Hrymak, N., Huang, C. F., 240 Huang, C. R., Huang, D., Huppler, J. D.,80 Hutton, J. F., Hylton, D.C., Ide, Y., Immergut, E. H., Isayev, I., Ishida, M., Ishida, N.,

80

444

Ishigure, Y., Isner, J., Itadani, K, Ito, K., Jacovic, M. S., Jakopin, S., James, W. H., Janeschitz-Kriegel,H., Janssen, L. P. B. M., Jeffrey, D. J., Jennings, C. N., Jepson, C. H., Jerman, R. E., Jinescu, V. V., John, F. W., Johnson, C.F., Johnson, J. F., 88, Johnson, R. Jones, R. K., 250 Jung, Juskey, V. P., Kaghan, W. S., Kamal, M. R., Kamamoto, T., Kambour, R. P., Kandyrin, L. B., Karnis, J., Kasajima, M., Kataoka, T., Kawasaki, H., Kearsley, E., Kelleher, P. G., Kendall, K., Kenig, S., Khadilkar, C.S., Kihira, H., Kikuchi, N., Kim, K U., Kim, K Y . , Kim, Y. W., King, R. G.,

Index

Kiparissides, C., Kishimoto, J., Kitagawa, K, Kitano, T., Klein, I., Klemn, H. F., Knappe, W., Knutsson, B. Kohan,M. I., Kondo, Kondu, J., Korcz, W. H., Kozichi, W., Kramer, M., Krassig, H., Kraus, G., Krause, S., Krumbock, E., Kuleznev, V. N., Kulichikhin, V. G., Kulkami, M. G., Kunii, D., La Mantia, F. P., Lamb, P., Lamonte, R. R., Landel, R. F., Lau, H.C., Lee, B. L., Lee, T. S., Lee, W. M., Leider, P. J., Leidner, J., Leitlands, V. V., Lem, K. W., R. Lenz, J., Leonov, I., Lewis, G., Liaw, T. F., Lightfoot, E. N.,

445

Index Lin, Y-H., Lobe, V. M., Lodge, S., Longworth, R., Loppe, J. P. Lord, H. Lorntsen, J. M., Loshach, S., Luo, H. L., Lupton, J. M., Lyngaae-Jorgensen, J.,

Lynn, R. E., Lyons, J. W., ,

Macdonald, I. F., Macosko, C. W.,

Maldenado, Malkin, Ya., Manaresi, P., Mangels, J. Manson, J. Marieta, T., Mark, H. F., Marker, L., Markovitz, H., Marmcci, G., Marshall, D.I., Martinez, C. B., Mashelkar, R. Masuda, T., Matsui, M., Matthews, G., Maxwell, B., McAUister, R. McCormick, J. M., McGrath, J. E., McKelvey, J. M., 23, Meares, P.,

Meissner, J., Meister, B. J., Mendelson, R. Menges, G., Mennig, G., Mertz, E. H., Metzner, B., Metzner, P., Michaeli, W., Middleman, S., Mihara, S., Mijovic, J., Milgram, J., Mills, N. J., Minagawa, N., Minnick, L. Minoshima, W., Mitterhofer, F., Mnatsakanov, S. S., Modest, M. F., Mob, W. D., Molau, G. E., Monte, S. J., Mooney,M., Moore, L. D., Morawetz, H., Morneau, G. Moroni, Mortimer, G. Munari, Munstedt, H., Mutel, T.,20 Mutsuddy, B. C., Naar, R. Z., Nadkami, V. M.,

