Rheology Of Filled Polymer Systems

Rheology Polymer of Filled Systems Aroon V. Shenoy Advisory Consultant Pune, India KLUWER ACADEMIC PUBLISHERS D...

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Rheology

Polymer

of

Filled

Systems

Aroon V. Shenoy

Advisory Consultant Pune, India

KLUWER ACADEMIC PUBLISHERS

DORDRECHT/BOSTON/LONDON

Library of Congress Cataloging-in-Publication data

ISBN 0-412-83100-7

Published by Kluwer Academic Publishers P.O. Box 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America by Kluwer Academic Publishers, 101 Philip Drive, Norwell, MA 02061, U.S.A. In all other countries, sold and distributed by Kluwer Academic Publishers, P.O. Box 322, 3300 AH Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved © 1999 Kluwer Academic Publishers No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without prior permission from the copyright owner.

Printed in Great Britain

Preface

Polymeric materials have been replacing other conventional materials like metals, glass and wood in a number of applications. The use of various types of fillers incorporated into the polymer has become quite common as a means of reducing cost and to impart certain desirable mechanical, thermal, electrical and magnetic properties to the polymers. Due to the energy crisis and high prices of petrochemicals, there has been a greater demand to use more and more fillers to cheapen the polymeric materials while maintaining and/or improving their properties. The advantages that filled polymer systems have to offer are normally offset to some extent by the increased complexity in the rheological behavior that is introduced by the inclusion of the fillers. Usually when the use of fillers is considered, a compromise has to be made between the improved mechanical properties in the solid state, the increased difficulty in melt processing, the problem of achieving uniform dispersion of the filler in the polymer matrix and the economics of the process due to the added step of compounding. It has been recognized that addition of filler to the polymer brings a change in processing behavior. The presence of the filler increases the melt viscosity leading to increases in the pressure drop across the die but gives rise to less die swell due to decreased melt elasticity. The decrease in melt elasticity can raise the critical shear rate at which melt fracture during extrusion starts to occur and hence one could often consider increasing throughput rate in the case of filled polymer melt processing. The purpose of the present book is to treat the rheology of filled polymer systems in as much detail as possible. With the idea of addressing readers of this book who may come from different

backgrounds, a concerted effort has been made to provide the initial three chapters with material needed for familiarizing with the basics about polymers, fillers, physico-chemical interactions between the two, rheology and rheometry. The first chapter introduces the subject and gives an overview. It briefly discusses the various types of polymers and fillers that can go into the formation of filled polymer systems. It also gives an outline about the physico-chemical interactions between polymers and fillers. The second chapter deals with the fundamentals of rheology and provides definitions of all the basic rheological parameters. It dwells on the non-Newtonian character of filled polymeric systems and explains the various anomalies that are encountered during the flow of viscoelastic materials. Various viscoelastic phenomena are depicted and these give an idea about the complexities involved in the flow of polymeric materials, which gets further complicated in the presence of fillers. The third chapter presents some of the different methods of rheological measurements. The entire range of rheometers has not been explained here as the focus in this chapter has been to include only those that find relevance to filled polymer characterization. The fourth chapter presents some constitutive theories and equations for suspensions. Suspension rheology normally deals with the flow behavior of two-phase systems in which one phase is solid particles like fillers but the other phase is water, organic liquids or polymer solutions. Literature on suspension rheology does not include flow characteristics of filled polymer systems. Nevertheless, this chapter needs to be included as the foundations for understanding the basics of filled polymer rheology stem from the flow behavior of suspensions. In fact, most of the constitutive theories and equations that are used for filled polymer systems are borrowed from those that were initially developed for suspension rheology. Chapter 5 goes into the details of how to prepare filled polymer systems. It discusses the criteria for good mixing and the various mixing mechanisms by which fillers get compounded with polymers. The compounding techniques are discussed and compounding/mixing variables are highlighted so that the sensitivity of these variables is understood in order to obtain well-dispersed filled polymer systems under optimum conditions. Chapters 6 to 9 discuss the steady shear viscous properties, steady shear elastic properties, unsteady shear viscoelastic properties and extensional flow properties, respectively. The effect of filler type, size, size distribution, concentration, agglomerates, surface treatment as well as the effect of polymer type are elucidated. The tenth chapter has been

included to recapitulate the important aspects discussed in the presented work. It is hoped that this book will provide all the necessary background needed to understand the various aspects relating to the rheology of filled polymer systems so that even new entrants to this exciting field may benefit from the information. For those who have already whetted their appetite with a taste for this research area, it is hoped that this book will provide complete details under one cover and entice them to probe into vacant areas of research that may become obvious to them on reading this book. Aroon V. Shenoy

Contents

Preface ............................................................................

ix

1.

Introduction .............................................................

1

1.1 Polymers ......................................................................

1

1.1.1

Thermoplastics, Thermosets and Elastomers ..................................................

1

1.1.2

Linear, Branched or Network Polymers .......

2

1.1.3

Crystalline, Semi-Crystalline or Amorphous Polymers ..................................

5

1.1.4

Homopolymers ............................................

6

1.1.5

Copolymers and Terpolymers .....................

7

1.1.6

Liquid Crystalline Polymers .........................

9

1.2 Fillers ............................................................................

9

1.2.1

Rigid or Flexible Fillers ................................

10

1.2.2

Spherical, Ellipsoidal, Platelike or Fibrous Fillers .............................................

10

Organic or Inorganic Fillers .........................

11

1.3 Filled Polymers ............................................................

11

1.4 Filler-Polymer Interactions ...........................................

16

1.2.3

1.4.1

Filler Geometry ...........................................

18

1.4.2

Volume Fraction ..........................................

19

1.4.3

Filler Surface ...............................................

19

1.4.4

Wettability ...................................................

19

1.4.5

Filler Surface Treatment ..............................

21

This page has been reformatted by Knovel to provide easier navigation.

v

vi

2.

Contents 1.5 Rheology ......................................................................

39

References ..........................................................................

43

Basic Rheological Concepts ..................................

54

2.1 Flow Classification .......................................................

55

2.1.1

Steady Simple Shear Flow ..........................

55

2.1.2

Unsteady Simple Shear Flow ......................

59

2.1.3

Extensional Flow .........................................

62

2.2 Non-Newtonian Flow Behavior ....................................

66

2.2.1

Newtonian Fluids ........................................

66

2.2.2

Non-Newtonian Fluids .................................

67

2.2.3

Viscoelastic Effects .....................................

71

2.3 Rheological Models .....................................................

79

2.3.1

Models for the Steady Shear Viscosity Function ......................................................

79

Model for the Normal Stress Difference Function ......................................................

84

Model for the Complex Viscosity Function ......................................................

86

Model for the Dynamic Modulus Functions ....................................................

90

Models for the Extensional Viscosity Function ......................................................

93

2.4 Other Relationships for Shear Viscosity Functions .....................................................................

99

2.3.2 2.3.3 2.3.4 2.3.5

2.4.1

Viscosity-Temperature Relationships ..........

2.4.2

Viscosity-Pressure Relationship .................. 103

2.4.3

Viscosity-Molecular Weight Relationship ................................................ 104

References ..........................................................................


99

104

Contents 3.

Rheometry ............................................................... 112 3.1 Rotational Viscometers ................................................

113

3.1.1

Cone and Plate Viscometer ......................... 115

3.1.2

Parallel-Disc Viscometer ............................. 117

3.2 Capillary Rheometers ..................................................

118

3.2.1

Constant Plunger Speed Circular Orifice Capillary Rheometer ................................... 119

3.2.2

Constant Plunger Speed Slit Orifice Capillary Rheometer ................................... 124

3.2.3

Constant Speed Screw Extrusion Type Capillary Rheometers .................................. 124

3.2.4

Constant Pressure Circular Orifice Capillary Rheometer (Melt Flow Indexer) ....................................................... 126

3.3 Extensional Viscometers .............................................

128

3.3.1

Filament Stretching Method ........................ 128

3.3.2

Extrusion Method ........................................ 130

References ..........................................................................

4.

vii

131

Constitutive Theories and Equations for Suspensions ........................................................... 136 4.1 Importance of Suspension Rheology ..........................

136

4.2 Shear Viscous Flow .....................................................

137

4.2.1

Effect of Shape, Concentration and Dimensions on the Particles ........................ 137

4.2.2

Effect of Size Distribution of the Particles ...................................................... 147

4.2.3

Effect of the Nature of the Particle Surface ....................................................... 150

4.2.4

Effect of the Velocity Gradient ..................... 150


viii

Contents 4.2.5

Effect of Flocculation ................................... 151

4.2.6

Effect of the Suspending Medium ................ 153

4.2.7

Effect of Adsorbed Polymers ....................... 154

4.2.8

Effect of Chemical Additives ........................ 160

4.2.9

Effect of Physical and Chemical Processes ................................................... 160

4.2.10 Effect of an Electrostatic Field ..................... 162

5.

4.3 Extensional Flow ..........................................................

164

References ..........................................................................

167

Preparation of Filled Polymer Systems ................ 175 5.1 Goodness of Mixing .....................................................

175

5.2 Mixing Mechanisms .....................................................

183

5.3 Compounding Techniques ..........................................

186

5.3.1

Selection Criteria ......................................... 186

5.3.2

Batch Mixers ............................................... 189

5.3.3

Continuous Compounders ........................... 192

5.3.4

Dump Criteria .............................................. 218

5.4 Compounding/Mixing Variables ..................................

221

5.4.1

Mixer Type .................................................. 223

5.4.2

Rotor Geometry .......................................... 224

5.4.3

Mixing Time ................................................. 225

5.4.4

Rotor Speed ................................................ 229

5.4.5

Ram Pressure ............................................. 229

5.4.6

Chamber Loadings ...................................... 231

5.4.7

Mixing Temperature .................................... 232

5.4.8

Order of Ingredient Addition ........................ 236

References ..........................................................................


237

Contents 6.

7.

8.

ix

Steady Shear Viscous Properties .......................... 243 6.1 Effect of Filler Type ......................................................

244

6.2 Effect of Filler Size .......................................................

246

6.3 Effect of Filler Concentration .......................................

248

6.4 Effect of Filler Size Distribution ....................................

262

6.5 Effect of Filler Agglomerates .......................................

272

6.6 Effect of Filler Surface Treatment ................................

273

6.7 Effect of Polymer Matrix ..............................................

279

6.8 Unification of Steady Shear Viscosity Data .................

287

References ..........................................................................

303

Steady Shear Elastic Properties ............................ 312 7.1 Effect of Filler Type ......................................................

313


315


317


321


321


323


330

References ..........................................................................

332

Unsteady Shear Viscoelastic Properties .............. 338 8.1 Effect of Filler Type ......................................................

344


344


345


350


356


360


372


x

9.

Contents 8.8 Effect of Matrix Additives .............................................

387

References ..........................................................................

390

Extensional Flow Properties .................................. 395 9.1 Effect of Filler Type ......................................................

396


400


402


405

References ..........................................................................

409

10. Concluding Remarks .............................................. 416 Appendices .................................................................... 425 Appendix A Glossary ..........................................................

425

Appendix B ASTM Conditions and Specifications for MFI ...............................................................................

430

Appendix C Data Details and Sources for Master Rheograms ..................................................................

433

Appendix D Abbreviations ..................................................

439

Appendix E Nomenclature .................................................

441

Appendix F Greek Symbols ...............................................

449

Author Index .................................................................. 455 Index ............................................................................... 469


Introduction

I

1.1 POLYMERS Polymers are high molecular weight organic substances that have usually been synthesized from low molecular weight compounds through the process of polymerization, using addition reaction or condensation reaction. 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, and reasonably high molecular weight polymers can be formed in a short time by this polymerization. In condensation polymerization, the reaction takes place between two polyfunctional molecules to produce one larger polyfunctional molecule with the possible elimination of a small molecule such as water. Long reaction times are essential for forming high molecular weight polymers by this step reaction. An elementary introduction to polymers is given here and those wishing to gain more knowledge about the physics, chemistry and engineering aspects of polymers should consult some of the standard references [1-13] on the subject. Polymers formed through the polymerization processes discussed above can be classified in a number of different ways based on certain chosen characteristics for comparison. 1.1.1

THERMOPLASTICS, THERMOSETS AND ELASTOMERS

Thermoplastics are those polymers that can be made to soften and take on new shapes by application of heat and pressure. The changes that occur during this process are physical rather than chemical and hence products formed from such polymers can be remelted and reprocessed.

Table 1.1 Some candidate polymers used in the formation of filled polymer systems Thermosets

Elastomers

Thermoplastics

Epoxies

Neoprenes

Nylons

Phenolics

Nitriles

Polypropylene

Unsaturated polyesters

Styrene butadienes

Polystyrene

Thermosets are materials that have undergone a chemical reaction, known as curing in A, B and C stages depending on the degree of cure by the application of heat and catalyst. The A stage is the early stage, B stage is the intermediate stage and C stage is the final stage of the curing reaction. The crosslinked structure that forms in the polymer during the reaction is stable to heat. Hence products formed through these polymers cannot be made to flow nor can they be melted and thus are not reprocessible. Not all thermosets go through A, B and C stages, and in fact, processors are often interested in the flow behavior of those that have not undergone cure. Elastomers are rubbery polymers that deform upon the application of stress and revert back to the original shape upon release of the applied stress. They are lightly crosslinked molecular networks above their glass transition temperatures. They are often capable of rapid elastic recovery. They are available as natural rubbers or synthetic rubbers. Natural rubbers are elastic substances that are obtained by coagulating the milky extracts from certain tropical plants; while synthetic rubbers are those that are artificially prepared by combining two or more monomers through a chemical reaction. Some of the candidate polymers from the above three categories which are used in the formation of filled polymer systems are given in Table 1.1.

1.1.2 LINEAR, BRANCHED OR NETWORK POLYMERS

A polymer can be classified as linear or branched depending upon its structure. The thermoplastic polyethylene serves as a good example because it exists with linear as well as branched structure as can be seen from Figure 1.1. Based on the pressure (low or high), the reaction temperature and the choice of the catalyst during the polymerization process, polyethylenes with different densities and structures are formed. High-density polyethylene (HDPE) has a linear molecular structure and a density ^0.94g/cm3, low-density poly-

LOW PRESSURE

HDPE

HIGH PRESSURE

LDPE

LOW PRESSURE

LLDPE

Figure 1.1 Comparison of structures of HDPE1 LDPE, and LLDPE.

ethylene (LDPE) has a branched structure and a density ^0.92g/cm3, whereas linear low-density polyethylene (LLDPE) with a density of 0.92-0.93 g/cm3, although branched is significantly different from LDPE due to the absence of secondary branching and presence of short branches. Polycondensation of compounds with a functionality of three or more with the addition of special hardening agents to form chemical crosslinks results in polymers with three-dimensional network structure. A classical example of the formation of a network polymer is the polycondensation of phenols with aldehydes. The reaction between phenol and formaldehyde in the absence of a catalyst is very slow and hence in all commercial synthesis, catalysts are always added to

accelerate the reaction. The nature of the end product and the reaction rate depend greatly upon the type of catalyst and the mole ratio of the two reactants. When one mole of phenol is reacted with 0.8-0.9 mole of formaldehyde in an acidic medium, the reaction product is a soluble, fusible resin which can be converted into an insoluble, infusible product only upon the addition of excess formaldehyde. These resins are therefore termed two-stage resins known as novolacs. On the other hand, when one mole of phenol is reacted with one or more moles of formaldehyde at a pH of 8 or above (i.e. alkali-catalyzed medium), then insoluble, infusible products are directly formed. These resins are termed one-stage resins and known as resols, which are linear or branched low molecular products.

Resols on further heating change into resitols, a three-dimensional network polymer of low crosslink density as

The last stage of the heating process results in the formation of resites, which is a network polymer of high crosslink density as

It should be noted that the structure of phenolics is much more random than that shown above. The pictorial representation is basically a simplified version only for the sake of exemplifying the formation of a three-dimensional network. Other thermosets besides phenol-aldehyde which are formed into network polymers by similar reactions are urea-aldehyde and melamine-aldehyde polymers. 1.1.3

CRYSTALLINE, SEMI-CRYSTALLINE OR AMORPHOUS POLYMERS

Polymers can also be classified as crystalline, semi-crystalline 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 primary transition temperature, when a crystalline polymer transforms from a solid to a liquid, is the melting temperature designated Tm. 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 the glass transition temperature and often designated Tg. Most thermoplastics have both Tg and Tm. This is because it is relatively difficult to get to the extreme case of a completely crystalline polymer with an ideal formation of single crystals 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 semi-crystalline polymer.

1.1.4 HOMOPOLYMERS

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

when R = H, then the result is the homopolymer polyethylene (PE); when R = CH3, it is polypropylene (PP); when R = C6H5, it is polystyrene (PS) and when R = Cl, it is polyvinyl chloride (PVC). The materials mentioned above namely, PE, PP, PS and PVC are among the largest volume thermoplastics utilized. They are used in a maximum number of applications, mostly in those which do not require high performance or special properties. In terms of cost they are among the cheapest of the thermoplastics. Hence they are often referred to as commodity plastics. Another important class of polymers which are formed by addition polymerization like the above are based on one of the following three repeating units

1,4 addition

3,4 addition

1,2 addition

When X = H, the resulting polymer is polybutadiene; when X = CH3, it is polyisoprene and when X = Cl, it is polychloroprene. The double bond may be 'cis' or 'trans' and would thus give the cis or trans forms of these polymers. It is the 1,4 addition form that predominantly goes into the formation of commercial dienes which are all elastomers. Typical examples of homopolymers that are formed by condensation polymerization are the polyamides and polyesters

When R E=(CH2)S, the resulting polymer is polyamide: nylon-6; when R'=(CH 2 ) 6 and R"=(CH 2 ) 4 , it is nylon-66; when X = (CH2), and X' = C6H5, it is polyester: polyethylene terephthalate (PET). These thermoplastics have properties which are superior to those of commodity plastics (namely, olefinics and styrenics). They go into a number of engineering applications and are termed engineering thermoplastics. Besides poly amides (nylons) and polyesters, some of the other homopolymers which fall into this category of engineering thermoplastics are the acrylics, acetals and polycarbonates. High performance engineering thermoplastics have recently assumed increasing importance due to their exceptional properties at elevated temperatures. A number of such specialty polymers have been introduced into the market for high temperature applications and examples of some of the outstanding ones are polyphenylene oxide (PPO), polyphenylene sulfide (PPS), polyether sulfone (PES), polyaryl sulfone (PAS), polyether ketone (PEEK), polyether imide (PEI) and polyarylate (PAr). Each of the above mentioned specialty polymers exhibits enhanced rigidity at high temperatures. This is a consequence of their high glass-transition temperatures, presence of aromatic ring structures in the backbone chain and relatively strong hydrogen bonds.

1.1.5

COPOLYMERS AND TERPOLYMERS

When two different monomers are used in the polymerization process, the result is a copolymer. The repeating units A and B both exist in the polymerized product and their varying configurations give different types of copolymers (i) Random copolymer: AA B AAA BB A BBB (ii) Uniform copolymer: AB AB AB AB AB (iii) Block copolymer: AAA 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 arms 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, the flow behavior of PVC is considerably improved by incorporating vinyl acetate as comonomer. Similarly, the thermal stability of polyoxymethylene is improved considerably by incorporation of -CH2-CH2-O 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, ethylenevinyl 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, while 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 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 [14-20] that are available. Polymerization of acrylonitrile and styrene in the presence of butadiene rubber results in a terpolymer called acrylonitrile-butadienestyrene (ABS). Besides grafting styrene and acrylonitrile into polybutadiene in latex form, ABS may also be produced by blending emulsion latexes of styrene-acrylonitrile (SAN) and nitrile rubber (NBR). Since ABS is a three component system, many variations are

possible. Acrylonitrile imparts chemical resistance while butadiene provides increased toughness and impact resistance. A variety of grades are available - some for general purposes, some for various levels of impact resistances and others for ease of plating. The major applications include plumbing systems, telephone housings and automobile grills (either painted or electroplated). 1.1.6

LIQUID CRYSTALLINE POLYMERS

Polymers in which rigid, anisotropic moieties are present in the backbone of the polymer chain are known to give rise to liquid crystalline behavior and are therefore known as liquid crystalline polymers (LCP) [21]. Such types of polymer have attained immense importance due to the possibility of producing ultrahigh modulus fibers and plastics. The main interest in the subject of LCP was kindled by the commercialization of the aromatic polyamide fiber, namely, Kevlar which was as stiff and strong as steel but at one-fifth the density of steel, and with excellent chemical and heat resistance. Kevlar is a lyotropic liquid crystal, that is, it attains liquid crystalline order only when dissolved in an appropriate solvent. Since the removal of the solvent is a necessary step during fabrication of the product, such lyotropic LCPs are restricted to the formation of thin fibers and films. For thicker products, however, polymers are needed which become liquid crystalline upon heating i.e. 'thermotropic' LCPs. There has been increasing interest in using LCPs as reinforcing additives in polymers to form blends and composites [22,23]. But moldable LCPs, regrettably, do not have immense commercial importance. 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. Thus understanding the rheological behavior associated with liquid crystallinity is undoubtedly essential for processing the LCPs into the appropriate structure to exhibit their desirable properties [24]. 1.2

FILLERS

The term 'filler' in the present context is used for describing those inert, solid materials which are physically dispersed in the polymer matrix, without significantly affecting the molecular structure of the polymer. Further, the term is restricted to those materials which are in the form of discrete particles or of fibers not exceeding a few inches in length. Continuous filaments or fabrics either woven or nonwoven are not included in this category of fillers discussed

Table 1.2 Examples of rigid and flexible fillers Filler type Rigid

Flexible

Aluminum oxide Barium carbonate Calcium carbonate Calcium hydroxide Calcium silicate Clay Glass fiber Magnesium hydroxide Metal fiber Mica Talc Wollastonite

Asbestos fiber Cotton flock Cotton !inters Jute fiber Kevlar fiber Nylon fiber Polyester fiber Sisal

herein. Readers wishing to know more about fillers may refer to the excellent handbook on the subject [25]. Filler categorization 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. 1.2.1

RIGID OR FLEXIBLE FILLERS

Rigid fillers are those fillers that do not change their shape or spatial configuration within the polymer matrix. An example of such type of filler is glass fiber. On the other hand, flexible fillers are those fillers whose spatial configuration within the polymer matrix is not rigidly defined. For example, asbestos fibers, nylon fibers, polyester fibers, etc. would lie in folded, coiled or twisted positions within the polymer matrix. This type of filler classification is shown in Table 1.2. 1.2.2

SPHERICAL, ELLIPSOIDAL, PLATELIKE OR FIBROUS FILLERS

Fillers can be classified based on their physical form and shape as shown in Table 1.3. Among the classifications shown, only spherical fillers are symmetric in physical form and hence provide symmetric changes in properties in all three spatial directions. It is normally rare to find exactly spherically well-formed fillers. There is always a slight defect in shape, especially for finer size particles. For instance, even when controlled conditions are used in the preparation of monodisperse silica spheres in the micron size range [26,27], all particles

Table 1.3 Filler classification by physical form Filler form

3-dimensional

2-dimensional

Spherical

Ellipsoidal

Flakes

Platelets

Glass beads

Wood flour

Mica

Clay

1-dimensional

Fibers

Whiskers

Glass fibers Wollastonite

produced are not exactly spherical when viewed under the scanning electron microscope. Thus, the terms spherical or ellipsoidal can be viewed as those referring to nearly spherical or nearly ellipsoidal fillers. When the physical form of the filler is two-dimensional, the fillers may be available as flakes (larger size plates) or platelets (smaller size plates). Thus, mica particles exist as flakes whereas clay particles exist as platelets. In the one-dimensional form, filler may be available in the thicker variety as a fiber or thinner (acicular-needle-shaped) variety as a whisker. Fillers available as fibers are glass, nylon, polyester, carbon and so on. Wollastonite stands out as a good example of an acicular filler. 1.2.3 ORGANIC OR INORGANIC FILLERS

Classification of fillers can also be based on their chemical form [28,29] as shown in Table 1.4. Organic fillers fall within the subcategory of cellulosics, lignins, proteins and synthetics. On the other hand, inorganic fillers include carbonates, oxides, silicates, sulfates, carbon, metal powders and so on. 1.3

FILLED POLYMERS

The use of fillers in polymers has been going on for years. In the early history of filled polymers, fillers were added to the polymers rather empirically. Woodflour was one of the first fillers used in thermosetting phenol-formaldehyde resins because the combination was found to be valuable in enhancing certain properties whereas the addition of some other finely divided material to such resins conferred no benefit at all and hence was never done. The presence of the woodflour increased

Table 1.4 Different types of fillers Organic Cellulosics Alpha cellulose Cotton flock Sisal Jute Wood flour Shell flour Cotton-seed hulls Cotton !inters Cork dust

Lignins Processed lignin Ground bark

Inorganic Proteins

Synthetics

Carbonates

Hydroxides

Soybean meal Keratin

Acrylics Nylons Polyesters

Calcium carbonate Barium carbonate Magnesium carbonate

Calcium hydroxide Magnesium hydroxide

Inorganic Oxides

Silicates

Sulfates

Carbon

Metals powders/fibers

Miscellaneous

Aluminum oxide Antimony trioxide Zinc oxide Magnesium oxide Quartz Diatomaceous earth Tripoli Hydrogel Aerogel

Calcium silicate Magnesium silicate Clay Talc Mica Asbestos Feldspar Wollastonite Pumice Vermiculite Slate flour Fuller's earth

Calcium sulfate Barium sulfate

Carbon black Graphite

Aluminum Copper Bronze Lead Steel Zinc

Barium ferrite Magnetite Molybdenum disulphide

Source: Refs 28 and 29 (Reprinted with kind permission from Society of Plastics Engineers, Inc., Connecticut, USA and Gulf Publishing Co., Houston, Texas, USA).

Table 1.5 Some typical examples of filled polymer systems Polymer

Filler

Thermoset: Phenol-formaldehyde resin Elastomer: Styrene butadiene rubber Thermoplastic: Polypropylene

Wood flour/cotton flock Carbon black Calcium carbonate/talc

strength and prevented cracking of the resin by reducing the exotherm in the curing reaction. Similarly, the use of carbon black as a reinforcing agent in rubber has been going on since early in the century as it was a major factor in the development of durable automobile tyres. Glass fiber in nylon and asbestos in polypropylene confer useful properties but, if the filler and polymer are switched, i.e. asbestos is put into nylon and glass into polypropylene the results are not nearly so good unless the fillers are treated with appropriate coupling agents. Polypropylene is also often filled with calcium carbonate and talc with constructive results. Little thought was given in the early days towards the reasons for the observed behavior. Nowadays, however, the marriage of filler to polymer is done on a scientific basis and the reason for the addition of the specific filler can be elucidated on the desired property it imparts. Some of the typical examples of filled polymer systems using a thermoset, elastomer and thermoplastic are given in Table 1.5. The escalating cost of engineering thermoplastics over the last couple of decades and the awareness of dwindling supply of petrochemicals has created renewed incentives to restrict the quantities of resins used through the addition of fillers to the polymer matrix. Besides savings in cost, certain fillers provide the added advantage of modifying specific mechanical, thermal and electrical properties of thermoplastic products as can be seen from Table 1.6. When stiffness, strength and dimensional stability are desirable, the polymers are extended with rigid fillers; for increased toughness as

Table 1.6 Reasons for the use of fillers in thermoplastics 1. 2. 3. 4. 5. 6. 7.