Nagatsuka, Y., Nakajima, N., Nakatsuka, T., Nation, R. G.,

Index Nazabal, J., Nazem, F., Nesterov, V. V., Neverov, N., Nevin, Newnham, R. E., Nicolais, L., Nielsen, L. E., Nikiton, Yu. V., Nikolayeva, N.E., Nishijima, K., Nishimura, T., Nissan, H., Noel, F., Noll, W., Norwood, D.D., Noshay, Oda, K., Ojama, T., Okada, T., Okubo, S., Oliver, D.R., Onogi, S., Orbey, N., Ostwald, W., Oyanagi, Y., Padget, J. C., Palmgren, H., Pao, Y. -H., Pappano, W., Parrott, R. G., Patel, R. D., Patton, P. Paul, D.R., Paulson, D.C., Peacock, D.G., Pearson, J. R. Pechoc, V., Pelagatti, U., Penwell, R. C., Peticolas, W. T., Petrie, C. J. S., Philippoff, W.,

Pilati, F., Platzer, N. Plochocki,

J., P.,

Plotnikova, E. P., Polinski, J., Pollet, W. F. O., Pollock, D., Porter, R. S.,

Powell, R. L., Prest, Jr., W. M., Price, M. B., Pritchard, J. H., Prokunin, N., Pryde, C. Puglia, L., Qureshi, S., Raadsen, J., Rabinowitsch, B., Radushkevich, B. V., Raible, T., Rajora, P., Ram, Ramamurthy, V., Ranalli, F., Rao, D. Ray, Reddy, K.R., Reiner, M., Rhi-Sausi, J., Richardson, P. N., Richter, C. Rideal, G.R., Riley, D.W., Robeson, L.M., Roff, W. J., Rogers, M. G., Rokudai, M., Romanini, D., Roovers, J.,

Index Rosato, D., Rosato, D. V., Rosen, M. R., Rosevear, J., Rubin, I., Ruckenstein, E., Rudd, J. F., Rudin, Runt, J., Russell, R. J., Russo, S., Ryan, M. E.,

Schurtz, J. F., Scott, G., J. R., Scott Blair, G. W., Selk, S., Sen, M., Seshadri, S. G., Severs, E.T., Shah, P. L., Shanks, R. A., Sharma, Y. N., Shekhtmeister, I. E., Shenoy, V.,

Sabia, R., Sabsai, 0.Yu., Sacks, M. D., Saechtling, H., Saini, D. R.,

Sakai, T., Sakamoto, K, Salvadori, M. G., 255 Samulski, E.T., Sandford, C., Sasahara, M., Savadori, Saxton, R. L., Scheiffele, G. W., Schenkel, G., Schmidt, W. P., Schmitz, O., Schott, H., Schowalter, W. R., Schramm, G., Schreiber, H. P., Schulken, R. M., Schultz, J. M.,

Shenoy, U. V., Sherliker, F. R., Shete, P., Sheu, R. S., Shida, M., Shirata, T., Shroff, R. N., Shusman, L., Sieglaff, C. L., Siegmann, A., Simon, R. H. M., Singleton, C. J., Siskovic, N., Sitzgber, K., Skaar, E. C., Skelland, H.P., Skinner, S. J., Slee, J. D., Slonaker, D. F.,

448

Index

Smeykal, J. P., Smith, D.J., Smith, H. V., Smith, T. G., Southern, J. H., Sowa, C. J., Speed, C. S., Spencer, R. S., Sperati, C. Sperling, L. H., Sprigs, T. W., Springer, P. W., Spruiell, J. E.,

Thomas, D.G., Thorne, J. L., Thornton, B. Tichy, J. Tiu, C., Tobolsky, V., Todd, D.B., Tolstukina, F. S., Tomis, F., Toor, H. L., Tordella, J. P., Toth, T., Trela, W., Tremayne, P., 20 Trishman, Jr., C. Trottnow, R., Truesdell, C., Tsutsui, M.,87

Sroog, C. E., Stade, K, Stade, K.H., Stamhuis, J. E., Starkweather, H. W., Stefan, J., Steingiser, S., Stephenson, S. E., Stephenson, T., Stevenson, J. F., Stewart, W. E.,

Uebler, E. Uhland, E., Uhlherr, P. H. T., Upadhya, R. S., Usagi, R., Utracki, L.