To To To To To To To

increase stiffness, strength and dimensional stability increase toughness or impact strength increase heat deflection temperature increase mechanical damping reduce permeability to gases and liquids modify electrical properties reduce the cost of the product

in the case of high-impact polystyrene or polypropylene, deformable rubber particles are added; asymmetric fillers such as fibers and flakes increase the modulus and heat distortion temperature; and electrical and thermal properties are modified by the use of metallized fibrous fillers. With fibrous fillers, the improvements can be further magnified due to the influence of the fiber aspect ratio and anisotropy as well as fiber orientation. The most effective reinforcing fillers are fibers of high modulus and strength. Glass fibers, which are non-crystalline in nature, or asbestos - a crystalline fiber - provide the reinforcement in most commercial fiber-reinforced thermoplastics. Carbon fibers or whiskers, single crystal fibers, are the other crystalline fibers used as reinforcement. The improvements in mechanical properties through the use of fillers acting as reinforcing agents have been discussed in detail by Nielsen and Landel [3O]. Such filled systems wherein the fillers provide reinforcements are often referred to as reinforced polymer systems or reinforced plastics. However, in the present book, the term 'filled polymer systems' is used in the most general sense and includes all systems wherein fillers are present as cost reducing agents as well as reinforcing or property modifying agents. One or more of the physical, mechanical and thermal properties of polymers can be effectively modified by the use of different types of fillers. For example, in the tyre industry, the presence of the filler carbon black in vulcanized rubber enhances properties like elastic modulus, tear strength and abrasion resistance [31-33] and also influences extrusion characteristics like extrudate distortion, extensional viscosity and die swell behavior [34-^4O]. Thus, carbon black functions as a reinforcing agent and a processing aid in the rubber industry. Different types of fillers serve different types of purposes. For example, titanium dioxide acts as a delustering agent in the fiber industry and aluminum trihydroxide as an economic flame retardant and smoke suppressing agent. In most applications, the proper balance of properties is no less important than an improvement of an individual property. It must be accepted that an improvement in one property can in all likelihood lead to deterioration of others and consequently, it is the overall performance of the filler in a given formulation that determines its choice. The predominant function of some typical fillers is given in Table 1.7. Selection of a filler is not just an art but a science and various factors would have to be considered in the choice such as, 1. 2. 3. 4.

Cost and availability Wettability or compatibility with the polymer Effect on polymer flow characteristics Physical properties

Table 1.7 Predominant function of some typical fillers Function

Typical fillers

Cost savings Reinforcement Hardness Thermal insulation

Wood flour, saw dust, cotton flock Glass fibers, cellulosic fibers, synthetic fibers, asbestos fibers Metallic powders, mineral powders, silica, graphite Asbestos, ceramic oxides, silica

Chemical resistance

Glass fibers, synthetic fibers, metallic oxides, graphites

5. 6. 7. 8. 9.

Thermal stability Chemical resistivity Abrasiveness or wear Toxicity Recyclability

Undoubtedly the idea of adding the fillers is to achieve reduction in cost. However, there are some special type of fillers which are used purely on a functional basis with an accepted trade-off in the cost reduction, for example, some fiber glass reinforcements for polyesters, barium ferrite as a magnetizable filler, metallic powders for electrical and thermal conductivity improvement. In fact all these specialty fillers are more expensive than general purpose fillers and in some cases even more expensive than the polymer which they fill. In any case, the cost-effectiveness of the filler ought to be determined. The objective should be to compare the full cost of the completed product with and without the filler. The first step involves obtaining the raw material costs which must be converted from cost per pound to cost per volume. This is because cost per pound of the filler is meaningless unless adjusted for specific gravity differences. The volume of the polymer that is displaced by the filler becomes the main consideration. A three-step calculation method [41] can be used to get the polymer saved and thus to determine the cost-effectiveness of filled systems. If the filled polymer system is compounded in-house, then that cost has to be included. Similarly, added labour cost or savings due to the use of filled polymer systems must be considered. Often it is found that a minimum of 30 volume percent of low-cost filler is required to get a cost benefit when switching from unfilled polymer to filled polymer system. When selecting a filler, it is important to bear in mind that for adequate stress transfer, wettability and good adhesion between the filler and the polymer is essential. Physical properties like, for example, the density should be low so that the filler stays in suspension or at

worst is able to be resuspended with minimum mixing. Thermal stability and chemical resistivity are also very important so that the filler does not change characteristics during the preparation of the filled polymer system. Fibrous materials and non-symmetrical fillers are more abrasive than others and could cause increased wear to the processing equipment. Hence care has to be taken when selecting such fillers as they may not turn out to be cost-effective due to excessive damage to the equipment. Also the effect of fillers on polymer flow characteristics, namely, the rheology must be carefully assessed as that determines its processibility and hence is a very important parameter. 1.4 FILLER-POLYMER INTERACTIONS When a filler is added to a polymer with the specific idea of reinforcement, it is expected that the reinforcing filler component which is strong and stiff should bear most of the load or stress applied to the system while the polymer which is of low strength, fairly tough and extensible should effectively transmit the load to the filler. Maximum reinforcement benefits would be achieved from fillers when conditions occur in accordance with this concept [42]. In order that the load transfer takes place effectively, the matrix must have sufficiently high cohesive and interfacial shear strength. Thus, apart from the filler and the polymer, it is the inevitable region between them, namely, the interphase which plays a vital role in the fabrication and subsequent behavior of the filled polymer systems in service. The interphase is that region separating the filler from the polymer and comprises the area in the vicinity of the interface. It would be synonymous with the words 'interfacial region' but different from the term 'interface' which would be the contacting surface where two materials under consideration meet. Thus, for some filled polymer systems, there could exist more than one interface as in the case of coated fiber-filled polymer. In such cases, the fiber-coating interface and the coating-polymer interface would have characteristics of their own. However, normally a less atomistic view is taken and the characteristics of the 'interfacial region' as a whole are generally investigated. Good mechanical strength can be achieved only by uniform and efficient stress transfer through a strong interfacial bond between the filler and the polymer. It is important that the bond is uniform on a fine scale rather than unevenly strong in local regions as areas of the filler-polymer interface which are not in contact begin to act as cracks under an applied stress. In the absence of a good interfacial bond, fibrous fillers will pull out of the polymer and result in an annulment of the reinforcing effect [43]. Controlled debonding at the Next Page

Basic Theological

r\

concepts

4—

Filled polymer theology is basically concerned with the description of the deformation of filled polymer systems under the influence of applied stresses. Softened or molten filled polymers 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 of heat and 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. Softened or molten filled polymer may behave as a viscous liquid or elastic solid during processing operations depending upon the relationship between the time scale of deformation to which it is subjected and the time required for the time-dependent mechanism to respond. The ratio of characteristic time to the scale of deformation is defined as the Deborah number by Reiner [1,2] as De = /lc//ls where Ac is the characteristic time, A8 is the time scale of deformation. The characteristic time, Ac/ for any material can always be defined as the time required for the material to reach 63.2% or [1 — (l/e)] of its ultimate retarded elastic response to a step change. If De > 1.0, elastic effects are dominant while if De < 0.5, viscous effects prevail. For any values of Deborah numbers other than these two extremes given above, the materials depict viscoelastic behavior. Filled polymer systems display the ability to recoil by virtue of their viscoelastic nature. However, they do not return completely to their original state when 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 bygone past, thus lending it the characteristics of a fading memory. Meissner [3] found that a filament of low density polyethylene (LDPE) at 423 K, which is stretched rapidly from 1 to 30cm length, and then suddenly set free, recovers to a length of 3 cm, thereby giving a recovery factor of 10. If the filament were made of filled LDPE, the recovery factor would be much smaller because the presence of the filler greatly reduces the stretchability as well as the recoil of the material. 2.1 FLOW CLASSIFICATION Flow is broadly classified as shear flow and extensional flow. A catalog of various types of shear flow has been given by Bird et al [4]. In the present book, the discussion is restricted to only 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. Similarly extensional flow could be steady or unsteady. In the case of extensional flow, it is often difficult to keep the measuring apparatus running for a long enough time to achieve steady state conditions and therefore unsteady conditions are quite often encountered. Thus, flow is classified here under three headings: 1. Steady simple shear flow 2. Unsteady simple shear flow 3. Extensional flow. Extensional flow (steady and unsteady) is treated under one heading for convenience. The definitions of important Theological parameters under each of the three headings are given below. 2.1.1 STEADY SIMPLE SHEAR FLOW

Fluid deformation under steady simple shear flow can be aptly described by considering the situation in Figure 2.1 wherein the fluid is held between two large parallel plates separated by a small gap dx2 and sheared as shown. The lower plate is moving at a constant velocity V1 while the upper plate is moving at a constant velocity of V1 H- Au1 under the action of a force / 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 V1 and V1 + Av1. Under steady-state conditions, the force / required to produce the motion becomes constant and is related to the velocity. The velocity profile of the fluid within the gap is given by dt^ = y dx2 where y is a constant.

Figure 2.1 Simple shear flow of a fluid trapped between two parallel plates. A.

Shear rate

The velocity gradient [Av1JdX2], which is termed the shear rate y can also be written as f 7 = ^ = Afel=lfel

dx2

dx2[dt\

dt[dx2\

l(21) ;

The term [dxi/dx2] 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 (sec"1). 6.

Shear-stress and extra stress tensor

The force per unit area [f/A] required to shear the material between two parallel plates is defined as the shear stress T2i, and it is basically a function (/cn) of the velocity gradient. Thus,

The units of shear stress are dynes/cm2 or Newtons/m2 (i.e. Pascals). It must be noted that i2i 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: f =-pf + f + ijv[trl5]I

(2.3)

where T(X, t) denotes the symmetric Cauchy stress tensor at positron x and time t, p(x, t) is the pressure in the fluid [T being the unit tensor), f is the extra stress tensor, T/V is the volume viscosity and D is the symmetric part of the gradient tensor of the velocity field v(x, t):

^••^[1+5] Note that Cartesian coordinates are used and vectors are denoted by single bar [—] above the letter while 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 tr 15 = Jp = O

(2Ab)

In such cases, the last term on the right-hand side of equation (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) fpr an incompressible fluid connects only the 'extra stress tensor' f or T -f pi uniquely with the local motion of the fluid but always leaves the pressure p indeterminate. A general form of the constitutive equation can be written as f = ri(l H, IH)H

(2.5a)

The apparent viscosity Y] in the above equation is a function of the first, second and third invariants of the rate of deformation tensor. For incompressible fluids, the first invariant I becomes identically equal to zero. The third invariant III vanishes for simple shear flows and is normally neglected in non-viscometric flows as well. The apparent viscosity then is a function of the second invariant II alone. Hence equation (2.5a) is written in the simplified form as

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: T11

f=

T12

O I

T21 T22 O O O T 33 1

(2.6)

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 Figure 2.2. All the components are not shown in the figure in order to maintain clarity. Note that in steady shearing flow, the stress components T13, T23, T 315 T 32 , are identically equal to zero. T12 = T21 is called the shear stress and T11, T22, T33, are called normal stresses.

Figure 2.2 Various stress components on a representative cubic volume of fluid (stress components T12, T13, T31, T32, have not been shown in order to maintain clarity of the figure).

C. Normal stress difference The absolute value of any particular component of normal stress is of no rheological relevance, whereas the values of the normal stress differences T11 — T22 and T22 — T33 do have considerable rheological significance. The first is termed the primary normal stress difference while the latter is termed the secondary normal stress difference. Thus, N1 - T11 - T22

(2.7)

N2 = T22 - T33

(2.8)

For most fluids, N1 ^> N2 and hence the latter is often excluded in rheological discussions. Attempts to determine the value of secondary normal stress difference experimentally have been made by several rheologists but without success. 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 in providing the necessary restoring force for stabilizing the wire position whenever it becomes off-centered. D. Viscometric functions The viscosity function r\ (referred to as the steady shear viscosity), the primary and secondary normal stress coefficients ^1, and ^2, respectively, are the three viscometric functions which completely determine the state of stress in any Theologically steady shear flow. They are defined as follows: T12 = T21 = ri(y)y

(2.9)

Tn-T 2 2 = iAi(7)y2

(2.10)

2

(2.11)

T 22 - % = *2(y)y

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 flows [4]. In the following, small-amplitude oscillatory flow is treated in sufficient detail while others are briefly described

and readers are encouraged to refer to Bird et al. [4] for more information. A. Small-amplitude oscillatory flow Small-amplitude oscillatory 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 dx2 between two large parallel plates of which the upper one undergoes small amplitude oscillations in its own plane with a frequency CD. The velocity field within the gap can be given by Av1 — y dx2 but y is not a constant as in steady simple shear. Instead it varies sinusoidally and is given by y(t) = J0COSCDt

(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, ^21 (O = Jo[G'(cD) sin CDt + G'(CQ) cos cot] = r°2l sm[cDt + 6]

(2.13) (2.14)

where d is the phase angle, y0 and T^ are the amplitudes of the strain and stress, respectively, and G, G" are linear viscoelastic material functions, respectively, referred to as the dynamic storage modulus and dynamic loss modulus. To Dynamic storage modulus: G(CD) = -^- cos d (2.15) Vo

Dynamic loss modulus:

To G"(CD) — -^- sin d

7o

(2.16)

Another term of importance is the ratio of loss to storage modulus, defined as Loss tangent: —^- = tan 6 G(CD)

(2.17)

It is also possible to define a dynamic complex viscosity in terms of G and G" as follows: /•"•/// \ Dynamic viscosity: rj'(cD) = —— (2.18) Imaginary part of the complex viscosity: Y\"(CD) =

(2.19)

Complex viscosity function:

(2.20)

jf(ico) = ^(CD) — irj"(cD)

In the same manner as above, a complex modulus can be defined as below: Complex modulus: G*(ico) = G(CO) + iG"(co)

(2.21)

The storage modulus G(CO) and the imaginary part of the complex viscosity, i.e. rjf/(co), are to be considered as the elastic contributions to the complex functions. They are both measures of energy storage. Similarly, the loss modulus G"(CD) and the dynamic viscosity rj'(co) 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 they approach their steady shear flow values. This is done by assuming that the fluid sample trapped in a small gap between two parallel plates is at rest for all times previous to t = O implying that there are no stresses in the fluid when steady shear flow is initiated at t = O. For t > O when a constant velocity gradient is imposed, the stress is monitored with respect to time till it reaches steady state value. C. Stress relaxation The aim of a stress relaxation experiment is to observe how the stresses decay with time (i) after cessation of steady shear flow or (ii) after a sudden shearing displacement. In case (i) the fluid sample trapped in a small gap between two parallel plates is allowed to maintain constant shear rate that was started long before t = O so that all the transients during the stress growth period have evened out. Then at t = O, the flow is stopped suddenly and the decay of the stress is monitored with respect to time till 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 (ii), 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 /e is defined as -y/T21. If the driving shear stress T21 is small enough then the value of the compliance is independent of the driving shear stress. E. Constrained recoil The aim of a constrained recoil experiment is to observe the shear displacement in a fluid sample trapped in a small 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 EXTENSIONALFLOW

Problems associated with fiber spinning, film blowing and 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 occurs in a single direction and hence the related flow is termed uniaxial extensional flow. Extension or stretching of polymer takes place in other processing operations as well, such as film blowing and flat-film extrusion. In such cases, the extension occurs in two directions simultaneously, and hence the flow is referred to as biaxial extensional flow in one case and planar extensional flow in the other. Extensional flows can orient polymer chains as well as fillers which are of the !-dimensional or 2-dimensional type and hence determine the performance and appearance of a product. A. Uniaxial extension 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 Figure 2.3. The velocity field in simple uniaxial extensional flow is given by V1 = SX1;

V2 = - \ £X2;

V3 = - \ £X3

(2.22)

(a) BEFORE EXIENSIOKAL DEFORMATION

(b) AFTER EXTENSIONAL DEFORMAUON

Figure 2.3 Schematic diagram of a fluid element in uniaxial extensional flow.

where e is the uniaxial extensional rate. For such a flow field, the rate of deformation tensor is given as _ B O U = O -s/2

O O

O

-s/2

O

in which

'-£

(2.23)

The uniaxial extensional rate may be constant or vary in the X1 direction of flow. When s is constant, i.e. when the axial velocity is proportional to X1, the resulting flow is steady uniaxial extensional flow. In such a flow situation, a cylindrical rod of length / is stretched along its axis according to the following equation: ft = el

(2.25)

Integrating this equation for a constant strain rate gives, / = /oexp(gt)

(2.26)

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

6. Biaxial extension In biaxial extensional flow, too, the dimensions of the fluid elements change drastically but they change in two directions as against the onedirection 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 Figure 2.4. The velocity field in simple biaxial extensional flow is given by, V1 = E3X1;

V2 = 63*2;

^3 = -2eB*3

(2.27)

where £B is the biaxial extensional rate. For such a flow field, the rate of deformation tensor is given as,

(a) BEFORE EXTENSIONAL DEFORMATION

(b) AFTER EXTENSIONAL DEFORMATION

Figure 2.4 Schematic diagram of a fluid element in biaxial extensional flow.

C. Planar extension Planar extensional flow is the kind of flow where there is no deformation in one direction and the velocity field is represented as follows: V1 = SpX1-,

V2 = -SpX2;

V3=O

(2.29)

where sp is the planar extensional rate. In this case, the rate of deformation tensor is given as, 8P

5=

O

O

O -fip O 0 0 0

(2.30)

Extensive reviews [6-10] and a monograph [11] 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. D. Material functions in extensional flow 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 TZ; are non-zero (i.e. i{j = O for i ^j). In the case of uniaxial extension, T11 is the primary stress that can be measured, while T22 and T33 are generally equal to the pressure of the environment. Thus, the uniaxial extensional viscosity rjE is defined by, T11 - T22 = T11 - T33 = rjE(s)s

(2.31)

By the same token, the biaxial extensional viscosity t]EB can be defined as, ^33 - T11 = T33 - T22 = -*?EB(£B)£B

(2.32)

and further, the planar extensional viscosity rjEP can be written as, T11 - T22 = J?Ep(eP)£p

(2.33)

2.2 NON-NEWTONIAN FLOW BEHAVIOR The viscoelastic nature of polymers (filled or unfilled) and their peculiarities in the viscous as well as elastic response to deformation under applied stresses bring them under the category of nonNewtonian fluids. There is distinctive difference in flow behavior between Newtonian and non-Newtonian fluids to an extent that, at times, certain aspects of non-Newtonian flow behavior may seem abnormal or even paradoxical [12-16]. An interesting movie about polymer fluid mechanics has been prepared [17] which clearly depicts certain peculiarities of such fluids. The dramatic differences between the qualitative responses of Newtonian and non-Newtonian fluids grossly affect their industrial and practical applications. 2.2.1

NEWTONIAN FLUIDS

Isaac Newton was the first to propose the basic law of viscosity describing the flow behavior of an ideal liquid as, ? = i?0D

(2.34)

where the constant rj0 is termed the Newtonian viscosity. Fluids, whose flow behavior follows the above constitutive equation, are known as Newtonian fluids. Some of the common Newtonian fluids with which most people are familiar are water Oy0 ^ 1 mPa.sec), coffee cream (rjQ % lOmPa.sec), olive oil (rjQ ^ 102mPa.sec) and honey Of0 % 104 mPa.sec). For a Newtonian fluid, equation (2.34) yields the following stress components in simple shear flow: T n = T22 = T33 = -p

(2.35)

T = T12 = T21 = rj0y

(2.36)

All other stress components vanish. According to equation (2.35), it can be seen that the three normal stress components are equal. The nonvanishing shear stress T varies linearly with shear rate and has a proportionality constant f/ 0 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 rj0 and the density p. Once these quantities are measured, the velocity distribution and the stresses in the fluid can, in principle, be found for any flow situation. In other words, different isothermal experiments on a Newtonian fluid would yield a single constant material property, namely, its viscosity. On the other hand, a variety of flow experiments performed on a softened or molten filled polymer system, which is a

non-Newtonian fluid, would yield a host of material functions that depend on shear rate, frequency and time. When the viscosity is a function of shear rate, then the relationship between shear stress and shear rate is given by equation (2.9). Since its form is similar to equation (2.36) except for the shear rate dependent viscosity, the equation is said to represent a Generalized Newtonian fluid. In such a fluid, the presence of normal stresses defined by equations (2.10) and (2.11) is considered to be negligible for a specific flow situation. In effect, equation (2.5b) represents the constitutive equation for a Generalized Newtonian fluid. The hypothesis of a Generalized Newtonian fluid differs from the simple Newtonian case by the assumption that the functional relationship between the stress tensor and the kinematic variable need not be only linear. It holds, however, the suggestion that only the kinematic variable of the first order can influence the state of stress in the fluid and no attempt is to be made to describe the normal stresses in it. 2.2.2 NON-NEWTONIAN FLUIDS

Non-Newtonian fluids are Theologically complex fluids that exhibit one of the following features: (a) Shear rate dependent viscosities in certain shear rate ranges with or without the presence of an accompanying elastic solid-like behavior. (b) Yield stress with or without the presence of shear rate dependent viscosities. (c) Time-dependent viscosities at fixed shear rates. The definitions of various types of non-Newtonian fluids along with examples of common real systems falling into each category are given in Table 2.1. Detailed discussions relating to non-Newtonian fluids are available in a number of books [18-27] as well as other review articles [28-33]. From Table 2.1, it can be seen that filled polymer systems fall within the non-Newtonian category of pseudoplastic fluids, pseudoplastic fluids with a yield stress, thixotropic fluids and viscoelastic fluids.

For pseudoplastic fluids, the shear rate at any given point is solely dependent upon the instantaneous shear stress, and the duration of shear does not play any role so far as the viscosity is concerned. The shear stress vs. shear rate pattern for a pseudoplastic fluid with and without yield stress is shown in Figure 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

Table 2.1 Various types of non-Newtonian fluids Fluid type

Definition

Typical Examples

• Pseudoplastic

• These fluids depict a decrease in viscosity with increasing shear rate and hence are often referred to as shear-thinning fluids.

• Dilatant

• These fluids depict an increase in viscosity with increasing shear rate and hence are often referred to as shear-thickening 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 vs. shear rate relationship.

• • • • • • • • • • •

• Bingham Plastics

• Pseudoplastic with a yield stress

• Thixotropic

• Rheopectic

• Viscoelastic

• These fluids have a nonlinear shear stress vs. 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 inbetween those of viscous liquids and elastic solids.

• • • • • • • • •

Filled polymer systems 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 clay/ fly ash/metallic oxides Sewage sludges Jellies Tomato ketchup Toothpaste Filled polymer systems Heavy crude oils with high wax content

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

• Filled polymer systems • Polymer melts • Polymer solutions

FSEDDOPLASTIC FLlTD WnU YIELDSTlESS

PSEUDOPLASHC FLUID

NEWTONIAN FLUID

Figure 2.5 Variation of shear stress vs. shear rate for pseudoplastic fluids with and without yield stress.

stress pattern with time for such fluids is shown in Figure 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 shown in Figure 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 deforming stress. At every instant during the deformation process, a viscoelastic fluid tries 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 polymer processing), in high shear rate flows (e.g. in processes like extrusion and injection molding) and in flows where changes in cross-section are encountered (e.g. entry into the mold cavity during injection molding). Some of the common encountered effects due to viscoelasticity are discussed below.

mxoiROPic FLHID

Figure 2.6 Variation of shear stress with time for a thixotropic fluid.

THIXOTROPIC JLUID HYSTERESIS LOOP

Figure 2.7 Variation of shear stress with shear rate (which is steadily increased from zero to maximum and brought down) for a thixotropic fluid.

Figure 2.8 Weissenberg effect showing how the viscoelastic fluid climbs up the stirrer-rod when stirred at moderate speeds. (Reprinted from Ref. 34 with kind permission from Chapman & Hall, Andover, UK.) 2.2.3 VISCOELASTIC EFFECTS A. Weissenberg effect 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 as shown pictorially in Figure 2.8. The first normal stress difference is much larger than the shear stress and hence gives rise to this startling effect. This type of phenomenon is commonly termed 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 [34-36]. In actuality, this effect was observed earlier by Garner and Nissan [37]. B. Extrudate 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 known variously as extrudate swell, die swell, jet swell, Barus effect or Merrington effect. Metzner [38] discusses the history of extrudate swell

Figure 2.9 Extrudate swell effect showing how the viscoelastic fluid swells in diameter when it exits from a die or orifice. (Reprinted from Ref. 34 with kind permission from Chapman & Hall, Andover, UK.) and argues against using the last two names. A review on extrudate swell has been given by Bagley and Schrieber [39]. Extrudate diameter (DE) of up to three or four times the orifice diameter (D0) is possible with some polymers. The swell ratio Sw (i.e. D E /D 0 ) 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, Sw should ultimately approach unity. But it is known [40] 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 pictorially in Figure 2.9. Theoretical analyses of this phenomenon, for flow in round capillaries, are available [41-45] in which the most basic [44] of them is built upon the free recovery calculations set down by Lodge [13] using the theory of Berstein, Kearsley and Zapas [46]. The developed expression for die swell Sw in which the elastic strain recovery SR is balanced by the shear stresses arising in the die, is given by, Sw = (l + iS2R)1/6 + 0.1 where,

(2.37)

SR=^

(2.38)

The above analysis does not include the rearrangement of the stress and velocity fields at the die exit, and consequently, it was found necessary [44] to 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 (T11 — T22)w/T2i,w and the ratio (TU - T22)/^ has been taken to be constant. Later work [47] 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 ^2/1Ai as well. The influence of this phenomenon in the filled polymer industry can hardly be overlooked. The industrial problems involving extrudate swell are particularly complex and challenging because the diameter increase depends not only on the particular type of polymer but also on the type and amount of filler as well as on the operating conditions such as temperature and flow rate. C. Draw resonance Draw resonance or surging is defined as the non-uniformity in the diameter of the extrudate when a polymer is stretched at different take-up speeds as it comes out of an orifice. This phenomenon is shown schematically in Figure 2.10. When take-up speed is small or when there is no stretching, only die swell is observed as can be seen from Figure 2.10(a). When take-up speed is higher and the stretched extrudate is solidified by quenching, then the contour appears as shown in Figure 2.10(b). Now 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 VQ at the die swell region. When the draw ratio DR goes beyond a critical value DRC, then the resulting phenomenon is draw resonance as shown in Figure 2.10(c). The theory of draw resonance has been developed and a method for calculating the critical draw ratio is also available [48]. Once draw resonance occurs its severity enhances with increasing take-up speed. D. Melt fracture When softened or molten polymer flows out of a capillary, a striking phenomenon of the distortion of the emerging stream is observed at shear stresses beyond a critical higher value and this is termed melt fracture [49,5O]. The extrudate distortion is a result of polymer molecules reaching their elastic limit of storing energy, thus causing

Figure 2.10 Draw resonance effect occurring when polymer melt is extruded from an orifice at various take-up speeds, (a) Extrudate without stretching, (b) extrudate with stretching DR DRC showing draw resonance.

melt fracture as a means of stress relief either at the capillary wall or at the capillary entrance. Another view [51] 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 [52] 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 [53-65] have been suggested for melt fracture. Based on a stick-slip mechanism, it is purported [53] that, above a critical shear stress, the polymer experiences intermittent slipping due to a lack of adhesion between itself 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 [53-63], both theoretically and experimentally. The other school of thought [64,65] is based on thermodynamic argument, according to which, melt fracture 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 of thermodynamics is violated and flow instability is induced [64]. It is important to distinguish between melt fracture, which is a gross distortion or waviness, and a fine scale high frequency surface

Figure 2.11 Difference between the phenomenon of matte and melt fracture (on distorted extrudates of different polymers): (1) rigid polyvinyl chloride, (2) polyethylene, (3) polypropylene, (4 & 5) polypropylene viewed from two angles, (6) polymethymethacrylate, (7) polytetrafluoroethylene. (Reprinted from Ref. 66 with kind permission from Society of Plastics Engineers, Inc., Connecticut, USA.)

roughness [4O]. The latter may commence at output rates below those at which melt fracture is observed and is termed 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 [66] as shown pictorially in Figure 2.11.

E.

Capillary entry flow patterns

A characteristic flow pattern at the capillary entrance develops when a polymer flows at high shear rates from a cylindrical reservoir through a capillary or die as shown in Figure 2.12. The qualitative difference between the capillary entry flows of linear and branched polyethylenes has been convincingly presented by Tordella [50] and discussed by others [67-7O]. For linear polymers, the converging flow at the die entry fills the available space, while for branched polymers there is a large

Figure 2.12 Capillary entry flow pattern for a branched polymer showing the flow cone and the recirculating vortex.

dead space filled by recirculating vortices. Vortices are induced by the viscoelastic characteristics of the converging fluid [71,72]. Polymers exhibiting larger extensional viscosities have been observed [71] to exhibit larger vortices and vice versa. The vortex or 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.

F. Abnormal fringe patterns in calendering During the process of calendering, very stable abnormal fringe patterns may appear on the roll surface at regular intervals. Though 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 on account of the effect of such changes on the viscoelastic response of the calendered material.

PRESSURE TRANSDUCER

NEWTONIAN FLUID

VISCOELASTEC FLDTO

Figure 2.13 Pressure hole error occurs in a viscoelastic fluid while it is absent in a Newtonian fluid. (Reprinted from Ref. 23 with kind permission from John Wiley & Sons, Inc., New York, USA.) G. Pressure hole error For Newtonian fluids, the pressure measured at the bottom of the pressure hole pM is the same as the true pressure p at the wall. For a viscoelastic fluid, on the other hand, the pressure (p + T22)M measured at the bottom of the pressure hole is always lower than the true pressure (p + T22) at the wall, no matter how small the hole is. 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 + T22)M -(p + T22). This effect is illustrated in Figure 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 for polymer melts with or without fillers. H. Parallel plate separation When a viscoelastic fluid is trapped between two parallel plates with one of the plates rotating, then there is a non-zero pressure p due to

NEWTONIAN FLUID

VISCOELASTIC FLUID

Figure 2.14 Parallel plate separation occurs in a viscoelastic fluid while it is absent in a Newtonian fluid. (Reprinted from Ref. 33 with kind permission from Gulf Publishing Co., Houston, Texas, USA.)

elasticity which tends to separate the two plates. This effect is illustrated in Figure 2.14. I. Tubeless siphon During the siphoning process, when the siphon tube is lifted out of the fluid, a Newtonian fluid will stop flowing whereas a viscoelastic fluid will continue unabated. At times, even 75% of the viscoelastic fluid in the container may get siphoned out in this manner. This effect is illustrated in Figure 2.15.

NEWTONIAN FLUID

VISCOELASIIC FLUID

Figure 2.15 Tubeless siphoning can be done for a viscoelastic fluid but not for a Newtonian fluid. (Reprinted from Ref. 23 with kind permission from John Wiley & Sons, Inc., New York, USA.)

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 spinnability of polymers. J. Uebler effect It has been observed [74,75] that when a polymeric fluid flows in a tube with a sudden contraction, large bubbles of the order of 1/6 to 1/8 of the small tube diameter, come to a sudden stop right at the entrance of the contraction along the centerline before finally passing through after a hold-time of about one minute. This particular behavior has been termed the Uebler effect [74,75]. This phenomenon has implications in the production of foamed plastics wherein a gas, normally nitrogen, is added to polymers such as PE, PP and PS during two-phase processing. 2.3 RHEOLOGICAL MODELS There have been a number of rheological models proposed for representing the flow behavior of softened or molten polymer and these are readily available in a number of books [18-27] and review articles [10,28,29,31,32,76]. The constitutive equations, which relate shear stress or apparent viscosity 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 vs. shear rate curve for unfilled polymer shown in Figure 1.2, it can be seen that in the low shear rate range, the material is essentially Newtonian in flow behavior with a constant apparent viscosity, which at zero shear rate, is termed the zero-shear viscosity rj0. In the medium shear rate range, the apparent viscosity r\ begins to decrease, depicting the shear-thinning characteristic until it stabilizes to a constant value ^00 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 an unfilled polymer over the entire shear rate range. However, when dealing with processing problems, only certain shear rate ranges attain significance and hence only portions of the flow curve need to be described by the constitutive equations thereby requiring less parameters. As a matter of fact, the very high shear rate range is invariably never reached and Next Page

Rheometry

O

Rheometry is the measuring arm of rheology and its basic function is to quantify the rheological material parameters of practical importance. 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 mode to the material and measure the subsequent force generated, or 2. It can apply a force mode to a material and measure the subsequent deformation. The best designs of rheometers use geometries so that the forces/ deformation can be reduced by subsequent calculation to stresses and strains, and so produce material parameters. It is very important that the principle of material independence is observed when parameters are measured on the rheometers. The flow within the rheometers should be such that the kinematic variables and the constitutive equations describing the flow must be unaffected by any rigid rotation of both body and coordinate system - in other words, the response of the material must not be dependent upon the position of the observer. When designing rheometers, care is taken to see that the rate of deformation satisfies this principle for simple shear flow or viscometric flow. The flow analyzed can be considered as viscometric (simple shear) flow if sets of plane surfaces (known as shear planes) are seen to exist and each is moving past the other as a solid plane, i.e. the distance between every two material points in the plane remains constant. The importance of viscometric flows becomes apparent when one appreciates that the equation of motion for most viscometric flows can be solved analytically. This is the reason why viscometric flows have been used for evaluating the viscosity function from viscometric data and this fact has brought about the alternative name for simple shear

flows. All flows that do not conform to the viscometric behavior as described above are termed non-viscometric flows. All rheometers have viscometric flows or at least 'near-viscometric' flows in them and hence are amenable to produce reliable material functions. Rheometers used for determining the material functions of filled polymer systems can be divided into two broad categories - (a) rotational type and (b) capillary type. Further subdivisions are possible and these are shown in Table 3.1. In what follows only those rheometers which are popularly used for rheological characterization of filled polymer systems are described and discussed in detail. For example, though the bob and cup rotational viscometer has been used [1] in the fifties for polyethylene melts, it has not been included in further detail. This is because this geometry is not at all popular even for unfilled thermoplastic melt studies, though Cogswell [2] did suggest it in the seventies for measuring shear viscosities under conditions of controlled pressure. Similarly, other rheometers which were developed for rheological measurements of filled systems, particularly suspensions such as cement [3], red mud [4] or other slurries [5,6], sealants [7], paints, foodstuffs or greases [8], dental composites [9-11], propellants [12], etc. are also not described here, as they are considered to be beyond the scope of this book. 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 [13-18]. 3.1 ROTATIONAL VISCOMETERS For filled polymer studies, rotational viscometers with either the coneplate or parallel-disc configuration are used. The major advantages of cone and plate viscometers are: (i) a constant shear rate is maintained throughout the melt sample, (ii) a small quantity of sample is required for measurement. On the other hand, the chief advantage of the parallel disc configuration is that it can be used for filled polymer systems of extremely high viscosity and elasticity. The basic limitation in rotational viscometers is that they are restricted in their use only to low shear rates for unidirectional shear and low frequency oscillations during oscillatory shear. At higher shear rates as well as at higher frequencies, a flow instability normally sets in the polymer sample which then begins to emerge out of the gap between the cone and plate or parallel-disc [19,20], thereby giving erroneous results. As a consequence of the above, the measured material functions do not actually conform to the

Table 3.1 Rheometers for filled polymer systems Capillary

Rotational

Unidirectional shear

Oscillatory shear

Constant speed

Plunger type Cone-n-plate

Screw extrusion type

Circular orifice

Circular orifice

* Commercial instrument.

Plunger type

Parallel disc Circular orifice

(a) Rheometrics mechanical spectrometer* (b) Sangamo Weissenberg rheogoniometer*

Constant pressure

(a) Monsanto automatic rheometer* (b) lnstron capillary rheometer*

Slit orifice

Slit orifice

Han's slit rheometer

(a) Haake rheocord* (b) Brabender plasticorder*

Melt flow indexer (a) Kayeness* (b) Ceast* (c) Davenport*

higher deformation rates which are normally prevalent in processing operations. Commercially available rotational instruments, such as the Mechanical Spectrometer (Rheometrics Inc., Piscataway, NJ, USA) and Weissenberg Rheogoniometer (Carri-Med Ltd, Dorking, England) can be used for unidirectional rotational shear as well as oscillatory shear and come with interchangeable cone and plate/parallel-disc configurations. 3.1.1

CONE AND PLATE VISCOMETER

The cone and plate viscometer is a widely used instrument for shear flow rheological properties of polymer systems [21-32]. The principal features of this viscometer are shown schematically in Figure 3.1. The sample whose rheological properties are to be measured is trapped between the circular conical disc at the bottom and the circular horizontal plate at the top. The cone is connected to the drive motor which rotates the disc at various constant speeds while the plate is AXIAL IHKUST MEASDHWGDEVICE

TORQUE MEASDWNG DEVICE

STATIONARY FLAT DISK

POLYMER MELT

RQTA3TNG CONICAL DISK

Figure 3.1 Schematic diagram showing the principal features of a cone and plate rotational viscometer.

connected to the torque-measuring device in 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 (O0 < 4° or 0.0696 radius) so as to maintain [14] cosec2 O0 = 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 of the conical disc. Similarly, the shear rate can be assumed to be constant for any point within the gap because of the predesigned method of gap adjustment as described earlier. The flow curve for a sample held between the cone and plate is generated from measurements of the torque experienced by the plate when the cone is rotated unidirectionally at different speeds. The various parameters of relevance are determined as follows. A. Shear rate For a constant speed of rotation of N rpm, the linear velocity (v = cor) is 27rrN/60m/sec where co is the angular velocity (rad/sec) and r is the radial position in meters. The gap height at r is rtan90 where 90 is the cone angle. Hence shear rate in reciprocal seconds at r can be written as, _ __ __ . _ 2nrN _ nN ^ nN y ~ 60rtan0 0 ~ 30tan00 ^ 306^ ' Since the cone angle is always maintained to be very small, the approximation of tan O0 = O0 does hold good. B. Shear stress The following expression defines the relationship measured torque and the shear stress:

between

the

f* -* T = 2TiT21 / r2 dr = f nRi2l

(3.2)

Tf T 2 I=-Z 5

(3-3)

Jo

Thus,

2nR The shear stress is then obtained in pascals when T is expressed in newtons.m and R in meters. The ratio of equation (3.3) to equation (3.1) results in the apparent viscosity expressed in Pa.sec.

C. Normal stress difference The cone and 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 atmospheric pressure, then it can be shown [14] that the total normal force NF on the plate is given by, Np = f 2nrpdr

(3.4)

JO

which on integration gives

nR2 NF=-J-Ni

(3,5)

Thus

N,=^ (3.6) nR Using equations (3.1) and (3.6), a plot of primary normal stress vs. shear rate can be generated. The shear stress and primary normal stress measurements can be done simultaneously on the sample when it is subjected to unidirectional rotational shear in the gap of a cone and plate viscometer. D. Oscillatory shear The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the sample is deformed by an oscillatory 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 is attached a stress transducer. On account of the energy dissipated by the viscoelastic polymer system, a phase difference develops between the stress and the strain. The complex viscosity behavior is determined from the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions, */', G', G" and others [33-4O]. 3.1.2

PARALLEL-DISC VISCOMETER

The parallel-disc viscometer used for measuring the shear stress and normal stress difference of filled polymer systems is similar in principle

to the cone and plate viscometer except that the lower cone is replaced by a smooth circular disc. This type of viscometer was initially developed for measuring the rheological properties of rubber [41-45] and hence made use of serrated discs placed in a pressurized cavity to prevent rubber slippage. When it was adapted for other polymeric systems [27,46,47], measurements were performed using smooth discs and without pressure. The rheological properties in the parallel-disc viscometer are based on the shear rate at the outer radius of the disc. Thus, ya = coR/H

(3.7)

where co is the angular velocity (rad/sec), R is the radius of the disc (m) and H is the gap between the two parallel discs (m). Shear stress and normal stress differences are given by the following relationships: ,R=jr/1+Î) 2nR\ 3dln yJ

(3.8)

(,,-„) -(*- *>=SH^g)

M

Oscillatory shear measurements can be done with the parallel-disc arrangement in a similar manner as in the case of the cone and plate viscometer and similarly the material functions, r\ ,G, G" and others can be generated. However, a slightly different technique [48] is at times used wherein the polymer sample is deformed between two oscillating parallel eccentric discs as shown in Figure 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 disc are thus interpreted in terms of G and G" [14]. 3.2

CAPILLARY RHEOMETERS

Capillary rheometers of various types are used for determining the rheological properties of polymer melts as can be seen from Table 3.1. The principal feature is that these rheometers are capable of extruding polymer samples at different speeds through the capillary of appropriate size. They are broadly categorized as (i) those operating at constant speed and (ii) those operating at constant pressure. A further categorization is possible based on the melt transport mechanism being of the plunger or the screw type and on the orifice

AXIALIHRUST MEASDMNGDEVICE

TORQUE MEASURING DEVICE

STAUQNARY FLATDISK

POLYMER MELT ROTAUNG FLATDISK

Figure 3.2 Schematic diagram showing the principal features of a parallel eccentric discs rotational viscometer.

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 Automatic Rheometer and the Instron Capillary Rheometer are examples of equipment which extrude the polymer through a capillary with a circular orifice using a plunger at constant speeds. The principal features of this rheometer are shown schematically in Figure 3.3. The major advantage of this type of capillary rheometer is that higher shear rate 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 processing operations, thus making the rheological data more meaningful for simulating processing behavior. Of course, the highest attainable shear rate data

V-CONSTANT PLUNGER.

BESERVOIR

CAPILLARY DEB

POLYMERMELT

Figure 3.3 Schematic diagram of a constant plunger speed circular orifice capillary rheometer.

are limited due to the occurrence of flow instabilities resulting in extrudate distortion or melt fracture at die wall shear stress levels greater than 1O5Pa [49-53]. The die wall shear stress TW can be easily calculated by taking a force balance across the capillary die as, 7ÂPdie = 27TRN/NTW

(3.10)

TW = ^^

(3.11)

or ^N

where RN and /N are the radius and length of the capillary die, while APdie is the pressure drop required to extrude the polymer melt. Since the polymer flows from a wide reservoir into a capillary die in a converging stream and then exits into open air or another wide reservoir in a divergent stream, it is necessary to correct the shear stress value for these entrance and end effects. The use of long capillaries in the vain hope that the end effects might be negligible is not recommended and in fact, should be discouraged. In capillaries longer

than D0, pressure dependence effects become significant. Hence, end effects can never be assumed to be negligible. The customary method of incorporating end effects correction is through the use of an effective capillary length (/N + £RN) as suggested by Bagley [54]. It must be emphasized here that basically there is no alternative but to carry out the Bagley procedure to make end corrections. The wall shear stress for fully developed flow over the length (/N + £#N) is then written as, w

_

^NAPdie

~~ A'N in + _i_rj? C^N)^t

^

>

The shear rate at the die wall is expressed by the RabinowitschWeissenberg [55] equation for steady laminar flow of a timeindependent fluid as, w

,4QT3 " ^3N [4

ldln(4Q/^N)1 4 dlnr w J

(6 L6)

'

The term d In (^Q/nR^)/d In TW is basically equal to l/n where n is the power-law index depicting the non-Newtonian character of the polymer system. Thus, from equations (3.12) and (3.13), the following relationship is written RNAPdie

=

/4Q\

2(/^Ki^ H^J

(3 14a)

'

or, 'N

=

r .

c

&Pdie

^ ~ 57M /cn

/oi/iu\

(

}

Uy

The above equation is a straight line when a plot of / N /K N vs. APdie is constructed at different constant values of (4Q/nR^) as shown in Figure 3.4(a). This is done using dies of various / N /# N ratios and the intercept on the / N /^N ordinate at APdie = O determines the value of —(. There are possibilities of observing slight non-linearity in the plots as can be seen for data at 3.6 and 10.8 s'1 in Figure 3.4(a). These are probably due to the breakdown of the assumptions made during the derivation of equation (3.14) of time-independence and no wall slip. True mechanical wall slip can occur during polymer flow when the shear stresses are large enough to overcome the static friction between the wall and the flowing material [56-62]. Mechanical slip can occur as either a steady-state phenomenon or as an unsteady phenomenon known as 'stick-slip' [62-64]. This wall slip may induce the slight nonlinearity in the plots shown in Figure 3.4(a). It must be shown that the Bagley plot is linear before any capillary viscometry data are regarded

PRESSUREDROP (1O6PASCAIS)

INTERCEPT

Figure 3.4(a) Plot for determination of the Bagley correction term during polymer melt flow through a capillary rheometer.

as meaningful. Hence, only those plots which are basically linear in Figure 3.4(a) are to be used. Once the plots have been shown to be linear for a particular capillary length and class of material, it is only then the capillary can be selected for viscometric measurements. From a linear regression of these plots, the correction term is determined. Using equation (3.12), the corrected shear stress value at the wall is estimated. It should be noted that, since polymers 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 [65] by modifying equation (3.12) to include the recoverable shear term as follows: ^

APdie

,gjgx

2(/ N /R N + C + SR/2)

^1DJ

Thus, the elastic energy stored at the capillary entrance is related to the correction term by the following expression [65]. ec = C + y

(3.16a)

Assuming Hooke's law in shear, TW = G x SR where G is the apparent melt shear modulus, the correction term is rewritten as

GLASS BEAD FILLED POLYPROPYLENE

UNITS

Figure 3.4(b) Variation of capillary correction term with true wall shear stress for glass bead filled polypropylene. (Reprinted from Ref. 66 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

ec = C + ^

(3.16b)

This suggests that when ec is plotted against TW, a straight line should emerge with a slope of ^G. When such a plot is prepared in the case of filled polymer systems, an interesting behavior is observed [66] as can be seen from Figure 3.4(b). The corrections for various concentrations of glass beads in polypropylene have been plotted. It can be seen that the correction term decreases with increasing filler volume concentration at constant shear, with the exception of the filled polypropylene system containing 26 vol.% of glass beads. The decreasing trend of the correction term with increasing glass beads is consistent with studies such as the one using glass bead filled styrene acrylonitrile (SAN) systems [67]. The decreasing trend indicates that the amount of stored energy must be decreasing and hence the recovered energy or die swell would also decrease with increasing glass bead volume fraction. This was indeed found to be the case [66] when a few measurements of die swell were qualitatively compared. The slope ec vs. TW lines are seen to be constant, except for = 0.21 and hence can be assumed to be independent of glass bead

concentration [66]. In the case of glass bead filled SAN systems, however, the ec vs. TW lines are highly non-linear [67]. The capillary rheometer can be used for estimating the normal stress difference using the total ends pressure loss [65,68] and the exit pressure loss [69-71], 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 [72] 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 [69,71,72] though others [73] 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 [69] as follows: TW =fro^ (3.17) dX where b0 = half thickness of the channel. The wall shear rate is determined from the following expression given in Refs 14 and 69:

y 7w

3Q T2 lln(3Q/4^)1 ~4fl 0 *d3 + 3

lnr w

J

(3 18)

'

where U0 is the half width of the channel. In general, this instrument is capable of providing data in the higher shear rate 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 determination of normal stresses. However, the probable velocity-profile distortions at the exit may introduce errors that may not be negligible though experimental evidence based on limited data [26,71] suggests otherwise. 3.2.3 CONSTANT SPEED SCREW EXTRUSION TYPE 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

HOPPER

POLYMEEt POWDER OIL PELLETS

POLYMERMELT

MELT TEMPERATDRS THERMOCOUPLE

PRESSURE TRANSDUCER EXTJLUSlDN SCREW

CAPULARy BODY

CAPILLARY DIE

POLYMER MELT

Figure 3.5 Schematic diagram showing the principal features of a constant speed screw extrusion type capillary rheometer.

of the screw extrusion type rather than the plunger type discussed earlier. A schematic diagram of an extrusion capillary rheometer is shown in Figure 3.5. Commercially available extrusion capillary rheometers are the Haake Rheocord (Haake Buchler Instruments Inc., Saddle Brook, NJ, USA) and the Brabendar Plasticorder (Brabendar, Duisburg, Germany). The rheological property measurements can be done using a circular or slit orifice as these are separate attachments for the miniaturized single screw extruder. These types of capillary rheometer 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 are automatically corrected for the Bagley correction and the RabinowitschWeissenberg correction through a computer software program [74]. The screw extrusion type capillary rheometers have been used for rheological studies of polymers [75,76] but have not become as popular as the plunger type capillary rheometers because they need a much larger quantity of feed. Care has to be taken that the material completely fills the extruder screw during 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 Brabendar 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 too has at times been used for assessing the rheological properties of polymer systems. The

torque vs. rpm data generated by internal mixer can be easily converted [77-79] to shear stress vs. shear rate data. A more detailed understanding of torque rheometry and instrumentation can be obtained from the excellent article by Chung [74]. 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 two differences. Firstly, the capillary used is of very short length and secondly, the polymer is extruded by the use of dead weights (i.e. constant pressure) rather than constant plunger speed. This instrument, popularly known as the Melt Flow Indexer, is very popular in the thermoplastics industry due to its ease of operation and low cost, which more than compensates for its lack of sophistication. The parameter measured through the melt flow indexer contains mixed information of the elastic and viscous effects of the polymer. Further, no end loss corrections have been developed for this capillary equipment nor can the melt flow index be easily related to the Weissenberg-Rabinowitsch shear rate expression. In most monographs and texts on polymer rheology, the Melt Flow Indexer has been treated in a very brief manner because it has generally been considered as an instrument meant only for quality control. It was specified as a standard rheological quality control test in the ASTM, BS, DIN, ISO and JIS (see Appendix D, Abbreviations list for complete forms of these standards). However, it has been shown in the recent past [80] that the Melt Flow Indexer provides more than just a quality control rheological parameter. In fact the book on Thermoplastic Melt Rheology and Processing [81] shows the multiple uses of the data from the Melt Flow Indexer, and treats this particular instrument in the utmost detail. Hence, in the present book the Melt Flow Indexer and the Melt Flow Index are discussed rather briefly; and readers are encouraged to refer to the other book [81] for more comprehensive discussion on the subject. 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 Figure 3.6. The cylinder is of hardened steel and is fitted with heaters, lagged, and controlled for operation at the required temperature with an accuracy of ±0.5°C. The piston is made of steel and the diameter of its head is 0.075 ± 0.015 mm less than that of the internal diameter of the cylinder, which is 9.5mm. The die (or 'jet') has an internal diameter of 2.095 ± 0.005 mm or 1.180 ±0.005 mm (depending on the procedure used) and is made of hardened steel. All

LOAD

PISTON BARBEL

HEATER A 3NSUIATEQN

REMOVABLE DIE

POLYMER Figure 3.6 Schematic diagram of the melt flow index apparatus showing a crosssectional view of the important parts.

surfaces of the apparatus which come into contact with the molten polymer are highly polished. MFI is basically defined as the weight of the polymer (g) extruded in lOmin 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 B for commonly used polymers. The test conditions include temperatures between 125 and 30O0C and different applied dead loads from 0.325 to 21.6kg giving pressures from 0.46 to 30.4kgf/cm2. The specifications have been selected in such a way as to

give MFI values between 0.15 and 25 for reliable results. ASTM D1238 gives the accuracy of the MFI value obtainable from a single measurement as carried out by different operators at different locations to be in the range of ±9 to ±15% depending upon the magnitude of the MFI. 3.3 EXTENSIONAL VISCOMETERS The rotational viscometers and the capillary rheometers described in sections 3.1 and 3.2 are those applicable for shear flows. However, there are 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 extend it. In general, measurement of steady-state extensional viscosity has proven to be extremely difficult. Steady extensional rate would be achieved by pulling the ends of the sample apart such that / = I0 exp(ef) 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 attempted for the measurement of extensional viscosity. 3.3.1 HLAMENT STRETCHING METHOD

The most common method for measurement of extensional viscosity is to stretch the filament of material shown in Figure 3.7 vertically as done by Ballman [82] or horizontally as done by Meissner [83]. The polymer must have a high enough melt viscosity of 104 Pa.sec 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 and hence the deformation rates are limited to 5 sec"1 or less. In the method of Ballman [82], which has been used by others [84,85], a vertical thermostated filament is clamped at both ends and stretched at the rate dl/dt such as to maintain a constant deformation rate. Thus,

,-i*

(a) VERTICAL FILAMENT STKBTCHING

CB) HORIZONTAL FILAMENT STKCTCHINO

Figure 3.7 Schematic diagram showing the principal features of the filament stretching method for extensional viscosity measurements: (a) vertical filament stretching; (b) horizontal filament stretching.

In the method of Meissner [83], a horizontal filament immersed in thermostated immiscible oil is held at both ends between pairs of toothed wheels rotating with a linear velocity of V/2. Thus, deformation rate is written as,

i= V/2=

V

ik J

(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 [86,87]. 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 [87] a better constancy in the extensional rate is obtained. Extensional viscosity based on constant stress measurements [88] has also been reported [89,9O]. In one case [89], the filament is extended vertically on top of a bath whereas in the other case [90], the vertical sample is immersed in the bath. The commercial equipment available for the measurement of extensional viscosity from rheometrics is based on the latter [9O]. A new universal extensional rheometer for polymer melts has been described by Munstedt [91]. It was specifically designed with the idea of making measurements on small samples possible 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 of extensional rheometer described earlier. The fact that the design is amenable for use in uniaxial and biaxial extensional rheometry has been shown by Meissner et al. [92]. Other biaxial extensiometers have also been described [93,94] by other researchers. A method for measurement of viscoelastic properties of polymers in the prestationary extensional flow has been investigated by Leitlands [95]. A 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 [96]. In terms of uniform extensional flow of polymers, a rather comprehensive review is that of Petrie and Dealy [97] which may be referred to for further information on the subject. 3.3.2 EXTRUSIONMETHOD

A typical example of extensional flow is the flow at the entrance of a capillary die. Besides the converging flow analysis of Cogswell [98,99],

there have been other analyses [100,101] in more recent times which are improved versions of the same ideas, and these can be used as better alternatives especially when dealing with filled polymer systems. Cogswell [102] has shown that the pressure losses through such dies can be used as a measure for the extensional viscosity. This method has not gained 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 [103] did suggest later that the die ought to 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 [104]. The extrusion method using a lubricated die [104,105] allows the measurements of systems with viscosity levels as low as 102Pa.sec. Thus, it can be used for extensional viscosity determinations in the case of nylon and polyester which are often spun to make synthetic fibers. Higher extensional rates, even 200 sec"1 are also achievable in this apparatus [104,105], thus making the information relevant for the polymer processing industries involved in fiber spinning.