Story, v., 250 Suetsugu, Y.,

Suganuma, Sugerman, G., suzuki, S., Swan, D., Swanborough, Tadmor, Z., Tager, Takahashi, J., Takahashi, M., Takayanagi, M., Tanaka, H., Tanford, C., Tanner, R. I., Taylor, N. H., Taylor, R., Teutsch, E. O.,

-

Valle, Jr., C. F., Van der Weghe, T., Van Krevelan, D.W., 5, Van Oene, H., Van Rijckevorsel, J., Van Wazer, J. R., Vasil’eva, N.P., Vaughan, G. Viilasenor, R. G., Vinogradov, G. V., Vlachopoulos, J., Wagner, M.H., Wales, J. L. S., Walker, B. M., Walker, J., Walters, K.,

Index Ward, I. M., Wasiak, Watkins, J. M., Watts, M. P. C., Weemes, D. E., Weill, A., Weiss, Y., Weissenberg, K., Wentz, R. P., Westover, R. F., Whalen, T. J., Whelan, J. P., White, J. L.,

Whorlow, R. W., Wild, L., Wilkinson, W. L., Williams, G., 240 Williams, M. C., Williams, M.L., Williams, R. M., Willmouth, F. M.,

Winter, H. H., Wissbrun, K.F., Woldering, J. F., Wolff, D., Wong, W. M., Wooten, Jr., W. C., Wortberg, J., Wright, B., Wright, D. G. M., Wu, P. C., WU, S.,

Yamada, M., Yamashita, S., Yanovsky, Yu. G., Yoo, H. J., Yu, T.C., Zabugina, M. P., Zabusky, H.H., Zachariades, E., Zahorski, S., Zaikov, G. E., Zakharenko, N. V., Zapas, L., Zavadsky, E., Ziabicki, A., Zingel, U.,

.

Subject Index

ABS

Acrylonitrile butadiene styrene)

Acetal, master rheogram, Acrylic, master rheogram, unified curve, viscous dissipation, Acrylonitrile butadiene styrene, master rheogram, Activation energy, tabulated values, Addition polymerization Polymerization) Amorphous polymer, Antioxidants, Antistatic agents, Apparent viscosity, Arrhenius equation, modified, Arrhenius-Erying equation, ASTM specifications,

Bagley correction, also Extrudate Barus effect, swell) Biaxial extension, Binders, Bingham plastic, Blend, master rheogram, HDPEPMMA, PMWOM, PP/HDPE, PSPMMA, PSPOM, Blending, also Polyblending) Block copolymer, Blow molding, Branched polyethylene, Branched polymer, Branching, degree of, frequency, secondary, Breaking stretch ratio,

Index Brittle temperature, BSR Breaking stretch ratio) Calendering, Capillary type rheometer, circular orifice, slit orifice, Carreau model, altered modified, modified, Catalyst activation temperature, Caution in MFI measurements, Cavity, filling time, thickness, Cellulosics, master rheogram, unified curve, viscous dissipation, Characteristic time, Clamping force, Clarity, Coalesced plot, also Master rheogram) complex viscosity, extensional viscosity, normal stress difference, storage modulus, viscosity upgraded, Commodity plastic, Compatibility, thermodynamic, Complex modulus, Complex viscosity, coalesced plots, Compliance, Compounding, shear-rate, temperature, Compression molding, Condensation polymerization Polymerization) Cone-n-plate viscometer, Consistency index values, Constitutive equation,

Converging flow, Copolymer, ABS, master rheogram, activation energy, block, EVA, master rheogram, graft, HBMET, master rheogram, random, -

SAN, master rheogram, SBS, master rheogram, uniform, VCVA, master rheogram, Cox-Mertz method or rule, Creep, Critical shear rate, Cross-linking, Crystal, Crystalline polymer, Crystallinity, degree of, Crystallization temperature, CSR (see Critical shear rate) Curing, Damping constant, Dart impact, Data in master curves, blends, copolymers, iilled polymers, homopolymers, PVC formulations, recycled polymers, Deborah number, Defects, fringe patterns, matte,