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83. Meissner, J. (1969) A rheometer for investigation of deformationmechanical properties of plastic melts under defined extensional straining, Rheol Acta, 8, 78-88. 84. Vinogradov, G.V., Radushkevich, B.V. and Fikham, V.D. (1970) Extension of elastic liquids; polyisobutylene, /. Polym. ScL1 A2 (8), 1-17. 85. Stevenson, J.F. (1972) Elongational flow of polymer melts, AIChE /., 18, 540-7. 86. Macosko, C.W. and Lorntsen, J.M. (1973) The rheology of two blow moulding polyethylenes, SPE Antec Tech. Papers, 461-7. 87. Ide, Y. and White, J.L. (1978) Experimental study of elongational flow and failure of polymer melts, /. Appl Polym. ScL, 22,1061-79. 88. Dealy, J.M. (1978) Extensional rheometers for molten polymers: a review, /. Non-Newtonian Fluid Mech., 4, 9-21. 89. Cogswell, F.N. (1968) The rheology of polymer melts under tension, Plastics & Polymers, 36,109-11. 90. Miinstedt, H. (1975) Viscoelasticity of polystyrene melts in tensile creep experiments, Rheol. Acta, 14,1077-88. 91. Miinstedt, H. (1979) New universal extensional rheometer for polymer melts measurements on a polystyrene sample, /. Rheol., 23, 421-36. 92. Meissner, J., Raible, T. and Stephenson, S.E. (1981) Rotary clamp in uniaxial and biaxial extensional rheometry of polymer melts, /. Rheol., 25, 1-28. 93. Denson, C.D. and Hylton, D.C. (1980) A rheometer for measuring the viscoelastic response of polymer melts in arbitrary planar and biaxial extensional flow fields, Polym. Engg ScL1 20, 535-9. 94. Rhi-Sausi, J. and Dealy, J.M. (1981) A biaxial extensiometer for molten plastics, Polym. Engg ScL, 21, 227-32. 95. Leitlands, V.V. (1980) Investigations of unsteady elongational flow of polymer melts, Int. J. Polymeric Mater., 8, 233-42. 96. Prokunin, A.N. (1980) Some methods of experimental studies into the extension of polymeric liquids, Int. Polymeric Mater., 8, 303-21. 97. Petrie, C.J.S. and Dealy, J.M. (1980) Uniform elongational flow of molten polymers, in Rheology (ed. G. Astarita, G. Marrucci and L. Nicolais), 1, 177-94. 98. Cogswell, F.N. (1972) Converging flow of polymer melts in extrusion dies, Polym. Engg ScL, 12, 64-73. 99. Cogswell, F.N. (1972) Measuring extensional viscosities of polymer melts, Trans. Soc. Rheol., 16, 383-403. 100. Binding, D.M. (1991) Capillary and contraction flow of long glass fiber reinforced polypropylene, Composites, 2, 243-52; (1988) An approximate analysis for contraction and converging flows, /. Non-Newtonian Fluid Mech., 27,177-89. 101. Gibson, A.G. (1989) Die entry flow of reinforced polymers, Composites, 20, 57-64. 102. Cogswell, F.N. (1969) Tensile deformations in molten polymers, Rheol. Acta, 8,187-94. 103. Cogswell, F.N. (1978) Converging flow and stretching flow: a compilation, /. Non-Newtonian Fluid Mech., 4, 23-38. 104. Everage, A.E. and Ballman, R.L. (1978) The extensional flow capillary as a new method for extensional viscosity measurement, Nature, 273, 217-15. 105. Winter, H.H., Macosko, C.W. and Bennett, K.E., (1979) Rheol. Acta, 18, 323.

Constitutive theories and

equations for

suspensions

A

H"

4.1 IMPORTANCE OF SUSPENSION RHEOLOGY A suspension is a system in which denser particles, that are at least microscopically visible, are distributed throughout a less dense fluid and settling is hindered either by the viscosity of the fluid or the impacts of its molecules on the particles. In the present terminology of filled polymer systems, the fillers form a disperse phase and the softened or molten polymers form the continuous phase and together they could represent a suspension. However, suspension rheology does not normally refer to filled polymer rheology. In fact, it commonly discusses the rheological behavior of two-phase systems in which one phase is solid particles like fillers but the other phase is water, organic liquids (e.g. benzene, fuel oil) or polymer solutions. These systems are much easier to study than filled polymer systems as the preparation and rheological characterization of the systems can be done at room temperature. Further such systems are encountered in a number of areas other than polymer technology, namely, biotechnology [1-17], cement and concrete technology [18], ceramic processing [19], coal transportation [20-28], coating and pigment technology [29-32], dental research [33-38], propellants and explosive science [39], poultry waste handling [40,41], mineral processing [42-46], soil science [47] and various slurry flow technology [48]. Hence, the rheology of suspensions has received a lot of attention. There are a number of reviews [49-61] which discuss various aspects of the rheology of suspensions and may be referred to for detailed study. In the present chapter, the topic is touched upon rather briefly and only certain aspects are discussed in a limited manner just enough to lay the foundations for understanding the basics of filled polymer rheology. Suspensions basically show strong departures from simple Newtonian laws of fluid flow and hence form complex rheological

systems. When the dispersing medium is a Newtonian fluid or behaves like one under a given range of shear rates, then the suspension exhibits Newtonian behavior at low concentration of solids and non-Newtonian behavior with increasing concentration. When the suspending medium is a polymer solution which itself is non-Newtonian in character, the presence of solid particles magnifies the complexities of its rheological behavior. The important rheological properties which need to be studied and measured in order to be able to characterize suspensions are the same as those which have already been indicated in Chapter 2. However, only the viscous flow behavior in shear and extensional flow will be discussed in this chapter. In particular, shear viscosity will be dealt with in sufficient detail because of the wealth of information that exists on it. 4.2 SHEAR VISCOUS FLOW Several theoretical and empirical relationships have been proposed to describe the viscosity of suspensions in Newtonian or non-Newtonian viscous liquids. These relationships have also been used, with ranging degrees of success, to correlate viscosity data when the suspending medium is viscoelastic [62]. In the following various relationships are reviewed. The viscosity of Newtonian as well as non-Newtonian suspensions is affected by the characteristics of the solid phase such as shape, concentration and dimensions of the particles, its size distribution, the nature of the surface, etc. The influence of each of these factors is examined below. 4.2.1 EFFECT OF SHAPE, CONCENTRATION AND DIMENSIONS OF THE PARTICLES

Spherical filler particles have received more attention than nonspherical and asymmetric particles. In the following, the effect of concentration and dimensions of the particles are presented under the sub-sections of different shapes of particles. A. Spherical particles (a)

Dilute suspensions

Einstein [63-65] was the pioneer in the study of the viscosity of dilute suspensions of neutrally buoyant rigid spheres without Brownian motion in a Newtonian liquid. He proposed the following relationship between the relative viscosity of the suspension rjr and the volume fraction of the suspended particles 0

>7r = l + aE0

(4.1)

where rjr is the ratio of the viscosity of the suspension rjs to the viscosity of the suspending medium q0 and aE is Einstein's constant. aE equals 2.5 when the suspended particles are neutrally buoyant, hard and spherical in shape, the mean interparticle distance is large compared to the mean particle size, the particle movement is so slow that its kinetic energy can be neglected and there is no slip relative to the particle surface. Experimental determinations of Reiner [66] and Kurgaev [67] revealed that for filler concentrations of = 0.003-0.05, aE was indeed equal to 2.5. Rutgers [68] concluded through experimental evidence that equation (4.1) with aE = 2.5 was valid for values up to 0 < 0.1. However, a slight disagreement in the value of aE = 2.5 was established from the works of Hatschek [69], who theoretically found that aE — 4.5 for (j) = 0-0.4 and Andres [70] who found that for dilute magnetic suspensions aE = 4.5 for (/> = 0-0.09. In fact Kurgaev [67] showed that when (j) = 0.15-0.18, aE = 4.5-4.75 depending upon the nature of the solid particles. The disagreement was accentuated when Happel [71] suggested a value of aE — 5.5 and Pokrovskii [72] a value of 1.5. It was revealed by Kambe [73] that the lack of agreement among the experimental results was due, among other things, to differences between the dimensions of the particles under study and the velocity gradients used for the experiments. It appears that for solid spheres with diameters large enough compared to the molecular dimensions but small enough compared to the characteristic length of the measuring instrument and for no slip at the sphere surface, the value of aE = 2.5 is generally accepted though values ranging from 1.5 to 5.5 have been suggested. Based on the theoretical analysis of Simha [74] for concentrated suspensions. Thomas [75] proposed the following expression for dilute suspensions ($ < 0.1) n, = l

+ 2.5(l +^JjJ

(4.2)

where U1 is an empirical coefficient whose value lies between 1 and 2. Thomas [75] suggested that U1 = 1.111 for < 0.15. Simha and Somcynsky [76] suggested that the expression (4.2) proposed by Thomas [75] could be written as follows: i/r = 1+2.5^1(0) where

*-(>^>

(4.3)

when higher terms in 0 in the expression suggested [76] are dropped. When (j) = 0.10, however, higher terms in (/> are not negligible and to compensate for this, Thomas [75] had to use the value of ^1 — 1.111 when actually U1 = 1.85 in the unapproximated expression of A((/>) gave excellent results. Ford [77] modified Einstein's equation (4.1) using a binomial expression and wrote -=l-aE0

(4.4)

'Ir

where l/rjr is defined as the fluidity and is equal to zero when 4> = l/a E . Equation (4.4) has been shown to be valid for 4> < 0.15 by the experimental data of Cengel et al. [78]. Though there is varied opinion about the relationship between the relative viscosity of a suspension and the volume concentration of the spheres for dilute suspension, one could get a reasonable estimate on using the simplest equation (4.1) of Einstein for 0 < 0.1. When 0.1 < (f> < 0.15, Thomas's [75] equation (4.2) or Ford's [77] equation (4.4) could be used for a reliable estimate. Of course for 0 < 0.1 too, these equations could be used and the result averaged out with the prediction from equation (4.1) to obtain a good conservative estimation. (b)

Concentrated suspension

Generally, when 0.1 < 0 < 4>m the suspensions are considered to be concentrated and the above discussed equations do not apply. Here 0m is defined as the maximum attainable concentration and has the following form: 0m = 1 — e, where e is the void fraction or porosity, and is defined as the ratio of the void volume to that total volume. Theoretically, the value of m is 0.74 for equal spheres in compact hexagonal packing, but in practice it is more like 0.637 for random hexagonal packing or 0.524 for cubic packing ([79]). When the filler concentration is increased, various phenomena take place, for example (i) the number of particles per unit volume which come in contact during the flow increases, (ii) the interparticle attraction and repulsion effects become stronger due to electrostatic charges, which depend upon the polarity of the medium, (iii) the rotation of the particles during flow, as well as the formation of doublets and their rotation during flow, produces additional dissipative effects which lead to an increase in the viscosity. Unlike the dilute suspensions, the size of the filler drastically changes the viscosity behavior of concentrated suspensions. De Brujin [80] showed that when the filler diameter is less than 10 (am, a concentrated suspension exhibits non-Newtonian behavior and the viscosity

increases with a decrease in the filler diameter. Clarke [81] found that for a filler diameter greater than 10 jam, the viscosity increases linearly with the diameter. For spheres, with increasing diameter the lateral displacement of the particles towards the centre of the tube (central tube effect) increases, thereby increasing the energy dissipated resulting in a tendency for the viscosity of the suspension to increase with increasing diameter. As there are many-fold effects of increasing the concentration of the fillers, a variety of physical models have been proposed but most of them (theoretical or experimental) can be expressed by the nonlinear relationship between rjr and c/> in the following power series form as given in Thomas [75] */r = 1 + a a 0 + a2(/>2 ± a3(/>3 ± ...

(4.5)

where (X1 is generally assumed to have a value of 2.5 as given by Einstein [63], while the coefficients a2, a 3 ,... have been assigned different values by different authors. For example, the value of a2 was 14.1 as determined by Guth and Simha [82], 7.349 by Vand [83], 12.6 by Saito [84]. 10.05 by Manley and Mason [85] and 6.25 by Harbard [86]. These varied values of a2 are the result of taking into account one or several effects appearing due to the increase in solid concentrations. Similarly, a3 values of 16.2 and 15.7 have been proposed by Vand [83] and Harbard [86], respectively. Alfrey [87] has also developed relationships of the power series form (4.5) and enlisted values of a,based on the works of Arrhenius [88], Fikentsher and Mark [89], Bungenberg de Jong et al [90], Papkov [91], Hauwink [92] as well as Brede and De Boojs [93]. As an example of the use of equation (4.5) to determine the viscosity of suspensions, one can refer to the works of Mullins [94] and Feldman and Boiesan [95] on rubbers containing fillers which are chemically inactive like wood flour or chemically active like carbon black. When the filler introduced is chemically inactive (with any (/>) or chemically active (with 0 < 0.10), the quadratic form of equation (4.5) with Oc1 = 2.5 and oc2 = 14.1 could be used to give a good estimate of the viscosity of the suspension. For higher concentrations of the chemically active filler (carbon black), particle interaction begins and the viscosity of the suspension increases markedly and equation (4.5) as such cannot then be used for an estimate. However, if particle interaction leads to agglomeration, then Mullins [94] and Feldman and Boiesan [95] recommend the use of Oi1 = 0.670, and a2 = 1.620? in equation (4.5), where a{ is the index of asymmetry of the elastomer macromolecules. The main drawback of equation (4.5) is that the termination of the series after 02 term means an error of 10% or more in the relative viscosity for > 0.15-0.20. The validity of the series increases to

(j) cz 0.40 on the inclusion of 0m, r\r ->> oo and rightly so.

(4-9)

Frankel and Acrivos [100] did away with all empiricisms and artificial boundaries and provided an expression for highly concentrated suspensions of uniform solid spheres intending to complement the classical Einstein's equation (4.1) valid only for very dilute suspensions. Their final result is written as follows: 9

0Mn)178

"'-81-OMA n T

(410) (4 10)

'

With so many theoretical expressions (4.5) to (4.10), it is increasingly difficult to make a choice between them and decide which one would give the most reliable estimate for the relative viscosity of a concentrated suspension. For concentrated suspensions, it is necessary to account for the hydrodynamic interaction of particles, particle rotation, particle collisions, doublet and higher order agglomerate formation and mechanical interference between particles as packed bed concentrations are approached. Different authors have taken into account one or several aspects mentioned above during the derivation of their theoretical expressions. For concentrated suspensions of uniform solid spheres, the use of expression (4.6) of Thomas [75] is recommended for 0.15 < < 0.60 and the expression (4.10) of Frankel and Acrivos [100] for 0 -> m, it does not reduce to equation (4.1) when 0 ^ 0 . Further, the averaging process used for deriving equation (4.10) has been shown to be incorrect [101] and it has been argued that the dissipation in pair interactions is too small to explain the observed trends. But since equation (4.10) does fit experimental data rather well for high solids concentrations, it can be simply considered as yet another empirical equation. Attempts [102-104] to fit the entire range of volume fraction from (j) -* O to 0 -> 0m have resulted in equations which give a unique curve through the use of a plot of relative viscosity versus the ratio of 4>/4>m. The work of Chong et al. [102] has shown a good fit between experimental results and an equation of the following type:

'"Mi^fc)]' 0m is normally determined from the experimental data. It is to be noted that equation (4.11) reduces to equation (4.1) at low values of (/) when (J)n takes a value of 0.6. One of the best available empirical expressions which fits the entire range of volume fraction, is the Maron-Pierce type equation that was

carefully evaluated by Kitano, Kataoka and co-workers [103,104], and extensively tested by Poslinski et al [105,106]. rjr = [1 - 4>/(t>m}-2

(4.12a)

For suspensions of smooth spheres, a value of 0m = 0.68 has been suggested [107] and a value of 0m = 0.60-0.62 has been determined through liquid displacement experiments [105,106]. In reality, of course, using ^1n as 0.6 or 0.62 or 0.68 does not improve the data fit appreciably. But at times it may be best to view m as an adjustable parameter and then equation (4.12a) is rewritten as follows: iyr = (1 - 1. Thus, >7r = l + a rl 0 + ar22

(4.13)

where «"=) + 2

--)SIwhere Ji is an interaction parameter, /J0 is a rate constant for the equilibrium between free particles and floccules and Z0 is the degree of flocculation.

Hashin [120] used the flow-elasticity analogy to give the following equation which was valid only for parallel, randomly placed infinitely long fibers ^r = l +1-0 A

(4-20)

Nielsen [121] used the same analogy, but his equation has not been tested for concentrated fiber suspensions. The shape of the rod (whether straight or curved) does affect the relative viscosity of the suspension. The viscosity for curved fiber suspension is known to be higher than that for a straight fiber suspension and the difference increases with increasing concentration (Figure 4.4). 4.2.2 EFFECT OF SIZE DISTRIBUTION OF THE PARTICLES

Clarke [81] observed that mixed suspensions of mainly coarse particles and relatively few fine particles showed a marked decrease in the viscosity compared to an all coarse suspension. Contrarily, suspensions with mainly fine particles and few coarse particles showed very little change from an all fine suspension. It could thus be concluded that

STRAIGHT FIBERS CURVED FIBERS

Figure 4.4 Variation of the relative viscosity of suspensions with concentration for (a) curved fibers and (b) straight fibers.

smaller particles are interposed between larger particles, causing a reduction in the interparticle impact resulting in a decrease in viscosity. Ward and Whitmore [122], Ting and Luebbers [123] and Moreland [124] also noticed similar results using different techniques of measurement. Shaheen [125] suggested that the addition of a little amount of small particles acts as a lubricant to facilitate the rotation of larger particles, leading to a reduction in the relative viscosity. Experimentally, it was shown that the viscosity of a mixture of two different-sized particles goes through a minimum at about a volume fraction of small particles equal to 0.25. Shaheen [125] wrote the modified form of Mooney's equation (i.e. equation (4.8)) for a mixture of spherical particles of two different sizes as follows:

=

/ 2/5a \

/ 2.502 \

'~ Mr^Mi^^J where

(421)

, LtQOIO : MCOH

SHCAP RATE (»-')

RELATIVE VISCOSITY

CCRAMtC: 3O v*l% *>}Oj UOUID: 3:1. HMK'MCOM

SMEAA RATE U•') Figure 4.9 Variation of relative viscosity with shear rate for 30 vol% alumina suspensions prepared with (a) methanol and (b) 3:1 methyl isobutyl ketone/methanol with indicated poly(vinyl butyral) concentrations. (Reprinted from Ref. 19 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.)

polymer. In 3:1 MIBK/MEOH suspensions, PVB additions (0.5 to 2.0vol%) give much lower relative viscosity and almost Newtonian behavior. A slight dilatancy has been observed which is not unusual in highly loaded suspensions in which repulsive forces are large [153,154]. The level of loading makes a lot of difference in the adsorption of the polymer. It can be seen from Figure 4.10(a) that for 30 vol% AÔs, there is only a small difference in the shear stress versus shear rate flow curve for the two polymer concentrations of 0.25 vol% and 0.5vol%

SMC ARSTfIESS(Pa)

CERAMIC : 30 V0 H- K^t. 4.2.10 EFFECT OF AN ELECTROSTATIC FIELD When suspension particles are charged, the electroviscous effects that arise strongly influence the viscosity of the suspension as was shown by the experiments of Fryling [157] as well as Krieger and Eguiluz [158]. Pseudoplastic as well as dilatant behavior was observed in the data of Fryling [157] and when the electroviscous effects were at their maximum, the suspensions of Krieger and Eguiluz [158] were seen to have a yield stress. Electroviscous effects are essentially of three types first, second and third, and are discussed in detail by Conway and Dobry-Duclaux [159]. The combined effect of the three electroviscous effects on the viscosity of a suspension can be written as follows: fr = l + (*vl +**+**)*

(431)

where evl,ev2 and ev3 correspond to each of the three electroviscous effects. Separation of the constituent effects is difficult but was attempted by Dobry [16O]. The three effects are discussed below separately in order to appreciate the influence of each one of them on viscosity.

A. First electroviscous effect The first electroviscous effect is due to the electrostatic contribution of charged colloidal particles and its effect on the viscosity of a dilute suspension can be expressed in an extension of Einstein's equation (4.1) as follows: >/r = l + a E =1 k/*M"

MFI(T, ). In the case of filled systems, this condition is naturally satisfied when the polymeric matrix is taken as the reference medium. Equation (6.15) predicts that a plot of I/log 0MFI vs. 1 /(J) 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 filled systems are in the form of viscosity vs. shear rate or shear stress vs. shear rate curve. In each case the data are transformed into specific MFI values using the method discussed in Shenoy and Saini [99]. Figures 6.11-6.16 show plots of l/log0MFI vs. 1/0 for different filled

QTJARTZ FILLED LOW DENSTTY POLYEIHYLENE

Figure 6.11 Melt Flow Index variation with filler composition for low density polyethylene/quartz powder composite at 22O0C and 2.16kg test load condition for MFI using data from Ref. 16. (Reprinted from Ref. 51 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

CALCIUM CARBONAU FILLED POLYPROPYLENE

Figure 6.12 Melt Flow Index variation with filler composition for polypropylene/ calcium carbonate composite at 20O0C and 2.16kg test load condition for MFI using data from Ref. 5. (Reprinted from Ref. 51 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

polymer systems. The systems are so chosen as to include different generic types of polymers as the matrix, and to include fillers with different shapes and types. In all cases, despite the apparent diversity, there is a uniqueness in the altered free volume state model, adjudging from the straight line fits obtained in all plots. Though it is recommended that the relative viscosity rjT or the relative MFI value aum be used for estimating the rheological changes due to filler concentration, at times a simple plot of MFI vs. the weight fraction of the filler can certainly provide the same information as done by Arina et al [17]. The effect of fillers on the melt flow properties of polyethylene were investigated [17] by determining the melt flow indices of the compounded filled systems. It was found that finely divided fillers reduced the melt flow index of polyethylene more than coarsely divided fillers, a result similar to that discussed in section 6.2. As regards the effect of concentration, the melt flow index was not affected much at small concentrations but there was a sharp

CARBON BLACZ FILLED POLYSTYlENE

Figure 6.13 Melt Flow Index variation with filler composition for polystyrene/carbon black composite at 18O0C and 5.0kg test load condition for MFI using data from Ref. 27. (Reprinted from Ref. 51 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

TTTANIDM DIOXIDE FILLED POLYSTYBENE

Figure 6.14 Melt Flow Index variation with filler composition for polystyrene/titanium dioxide composite at 18O0C and 5.0kg test load condition for MFI using data from Ref. 27. (Reprinted from Ref. 51 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

GLASS FIBER FILLED POLYCARBONATE

Figure 6.15 Melt Flow Index variation with filler composition for polycarbonate/glass fibers composite at 29O0C and 1.2kg test load condition for MFI using data from Ref. 31. (Reprinted from Ref. 51 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

GLASS FIBER FELED POLYEmYlEOT TBREPHIHMAEB

Figure 6.16 Melt Flow Index variation with filler composition for poly(ethylene terephthalate)/glass fibers composite at 2750C and 2.16kg test load condition for MFI using data from Ref. 20. (Reprinted from Ref. 51 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

drop with increasing concentration as can be seen in Figures 6.17(a) and (b). 6.4

EFFECT OF FILLER SIZE DISTRIBUTION

In order to study the effect of filler size distribution, it is necessary to work with uniform monodisperse particles. These would have to be available in different sizes so that controlled mixtures of two or three different sizes can be studied. There are various methods of getting uniform-sized particles and these have been discussed in a number of articles [102-106]. The simplest technique of getting uniform mono-sized particles is by precipitation from solution due to the controlled generation of solutes by a single burst of nuclei [102]. This method is commonly used to form hydrated metal oxides by hydrolyzing the appropriate metal salt. Thus, spherical aluminum hydroxide can be obtained [102] from alum and spherical colloidal rutile can be prepared from TiCl4. Similarly, mono-

TALC/DOLOMTTE FHTFD LOW DENSITY POLYHHYLENE

UWTTS TALCA TALCB DOLOIgCTE TALCA

Figure 6.17(a) The influence of some fillers on the melt flow indices of B3024 and B8015 polyethylene grades. (Reprinted from Ref. 17 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

MICA FILLED LOW DENSITY POLYElHYLENE

UNITS

Figure 6.17(b) The influence of mica A on the melt flow index of B 8015 polyethylene at various temperatures. (Reprinted from Ref. 17 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

sized silica particles can be prepared by precipitation of solutes [103,104]. The other method of developing fine, spherical particles is through the use of plasma techniques which may involve a physical phenomenon or a chemical reaction [105]. In the former case, a simple spherodizing process takes place when irregular shaped powder is introduced into the plasma which melts it and then vaporizes it to form uniform spheres. When a chemical reaction is involved, the reactants in the solid or gaseous form in contact with the plasma are allowed to react in the vapor phase and the products are allowed to condense as deposits of fine powder on the cooler regions of the chamber after leaving the plasma. Colloidal spheres of aluminum, copper, aluminum nitride and silicon carbide are made in this way.

Uniform-submicron size polymer spheres can be made through emulsion polymerization [106]. When a sparingly water-soluble vinyl monomer like styrene that polymerizes by a free radical mechanism is dispersed in water in the presence of a surfactant and a water-soluble initiator, uniform-sized polystyrene beads can be obtained by controlling the number of micelles and ensuring that each micelle traps an initiator at about the same time. The various methods employed to get uniform-sized particles have a set goal to get model filler particles so that filled system experiments can be controlled effectively to isolate the effects of various parameters. This is especially important when studying filler size distribution effect on the rheology. Despite all efforts to get uniform-sized particles, there is often a distribution involved, however narrow it may be. Hence, to study the effect of a bimodal size distribution on the rheological behavior of filled polymer systems, glass spheres obtained from Potters Industries Incorporated were fractionated by Poslinski et al. [72] into narrow size ranges. Two size ranges of spheres were picked in the preparation of the bimodal mixtures, and Table 6.3 lists their particle size distribution, with the number of fractions of the zth component /j vs. the diameter of the zth component D1 being tabulated. Polydispersity tends to reduce the viscosity of filled systems at a fixed loading level [107-112]. For dilute suspensions having a volume fraction of solids less than 0.2, the effect of variation of particle size on filled system viscosity is minimal [108,111]. However, at high loading levels the viscosity can be reduced dramatically when the particle size Table 6.3 Average particle size and particle size distribution using digital image analyzer Small spheres

f\ 0.04 0.11 0.15 0.24 0.10 0.13 0.08 0.06 0.04 0.03 0.02

D

\ (^m) 2 5 9 12 16 19 21 25 29 32 35

Large spheres

f\ 0.01 0.04 0.04 0.15 0.10 0.20 0.21 0.15 0.06 0.03 0.01

D\ (Hm) 45 50 56 62 70 78 83 90 99 105 110

Source: Ret. 72 (reprinted with kind permission from Society of Rheology, USA).

modality is increased [98,113-115]. Henderson et al [114] reported a reduction in filled system viscosity as high as 96% when the modality was changed from a unimodal to a bimodal size distribution of spheres at a fixed volume fraction of 0.66. Theories on the viscosity of polydisperse systems have been developed [109,113] and a method to predict the viscosity of multimodal filled systems from the data for monomodal filled systems is also available [113]. Farris [113] has shown that the relative viscosity of the filled polymer system can be described by ^ = (1 - (t>Tk

(6.16)

where k is a constant which varies according to the particle size distribution and depends on the number of components making up the distributions. In practice, k varies from 21 for monomodal to 3 for infinite-modal distributions. Figure 6.18(a) graphically shows the relationships. For uniform spherical particles viscosity increases steeply after 0 = 0.5, approaching infinity at 0 = 0.74. However, if the total volume of particulate filler is split into 25% fines and 75% coarse, very high loading can be obtained without increase in viscosity; this is shown in Figure 6.18(b).

MQNOMODALk -21 BlMODALk -5.8 TRIMODALk. -3.6 MFINTTE MODAL k - 3

Figure 6.18(a) Comparison of calculated relative viscosity for the best multimodal system. (Reprinted from Ref. 113 with kind permission from Society of Rheology, USA.)

MONOMODAL

Figure 6.18(b) Variation of relative viscosity as a function of vol% solid spheres in monomodal and bimodal suspensions, with volume fraction of small spheres being 25%. (Reprinted from Ref. 113 with kind permission from Society of Rheology, USA.) If the ratio of the coarse to fine particle diameters is 7:1 the volume of filler can be increased to = 0.73 from the monomodal loading of (j) = 0.59 without increase in viscosity. Figure 6.18(c) shows the proportion of coarse and fine particles to give minimum viscosity for a range of total filler loading, indicating minima in the region of 30% fines. Further reductions may be possible with increasing modality; but for modalities greater than trimodal, the effects are not dramatic. A natural consequence of the above findings is that for a given level of filled system viscosity, it is always possible to increase the loading level of the fillers through a careful choice of particle size distribution. This particular point has a direct significance when dealing with ceramic and metal processing [48,66-69,73-75,116-127] as well as during the preparation of functional filler composites [60-63] where the prime intention is to have as high a filler loading as possible and yet maintain good processibility. Mangels [116,117] has made use of the reduced viscosity of wide size distribution powders to produce injection molding blends of high powder loading (73.5 vol%). Working with silicon powder, particle size distributions were obtained [116] by dry ball milling and air classifying, and it was subsequently shown that a 140 h dry ball-milled powder with the broadest particle size distribution yielded the best viscosity in

Figure 6.18(c) Comparison of calculated relative viscosity for bimodal suspension of various blend ratios and concentrations. (Reprinted from Ref. 113 with kind permission from Society of Rheology, USA.)

a spiral flow mold test [117]. In general, by altering the particle size distribution from a sharp, monomodal type [118] distribution to a very broad distribution, the solids content can be increased without increasing the viscosity of the system [119]. Similar requirements have been noted by Adams [120] for slip cast ceramics and there is a similarity with the requirements for achieving high compaction density in a pressed powder [121-123]. Chong et al. [98] were able to achieve volume concentrations close to Eiler's value of 0.74 for rhombohedral packing by using a bimodal mixture of spheres. They identified the diameter ratio of small to large spheres, 6, and volume percent of smaller particulates in the total solids mixtures, 0S, as two important parameters characterizing a bimodal solids mixture. By fixing 0S at 25%, Chong et al. [98] showed that the filled system viscosity was reduced as 6 decreased from 1.0 to 0.138; however, they surmised that no significant reductions would occur below a limiting particle size ratio of approximately 0.1 as the small spheres could easily migrate through the interstices of the large spheres. Calculations with varying s [98] suggested that there also exists an optimum volume percent of smaller spheres where the filled system viscosity is minimized.

Higher packing densities can be achieved if the particle sizes are not uniform. This enables the finer particles to fill the holes between the larger particles. The particles size range can be broadened in two ways. Mixtures of two particles sizes can be blended, for example, coarse and fine particles, or a continuous wide distribution of particle sizes can be selected. Increasing the number of particles sizes in the mixture can increase the calculated packing density [123]. Table 6.4 shows the maximum packing density attainable for random packing mixed spheres (with from one to four sizes) [124]. An extension of the idea of bimodal packing of spheres, is the packing of combined fibers and spheres. These have been well described and discussed in detail by Milewski [128]. It is pointed out that packing parameters change with respect to fiber length to diameter ratio and choosing proper size combinations of the mixed fillers optimizes the benefits from packing. Gupta and Seshadri [129] used Ouchiyama and Tanaka's results [130] to calculate the maximum packing parameter of polydisperse systems of spheres given the value for the monodisperse samples and taking into account particle size, size distribution and modality as follows m =

T^^ £(DZ - D3)3/;. + -[(D1- + D3)3 - (D1- - D3)3]/.

(6.17)

where 2 V(D LJ\î + ' D â/) Tl 1

'-'+* = 0.55 and m was taken as 0.62 based on an experimental evaluation. The experimental data in Figure 8.4 are fitted by two curves using the following theoretical expressions analogous to those of Kitano et al [22] and Chong at al. [89] for the steady shear case by substituting r(r for rjr. Thus, IJi = (I-^n)-2

(8.8a)

GLASS SPHERE FILLED POLYMERS CHEVRON GRADE - 18 CEEVUON GRADE - 24 CHEVRON GRADE - 32 CHEVRON GRADE - 122 THERMOPLASTIC @ 1300 C THERMOPLASTIC @ 150 0 C THERMOPLASTIC @ 1700C EQUATION (8.8a1 EQUAUON (8.8b)

Figure 8.4 Average relative dynamic viscosity at high frequencies as a function of the reduced volume fraction of filler. (Reprinted from Ref. 71 with kind permission from John Wiley & Sons, Inc., New York, USA.)

and

*-('*£&)'

It is seen from Figure 8.4 that equation (8.8a) consistently gives high values in comparison to (8.8b) but the fit provided by both equations is reasonably good. It is to be noted that equation (8.8a) is used for theoretical fit in Figure 8.4 instead of the expression of Kitano et al. [56] used by Poslinski et al [71]. Poslinski et al. [71] found that the storage modulus was far too low to be measurable for polybutene grades 18, 24 and 32 over the entire range of investigated frequencies. Hence, the relative storage modulus data

were available only for polybutene grade 122 and the thermoplastic polymer at three different temperatures. The average relative storage modulus at high frequencies was plotted vs. the reduced volume fraction /m as shown in Figure 8.5. The average values were obtained for each volume fraction from experimental data between 10 and lOOrad/sec. It is seen that experimental data fall on a reasonably unique curve fitted by the following theoretical expression [71] analogous to the equations (6.11) and (7.1) by simply substituting G'r for the material parameter as G; = (l-0/0 m )- 2

(8.9)

The effect of filler concentration on the complex viscosity and storage

GLASS SPHERE FILLED POLYMERS CHEVRON GRADE -122 THERMOPLASTIC @130°C THERMOPLASTIC @ 1500C THERMOPLASTIC @ 1700C EQUAHON (8.9)

Figure 8.5 Average relative storage modulus at high frequencies as a function of the reduced volume fraction of filler. (Reprinted from Ref. 71 with kind permission from John Wiley & Sons, Inc., New York, USA.)

modulus vs. frequency has been brought out by the data of Bigg [44] on alumina filled low density polyethylene systems as shown in Figure 8.6. For the same system but using a very low viscosity polyethylene, the effect of filler concentration on dynamic viscosity and loss modulus vs. frequency has been presented by Dow et al. [68] as shown in Figures 8.7(a) and (b). Independent of the base viscosities of the polyethylene (20Pa.sec and 0.2Pa.sec respectively), it is seen that the complex/ dynamic viscosity as well as storage/loss modulus increase by orders of magnitudes with increasing filler concentration, the effect being more prominent at lower frequencies. The storage and loss moduli begin to depict more solid like behavior at higher concentrations, as exemplified by its independence with respect to increasing frequency.

ALUMINA FILLKP LOW DENSITY POLYETHYLENE

UNITS PASCALS RAD/SEC

Figure 8.6 Effect of filler concentration on the complex viscosity and storage modulus variation with frequency for alumina-filled low density polyethylene. (Reprinted from Ref. 44 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

ALUMINA FILLED LOW DENSITY POLYETHYLENE UNITS

Figure 8.7(a) Plots of dynamic viscosity vs. frequency for alumina filled low density polyethylene prepared at the indicated concentrations of AI2O3. Mixing was carried out at 15O0C and the rheological measurements were made at 1250C. (Reprinted from Ref. 68 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.) 8.4

EFFECT OF FILLER SIZE DISTRIBUTION

The effect of filler size distribution is normally the most difficult to isolate due to added complexities in performing reliable controlled experiments. Hence, it often remains a neglected area. However, in the case of highly filled systems, Bigg [54] has used a very systematic procedure to investigate the effect of different particle size distributions on the packing behavior and the dynamic rheological properties of the filled systems. Filler particles were chosen very discretely to include both agglomerating and non-agglomerating types, as well as bimodal, narrow, and broad particle size distribution. Table 8.1 gives the characteristics of the particles investigated by Bigg [54] while Table 8.2

ALTJMINA FILLED LOW DENSITY POLYETHYLENE UNITS PASCALS

Figure 8.7(b) Plots of loss modulus vs. frequency for alumina filled low density polyethylene prepared at the indicated concentrations of AI2O3. Mixing was carried out at 15O0C and the rheological measurements were made at 1250C. (Reprinted from Ref. 68 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.) Table 8.1 Characteristics of filler particles investigated to study the effect of filler particle size distribution (PSD) Type

Supplier

Grade

Gf1GIm)

Stainless steel (SS) Stainless steel (SS) Alumina (AI2O3) Alumina (AI2O3) Zirconia (ZrO2) Silicon nitride (Si3N4)

Amdry

136F

15

Source: Ref. 54.

Alcoa

A16-SG

Zircoa GTE Sylvania

Type C SN 5

G(2Qim)

QU-cfie 30-9.3

6(36%)

54(64%)

0.6 2(30%) 3 s = 10 to 50% of the smaller spheres. The solid lines in Figure 8.10 represent the predictions of equation (8.8b) with the maximum packing fraction determined by equation (6.17) using the particle size distri-

GLASS SPHERE FILLED POLYBUIEHE TOIALSPHERES

EQUAHON

Figure 8.10 Average relative dynamic viscosity as a function of 0S, the volume fraction of the 15 urn glass spheres in the total solids mixture suspended in a polybutene grade 24 matrix at 220C. (Reprinted from Ref. 72 with kind permission from John Wiley & Sons, Inc., New York, USA.)

bution values of the bimodal components listed in Table 6.3. The agreement between theory and experiment can be seen to be quite adequate. In the case of the storage modulus, it was found that G values of the polybutene grade 24 matrix measured out as zero. Only at 40, 50 and 60% total glass spheres by volume were the filled polybutene G values measurable. Since a relative storage modulus could not be defined, the investigation [72] of the storage modulus was restricted to a bimodal size distribution at high concentrations. Figure 8.11 shows the storage modulus for 40, 50 and 60% volume fraction of total spheres for various volume percentages of the 15 jam spheres at a frequency of 1.0 rad/sec. It is seen that the storage modulus is also

GLASS SPHERE FILLED POLYBUTENE UNITS PASCALS RAD/SEC

TOTAL SPHERES

EXPERIMENTAL CURVES

Figure 8.11 Dynamic storage modulus as a function of ^8, the volume fraction of the 15^m glass spheres in a polybutene grade 24 matrix at 220C and a frequency of 1 rad/sec. The solid lines represent the best fit of the experimental data. (Reprinted from Ref. 72 with kind permission from John Wiley & Sons, Inc., New York, USA.)

reduced when two sizes of spheres are mixed together, and the lowest values are obtained for ^8 = 10 to 30% of the smaller spheres, which of course happens to be the same range when the maximum packing parameter is the highest. 8.5

EFFECT OF FILLER AGGLOMERATES

As already discussed in sections 6.5 and 7.5, the presence of agglomerates creates an apparent situation of higher filler loading than is actually present. This is because the agglomerates trap part of the surrounding liquid in their interparticle voids thereby decreasing the volume fraction of the liquid around it.

It is known that high shear mixing helps in reducing the agglomerates whereas low shear mixing may at times increase the number of agglomerates. This happens because during mixing the probability of particle-particle contact as well as particle-liquid contact is increased. If the mixing is done under low shear, then the agglomerates that are formed during particle-particle contact do not get an opportunity to break down. On the other hand, the high shear provides the energy to break the particle-particle bonds and once the polymer wets the particle, the bonds fail to reform, thereby giving a better dispersion. The effect of filler agglomerates is thus best illustrated by observing the unsteady shear response of the filled polymer system prepared under conditions of low shear and high shear mixing. Dow et al. [74] prepared alumina filled low density polyethylene under two different rotor speeds of lOrpm and 200 rpm. The mixing temperature was held at 15O0C which was found by them to be the optimum for the system under investigation as discussed in section 5.4.7. Figure 8.12 shows the torque curves in the mixing bowl during the mixing operation. It is seen that the maximum torque generated during mixing at 10 rpm was only 0.33 of the peak value obtained when mixing at a rotor speed of 200 rpm. Figure 8.13 shows the plots of dynamic viscosity vs. frequency for samples mixed at the two rotor speeds. As expected, the low torque generated during mixing at 10 rpm results in higher viscosities, indicating poorer dispersion and larger number of agglomerates. Quantitative microscopy measurements confirmed this [74]. These measurements were done on plasma etched samples. Experiments showed that polymer was preferentially removed from larger interparticle regions during etching. Thus, etched samples tended to have polymer remaining in the fine pores of agglomerated particles. These agglomerated regions appear as single particles in micrographs. Thus, measured particle sizes would be larger in poor-dispersed systems containing more agglomerates. The particle size measure used was the projection circumscribing diameter Dpc [9O]. The sample mixed at 10 rpm had an average Dpc value of 0.50 jam while the average Dpc was 0.43 jim for the sample mixed at 200 rpm. Besides using the small-amplitude oscillation response to understand the effect of agglomerates on rheology of filled systems, one could also use the shear stress growth and first normal stress difference growth function as was done by KaIyon [91]. Figures 8.14 and 8.15 show the comparison of the rheological behavior of a surface-treated calcium carbonate filler with an untreated silica-based filler having similar particle size distribution. The filler loading was 30% in both cases and they were incorporated under identical compounding conditions into low density polyethylene. The plots show T*2,r and i/^r which are the

TORQUE (g*m)

50 vol% Al2 O3 Uncalcined Mixing Conditions: 15O0C, 200 rpm

TORQUE (g-m)

TIME (min)

50 vol% Al2O3 Mixing Conditions: 15O0C, 10 rpm

TIME (min) Figure 8.12 Plot of torque vs. mixing time for 50vol% of alumina in low density polyethylene mixed at rotor speed of 200 rpm (A) and 10rpm (B)1 respectively. (Reprinted from Ref. 74 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.)

normalized values obtained by division of the respective parameters with the values for pure polymer at 0.1 sec"1. It is seen that the surface treated calcium carbonate filled system behaves similarly to the pure polymer and reaches the steady torque and normal force values quite instantly. On the other hand, steady-state shear stress and first normal stress difference values are not reached with the silica-based filler, particularly the first normal stress difference which exhibits a spectacular rise with the time of deformation. This suggests that agglomeration and strong interaction occur between agglomerates of the silica-based filler. These interactions convert the structure during simple shear into a network that approaches more solid-like behavior. The effect of agglomerates is no different from that

DYNAMIC VISCOSITY (Pa-s)

SOVOKAI 2 Q 3 Mixing Temperature = 15O0C

FREQUENCY (rad/s) Figure 8.13 Plot of dynamic viscosity vs. frequency at 1250C for 50vol% in low density polyethylene mixed at rotor speeds of 10rpm and 200 rpm using a fixed mixing temperature of 15O0C. (Reprinted from Ref. 74 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.)

FUJJED LOW DENSTTY POLYEIHYUENE UNTTS DYNES SEJCA

Figure 8.14 Shear stress growth function of a low density polyethylene filled with two different fillers. (Reprinted from Ref. 91 with kind permission from Gulf Publishing Co., Houston, Texas, USA.)

FELLED LOW DEHSTTY POLYETHYLENE UNITS DYNES.S[EC2;CM2

SILICA

Figure 8.15 First normal stress difference coefficient growth function of a low density polyethylene filled with two different fillers. (Reprinted from Ref. 91 with kind permission from Gulf Publishing Co., Houston, Texas, USA.) of filler concentration. With larger number of filler agglomerates, the system would behave rheologically in a manner similar to a system with a higher filler concentration than actually exists.

8.6

EFFECT OF FILLER SURFACE TREATMENT

Surface modifiers such as those listed in Table 1.5 are often used in order to achieve better filler dispersion and reduced agglomeration due to improvement in the wettability of the filler and on account of promotion of filler-polymer contact rather than filler-filler contact. The effect of surface treatment on unsteady shear viscoelastic properties has been studied in highly filled systems [43^5,54,92] and the available data do provide some basis for understanding the use of surface modifiers. The major thrust of the efforts of Bigg [43,44,54] and Althouse et al Next Page

Extensional flow

Q

properties

vy

The bulk of the extensive literature on the rheology of filled polymer systems [1-85] is focused on the flow behavior in shear. The extensional flow properties have been treated in a rather limited manner [1,4,14,27,29,86,87], despite the fact that knowledge of the rheology in shear mode generally does not allow prediction of the behavior in extension [88]. The reason for this is because steady extensional viscosity is in general difficult to measure, and also because filled polymers go less into applications involving the film blowing, fiber spinning and flat-film extrusion processes wherein the extensional flow is of importance. Extensional flow occurs when the material is not in contact with solid boundaries, as is the case during drawing of filaments, films, sheets or inflating bubbles. Converging flows at the inlets of dies are also extensional in nature. In extension, the material is stretched continually in a particular direction as already explained in section 2.1.3. The principal axis of strain keeps doubling in length at equal intervals of time during a steady extensional flow. For example, a circular filament having a length I0 initially and / at time t undergoes steady extensional flow when / = I0 exp(ef) where s is the extensional rate. There are a few different ways in which extensional flow can be measured as discussed in section 3.3. However, it is often difficult to keep the apparatus running for long enough time to achieve steady state extensional flow conditions for sure. Where such steady flows are achievable then the ratio of the tensile stress along the filament, to the extensional rate e gives the extensional viscosity rjE; or else, the ratio that results from such measurements basically depends in a rather complicated manner on the transient viscoelastic properties. The limited information on the extensional flow properties of filled polymer systems does not leave much room for extensive discussion on this subject. Thus, this chapter is rather restricted and though the

intention was to discuss the effects of various factors on the extensional flow properties as was done earlier for the shear flow properties in the preceding three chapters, the same could not be done due to lack of available information. Certain subcategories are absent and even in the subcategories that are covered in this chapter, the discussion is quite concise. 9.1

EFFECT OF FILLER TYPE

The experimental studies of White et al [29] illustrate the effect of the filler type on the extensional flow properties of filled systems. Though nine fillers were covered as given in Table 6.1 when studying shear flow properties, the extensional flow studies were restricted only to three of them, namely, titanium dioxide (TiO2), carbon black (CB) and calcium carbonate (CaCO3). The filled systems were prepared using a fixed grade of polystyrene Dow Styron 678U and the loading level was fixed at 30vol%. The extensional viscosity measurements were done using an extensional rheometer developed in house by Ide [89]. Figure 9.1 shows the variation of extensional viscosity with extensional rate for the three filled polystyrene systems containing TiO2, CB and CaCO3. A comparison between Figure 9.1 and Figure 6.1 indicates the fillers appear in the same sequence when their levels of increases are considered. The highest viscosity increase occurs in CaCO3 filled system, the lowest in TiO2 filled system and the medium in CB filled system both in extensional as well as shear flow. This naturally leads to the conclusion that the effect of filler type on the extensional viscous properties would be qualitatively akin to the effect on the shear viscous properties. Even though the available information on the effect of filler type on the extensional flow properties is not as extensive as in shear, it provides a sufficient premise to draw reasonable conclusions due to the qualitative behavioral similarity. Thus, the extent of extensional viscosity increase would always be lowest for three-dimensional spherical fillers, higher for two-dimensional platelet fillers and highest for one-dimensional fibrous fillers. Also when considering rigid and flexible fillers, the increase in the level of steady extensional viscosity would be more for rigid fillers than for flexible fillers because the rigid fillers would resist deformation to a greater extent. Where extensional flow occurs due to converging flow fields through dies, then orientation of the filler, particularly the one-dimensional fiber, and, to some extent, the two-dimensional platelet types, affects the extensional flow behavior. Theory for the flow of concentrated dispersions of chopped fibers in polymer melts in different extensional flow situations is available [90,91] and the equilibrium fiber orientation

FILLED POLYSTYRENE

UNTTS

Figure 9.1 Variation of steady state extensional viscosity with extensional rate for filled polystyrene melts at 30vol% of various types of fillers as indicated. (Reprinted from Ref. 29 with kind permission from American Chemical Society, Washington DC, USA.)

can be calculated. The limitation of the approach [90,91] is that the polymer matrix is considered as a second order fluid and only twodimensional flow has been considered. It has been shown elsewhere [92] that extensional strain is more effective than shear strain for aligning fibers. In capillary rheometers, when short fiber filled polymer melt enters the die [93-96], the fibers get aligned due to the extensional strain and a partial plug flow is at times observed [94,96,97]. Alternatively, the migration effect [98,99] is observed because the flow front is found to be deficient in polymer for the glass bead filled low density polyethylene in a spiral mold test [98] and for glass filled epoxy systems flowing in a rectangular section end-gated mold [99]. Observations of fiber orientation under conditions of converging, diverging and shearing flows are available [24]. Convergent flow results in higher fiber alignment along the flow direction, whereas diverging flow causes the fibers to align at 90° to the

major flow direction. It was also observed [24] that shear flow, on the other hand, produces a decrease in alignment parallel to the flow direction and the effect is pronounced at low flow rates. Contact microradiography was used [24] to study extrudates produced using a Davenport constant volume flow rate capillary rheometer. A variety of dies of different diameters was used, and in each case the entry angle was 180°. Contact microradiographs were made at various flow rates and the fiber orientation was found to depend strongly on flow rate. Figure 9.2 shows contact microradiographs of sections cut parallel to the cylinder axis in extrudates obtained from 2mm diameter dies for a commercially available glass fiber-filled polypropylene produced by ICI (Propathene HW60GR/20). This material was in the form of roughly spherical granules, containing 20% by weight of welldispersed glass fibers having a diameter of 10 jim and modal length of 500 jim. Figure 9.2(a) shows an extrudate collected at a shear rate of 1.5 sec"1 from a die of 100mm length. The fibers show little sign of alignment and appear to form a fairly random tangled mesh. In Figure 9.2(b) the shear rate is 24 sec"1 from the same die. The fibers are more highly aligned along the flow direction at this flow rate. Figure 9.2(c) shows an extrudate collected at a shear rate of 24 sec"1 from a 2 mm diameter die of length 0.3mm, and the alignment in the flow direction is more pronounced than for the 100 mm die. Figure 9.2(d) shows a section cut from the extrudate at a shear rate of 1430 sec"1 from a die of length 0.3 mm, and in this case the fibers are aligned almost completely in the flow direction. From these contact microradiographs it appears that fiber alignment increases with flow rate, but decreases with die length. In order to improve the tensile properties of low-density polyethylene, Mead and Porter [100] added high density polyethylene fibers and film strips. This resulted in an increase in the extensional viscosity and consequently, the tensile modulus of the composite was increased by a factor of 10. The effect of different mineral fillers (e.g. talc, mica, clay, dolomite) on the rheological properties of low density polyethylene films was studied by Arina et al. [17]. It was found that the fillers increased the extensional viscosity of a polymer matrix in concurrence with the earlier observations of Han and Kim [86] as well as Mead and Porter [10O]. Nakajima et al. [101] studied the viscoelastic behavior of butadieneacrylonitrile copolymer filled with carbon black. Capillary extrusion measurements with an Instron and dynamic oscillatory measurements with a Rheovibron suggested the occurrence of 'strain hardening7 in filled elastomer due to tensile extension causing structural changes in the carbon black filled elastomer. It is possible that the structure built by the carbon black in the elastomer increasingly jams against

Figure 9.2 Contact microradiography of extrudate from a capillary rheometer of commercially available glass fiber filled polypropylene produced by ICI (Propathene HW60GR/20). Extrudate was obtained at 21O0C dies of 2mm diameter: (a) 100mm long die, shear rate = 1.5sec"1; (b) 100mm long die, shear rate = 24sec"1; (c) 0.3mm long die, shear rate = 24sec~1; (d) 0.3mm long die, shear rate = 1430sec"1. (Reprinted from Ref. 24 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

extension until finally the structure yields and, thereafter, the sample breaks. The effect of adding carbon black in styrene-butadiene rubber (SBR) compounds has received the attention of Gotten and Thiele [102]. It was shown that, in extensional flow, the stress in carbon black filled SBR compounds continues to grow with increasing strain up to the point of rupture. Gotten and Thiele [102] evaluated their data using the DennMarrucci equation (2.66b) given in section 2.3.5. It was concluded that whenever stiffening of SBR compounds during extension was desired, low structure carbon black with high surface area ought to be used. Fedors and Landel [103] pointed out that stress-strain behavior of swollen elastomers can be determined experimentally more conveniently by measurements in uniaxial compression than uniaxial extension. In extension, strains of the order of a few hundred percent are required whereas, in compression, strains of the order of only a few percent provide sufficient data for analysis. SBR-glass bead composites cured by means of dicumyl peroxide were used for stress-strain measurements to estimate the concentration of the effective network chains per unit volume of the whole rubber. It was found that with decreasing volume fraction of the composite, the effective network density decreased linearly at first and then rather rapidly in an unexpected and inexplicable manner. 9.2

EFFECT OF FILLER SIZE

When the extensional viscosity is plotted as a function of the applied tensile stress [27] as shown in Figure 9.3, the effect of filler size becomes obvious. It is seen that filled polystyrene systems containing the two fillers TiO2 and CB at the same loading level of 30vol% show the existence of the yield stress during extensional flow. This behavior is identical to that observed for steady shear viscosity in Figure 6.1 for the same two systems. In fact, it is seen that the higher yield stress value is observed for CB filled system and the lower for TiO2 filled system for both shear and extensional flow. A look at the particle size of these fillers in Table 6.1 indicates that though both have small particle sizes, CB is the smaller of the two by a factor of 4. The smaller the particle size, the higher the yield stress, as was concluded in section 6.2, happens to hold for extensional viscosity as well. The smaller size fillers, especially those below a diameter of 0.5 |xm would have strong particle-particle interactions which would aid in forming a network of finite strength and manifest this by a display of yield stress. The yield stress values in shear and extensional flows have been given [29] in Table 9.1 for filled polystyrene systems containing three different

FILLED POLYSTYMNE UHTTS PASCALS

Figure 9.3 Variation of steady state extensional viscosity with tensile stress for filled polystyrene melts at 30vol% of two types of fillers as indicated. (Reprinted from Ref. 27 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.) Table 9.1 Elongational and shear flow yield values for some filled polystyrene systems at 18O0C Filler Carbon black Carbon black Titanium dioxide Titanium dioxide Calcium carbonate Calcium carbonate (untreated) Calcium carbonate (treated)

Loading (vol%)

Y6 x 10~3 (Pa)

Y5 x 10~3 (Pa)