Index

452 [Defects] melt fracture, sharkskin, splay mark, weld line, Degradation, hydrolytic, Degradation temperature, Die swell, (see also Extrudate swell) degree of, Dilatant fluid, Draw ratio, Draw resonance, Dynamic loss modulus, Dynamic shear flow, 58 Dynamic storage modulus, Dynamic viscosity, Effective cavity thickness, Elastic-energy correction, Elastic modulus, Elastic solid, Elasticity, Ellis model, modified, Elmendorf tear, Elongation, at break, ultimate, Energy for injection, Engineering thermoplastics, Entrance effect, Environmental stress cracking, ESC (see Environmental stress cracking) Ethylene-vinyl acetate, master rheogram, EVA (see Ethylene-vinyl acetate) Extenders, Extension, biaxial, planar, uniaxial, Extensional flow,

Extensional stresses, Extensional viscometer, Extensional viscosity, biaxial, coalesced plots, planar, uniaxial, Extrudate distortion, Extrudate swell, Extruder screw, Extrusion, pressure drop, cylindrical channel, polygonal channel, rectangular channel, truncated right cone, Fiber spinning, Filled polymer, master rheograms, Filler, calcium carbonate, carbon black, fibrous, glass fibers, particulate-type, platelike, quartz, titanium dioxide, Flame retardants, Flashing, Flexural stress, Flow, classification, converging, curve, dynamic, extensional, oscillatory, secondary, shear, Fluid, Bingham plastic, dilatant, Ellis,

Subject Index [Fluid] Newtonian, non-Newtonian, pseudoplastic, rheopectic, second-order, shear-thickening, shear-thinning, thixotropic, viscoelastic, Free-radical polymerization (see Polymerization) Free volume, Fringe patterns, Frozen-in stresses, Fungicides, General Rheological model, modified, Glass-transition temperature,

453 HIPS (see High impact polystyrene) Homopolymer, (see also Polymer) Hydrodynamic theory, with slip, without slip, Hydrolysis time, Hydrolytic degradation, Hydrolytic stability, Ideal liquid, Impact strength, Inherent viscosity, Injection molding, Interpenetrating polymer network, Intrinsic viscosity, IPN Interpenetrating polymer network) Irradiation, Isotactic index, Izod impact, Jet swell,

Gloss, Graft copolymer, HDPE (see High density polyethylene) Heat diffusionmefficient, Heat-distortion temperature, Heat shock failure, High density polyethylene,

master rheogram, coalesced plots, complex viscosity, extensional viscosity, normal stress difference, storage modulus, viscosity upgraded, unified curve, viscous dissipation, High impact polystyrene,

(see

also Extrudate swell)

LDPE (see Low density polyethylene) Linear low density polyethylene,

master rheogram, coalesced plot, normal stress difference, unified curve, viscous dissipation, Linear polyethylene, Linear polymer, Liquid-crystalline polymer, master rheogram, LLDPE (see Linear low density polyethylene) Low density polyethylene, 6,

master rheograms,

Subject Index

454 [Low density polyethylene] coalesced plots, normal stress difference, 187, 188 viscosity upgraded, 179, 192 unified curve, viscous dissipation, 271 Loss modulus, 59, 82, 84 Lubricants, 20, 21, 26 Master curve, 137, 344, 345 Master rheograrn for, also Coalesced Plot) ABS, 157-159 acrylic, 147, 148 cellulosics, 145-147 EVA, 159, 161 filled polymers, 167-169 HBAPET copolymer, 162, 163 HDPE, 142 HDPEPMMA blend, 164 LDPE, 138-141 LLDPE, 144 nylon, 148-150 PAr, 155, 157 PAS, 153, 155 PC, 119, 152 PEEK, 154, 156 PEI, 154, 156 PES, 153, 155 PET, 149, 151 P " A / P O M blend, 165 polyester elastomer, 160, 161 POM, 148, 149 PP, 144, 145 PP/HDPE blend, 162-164 PPO, 151 PPS, 151-153 PS, 145, 146 PSPMMA blend, 164, 165 PSPOM blend, 165, 166 PVC formulations, 165-167 PVDF, 150, 152 recycled polymers, 169-171 S A N , 157, 158 SBS, 157