YJ Y5

20 30 20 30 20 30

-4.5 17 ~1.6 4 ~5.0 22

25 9 ~1.0 2.2 3.0 12

~1.8 M.9 ~1.6 —1.9 —1.7 —1.8

30

-0.7

-0.4

-1.75

Source: Refs 27 and 29 (reprinted with kind permission from Society of Plastics Engineers Inc., Connecticut, USA and American Chemical Society, Washington DC, USA). types of filler - CB, TiO2 and CaCO3 at a loading level of 20vol%. The yield value in extension is seen to be 1.6 to 1.9 times greater than that measured in shear [27]. This is very close to the von Mises criterion [104] of 1.73 or approximately equal to the J3 suggested [105] for plastic yielding, which is referred to as a critical distortion strain energy in the interpretation by Hencky [106,107]. The existence of a von Mises criterion

equivalent to a critical distortional strain energy seems reasonable to explain the breakup of particle network structures formed due to interparticle forces. The yield stress in extension and shear can thus be understood to relate to the particle-particle interaction energy per unit volume, especially in the case of small size particles. Filled systems with larger particles would be non-interacting and hence would show no yield stress. In fact, their response to deformation is determined by hydrodynamic interaction and not by particle-particle interaction. Extensional viscosity measurements on styrene acrylonitrile (SAN) melts with large glass beads have been reported by Nazem and Hill [4]. They found that extensional viscosity is equal to three times the zero shear viscosity, thereby endorsing the fact that there is no particleparticle interaction when dealing with larger particles, especially at volume fractions of less than 20%. However, at higher volume fractions of 36%, it was found that the ratio of extensional viscosity to shear viscosity dropped to 1.7. 9.3 EFFECT OF FILLER CONCENTRATION One of the effects of increasing filler concentration is that constant extensional viscosities, namely, steady-state conditions are reached more easily and earlier in the filled systems than in unfilled systems and the values decrease with increasing extensional rate. This point has been brought out in the work of Lobe and White [19] who studied the influence of carbon black on the rheological properties of a polystyrene melt. Figure 9.4 shows the extensional viscosity vs. time curves for unfilled polystyrene melt at different extensional rates. It was found [19] that the extensional viscosity may tend to become constant at very low deformation rates, but become unbounded at higher and higher deformation rates. With filler concentration at low loading levels of 5 and 10% of carbon black filler, it was found [19] that the plots resembled those in Figure 9.4. However, at higher filler concentrations, constant extensional viscosities were achieved with time and these values were found to decrease with increasing extensional rate as shown in Figures 9.5 and 9.6 for 20 and 25 vol% carbon black loading. The extensional behavior of a polymer system containing particulate filler was studied experimentally by Han and Kim [86]. It was found that, at a fixed extension rate, the extensional viscosity increased with increasing filler concentrations because the solid particles of calcium carbonate did not deform under stretching and hence exerted more resistance to the flow of the molten threadline with an increase in concentration. It is natural that the effect of filler concentration on steady-state

TTKnRTTJJm POLYSTYRENE UNITS

Figure 9.4 Variation of extensional viscosity with time at different extensional rates for unfilled polystyrene melt at 17O0C. (Reprinted from Ref. 19 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

extensional viscosity is qualitatively not any different from that on steady shear viscosity [27,86]. From Figures 9.7-9.9, it can be seen that, as expected, extensional viscosity increases with increases in loading level and decreases rapidly with increasing extensional rate. In Figures 9.7 and 9.8, the data were generated using an in-house developed rheometer [89], while in Figure 9.9, the data were obtained using a melt spinning apparatus [86]. It should be noted that the melt spinning apparatus does not provide steady extensional flow conditions and hence the extensional viscosities determined from such an instru-

CARBON BLACK HTTED POLYSTYRENE UNITS

Figure 9.5 Variation of extensional viscosity with time at different extensional rates for 20vol% carbon black filled polystyrene melt at 17O0C. (Reprinted from Ref. 19 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

ment are at times referred to as spinning viscosities indicating the unsteady state conditions. In general, the relative influence of different kinds of particulate fillers on the extensional viscosity at the same loading level is the same as that of the shear viscosity [27]. In the case of fiber-filled polymer systems, the same comment cannot be made as can be seen from Figure 9.10. For unfilled high density polyethylene (HDPE), rjE/rj0 is 3 at low extensional rates, while it is much higher for glass fiber filled HDPE [14]. This result can be explained qualitatively through the theoretical

25% CARBONBLACK FILLED POLYSTYRENE UNITS

Figure 9.6 Variation of extensional viscosity with time at different extensional rates for 25vol% carbon black filled polystyrene melt at 17O0C. (Reprinted from Ref. 19 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

arguments put forth by Batchelor [108]. However, what is unusual in Figure 9.10 is that the values of riE/3rjQ are much higher for the melt containing 20wt% of glass fibers than for the one containing 40wt% of glass fibers [14]. Chan et al. [14] attribute this peculiar behavior to the small aspect ratios of the fibers used by them. They also found that using the theoretical expressions of Batchelor [108] gave them values which were far too large compared to their experimental data. Further, since the theory did not predict a decreasing trend for extensional viscosity with increasing extensional rate, it is not truly appropriate to seek explanations based on such theory. 9.4

EFFECT OF FILLER SURFACE TREATMENT

There is enough evidence, at least in the case of shear viscosity, that any increase due to filler addition can be significantly reduced through filler surface treatment. The same effect can be expected for extensional viscosity as well. In the case of shear viscosity, there have been a number of studies to support conclusions drawn on the effect of various

CARBON BLACK FILLED POLYSTYItENE UNITS

Figure 9.7 Variation of steady state extensional viscosity with extensional rate for carbon black filled polystyrene melt at 18O0C with different levels of filler loading as indicated. (Reprinted from Ref. 27 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

TTTANIDM DIOXIDE FILLED POLYSTYRENE UNITS

Figure 9.8 Variation of steady state extensional viscosity with extensional rate for titanium dioxide filled polystyrene melt at 18O0C with different levels of filler loading as indicated. (Reprinted from Ref. 27 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

CALCIDM CAKBONATC FILLED POLYPROPYLENE UNITS

Figure 9.9 Variation of apparent extensional viscosity with extensional rate for calcium carbonate filled polypropylene at 20O0C with different levels of filler loading as indicated. (Reprinted from Ref. 86 with kind permission from John Wiley & Sons, Inc., New York, USA.)

surface treating agents. However, in the case of extensional viscosity, the available information is limited [27]. Figure 9.11 shows the effect of surface treatment on extensional viscosity for 30% calcium carbonate filled polystyrene [27]. The data are presented in two forms, namely steady state extensional viscosity vs. extensional rate in Figure 9.11 (a) and steady state extensional viscosity vs. tensile stress in Figure 9.11(b). Irrespective of the type of data representation, it is seen that surface treated calcium carbonate reduces the level of extensional viscosity and brings it closer to that of the unfilled polymer. The yield stress value is reduced considerably though the values of the ratio of yield stress in extension to that of shear is still maintained nearer to the von Mises value of 1.73 as can be seen from Table 9.1. Surface treatment tends to modify the forces of particleparticle interaction and hence show reduced yield stress values due to lowering of the interaction forces [2,27]. The effects of titanate coupling agents on the rheological properties of particulate filled polyolefin melts were studied by Han et al. [15]. Experi-

GLASS FIBER FILLED HIGH DENSITY POLYEIHYLENE UNITS

Figure 9.10 Variation of relative extensional viscosity with extensional rate for glass fiber filled high density polyethylene at 18O0C with different levels of filler loading as indicated. (Reprinted from Ref. 14 with kind permission from John Wiley & Sons, Inc., New York, USA.)

CALCIDM CABBONAU FILLED POLYSTYRENE UNITS

UNTREATED (30% CaCO3) TREATED (30% CaCO3) UNFTTJ-KD

Figure 9.11 (a) Variation of steady state extensional viscosity with extensional rate for calcium carbonate filled polystyrene containing 30% untreated and treated filler. (Reprinted from Ref. 27 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

CALCIUM CARBONATE FILLED POLYSTYRENE UNITS PASCALS

UNTBEA1IED (30% CaCO3) TREATED (30% CaCO3)

Figure 9.11(b) Variation of steady state extensional viscosity with tensile stress for calcium carbonate filled polystyrene containing 30% untreated and treated filler. (Reprinted from Ref. 27 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

ments were carried out with systems like calcium carbonate-filled polypropylene and fiber glass-filled polypropylene with the addition of titanate coupling agents, and an increased extension of the filled systems was observed in the presence of these additives. In fact addition of titanate coupling agents to calcium carbonate-filled polypropylene decreased the extensional viscosity to such an extent that it almost equalled the extensional viscosity of pure polypropylene. Effect of the additives on fiber glass-filled polypropylene was the same but the decrease in extensional viscosity was to a much lesser extent.

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Concluding remarks

I U

This final chapter of the book is meant to provide a forum for recapitulation of some of the matters of importance discussed in all the preceding chapters. Chapter 1 introduced the various materials which go into the making of filled polymer systems. The polymer could be a thermoplastic, thermoset or elastomer. Further, it could be linear, branched, amorphous, crystalline or semi-crystalline. At the same time, it may fall into the category of a homoploymer, copolymer, or liquid crystalline polymer. Any type of such a polymer could be compounded with fillers to form filled polymer systems. The fillers, on the other hand, include those solid materials that are added as reinforcing agents to provide strength or mere extenders to reduce cost. There are various reasons for the use of fillers besides increasing stiffness, strength, toughness, impact strength or reduction in cost. They may be included to get better dimensional stability, increased heat deflection temperature, reduced permeability to gases or liquid, and to modify electrical/magnetic properties of the polymer matrix into which they are incorporated. Available fillers are of various types and forms, namely, rigid, flexible, spherical, ellipsoidal, flakes, platelets, fibers or whiskers. They may be organic or inorganic in nature. All varieties of fillers have been tried in polymers in order to impart various advantageous benefits. The presence of the fillers in the polymer matrix alters the rheological properties of the polymer. Rheological measurements are often used as an effective tool for quality control of raw materials, manufacturing process/final product and predicting material performance. The rheological properties of the filled polymers are dictated, not only by the type of the filler, but also by its size, size distribution and amount. A key factor in the use of fillers without adversely affecting material properties is the stress

transfer at the filler-matrix interface. The physico-chemical interactions between the filler and matrix then achieve a great deal of importance. The interfacial adhesion can be substantially enhanced through use of a surface modifying agent which is capable of adhering well to both the matrix and filler particles. Not all surface modifying agents can couple at both ends of their chemical moieties. In fact, some surface modifiers attach only to the filler, others react only with the polymers and some others do not react at all. In all these cases, they may be treated as lubricants. Only those surface modifiers which react with filler and polymer can be termed coupling agents, though in most available literature this term has been loosely used to describe any surface modifying agent. Chapter 2 discusses the fundamentals of rheology and classifies flow as shear and extensional. Definitions of various rheological parameters under the three subheadings of steady simple shear flow, unsteady simple shear flow and extensional flow are given. The important material parameters that burgeon out of the discussion are the steady shear viscosity, the normal stress difference, complex viscosity, dynamic viscosity, storage modulus, loss modulus and extensional viscosity. This chapter sets the platform for rheological discussions that are undertaken in Chapters 6, 7, 8, and 9 to study the effect of different parameters related to filler and polymer on the rheology of the filled polymer systems. Basic description of non-Newtonian fluids is provided so that concepts of shear rate dependent viscosities with or without elastic behavior, yield stress with or without shear rate dependent viscosities and time dependent viscosities at fixed shear rates get classified. The filled polymer systems fall into the category of pseudoplastic fluids with or without yield stress and also often depict the behavior of thixotropic fluids. Their viscoelasticity may give rise to various anamolous effects that are discussed in Chapter 2, such as the Weissenberg effect, extrudate swell, drawn resonance, melt fracture and so on. Rheological models have been described for steady shear viscosity function, normal stress difference function, complex viscosity function, dynamic modulus function and the extensional viscosity function. The variation of viscosity with temperature and pressure is also discussed. Chapter 3 deals with rheometry which is the method of measurement of the various rheological parameters described in Chapter 2. The rheometers may be of the rotational type or the capillary type for shear flows and the shear free type for extensional flows. Chapter 4 deals with constitutive theories and equations for suspensions and lays down the foundations for understanding the basics of filled polymer rheology. Starting from the simplest dilute

suspensions for spherical particles to the complex concentrated suspensions of fibrous fillers, the various relationships between relative viscosity with volume fraction are highlighted. The effect of filler shape, concentration, size, size distribution on the viscous behavior is discussed. The migration of particles towards the tube axis and the consequences of the wall effect on the rheological properties are described. Particle rotation is also shown to affect the rheology and result in an apparent display of increased viscosity. The effect of flocculation is seen to increase the viscosity of suspension rather sharply. The effect of the suspending medium and its interaction with the suspended particles results in an increase in effective particle size and decrease in molecular mobility. Similarly, the effect of physical processes such as crystallization and chemical processes such as polymerization modify the viscosity of suspensions with time. There is considerable effect of the electrostatic field and displays of increased viscosity due to the combined effect of the first, second and third electroviscous effects. Extensional viscosity is modified by the presence of suspended particles. Particularly long slender particles have a drastic effect on the extensional viscosity of suspensions as seen from the constitutive equations provided for these systems. Chapter 5 describes methods for the preparation of filled polymer systems. The quality of the mixture resulting from any compounding action of filler with polymer must be evaluated as there are various mixing mechanisms involved. The efficiency of the dispersive and distributive mixing determines the goodness of mixing. The compounding techniques that are used are the traditional two-roll mills or internal mixers and the modern-age single or twin-screw extruders. The techniques vary considerably in their method of operation and results. When setting up a compounding operation, it is important to use proper selection criteria to decide whether batch or continuous mixers are appropriate and pay attention to the dump criteria. It is known that the physical properties of the compounded filler system vary considerably with the selected compounding techniques. The goodness of mixing is best adjudged by the determination of the rheological properties of the filled polymer systems. There are a number of compounding/mixing variables that affect the final quality of the mix. Variables affecting the compounding operations could be machine variables or operating variables. The mixer type, rotor geometry, mixing time, rotor speed, ram pressure and chamber loadings (in the case of internal mixers), as well as the mixing temperature all have a considerable effect on the goodness of the final mix. It is important to understand the sensitivity to these variables so that the mixing can be carried out under optimum conditions.

The rheological properties of the filled polymer systems are discussed under various headings, namely, steady shear viscous properties, steady shear elastic properties, unsteady shear viscoelastic properties and extensional flow properties, in Chapters 6, 7, 8 and 9, respectively. The effect of filler type, size, concentration, size distribution, agglomerates, surface treatment and polymer matrix on the rheology of the filled systems is discussed in detail in most cases. Only where information is lacking, such as in the case of extensional flow properties in Chapter 9, are some of the effects missing, and the discussion is concise on the treated effects. In general, steady shear viscosity and extensional viscosity bear very similar results when the effects of the various parameters are considered. Addition of fillers increases the viscosity of the base polymer. The extent of viscosity increase is the lowest for 3-dimensional spherical fillers, higher for 2-dimensional platelet fillers and the highest for !-dimensional fibrous fillers. Rigid fillers show greater increases in the level of viscosity than flexible fillers. The size of the filler determines whether the filled systems would show unbounded viscosity buildup at low shear/extensional rates during shear and extensional flows, respectively. The smaller the particle size, the greater the yield stress. The ratio of the yield stress in extension to that in shear, is approximately quite close to the value of the von Mises criterion for plastic yielding of 1.732 or ^/3. Polydispersity reduces the viscosity of filled systems at a fixed loading level. For dilute suspensions having a volume fraction of solids less than 0.2, the effect of the variation of particle size on filled system viscosity is normally minimal. However, at high loading levels, the viscosity can be reduced drastically when the particle size modality is increased. Agglomerates occlude liquid in their interparticle voids, thereby increasing the relative viscosity value at any given solids loading. With increasing number of agglomerates, the maximum possible filler loading is decreased. Thus, for highly filled polymer systems, it is important to reduce the degree of agglomeration to a minimum level, in order to decrease the system viscosity for easier processing and also to increase the extent of filler loading if desired. With increasing concentration of filler, the interparticle interactions increase weakly at first and then rather strongly as the concentration becomes higher and higher. The concentration at which particle-particle interactions begin depends mainly on the geometry and surface activity of the filler. High aspect ratio fillers would begin to interact at much lower concentrations, while non-agglomerated large size spherical particles do not interact even up to 20vol%. Use of surface modifiers helps to decrease particle-particle interaction as the surface treatment

helps the polymer to wet the filler better. However, the action of the surface modifiers is system-specific and hence, it is very difficult to predict the performance of the surface modifiers a priori for any fillerpolymer combination. Thus, most often a surface modifier is selected rather empirically for the particular filler-polymer combination of interest. The amount of surface modifier should be small but adequate because too little of it does not give the desired effects and too much of it does not improve rheological or product properties to more than a certain extent besides adding extra cost to the product. The optimum surface modifiers to be used are most often in the range of 0.6-0.8% by weight of the filler. The method of surface treatment also affects the performance. Pretreatment is more efficient in imparting favorable improvements in rheological properties but adds to the cost due to the extra step of pretreatment. Direct addition of surface modifier during the filler-polymer compounding process saves on the cost of pretreatment but requires a greater amount of surface modifiers which offsets this cost benefit. Proper evaluation of which is more apposite for a particular situation must be done before opting for the preferred method of surface treatment. The chemical nature and the viscoelastic characteristics of the polymer matrix do have an effect on the final rheological properties of the filled polymer system. This is because the original characteristics of the polymer determine the level of the shear imparted during the shear mixing when the filled polymer systems are being prepared. Higher viscosity polymer would develop higher shear stress and may be able to break agglomerates better during mixing. Depending on the chemical nature of the polymer, the matrix-filler affinity would also control the level of the force to break filler-filler bonds during high shear mixing. Polymers with greater filler affinity would provide greater force. Such polymers, if they also have reasonably high viscosity which is not highly shear-thinning, would probably help the most in getting the best filler dispersion. It has been shown that the steady shear viscosity vs. shear rate flow curves of filled polymer melts at various temperature and filler loadings can be unified when plotted on a log-log scale in terms of a reduced viscosity parameter (77 x MFI) vs. a reduced shear rate parameter (y/MFI). The unified curves are independent of the filler type and shape, but depend on the polymer matrix. Thus, separate unification is achieved for each generic type of polymer. They do depend on the filler size and loading level. In cases wherein particleparticle interaction gives rise to yield stress, the master curve would not be unique in the low shear region. However, in the higher shear rate region beyond 10"1S"1, the master curve would be unique irrespective of the filler type, size, and amount as well as surface

modifier and amount. In fact, in this region, the master curve for the filled polymer system is not different from the unfilled polymer system. The master curves can be used to estimate the flow curves in the higher shear rate region at the temperature of interest, merely from the knowledge of MFI and the glass transition temperature of the specific system. There are certain precautions which should be borne in mind when determining the MFI of filled polymer systems. The filler particle size and shape may warrant modification of the MFI apparatus, and possible yield stress characteristics of the filled polymer system may demand a change from the standard ASTM temperature and load conditions given in Appendix A. Whatever changes are done in order to obtain reliable MFI data for the filled polymer systems, it is important that these values are converted to those at standard temperature and load conditions using equations from section 6.8 before using them in the master rheograms. During normal stress measurement of filled polymer systems, there are certain difficulties due to the abnormal effects of interactions between the measuring equipment and the yield stress of the filled systems. Further, there is always a problem of gap setting of the conen-plate viscometer due to the high residual stresses which do not relax for a long time and also due to the filler particle size or agglomerates which may interface with the gap. Hence generating normal stress data is often quite difficult. Nevertheless, reliable steady shear elastic data are available in the literature which has obviously been generated with great care. It is seen that spherical fillers like glass beads do not affect the normal stress difference. Particulate fillers like titanium dioxide, calcium carbonate and carbon black, reduce the normal stress difference whereas fibrous fillers like aramid, glass and cellulose fibers, increase it. The large increase in normal stresses of fiber filled polymer systems is explained on the basis of the hydrodynamic particle effect, associated with orientation in the flow direction. Of course, if the fiber diameter is very small then the increase in normal stresses is small and at times may even show a decrease. It is important to make plots of normal stress difference N1 vs. shear stress T12 rather than vs. shear rate y if correct data interpretation is intended. The former plots are independent of temperature and molecular weight of the polymer matrix (though not its distribution) and the rheological behavior is correctly interpretable by analogous comparison with the steady state compliance /e. It is obvious that the mobility of polymer chains under the influence of an applied stress is reduced by the presence of the filler, thereby decreasing the elastic response of particulate filled polymer systems

with increasing concentration. However, it was shown that a plot of \l/ltt = N1(C/), 7VN1(O, y) vs. /m was unique and independent of shear rate but an increasing function of filler concentration. This approach of presenting the normal stress difference vs. filler concentration may be acceptable for data representation but not for data interpretation. Using a filler size distribution, the normal stress difference of the filled polymer system can be altered. The relative primary normal stress coefficient is reduced when a bimodal distribution is used and gives the lowest values at about 10-30% volume fraction of the small particles. With increasing number of filler agglomerates for particulate fillers, the normal stress difference is lowered. The extent of lowering of the normal stress difference depends on the amount of occluded liquid by the agglomerates, the average number of particles in each agglomerate and hence the size of the agglomerates. When an agglomerate is formed or is present, it is as though the particle size of the filler has increased throughout the system. Larger particles in an unagglomerated system would lower the normal stress difference less. By analogy then, with increasing number of particles in the agglomerates, the extent of normal stress difference lowering decreases. Contrary to this, if more liquid is occluded in the interparticle voids of the agglomerates, then the extent of normal stress difference lowering increases. These two opposing factors determine the eventual extent of normal stress difference lowering due to agglomerates formed by particulate fillers. Agglomerates of fiber also would tend to decrease the normal stress difference. But, since unagglomerated fibers are known to increase normal stress difference, the extent of this increase would be reduced due to presence of agglomerates as the fibers which form the agglomerates are restrained and cannot orient during flow. Use of surface modifiers often helps to reduce the number of agglomerates, thus increasing the values of N1 as against that of the untreated system of a fixed filler loading at the same level of shear stress. The behavior is system-specific, and changing the filler or the polymer or the type of surface modifier can accentuate or reduce the effect that the surface treatment has on the normal stress difference. Viscoelastic properties of filled polymer systems under shear can also be studied through unsteady state data. Small amplitude oscillations for getting dynamic rheological data are truly appropriate when handling highly filled systems because they keep the samples in the gap of the measuring instrument intact. During dynamic data generation, it is important to use a low amplitude because the effect of strain on the rheological response is quite strong. Unsteady shear data in terms of thixotropic sweep responses and stress relaxation behavior also provide a good insight into the dispersion level of the filler in the polymer.

The complex viscosity vs. frequency behavior on different types of filler is qualitatively the same as that of shear viscosity vs. shear rate. Only the extent of the viscosity increases due to the filler addition would be different for the unsteady and steady state because the CoxMertz rule is known to fail for filled polymer systems. When storage modulus vs. frequency plots for different types of filler are considered, it is revealed that all fillers increase storage modulus at any frequency. On the other hand, the storage modulus for spherical fillers is known to decrease with increasing frequency. With increasing filler concentration, the complex/dynamic viscosity as well as storage/loss moduli show a continually increasing trend. However, the viscous response dominates the elastic response with increasing filler concentration. The storage and loss moduli begin to depict more solid like behavior at higher concentrations and show independence with respect to frequency. Using a filler size distribution, the filler loading can be increased. In fact, if changes in filler size distribution are made, a good method of tracking the packing arrangements is by observing the differences in the storage modulus-frequency response of the filled polymer systems during dynamic measurements in the low strain region. Bimodal distributions are more effective than broadly distributed powders in achieving higher filler loadings. Presence of filler agglomerates can be detected by using the torque vs. mixing time curves and the dynamic data in conjunction with each other. By tracking the maximum torque, it can be deduced whether the mixing was done under high shear stress conditions. When the peak torque is high, the shear force is strong enough to break agglomerates and this gets reflected in the dynamic viscosity and storage modulus being lower. Effect of filler surface treatment is quite dramatic, whether it is through the use of surface modifiers or by heat treatment of the filler. It is good to back up the dynamic data with other unsteady measurements like the thixotropic sweep and stress relaxation measurements and the torque curve when handling surface treated fillers. This is because the effect of surface treatment is quite complex and system-specific. Where the torque peak is high, while the complex viscosity and storage modulus are low and the stress relaxation time is short for a treated filler system when compared with the untreated one, then it is definite that the surface treatment has helped in dispersing the filler. This is because the high peak torque shows that high shear forces broke down the agglomerates and the surface treatment prevented any chance of their reformation because the dynamic data and stress relaxation measurements indicated so.

Effect of filler-matrix affinity can be quantitatively estimated through dynamic viscoelastic data. An interaction parameter has been defined which gives a measure of the postulated matrix immobilization at the interphase of the filler. It has been shown that there is a correlation between the interaction parameter in the melt state and the solid state at comparable frequency of deformation. Thus, it is possible to generate dynamic data in the melt state for quantitatively estimating the matrixfiller interaction and then extrapolating the affinity behavior to the solid state. Rheology is a powerful tool for studying the dispersion level of the filler and the matrix-filler affinity. The various methods that can be used to understand the rheological behavior of filled polymer systems have been elucidated in this book and it is hoped that it will serve as a useful guide when indulging in further research areas related to filled polymer systems in future.

Appendix A

Glossary

Addition polymerization is a chemical reaction in which simple molecules (monomers) are added to each other to form long chain molecules (polymers) without the formation of byproducts. Amorphous polymer is one that has no crystalline component and there is no order or pattern to the distribution of the molecules. Apparent viscosity is the ratio of shear stress by shear rate which has not been corrected for entrance length effects in a capillary rheometer. 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. Complex modulus consists of the real and imaginary part of the modulus. The real part is called the storage modulus and the imaginary part is called the loss modulus. Compounding involves the process in which polymers are softened, melted and intermingled with solid fillers and other liquid additives to form filled polymer systems. Condensation polymerization is a chemical reaction that takes place between the polyfunctional molecules with the possible elimination of a small molecule such as water. 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 between two coaxial cylinders in relative rotation. Crystalline polymer is one that has an ordered structural arrangement of molecules. Deborah number is defined as the ratio of characteristic time (or in other words, the relaxation time) of the material to the scale of deformation to which 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. Dielectric constant is a dimensionless factor derived by dividing the parallel capacitance of the material by that of an equivalent volume of vacuum. Dispersive mixing is defined as an operation which reduces the agglomerate size of the minor constituent to its ultimate particle size. Distributive mixing is defined as an operation which is employed to increase the randomness of the spatial distribution of the minor constituent within the major base with no further change in size of that minor constituent. Dump criteria is the standard taken in judging the moment when the mixing is deemed as complete. 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. Elastomer is a rubbery polymer that deforms upon the application of stress and reverts back to the original shape upon release of the applied stress. Equation of state or constitutive equation is an equation 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. Extrudate swell or die swell or Barus effect is the increase in diameter of the polymeric melt extrudate upon emergence from the die.

Filled polymer system is the softened or melted polymeric mass in which one or more fillers have been dispersed. Filler is the inert solid material added as cost reducing or reinforcing or property modifying agent to a polymer without significantly affecting the molecular structure of the polymer. Flexural modulus is the term relating to stiffness of the material and basically represents the force required to break a sample by bending or flexing. 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 stress or viscosity to shear rate. Glass transition temperature is the temperature at which increased molecular mobility results in significant change in properties. Heat distortion temperature is the temperature at which a material bends by a predetermined amount under a given load. Hysteresis 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. Interface is the contacting surface where two materials meet. Interphase is the region separating the filler from the polymer and comprises of the area in the vicinity of the interface. Loss modulus is the imaginary part of the complex modulus. Melt Flow Index (MFI) is the weight of the polymer in grams extruded in ten 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 fracture is the irregular distortion of a polymeric melt extrudate upon passing through a die due to improper melt or process characteristics.

Mixing describes the process of intimate intermingling of polymers with filler s/additives or two polymers without any specific restrictions. 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. 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 velocity as that of the surface. Normal stress coefficient is the ratio of the normal stress by the square of the shear rate. Normal stress difference is the difference between the normal stress components. Paraffin wax is a chemical substance obtained as a residue from the distillation of petroleum and is made up of higher homologues of alkanes with a melting range of 50 to 9O0C. 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 finished product. Power-law model is behavior characterized by a power (n) relationship between shear stress and shear rate. Relaxation time is the time taken for the stress to decrease to an exponentially inverse of its initial value under constant strain. Rheology is concerned with the description of the deformation of the material under the influence of stresses. Rheogram or flow curve is a curve relating shear stress or viscosity to shear rate. Rheometry is an instrumental technique for measuring rheological properties. 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. Suspension is a system in which denser particles, that are at least microscopically visible, are distributed throughout a less dense fluid and settling is hindered either by the viscosity of the fluid or by the impact of its molecules on the particles.

Tensile strength is the ability of a material to withstand forces tending to pull it apart. Thermoplastic is a polymer that can be made to soften and take on new shapes by the application of heat and pressure. Thermoset is a polymeric material that has undergone a chemical reaction, known as curing in A, B and C stages depending on the degree of cure by the application of heat and catalyst. 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. Yield stress is the stress corresponding to the transition from elastic to viscous deformation of the flow curve.

Appendix B

ASTM conditions

and specifications for MFI

Table B1a Standard testing conditions of temperature and load as per *ASTM D1238andtASTMD3364 Condition

*A *B *C *D *E *F *G *H *l *J *K *L *M *N *0 *P *Q *R *S *T t

Temp. (0C) 125 125 150 190 190 190 200 230 230 265 275 230 190 190 300 190 235 235 235 250 175

Load piston+ weight (kg)

Approximate pressure (kg/cm2)

Shear stress (x105 dynes/cm2)

(psi)

0.325 0.46 6.50 0.3 2.160 3.04 43.25 1.97 2.160 3.04 43.25 1.97 0.325 0.46 6.50 0.3 2.160 3.04 43.25 1.97 21.600 30.40 432.50 19.7 5.000 7.03 100.00 4.6 1.200 1.69 24.00 1.1 3.800 5.34 76.00 3.5 12.500 17.58 250.00 11.4 0.325 0.46 6.50 0.3 2.160 3.04 43.25 1.97 1.050 1.48 21.00 0.96 10.000 14.06 200.00 9.13 1.200 1.69 24.00 1.1 5.000 7.03 100.00 4.6 1.000 1.41 20.05 0.91 2.160 3.04 43.25 1.97 5.000 7.03 100.00 4.6 2.160 3.04 43.25 1.97 20.000 28.12 400.00 18.4

Note: An asterisk (*) denotes ASTM D1238 and a dagger (t) denotes ASTM D3364.

Table B1b Testing conditions for commonly used polymers Polymer

Condition

*Acetals *Acrylics *Acrylonitrile-butadiene-styrene 'Cellulose esters *Nylon *Polychlorotrifluoroethylene *Polyethylene *Polyterephthalate *Polycarbonate *Polypropylene *Polystyrene tPoly(vinyl chloride) *Vinyl acetal

E1M H11 G D1 E 1 F K1 Q, R1 S J A1 B1 D1 E1 F1 N T O L G, H 1 1 1 P C


Table B1c Test temperature summary Test temperature (0C)

Condition

*125

A1B

*150 |175

C

*190 *200 *230 *235 *250 *265 *275 *300

D1 E1 F1 M1 N1 P G H1 I1 L Q1 R 1 S T J K O


Table Bid Test load summary Load (kg) *0.325 *1.000

*1.050 *1.200 *2.160 *3.800 *5.000 *10.000 *12.500 t20.000 *21.600

Condition A1 D, K Q

M H1O B1C1E1L1R1T I G1 P1 E N I F


Table B1e ASTM specifications for piston and die dimensions Piston

Die

Diameter *, f (0.3730 ± 0.0003 in = 9.474 ± 0.007 mm)

*, f

(0.0825 ± 0.0002 in = 2.095 ± 0.005 mm)

Length *, f

*

(0.315 ± 0.0008 in = 8.00 ± 0.02 mm)

t

(0.916 ±0.0008 in = 23.26 ± 0.02 mm)

(0.250 ± 0.005 in = 6.35 ± 0.13 mm)


Appendix C

Data details and

sources for master rheograms

Table C1 Details of data used for master rheograms of filled polymers in Figures 6.32-6.39 (Source: Ch. 6 Refs [50] and [149]) Polymer

Grade

LDPE

P1 P1 P1 P1 P1 P1 P1 P1 P1 P1 10-2626 10-2626 10-2626 E115 E115 E115 E115 E115 Profax 6523 Profax 6523 Profax 6523 Profax 6301 Profax 6301

PP

Filler type, amount

Coupling agent, amount

Quartz powder, 33 phr Quartz powder, 100 phr Quartz powder, 200 phr Quartz powder, 11 phr Quartz powder, 33 phr Calcium carbonate I1 33 phr Calcium carbonate I1 66 phr Calcium carbonate II, 33 phr Calcium carbonate II, 66 phr Talc, 66 phr Talc, 66 phr Talc, 66 phr CaCO3, CaCO3, CaCO3, CaCO3,

11 phr 25 phr 66 phr 230 phr

CaCO3, 230 phr CaCO3, 230 phr Mica, 66 phr Mica, 66 phr

KR-TTS, 0.5 phf KR-TTS, 1.5 phf Z-6032, 0.5 phf

MFI (temp.°C/ load condition, kg)

Data temp. 0 C

No. of data points (shear rate range, sec"1

Data source Ch. 6 Ref.

4.9C (220/2. 16) 16.7C (220/2.16) 3.0C (220/2.1 6) 1.7C (220/2. 16) 2.2C (220/2.1 6) 7.4C (220/2.1 6) 2.1 c (220/2.1 6) 0.4C (220/2.1 6) 5.4C (220/2.1 6) 2.0C (220/2.1 6) 3.3a (200/2. 16) 5.5b (230/2. 16) 8.7a (250/2. 16) 5.4C (200/2.1 6) 3.0C (200/2.1 6) 2.3C (200/2. 16) 0.85C (200/2. 16) 0.18C(200/2.16) 15.7C (240/2. 16) 7.6C (240/2.1 6) 44C (240/2. 16) 30C (220/2. 16) 108C (220/2.1 6)

220 220 220 220 220 220 220 220 220 220 200 230 250 200 200 200 200 200 240 240 240 220 220

5(630-12000) 4(630-12000) 4(630-12000) 4(630-12000) 4(630-12000) 4(630-12000) 4(630-12000) 4(630-12000) 4(630-12000) 4(630-12000) 6(10-500) 6(10-500) 6(10-500) 4(5-40) 5(5^0) 5(4-40) 5(1-40) 5(0.3^0) 3(1-100) 4(0.1-100) 3(1-100) 3(3-60) 4(2-10)

[16] [16] [16] [16] [16] [16] [16] [16] [16] [16] [157] [157] [157] [5] [5] [5] [5] [5] [18] [18] [18] [9] [9]

Tabled (continued) Polymer

Grade

PS

STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U STYRON 678 U PET1 PET1 PET1 Makrolon 2805 Makrolon 9410 Makrolon 8324 Makrolon 8035 Makrolon 8344

PET

PC

Filler type, amount

Carbon black, 11 phr Carbon black, 25 phr Carbon black, 43 phr TiO2, 11 phr TiO2, 25 phr TiO2, 43 phr Glass fiber, 25 phr Glass fiber, 66 phr CaCO3, 50 phr CaCO3, 50 phr Glass fiber, 33 phr Glass fiber, 66 phr Glass fiber, 10 phr Glass fiber, 20 phr Glass fiber, 30 phr Glass fiber, 40 phr

Coupling agent, amount

Treated

MFI (temp.°C/ load condition, kg)

Data temp. 0 C

1.5C( 180/5) 0.93C( 180/5) 0.55C (180/5) 0.16c(180/5) 0.83C( 180/5) 0.53C( 180/5) 0.44C (180/5) 0.73C( 180/5) 0.36C( 180/5) 0.27C (180/5) 2.0C (180/5) 49C (275/2. 16) 11. 3C (275/2. 16) 4.9C (275/2. 16) 6.1 c (290/1 .2) 3.3C(290/1.2) 2.0C(290/1.2) 1.4C(290/1.2) 1.1° (290/1 .2)

180 180 180 180 180 180 180 180 180 180 180 275 275 275 290 290 290 290 290

No. of data points (shear rate range, sec"1 6(0.01-100) 6(0.01-50) 7(0.01-0.5) 6(0.01-0.5) 4(0.01-0.4) 4(0.01-0.4) 3(0.01-0.4) 4(0.01-100) 4(0.01-100) 4(0.01-0.4) 4(0.01-0.4) 4(10-5000) 5(6-5000) 5(6-5000) 3(10-1000) 4(1-1000) 4(1-1000) 4(1-1000) 4(1-1000)

Data source Ch. 6 Ret. [27] [27] [27] [27] [27] [27] [27] [14] [14] [18] [18] [20] [20] [20] [31] [31] [31] [31] [31]

Nylon

PEI

PEEK

a b c

Zytel Zytel Zytel Zytel Zytel Maranyl A100 Maranyl A190 ULTEM 1000 ULTEM 1000 ULTEM 1000 ULTEM 1000 PEEK 45G PEEK 453OA PEEK 453OG L PEEK 453OA

Glass fiber, 15phr Glass fiber, SOphr Glass fiber, 50phr Glass fiber, 50phr Glass fiber, 11 phr or 10w% Glass fiber, 25 phr or 20 w% Glass fiber, 43 phr or 30 w% Carbon fiber, 43 phr or 30 w% Glass fiber, 43 phr or 30 w% Glass fiber, 43 phr or 30 w%

40C (288/2. 16) 49C(291/2.16) 29.4C(291/2.16) 12.8C (288/2.1 6) 15.7C(291/2.16) 113C (280/2. 16) 22C (280/2. 16) 1 9.81 c (360/5) 5.9C (360/5) 3.96C (360/5) 2.45C (360/5) 1.9C (380/5) 16.7C (380/5) 2.96C (360/5) 4.2C (395/5)

MFI value calculated from equation (6.30) knowing the MFI value as per footnote b. MFI value given by manufacturer or measured under ASTM test conditions. MFI value read out from t vs. y curve using equations (6.26) and (6.27) by the method discussed in section 6.8.

288 291 291 288 291 280 280 360 360 360 360 380 380 360 395

3(50-6700) 4(10-10000) 4(10-1000) 3(10-4400) 4(10-10000) 5(100-10000) 5(100-1000) 5(5-5000) 5(5-5000) 5(5-5000) 5(5-5000) 5(0.1-1000) 5(0.01-1000) 7(15^1000) 7(15-1000)

[158] [158] [158] [158] [158] [25] [25] [136] [136] [136] [136] [159] [159] [159] [159]

Table C2 Parameters of the filled polymer systems covered in the data analyzed for master rheograms of filled polymers in Figures 6.32-6.39 (Source: Ch. 6 Refs [50] and [149]) Matrix

Filler type

Amount (phr)

LDPE

Quartz powder I Quartz powder Il Calcium carbonate I Calcium carbonate Il Talc Calcium carbonate Calcium carbonate

33 100 200 11 33 33 66 33 66 66 11 25 66 230 230

Mica Mica Carbon black Titanium dioxide Glass fiber Calcium carbonate Calcium carbonate Glass fiber Glass fiber Glass fiber Glass fiber Carbon fiber Glass fiber

66 66 11 11 25 50 50 15 10 33 11 43 43

PP

PS

Nylon PC PET PEI PEEK

Coupling agent

0.5 to 1.0 phf titanate

0.5 phf silane

25 43 25 43 66 Treated Untreated

50 20 30 40 66 25 43

Shape Particulate Particulate Prismatic Prismatic Platelet Prismatic Prismatic Platelet Platelet Particulate Particulate Fibrous Prismatic Prismatic Fibrous Fibrous Fibrous Fibrous Fibrous Fibrous

Table C3 Details of unfilled polymer data used for master rheograms in Figures 6.326.39 (Source: Ch. 6 Refs [50] and [149]) Polymer Grade

MFI (temp.°C/ Temperature No.of data points Data load condition, at which data (shear rate source kg) generated (0C) range, sec"1) Ch.6 Ref.

LDPE

0.2ab (190/2.16) 3.0 (175/2.16) 4.0aa (190/2.16) 5.0b (205/2.16) 20 (190/2.16) 3.9ba(210/2.16) 6.5 a(230/2.16) 10.3 (250/2.16) 7.5b a(200/5) 37.Oa (220/5) 130 (240/5) 15.4aa (210/5) 47.7a (230/5) 121 (250/5) 5.0C C(231/2.16) 13.7 (260/2.16) 29.5CC (288/2.16) 235C (280/2.16) 54 (275/2.16) 64°C (285/2.16) 86 C(295/2.16) 103 (305/2.16) 1.5CC (275/2.16) 1.6 C(285/2.16) 1.77C (295/2.16) 1.96 (305/2.16) 1.3° (250/1.2) 3.5Cc (270/1.2) 6.1 (290/1.2) 8.34C (355/5) 22.08°C (375/5) 39.25C (395/5) 10.8 (360/5) 22.6C (395/5)

PP PS

Nylon

PET

PC PEI PEEK a b c

lndothene 22FA002 lndothene 24FS040 lndothene 24FS040 lndothene 24FS040 lndothene 26MA200 10-6016 10-6016 10-6016 Styrene666U Styrene666U Styrene666U H5M H5M H5M Plaskon8201 Plaskon8201 Plaskon8201 Nylon 610 Fiber grade IV = 0.57 Fiber grade IV = 0.57 Fiber grade IV = 0.57 Fiber grade IV = 0.57 Bottle grade IV =1.044 Bottle grade IV =1.044 Bottle grade IV =1.044 Bottle grade IV =1.044 Lexan141 Lexan 141 Lexan 141 ULTEM ULTEM ULTEM PEEK 951GV PEEK 951GV

190 175 190 205 190 210 230 250 200 220 240 210 230 250 231 260 288 280 275 285 295 305 275 285 295 305 250 270 290 355 375 395 360 395

9(0.01-1000) 10(0.01-1000) 10(0.01-1000) 10(0.01-1000) 10(0.01-1000) 6(10-500) 6(10-500) 6(10-500) 10(5-5000) 10(5-5000) 10(5-5000) 6(10-500) 5(20-500) 6(10-500) 4(10-4000) 4(10-1000) 4(10-10000) 4(10-10000) 9(1-5000) 9(1-5000) 9(1-5000) 9(1-5000) 8(1-1000) 8(1-1000) 8(1-1000) 8(1-1000) 4(20-300) 4(20-300) 4(20-300) 3(200-7000) 3(200-10000) 3(200-10000) 4(15-1000) 4(^1000)

[144] [144] [144] [144] [144] [157] [157] [157] [160] [160] [160] [157] [157] [157] [161] [161] [161] [162] [163] [163] [163] [163] [163] [163] [163] [163] [164] [164] [164] [165] [165] [165] [159] [159]

MFI value calculated from equation (6.30) knowing the MFI value as per footnote b. MFI value given by manufacturer or measured under ASTM test conditions. MFI value read out from T versus y curve using equations (6.26) and (6.27) by the method discussed in section 6.8.

Appendix D

Abbreviations

STANDARDS ASTM BS DIN ISO JIS

American Society for Testing and Materials British Standards Deutsches Institut fiir Normung International Standards Organisation Japanese Industrial Standards

POLYMERS ABS C-ester C-ether EVA HDPE HIPS LDPE LLDPE PA POM PAr PAS PBT PC PE PEEK PEI PES PET PIB PMMA

Acrylonitrile-butadiene-styrene Cellulose ester Cellulose ether Ethylene-vinyl acetate High density polyethylene High impact polystyrene Low density polyethylene Linear low density polyethylene Polyamide Polyacetal Polyarylate Polyaryl sulfone Poly(butylene terephthalate) Polycarbonate Polyethylene Polyether ether ketone Polyetherimide Polyether sulfone Polyethylene terephthalate) Poly(isobutylene) Poly methyl methacrylate

PP PPO PPS PS PVA PVB PVC PVDF SAN SBR SBS SIS SMA TPE UHMWPE VCVA

Polypropylene Poly phenylene oxide Poly phenylene sulfide Polystyrene Poly(vinyl alcohol) Poly( vinyl butyral) Poly(vinyl chloride) Poly(vinylidene fluoride) Styrene acrylonitrile Styrene butadiene rubber Styrene-butadiene-styrene Styrene-isoprene-styrene Styrene-maleic anhydride Olefinic-type thermoplastic elastomer Ultra high molecular weight polyethylene Vinyl chloride-vinyl acetate

Appendix E

Nomenclature

Table E1 Symbol 1

Description 2

a

Volume fraction of component 1A' in a mixture Half width of channel Empirical coefficient whose value lies between 1 and 2 MFI ratio Model parameter m2 Surface area of plate in Figure 2.1 Frequency term depending on the Pa sec entropy of activation for flow Constant Constant Hamaker's constant Adjustable parameter Constant dependent on the nature of the continuous phase Volume fraction of component 1B' in a mixture Half thickness of channel m Overall width of the residence time distribution curve in Figure 5.28 Filler-polymer interaction parameter Constant Coefficients Function dependent on volume fraction and power-law index Function dependent on volume fraction and power-law index

a0 ai ^MFl a A A0 A'0 A0 A* \ A"

b bQ b B Bo B1, B11 B1(^n) 82(0, n)

Units 3

Equation 4 (5.2a), (5.3b) (3.18) (4.2) (6.15) (2.53)-(2.56) (2.69) (2.74) (2.73) (6.5a), (6.6) (4.12b) (6.12), (6.13) (5.2b), (5.3c) (3.17) (5.14) (8.10)-(8.14) (2.74) (4.27) (6.3) (6.3)

Table E1 (continued) Symbol 1

Description 2

Arbitrary adjustable parameter Constants Positive constant representing the total number of nearest neighbors of each sphere C^ Constant C2 Constant ofp Particle diameter in Figure 4.4 c/ Particle diameter in Figure 4.5 D Diameter of sphere, rod or plateletshaped particle D3 Average diameter of different sizes of particulates D1 Diameter of the /th component D0 Orifice diameter DE Extrudate diameter Dp1 , Dp2, Dp, Average diameter of suspending particles 1, 2 and /th fraction, respectively Dpc Projection circumscribing diameter in Table 8.4 from section 8.6 D1 Diameter of tube or pipe DR Draw ratio DRC Critical value of draw ratio D Symmetric part of the gradient or deformation tensor

Units 3

c C 1 5 C 2 , C3 C

e e ev evi . ev2> ev3 E Ec Ev Eint E°s Evis E0 E00

Exponential (where e' = 2.71828) Independent parameter Parameter contributing the electroviscous effects Coefficients corresponding to each of the three electroviscous effects Activation energy for viscous flow Coulombic interaction energy van der Waals interaction energy Interaction energy Viscous dissipation energy for matrix Viscous dissipation energy for filled system Activation energy of the flow process for unfilled polymer system Activation energy of the flow process for filled polymer system containing $ volume fraction of filler

Equation 4 (2.57) (2.63) (6.5a), (6.6) (2.68) (2.68)

m m m m m m m m

(6.5a), (6.6), (8.11H8.13) (6.17)-(6.20) (6.17H6.21) (4.22), (4.23), (4.25)

jim

m

(4.22), (4.23)

sec~1

(2.3), (2.4), (2.23), (2.28), (2.30) (2.57) (4.32) (4.31)

kJ/mole J J J J J

(2.69), (2.70) (6.2) (6.2) (6.1) (6.1), (6.3) (6.1), (6.3)

kJ/mole

(6.8), (6.10)

kJ/mole

(6.9), (6.10)


Description 2

Units 3

Equation 4

ET

Value of E determined under constant shear stress conditions Value of E determined under constant shear rate conditions Dynamic loss modulus for filled polymer in solid state Dynamic loss modulus for unfilled polymer in solid state Difference between the activation energy for viscous flow of filled and unfilled polymer system Force in Figures 2.1, 2.3, 2.4 Function of the volume fraction of the floccules in a suspension Free volume of the continuous phase at temperature T but without the presence of any other dispersed phase Free volume of the multi-component system at temperature 7 containing <j> weight fraction of the dispersed phase Number fraction of the /th component defined by eq. (6.21) Force exerted by the test load L on the polymer in the melt flow indexer Maximum force tending to divide a dumbbell shaped agglomerate in a fluid Apparent melt shear modulus Storage moduli for the filled and unfilled polymer, respectively Dynamic storage modulus Dynamic storage modulus at frequency of a> for unfilled polymer system Dynamic storage modulus at frequency of co for filled polymer system containing $ volume fraction of filler Relative dynamic storage modulus defined as the ratio of the dynamic storage modulus for the filled to unfilled polymer system Dynamic loss modulus

kJ/mole

(2.71)

kJ/mole

(2.72)

Pa

(8.14)

Pa

(8.14)

E) Eg Ep AE f f(4>r) /(7, O)

f(T, 4>)

^1 F Fmax G Gc, Gmo G' G' (O, o>) G'((/>, CD)

G'r

G"

kJ/mole Newtons

(2.2) (4.28)

m3

(6.13), (6.14), (6.15)

m3

(6.12), (6.14)

(6.17)-(6.19) Newtons

(6.24)

Newtons

(5.11)

Pa

(3.16b) (6.23)

Pa Pa

(2.15) (8.7a)

Pa

(8Ja)

Pa

(8.9)

Pa

(2.16)


Description 2

Units 3

Equation 4

G" (O, co)

Dynamic loss modulus at frequency of co for unfilled polymer system Dynamic loss modulus at frequency of co for filled polymer system containing 0 volume fraction of filler Dynamic loss modulus for filled polymer melt Dynamic loss modulus for unfilled polymer melt Complex modulus Gap between two parallel discs of viscometer Intensity of segregation Steady state compliance Constant power index Thickness of the electrostatic interaction layer Hydrodynamic interaction coefficient Constant Consistency index whose values are tabulated in Tables 6.8 and 6.9 Rate constant for volume increase of the crystals Coefficient Consistency index

Pa

(8.7b)

Pa

(8.7b)

Pa

(8.10)

Pa

(8.10)

Pa m

(2.21) (3.7)

G"($, co)

GC Gp G* 77 / J6 k ke kj kQ K K0 KJe K / /o /N /P L L1 L2 m Tf? m' m" M MFI

Variable length of cylindrical rod in Figure 2.3 Initial length of cylindrical rod Length of nozzle Length of rod-shaped particle Test load, i.e. dead weight+ piston weight Test load 1 Test load 2 Damping constant Number of thread starts per screw Adjustable parameter Adjustable parameter Total number of components in a blend of suspended particles Melt flow index

Pa"1

g/cmsec2"" (g/10 min)n

(5.10) (2.50) (6.16) (6.5a) (4.25) (2.75) (6.35), (6.36) (4.30)

kg/m sec2'"

(2.49), (2.50) (2.39), (2.40), (2Ma), (4.29), (6.31)

m m m m kg

(2.26) (6.24) (8.12) (6.26)

kg kg

(6.32) (6.32) (2.47), (2.48) (5.13) (2.60) (2.61)-(2.63) (4.27)

g/10min


Description 2

Units 3

Equation 4

MFI(T, O)

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 (/> volume fraction of the dispersed phase Melt flow index determined under test load 1 Melt flow index determined under test load 2 Melt flow index at ASTM recommended test temperature Melt flow index at required temperature Mixing index Weight average molecular weight

g/10min

(6.13)

g/10min

(6.12)

g/10min

(6.32)

g/10min

(6.32)

g/10min

(6.30)

g/10min

(6.30)

MFI(T, 0)

MFI1 MFI2 MFI11 MFI12 /7?i,7r?| Mw MWc ~M2 n n' n" 7? N

(5.7) (2.49), (2.50), (2-75) (2.75)

Critical weight average molecular weight z average molecular weight Power-law index whose values are given in Tables 6.8 and 6.9 Power-law index Power-law index Power index Power index in the Carreau model

(2.49), (2.50)

Pa Pa Newtons

A/'r N" /V

Primary normal stress difference Secondary normal stress difference Normal force Number of particles in sample Number of ultimate particles of the major component Number of ultimate particles of the minor component Number of floccules Number of samples of same size Speed of rotation

p PH pm

Pressure Pressure hole error Measured pressure

Pa Pa Pa

/V1 /V2 A/F N' /V1 A/2

(2.52b), (2.62b) (2.52b), 2.62b) (8.10), (8.14) (2.42), (2.43), (2.48), (2.52a), (2.62a), (6.33) (2.7) (2.8) (3.4H3.6) (5.1) (5.1)

rpm

(4.28) (5.4H5.6) (3.1), (5.12), (5.13) (2.3)


Description 2

P

APdie

Power-index in the General Rheological model whose values are given in Table 6.9 Particle size frequency function of the /th and /th particle Probability distribution for finding XA concentration in a sample Pressure drop in extrusion die

Pa

O

Volumetric flow rate

nrrVsec

Oc

Leakage flow rate between screw flight and other screw Leakage flow rate between screw flight and barrel wall Leakage flow rate Leakage flow rate between flanks perpendicular to the plane through screw axis Leakage flow rate between flanks of screw flights Theoretical flow rate Radial position Radius of particle 1 in an agglomerate Radius of particle 2 in an agglomerate Ratio of length of diameter of a rodshaped particle

m3/sec

(3.10H3.12), (3.14), (3.15) (5.12), (5.13), (5.15), (6.25) (5.13)

m3/sec

(5.13)

rrvVsec m3/sec

(5.13) (5.13)

m3/sec

(5.13)

rrvVsec m m

(5.13) (3.1) (5.11)

m

(5.11)

PI, Pj P(XA)

Of O1 O3 Ot Qth r T1 r2 ra re R R(r) RN RP fl s S S2 SL SR

Ratio of the semi-axis of the ellipsoid of rotation Gas constant = 8.314 Coefficient of correlation Radius of nozzle Radius of piston Radius of disc or cone of viscometer Self-cleaning time Standard deviation of composition of spot samples Variance of composition of spot samples Scale of segregation Elastic strain recovery

Units 3

Equation 4 (2.44), (6.36) (4.26) (5.3a)

(4.15), (4.16), (4.18), (4.19), (4.33)-(4.36) (4.14a), (4.14b) J/molK

m m m sec

(2.69)-(2.72) (5.9) (6.24), (6.25) (6.24) (3.2)-(3.9) (5.14) (5.6), (5.7) (5.5) (5.8) (2.38), (3.15), (3.16a)


Description 2

Sw

Die swell ratio of extrudate diameter to die diameter Time Average residence time Loss tangent = G" /G' Trace of the deformation tensor Polymer melt temperature ASTM recommended test temperature Temperature at which MFI is required Glass transition temperature of polymers Characteristics glass transition temperature for filled composite and unfilled polymer Standard reference temperature equal to T9 +50 Measured torque Symmetric Cauchy stress tensor Volume fraction of the /th component Velocity components along X1, X2, X3 axis Volume of C-shaped channel between flanks of successive flights Free volume Volume of sample Distance between the points of inflection on the residence time distribution curve in Figure 5.28 Distances along the X1 , X2, X3 axis Weight proportion of the /th and /th component in a blend of suspended particles Positional vector Thickness of platelet-shaped particle Concentration of component 1A' in a sample Measured value of XA for the /th sample Concentration of component 'A' at point 1 in the /th sample

t ~t tan 6 tr D T T1 T2 T9 7

Qc' 7Q0

T3 T T Vj V1, V2, V3 Vc Vf V5 w X1 , X2, X3 xh X1 x X XA XA. X A/

Units 3

Equation 4 (2.37)

sec sec 0

(5.14), (5.15) (2.17) (2.5)

C K

(6.30)

K

(6.30)

K K

(6.23)

K

(2.68), (2.70), (6.30) (3.2), (3.3) (2.3) (6.21)

Newtons Pa m/sec

m3

(5.13)

m3 m3 m

(5.15) (5.3c) (5.14)

m (4.27)

m m

(2.4) (8.13) (5.3a) (5.4), (5.5) (5.9)


Description 2

X^.