[Master rheogram for] TPE, 160-162 UHMWF'E, 142, 143 VCVA, 159, 160 Material functions, 58, 59, 63 Matte, 72 Matteness Matte) Melt Flow Index, 115-117 basic principle, 116 definition, 116, 123, 394 load relationship, 123 measurements-caution, 390 of rubber mixes, 338, 339 origin, 115 read-out method, 123, 124 temperature relationship, 125 utility, 120, 121 viscosity relationship, 124 Melt Flow Indexer, 107, 394 manufacturers/suppliers,424-426 Melt Flow Index Recovery, 329, 332, 394 Melt fracture, 72, 395 Melt rheology, 27, 37, 40, 41, 53 Melt strength, 330, 331, 395 Melting temperature, 10, 12, 17 Merrington effect, 69 (see also Extrudate swell) MFI Melt Flow Index) MFTR Melt Flow Index Recovery) MFI123, 227-237 Mixing, 36,37 Model, Carreau, 78 altered modified, 214, 215 modified, 207,209,221,222 Ellis, 78 modified, 207, 210 General Rheological, 78, 79, 207, 208, 227 modified, 222 Ostwald-de Waele power-law, 77 modified, 207,211, 222, 241, 252 power-law, 77

Subject Index [Model] modified, 207, 211, 222, 241, 252 rheological, 76 Modulus, dynamic loss, 59, 82, 84 dynamic storage, 59, 82, 84 Moisture-sensitive polymer, 120 Mold filling, 336 Molding, 29-36 blow, 33 compression, 31, 32, 244-250 conditions, 40 injection, 29, 238-244 rotational, 36 slush, 36 transfer, 34, 35 Molecular mass, 3 Molecular weight, 3-6, 27-29, 395 average, 3-5 number, 4, 320, 322, 323 weight, 5, 88, 314, 317, 319 2, 5 z+l, 5 distribution, 5-6, 27-29, 324 effect on viscosity, 124-125 Mooney value, 338,339, 395 MS (see Melt strength) Multicomponent polymeric systems, 281307 Newtonian fluid, 65, 67 Non-Newtonian flow, Non-Newtonian fluid, 64-66 Normal stress coefficient, 58, 79, 395 Normal stress difference, 56, 57, 79, 82, 99, 101, 215, 249,395 coalesced plots, 184-193 unified plots (predicted), 217-221 Noryl, 14 (see also Polyphenylene oxide) Number-average molecular weight, 4, 320, 322,323 Nylon, 13, 19, 22 master rheogram, 148-150 coalesced plot, normal stress difference, 188, 190

[NYW unified curve, viscous dissipation, 278 Nylon-12, 19 Nylon-6, 6, 13, 19, 22 Nylon-66, 11 Olefmics, 13 Olefinic-type thermoplastic elastomer, master rheogram, 160, 162 Oscillatory flow, 58 Oscillatory shear, 100 Ostwald-de Waele power-law model, 77 modified, 207, 211, 222, 241, 252 PAr (see Polyarylate) Parallel-disk viscometer, 100 PAS (see Polyaryl sulfone) PBT (see Poly(buty1ene terephthalate)) PC (see Polycarbonate) PE (see Polyethylene) PEEK (see Polyether ether ketone) PE1 (see Polyetherimide) PES (see Polyether sulfone) PET (see Poly(ethy1ene terephthalate)) Phase angle, 58 PIB (see Poly(isobuty1ene)) Pigments, 20, 21 Planar extension, Plastic, commodity, 13 Plasticization, 329 Plasticizers, 20, 21, 205, 329, 395 PMMA (see Polymethyl methacrylate) Polyacetal, 19 master rheogram, 148, 149 Polyamide, 293, 294, 297 Polyarylate, 15 master rheogram, 155-157 Polyaryl sulfone, 14 master rheogram, 153, 155 Polyblending, 284, 298 Polybutadiene, 133 Poly(buty1ene terephthalate), 318, 321, 327, 347, 349, 351

Subject Index

[Poly@utylene terephthalate)] unified curve, viscous dissipation, 279 Polycarbonate, 13, 14, 305, 347 master rheogram, 149, 150, 152 unified curve, viscous dissipation, 280 Polydisperse, 4 Polydispersity index, 5 Polyester, 13 Polyester elastomer, master rheogram, 160, 161 Polyether ether ketone, 14 master rheogram, 154, 156 unified curve, viscous dissipation, 282 Polyetherimide, 15 master rheogram, 154- 156 Polyether sulfone, 14 master rheogram, 153-155 unified curve, viscous dissipation, 281 Polyethylene, 13, 16, 19, 76, 137, 313315, 324, 326,332-336, 340342, 368, 371, 375, 376,384 Poly(ethy1ene terephthalate),11, 236, 306, 347, 349, 352 copolymer of, 16 master rheogram, 149, 151 unified Curve, viscous dissipation, 279 Poly(isobutylene), 137 Polymer, 6, 9-12 amorphous, 1 0 , l l blends, 17-19 master rheograms, 162-165 branched, 6, 72 classification, 8 cross-linked, 18 crystalline, 9, 11 degradation, 32, 40 filled, 20, 120 master rheograms, 167-168 interpenetrating network, 18 linear, 72

[Polymer] liquid crystalline, 16, 17 master rheogram, 162, 163 recycled, 26, 27 master rheograms, 169-171 semicrystalline, 11 specialty, 15 structure, 12 waste, 26, 27 Polymerization, 1-2 addition, 1, 392 catalyst for, 6, 9 condensation, 2 degree of, 3, 27 free-radical, 2 method, 7 step-growth, 2 Polymer processing, 29-37, 238-307 Polymer processing problems, 38 Polymethyl methacrylate, 11, 13, 17, 19, 40, 85, 288,291,292,294, 297 Polyphenylene oxide, 14, 17 master rheogram, 151 Polyphenylene sulfide, 14, 340, 341 master rheogram, 151, 153 Polypropylene, 11, 13, 19, 26, 76, 137, 269, 290,297, 302,335-340, 343-346,348-350, 356,358361,376 master rheogram, 144, 145 coalesced plots, extensional viscosity, 199 normal stress difference, 185-187 viscosity upgraded, 178-181 unified curve, viscous dissipation, 274 Polystyrene, 11, 13, 19, 76, 133, 137, 203, 204,269, 292,293,297, 303,304,325, 329, 330, 359 master rheogram, 145, 146 coalesced, plots, complex viscosity, 195, 197, 198 extensional viscosity, 195-197, 199, 200 storage modulus, 195, 197, 198

Subject Index .[Polystyrene] unified curve, viscous dissipation, Poly(viny1 chloride), calendering, power-input, pressure, torque, master rheogram, pressure profiles, Poly(viny1idene fluoride), master rheogram, POM (see Polyacetal) Power-law index values, Power-law model, modified, PP (see Polypropylene) PP0 (see Polyphenylene oxide) PPS (see Polyphenylene sulfide) Pressure gradient, Pressure-holeerror, , Processing, ceramic, metal, polymer, problems, Process time, PS (see Polystyrene) Pseudoplastic fluid, PVC (see Poly(viny1 chloride)) PVDF (see Poly(viny1idene fluoride))

Rabinowitsch-Weissenberg equation, Random copolymer, Reaction temperature, Recoil, constrained, Recoverable shear, Recycled polymer, Relaxation time, Residence time, Rheological models, Carreau,

factor,

[Rheological models] altered modified, modified, Ellis, modified, General Rheological, modified, Ostwald-de Waele power-law, modified, Rheology melt, 40, Rheometer, (see also Viscometer) capillary type, circular orifice, slit orifice, screw-extrusion type, Rheometry, Rheopectic fluid, Rotational molding, Rotational viscometer,

SAFT (see Shear adhesion failure temperature) S A N (see Styrene acrylonitrile) SBR (see Styrene butadiene rubber) SBS (see Styrene butadiene styrene) Screw-extrusion type rheometer, Secondary flow, Second-order fluid, Semicrystalline polymer, Shear adhesion failure temperature, Shear flow, steady simple, unsteady simple, Shear rate, compounding, critical, modified, Shear strain, 55 Shear stress, 55, Shear-thickening fluid, Shear-thinning fluid, Shift

Subject Index

'

Silanes, Simultaneous interpenetrating network, SIN (see Simultaneous interpenetrating network) Slip, Slush molding, SMA (see Styrene maleic anhydride) Solution viscosity, Specialty polymer, Spiral flow length, Spreadsheet program, Spriggs model, Stability, hydrolytic, Stabilization, Stabilizers, Standard testing conditions, Step-growth polymerization(see Polymerization) Stick-slip mechanism, Stiffness, Storage modulus, coalesced plots, Stress extensional, frozen-in, shear, tensor, transverseflongitudinal, Stress cracking, Stress crack resistance, Stress growth, Stress relaxation, Styrene acrylonitrile, master rheogram, Styrene butadiene rubber, Styrene butadiene styrene, master rheogram, Styrene maleic anhydride, Styrenics, Surface-treatment, 23-25 Surging, (see also Draw resonance) Swelling ratio, Swelling test,

Tear strength, Temperature, brittle, catalyst activation, compounding, crystallization, degradation, glass-transition, heat-distortion, highest processing, melting, reaction, shear adhesion failure, Tenacity, Tensile strength, at break, at yield, ultimate, Terpolymer, Test load, Thermoforming, Thermoplastics, engineering, Thermostabilizers, Thixotropic fluid, Time, cavity filling, characteristic, hydrolysis, process, relaxation, residence, scale of deformation, Time constant, Titanates, TPE (see Olefinic-type thermoplastic elastomer) Transfer molding, Tubeless siphon, Uebler effect,

UHMWPE (see Ultra high molecular weight polyethylene) Ultimate elongation,

Ultimate tensile strength, 352, 354, 357, 396 Ultra high molecular weight polyethylene, 27, 31 compression molding, 244-250 master rheogram, 142, 143 Uniaxial extension, 61 Unification technique, 136 extensions, 181, 184-200 upgradation, 177- 181 Uniform copolymer, 15-17 VCVA (see Vinyl chloride-vinyl acetate) Vinyl chloride-vinyl acetate, 134 master rheogram, 159, 160 Viscoelastic behavior, 41, 53 Viscoelastic fluid, 66, 68-76 Viscometer, 96 (see also Rheometer) cone-n-plate, 98 extensional, 107-110 parallel-disk, 100 rotational, 96 Viscometric functions, 58 Viscosity, 58 coalesced plots, 138-171, 179-183 complex, 81 dynamic, 59, 393 inherent, 124, 325, 326 intrinsic, 124, 325, 329, 394 MFI relationship, 124 shear, 77-79

[Viscosity] solution, 324 zero-frequency, 81 zero-shear, 76, 88 modified function values, 209, 210, 212, 213 Viscosity curve, 28, 77 Viscous dissipation, 33, 263, 269-282 Viscous liquid, 27, 53, 66 Volume viscosity, 55, 56 Vortex, 73, 396 Vortices (see Vortex) Wagner’s relationship, 80 altered modified, 219 modified, 215, 223 Weight-average molecular weight, 5, 88, 314, 317, 319 Weissenberg effect, 68, 396 Wetting agents, 25 W-L-F equation, 85, 326, 327 WF-type equation, modified, 136 Yield stress, 65, 396 z average molecular weight, 5

z + l average molecular weight, 5 Zircoaluminates, 25 Zirconates, 24