Concentration of component 'A' at point 2 in the /th sample Actual mean concentration of component 'A' in a sample Quantity calculated for checking mixing quality Degree of flocculation Model parameter

XA Z Z0 Z(a)

Units 3

Equation 4 (5.9) (5.4H5.6) (5.6) (4.19) (2.53)-(2.56)

Appendix F

Greek symbols

Table F1 Greek symbols Symbol 1

Description 2

a a0

Ultimate particle size of component A Rate of energy dissipation within the neighbourhood of a typical sphere in a suspension Coefficients Coefficient in the crowding factor expression Crowding factor coefficient Function of the axis ratio re Function of the axis ratio re Einstein's constant

0^,Of 2 Ja 3 ac a/v Ocn OLf2 aE a' a'c «c a a /? P(T) j50 /? y

Ellis model parameter Adjustable factor Adjustable factor Constant Coefficient Term defined by equation (6.18) Difference between free volumes of the polymer and the filled polymer system Rate constant for the equilibrium between free filler particles and floccules Constant Shear rate

Units 3

Equation 4 (5.3c) (4.29) (4.5) (4.8), (4.18)

m3

(4.26) (4.13), (4.14b) (4.13), (4.14b) (4.1), (4.4), (4.15), (4.17) (2.41), (6.34) (4.21) ,(4.22) (4.21), (4.23) (5.12) (4.17) (6.17) (6.14), (6.15) (4.19)

see'1

(5-12) (2.1), (6.23), (6.27), (6.29)

Table F1 Continued Symbol 1

Description 2

y0

Amplitude of the sinusoidal variation of shear rate Shear rate at outer radius of disc Amplitude of oscillation in the maximum (or effective) shear rate (yj Wall shear rate lnterfacial energy between liquidsolid phase lnterfacial energy between liquidvapour phase lnterfacial energy between solidvapor phase Phase angle Diameter ratio of small particle to large particle in Figure 6.1 8(b) Void fraction or porosity Dielectric constant Degree of fill Uniaxial extensional rate

ya ym yw y'LS y'LV 7sv 6 yrmjx rjs rj-f T/V rjQ

riQJ rjOJn rjjj

Tj^jn

r\ rj'r Y]' r\* Y]I rj*((f), y) ^00 9

Steady shear viscosity of unfilled polymer system at a temperature T Steady shear viscosity of unfilled polymer system at a reference temperature 7R Steady shear viscosity of filled polymer system containing volume fraction of filler at a temperature T Steady shear viscosity of filled polymer system containing <j> volume fraction of filler at a temperature 7"R Dynamic viscosity Relative dynamic viscosity defined as the ratio of dynamic viscosity of filled to unfilled polymer system Imaginary part of complex viscosity Complex viscosity Zero-frequency viscosity function Viscosity function dependent on volume fraction and shear rate Shear viscosity value at very high shear rate Spherical coordinate giving the orientation of the rod as shown in Figure 4.2

(6.7) (4.21) Pa.sec

(4.29)

Pa.sec

(2.65)

Pa.sec Pa.sec

Pa.sec

(2.3) (2.34), (2.36), (4.22), (4.23), (4.28), (4.33)(4.36) (6.8), (6.10)

Pa.sec

(6.8), (6.10)

Pa.sec

(6.9), (6.10)

Pa.sec

(6.9), (6.10)

Pa.sec

(2.18) (8.8)

Pa.sec Pa.sec

(2.19) (2.20), (2.51), (2.52a), (2.52b) (2.52a) (6.5b)

Pa.sec Pa.sec Pa.sec

Table F1 Continued Symbol 1

Description 2

Units 3

Equation 4

9Q & 9

Cone angle in Figure 3.1 Contact angle Half angle of convergence of stream line at the entrance of die Relaxation time; also time constant and model parameter

radians radians radians

(3.1) (1.1) (2.67)

sec

(2.42), (2.43), (2.48), (2.56), (2.62a), (6.33) (2.66a)

A A1 , A 2 Ac A3 A(0) IJL /Z v v p P0 a a2 (T1 , C2 03

Initial volume of the crystals Volume concentration of suspended particles for 1, 2 and /th size fraction, respectively Critical concentration below which the interaction between the rods can be neglected Volume fraction of floccules in the suspension Volume fraction of larger particles Maximum attainable concentration of fillers defined as 1 - s or maximum packing fraction Maximum packing of spheres of uniform size r Volume fraction of smaller particles Spherical coordinate giving the orientation of the rod as shown in Figure 4.2 Surface potential of particulates Primary and secondary normal stress coefficients Primary normal stress coefficient function Secondary normal stress coefficient function Relative normal stress difference coefficient defined as the ratio of normal stress of filled to unfilled polymer system at the same shear rate Enclosed angle for a curved fiber in Figure 4.4 Frequency of oscillations or angular frequency Probability

m3

(4.30) (4.21), (4.24)

0cr c/>f 0L 0m 0° s \l/ \I/Q ^1, \l/2 ^1(JO ^2Cx) ^1 y

1

F1

co Qmixed

(4.16) (4.28)

(6.18)

J kg/m

(6.5a)

kg/m

(2.10), (2.46), (2.47) (2.11)

kg/m

(7.1)

rad/sec

(2.12H2.21) (5.1)

Author Index

Abbas, K.A. 41, 49, 243, 292, 305, 312, 313, 315, 331, 334, 338, 391, 395, 411 Abdel-Khalik, S.I. 83-5, 107, 108 Abe, D.A. 41, 49, 243, 273, 305, 312, 334, 338, 391, 395, 411 Acierno, D. 9, 44, 166, 174, 397, 414 Acrivos, A. 136, 142, 170, 171, 247, 308 Adams, E. F. 266, 309 Adarns, J.W.C 117, 133,267 Adams- Viola, M. 136, 168 Adler, P.M. 136, 170 Advani, S.G. 41, 51, 243, 308, 312, 337, 338, 394, 395, 414 Agarwal, P.K. 98, 109 Aggarwal, S.L. 8, 44 Agur, E.E. 194, 239 Ahu, C.W. 262, 263, 309 Akay, G. 74, 106 Alfrey, T., Jr. 140, 171 Allen, P.W. 1, 44 Allen, T. 357, 394 Allport, D.C. 8, 44 Alter, H. 18, 45 Althouse, L.M. 41, 49, 243, 273, 306, 312, 335, 338, 360, 361, 387, 388, 392, 394, 395, 412 Anderson, P.C. 136, 168 Andres, U.T. 138, 170 Andrews, R.D. 117, 133 Angerer, G. 121, 133 Aoki, Y. 117, 133 Aral, E. 41, 51, 243, 307, 312, 336, 338, 393, 395, 413 Aral, B.K. 338, 390

Arina, M. 41, 48, 243, 259, 304, 312, 333, 338, 390, 395, 398, 410, 414 Armstroff, O. 208, 240 Armstrong, R.C. 55, 59, 60, 67, 79, 104, 105 Arrhenius, S. 140, 171 Ashare, E. 85, 108 Astarita, G. 67, 79, 81, 84, 105, 107, 108, 395, 414 Ayer, J.E. 266-8, 309 Bachmann, J.H. 18, 45 Baer, A.D. 142, 171, 257, 265, 267, 269, 308, 346, 394 Bagley, E.B. 72, 74, 105, 106, 121, 133 Baily, E.D. 117, 133 Baird, D.G. 9, 44, 331, 337 Baker, F.S. 113, 132, 136, 169 Ballenger, T.F. 85, 108 Ballman, R.L. 100, 110, 128, 131, 134, 135 Balmer, J. 74, 106 Bamane, S.V. 81, 82, 86, 87, 107, 301, 302, 311 Bandyopadhyay, G. 41, 52 Bares, J. 1, 44 Barnes, H.A. 41, 47, 67, 79, 84, 105, 107, 136, 170 Barringer, E.A. 266, 267, 309 Bartenev, G.M. 14, 41, 45, 285, 310 Batchelor, G.K. 136, 164-6, 170, 174, 405, 414 Bauer, L.G. 136, 168 Baumann, O.K. 189, 238 Baumann, G.F. 104, 111 Bayliss, M.D. 136, 153, 169

Becker, E. 67, 79, 105 Beek, W.I. 194, 239 Belcher, H.V. 100, 109 Bell, J.P. 166, 174, 397, 414 Benbow, JJ. 74, 75, 106 Bennett, K.E. 131, 135 Berger, S.E. 35, 46 Beris A.M. 88, 90, 108 Berlamont, J. 136, 168 Bernier, R. 136, 169 Bersted, B.H. 74, 106 Berstein, B. 72, 85, 106 Bestul, A.B. 100, 109 Bhardwaj, LS. 41, 49, 243, 273, 277, 278, 305, 312, 334, 338, 391, 395, 411 Bhattacharya, S.K. 41, 52, 266, 268, 310 Bhavaraju, S.M. 136, 168 Bierwagon, G.P. 19, 45 Bigg, D.M. 41, 47, 49, 50, 196, 219, 225, 236, 237, 239, 241, 243, 273, 277, 278, 290, 304, 306, 312, 333, 335, 338-40, 349, 350, 354, 360, 361, 387, 388, 390, 392, 394, 395, 410-12 Billmeyer, Jr. F.W. 1, 43, 44 Binding, D.M. 98, 109, 131, 135 Birchall, J.D. 190, 238 Bird, R.B. 55, 59, 60, 65, 67, 79-81, 83-5, 97, 104, 105, 107-9 Birks, A.M. 74, 106 Blake, W.T. 125, 134 Blanch, H.W. 136, 168 Blankeney, W.R. 146, 147, 172 Bludell, DJ. 103, 110 Blyler, L.L. 118, 126, 133, 134 Boger, D.V. 113, 131, 136, 170, 312, 337 Bogue, D.C. 85, 91, 108, 117, 133 Bohn, E. 10, 44, 262, 263, 309 Boiesan, V. 102, 110, 140, 171 Boira, M.S. 41, 47, 243, 273, 275-7, 292, 304, 312, 333, 338, 390, 395, 410 Boonstra, B. B. 226, 241 Booth, F. 163, 173 Booy, M.L. 208, 240 Borghesani, A.F. 136, 168 Botsaris, G.D. 136, 168 Boudreaux, Jr. E. 75, 106 Bourne, R. 299, 311 Bowen, B.D. 10, 44

Bowen, H.K. 262, 263, 266, 267, 272, 309, 310 Bowerman, H. H. 398, 414 Bradley, H.B. 35, 46 Brandrup, J. 1, 44 Brauer, G.M. 136, 169 Braun, D.B. 136, 169 Brede, H.L. 140, 171 Brenner, H. 136, 170 Bretas, R.E.S. 41, 50, 243, 306, 312, 335, 338, 393, 395, 412 Bright, P.P. 41, 48, 243, 305, 312, 334, 338, 391, 395, 397, 398, 410, 414 Brodnyan, J. 147, 172 Broutman, LJ. 17, 45 Browned, W.E. 136, 169 Bruch, M. 312, 332 Brydson, J.A. 41, 47 Bulkley, R. 83, 107, 247, 308 Bungenberg de Jong, H.G. 140, 171 Burgers, J.M. 145, 172 Burke, JJ. 8, 44 Cameron, G.M. 36, 46 Carley, J.F. 103, 110, 196, 239 Carr, R. 136, 167 Carreau, PJ. 41, 51, 74, 81, 85, 106-8, 243, 284, 287, 308, 312, 337, 338, 341, 394, 395, 414 Carruthers, J.M. 113, 132, 222, 241 Carter, R.E. 113, 132, 136, 169 Caso, G.B. 136, 169 Castillo, C. 136, 168 Cengel, J.A. 139, 171 Cessna, L.C. 16, 45 Chaffey, C.E. 41, 47, 49, 50, 243, 273, 275-7, 292, 304-6, 312, 333-5, 338, 390, 392, 395, 410-2 Chan, C.F. 79, 107 Chan, F.S. 163, 173 Chan, Y. 41, 48, 163, 166, 173, 243, 292, 304, 312, 315, 333, 338, 390, 395, 405, 410 Channis, C.C. 18, 45 Chapman, P.M. 41, 47, 115, 132, 151, 172, 243, 246, 303, 312, 332, 338, 390, 395, 407, 409 Charles, M. 124, 134, 136, 168 Charley, R. V. 74, 106 Charrier, J.M. 166, 174

Chartoff, R.P. 118, 133 Chattopadhyay, S. 290, 299, 310 Chen, IJ. 85, 108, 117, 133 Chen, SJ. 179, 237 Chen, Y.R. 136, 169 Cheng, D.C-H. 136, 170 Cheremisinoff, N.P. 67, 79, 105, 136, 169, 189, 194, 238, 239 Chhabra, R.P. 81, 107 Chipalkatti, M.H. 41, 49, 243, 273, 305, 312, 334, 338, 391, 395, 411 Chong, J.S. 142, 171, 257, 265, 267, 269, 308, 346, 394 Christiansen, E.B. 142, 171, 257, 265, 267, 269, 308, 346, 398 Chung, C.I. 194, 239 Chung, J.T. 67, 105, 125, 126, 134 Chung, K.H. 194, 239 Churchill, R. W. 81, 82, 107 Churchill, S. W. 81, 82, 107 Clark, E.S. 86, 108, 316, 337 Clarke, B. 140, 145-7, 150, 171 Clegg, D.W. 43, 52 Clegg, P.L. 72, 75, 105 Code, R. K. 136, 168 Cogswell, F.N. 41, 47, 65, 76, 98, 103, 104, 107, 109, 110, 113, 130, 131, 135 Cohen, L.B. 273, 277, 310 Cohen, M. 103, 110 Cokelet, G.R. 136, 167 Coleman, B.D. 43, 52, 66, 104 Coleman, G.N. 92, 109, 136, 168 Collins, E. A. 1, 44, 103, 110, 398, 414 Collyer, A.A. 9, 43, 44, 52 Colwell, R.E. 43, 53, 113, 132 Connelly, R. W. 113, 132, 222, 241 Conway, B.E. 162, 173 Cook, J. 17, 45 Cooper, E.R. 8, 37, 44, 47 Cooper, S.L. 8 Cope, D.E. 41, 49, 243, 273-5, 305, 312, 334, 338, 391, 395, 411 Copeland, J.R. 30-33, 46, 243, 273, 304, 312, 333, 338, 390, 395, 410 Cotten, G.R. 14, 45, 226, 241, 319, 337, 400, 414 Cox, W.P. 86, 108 Crabbe, P.G. 113, 131 Crabtree, J.D. 36, 46

Crackel, P.R. 136, 168 Craig, R.G. 136, 169 Crawley, R.L. 72, 105 Cross, M.M. 81, 107, US, 132 Crowder, J.W. 14, 41, 45, 85, 108, 243, 304, 312, 319, 333, 338, 390, 395, 410 Crowson, RJ. 41, 48, 243, 292, 305, 312, 334, 338, 391, 395, 397, 398, 410, 411, 414 Cuculo, J.A. 75, 106 Czarnecki, I. 41, 48, 49, 243, 244, 305, 312, 313, 315, 334, 338, 344, 391, 395, 396, 400, 411, 414 Daane, J.H. 126, 134 Daley, L.R. 41, 50, 243, 306, 312, 335, 338, 392, 395, 412 Danckwerts, P.V. 175, 181, 183, 237, 238 Danforth, S.C. 88, 90, 108 Darby, R. 43, 52, 67, 79, 105, 136, 168, 170 Daroux, M. 81, 107 Davis, J.H. 19, 45 Davis, P.K. 136, 168 De Boojs, J. 140, 171 De Brujin, H. 139, 171 De Cindio, B. 166, 167, 174 De Kee, D. 81, 107, 136, 168 De Simon, L.B. 113, 132, 136, 169 De Waele, A. 80, 107 Dealy, J.M. 41, 43, 47, 53, 65, 104, 113, 130, 132, 135 Den Otter, J.L. 117, 124, 133, 134 Denn, M.M. 73, 97, 106, 109, 142, 171, 312, 337 Denson, C.D. 130, 135 Derjaguin, B. 247, 308 Deryagin, B.V. 186, 238 Dhimmar, LH. 41, 49, 243, 273, 277, 278, 305, 312, 334, 338, 391, 395, 411 Dibenedetto, A. T. 285, 286, 310 Dillon, R.E. 74, 106, 120, 133 Dintenfass, L. 136, 167 Dobry, A. 162, 173 Dobry-Duclaux, A. 162, 173 Donovan, R.C. 194, 239 Doolittle, A.K. 257, 308

Doraiswamy, D. 88, 90, 108 Dougherty, TJ. 151, 153, 173 Dow, J.H. 41, 51, 136, 154, 157, 160, 168, 173, 229, 232, 236, 237, 241, 243, 266, 307, 312, 336, 338, 349, 357, 360, 367, 369-71, 393-5, 413 Driol, E. 36, 46 Droste, D. H. 285, 286, 310 Duffey, HJ. 72, 105 Dufresne, A. 286, 287, 310 Dulik, D. 136, 169 Edirisinghe, MJ. 41, 51, 52, 189, 238, 243, 266, 268, 307, 310, 312, 336, 338, 393, 395, 413 Eguiluz, M. 162, 173 Eilers, H. 141, 171 Einstein, A. 137, 140, 170, 247, 308 Eise, K. 186, 238 Eisenlauer, J. 155, 173 Eisenschitz, R. 121, 133 Epstein, N. 10, 44, 262, 263, 309 Erdmenger, R. 208, 240 Erenrich, E.H. 136, 168 Erickson, P. W. 35, 37, 38, 46, 47 Erwin, L. 196, 215, 240 Espesito, R. 200, 240 Ester, G.M. 8, 44 Evans, J.R.G. 41, 51, 52, 189, 238, 243, 266, 268, 307, 310, 312, 336, 338, 393, 395, 413 Evans, R.L. 136, 167, 223, 224, 241 Everage, A.E. 131, 135 Eveson, G.F. 264, 309 Eyring, E.M. 141, 171 Eyring, H. 99, 109, 141, 171, 255, 308

Fan, L.T. 179, 237 Farooqui, S.I. 136, 169 Farris, RJ. 265, 309 Faruqui, A.A. 139, 171 Faulkner, D.L. 41, 47, 123, 124, 134, 243, 304, 312, 333, 338, 345, 390, 395, 410 Fedors, R. F. 400, 414 Feldman, D. 102, 110, 140, 171 Fenner, R.T. 196, 239 Ferguson, J. 100, 109, 136, 153, 169 Ferraro, C.F. 151, 173 Ferry, J.D. 41, 47, 99, 109

Fiekhrnan, V.D. 97, 109 Fielding, J.H. 118, 133 Fikentscher, H. 140, 171 Fikham, V.D. 128, 135 Filymer, Jr., W.G. 136, 168 Fink, A. 10, 44, 262, 263, 309 Finnigan, J.W. 139, 171 Fisa, B. 41, 49, 243, 305, 312, 334, 338, 392, 395, 411 Fischer, E.K. 151, 173 Fiske, TJ. 395, 409 Flory, PJ. 1, 43 Folkes, MJ. 9, 41, 44, 48, 243, 292, 305, 312, 334, 338, 391, 395, 397, 398, 410, 411, 414 Ford, R.G. 41, 52 Ford, T.F. 139, 141, 171 Forger, G. 15, 45 Fox, T.G. 104, 111 Frados, J. 1, 44 Franked N.A. 142, 171, 247, 308 Frechette, FJ. 41, 51, 83, 107, 143, 171, 243, 250, 255, 257, 264, 269, 307, 308, 312, 316, 319, 321, 336-8, 345-8, 353, 355, 393, 395, 413 Fredrickson, A.G. 66, 104 Freestone, A.R.I. 113, 131 French, K.W. 41, 52 Friedrich, C. 312, 332 Fryling, C.F. 162, 173 Fujita, H. 257, 308 Fu-lung, L. 136, 167 Fukase, H. 194, 239 Fukusawa, Y. 118, 133 Galgoci, E.G. 41, 52 Garcia, R.R. 272, 310 Gatner, F.H. 71, 105 Gaskins, F.H. 100, 109, 113, 122, 124, 131, 134 Geisbusch, P. 41, 48, 243, 292, 304, 312, 333, 338, 390, 395, 410 George, H.H. 166, 174, 397, 414 German, R.M. 41, 52, 266, 268, 310 Gerson, Ph.M. 194, 239 Gibson, A.G. 98, 109, 131, 135 Gillespie, T. 151, 173, 264, 309 Glasscock, S.D. 72, 105 Glazman, Yu. M. 136, 168 Goddard, J.D. 165, 166, 174

Godfrey, J. 200, 240 Goel, D.C. 41, 48, 151, 172, 243, 305, 312, 334, 338, 391, 395, 411 Goettler, L.A. 166, 174, 397, 414 Gogos, C.G. 180, 183-5, 194, 237 Goldman, A. 266, 267, 309 Goldsmith, H.L. 150, 172 Goodrich, J.E. 126, 134 Gordon, J. 17, 45 Goring, D.A. 163, 173 Govier, G.W. 151, 173 Graessley, W. W. 72, 105 Grateh, S. 104, 111 Greener, J. 113, 132, 222, 241 Groto, H. 264, 309 Gruver, J.L. 104, 111 Gunberg, P. F. 226, 241 Gupta, R.K. 41, 51, 83, 107, 143, 171, 243, 250, 255, 257, 264, 268, 269, 307, 308, 310, 312, 317, 319, 321, 336-8, 345-8, 353, 355, 393, 395, 413 Gurland, J. 248, 308 Guth, E. 140, 143, 171, 172 Hagler, G.E. 85, 108 Hallouche, M. 201, 240 Harnaker, C. 247, 308 Han, C.D. 41, 47-51, 91, 108, 115, 124, 132, 134, 243, 273, 276-8, 292, 304-7, 312, 313, 315, 316, 319, 323, 332-8, 361, 390-2, 394, 395, 398, 402, 403, 407, 410-4 Hancock, M. 41, 48, 243, 305, 312, 334, 338, 391, 395, 411 Hanks, R.W. 136, 168 Hanna, R.D. 16, 45 Happel, J. 138, 170 Harbard, E.H. 140, 171 Harmsen, GJ. 163, 173 Harper, J.C. 151, 172 Harris, J. 43, 52 Harris, S.L. 194, 239 Hartlein, R.C. 29, 36, 38, 46, 47 Hartnett, J.P. 84, 107 Harwood, J.A.C. 38, 47 Hashimoto, A.G. 136, 169 Hashin, Z. 147, 172 Hassager, O. 55, 59, 60, 67, 79, 83-5, 104, 105, 107, 108

Hatshek, E. 138, 170 Hauwink, R. 140, 171 Haw, J.R. 41, 49, 243, 273, 276, 277, 305, 312, 313, 319, 323, 334, 338, 391, 395, 411 Haward, R.N. 100, 109 Hayashi, K. 41, 51, 243, 266, 307, 312, 336, 338, 393, 395, 413 Heath, D. 136, 155, 169 Heertjes, P.M. 136, 169 Hencky, H. 401, 414 Henderson, C.B. 265, 309 Henderson, D. 141, 171 Herrick, J. 22, 46 Herschel, W.H. 83, 107, 247, 308 Heywood, N.I. 136, 151, 169 Higashitani, K. 77, 107 Highgate, DJ. 154, 173 Hildebrand, J. H. 103, 110 Hill, C.T. 41, 47, 145, 172, 243, 304, 312, 333, 338, 390, 395, 402, 409 Hill, J. W. 65, 104 Hinkelmann, B. 41, 49, 50, 243, 305, 306, 312, 334, 335, 338, 392, 395, 411, 412 Hlavacek, B. 74, 106 Hlavacek, V. 262, 263, 309 Hodgetts, G.B. 113, 131 Hoffman, DJ. 103, 110 Hofman-Bang, N. 117, 133 Hold, P. 175, 179, 180, 189, 190, 195, 215, 217, 237, 238, 240 Holderle, M. 312, 332 Holdsworth, PJ. 103, 110 Holmes, L. A. 85, 108 Honkanen, A. 41, 48, 243, 259, 304, 312, 333, 338, 390, 395, 398, 410, 414 Hooper, R.C. 37, 46 Hope, P.S. 9, 44 Hopper, J.R. 14, 45, 319, 337 Hori, Y. 74, 106 Horie, M. 113, 131 Hornsby, P.R. 224, 241 Howards, AJ. 190, 238 Howland, C. 215, 240 Hsieh, H.P. 149, 172 Hu, R. 84, 107 Huang, C.R. 115, 124, 132 Hudson, N.E. 136, 153, 169

Huget, E.F. 113, 132, 136, 169 Hugill, H.R. 266, 267, 309 Hunt, K.N 41, 51, 223, 224, 241, 243, 266, 307, 312, 336, 338, 393, 395, 413 Huppler, J.D. 85, 108 Hutton, J.F. 41, 43, 47, 52, 67, 79, 105 Hylton D.C. 130, 135

Ide, Y. 130, 135, 396, 403, 414 Immergut, E.H. 1, 44 Insarova, N.I. 165, 174 Irving, H.F. 198, 240 Ishida, N. 118, 133 Ishigure, Y.41, 51, 243, 266, 307, 312, 336, 338, 393, 395, 413 Itadani, K. 41, 49, 243, 273, 305, 312, 334, 338, 391, 395, 411 Ito, K. 103, 110 Jacobsen, P.H. 113, 131, 136, 169 Jakopin, S. 186, 193, 196, 202, 227, 238 James, A.E. 81 44, 136, 169 Janacek, J. 286, 310 Janeschitz-Kriegl, H 41, 47, 117, 124, 133, 134 Janssen, L.P.B.M. 74, 106, 202, 214, 240 Jarzebski, GJ. 247, 250, 308 Jastrzebski, Z. 151, 172 Jeffrey, DJ. 136, 170 Jeffrey, J.B. 143, 172 Jepson, C.H. 175, 196, 237 Jerdrzejczyk, H. 136, 167 Jewmenow, S.D. 208, 240 Jinescu, V. V. 103, 110, 136, 170 Johnson, A.F. 396, 397, 414 Johnson, C.F. 41, 51 Johnson, J.F. 100, 101, 104, 109, 111 Johnson, R.O. 41, 50, 279, 292, 310 Jung, A. 103, 110 Juskey, V.P. 41, 49, 243, 273, 305, 312, 334, 338, 392, 395, 411 Kaghan, W.S. 100, 110 Kaloni, P.N. 67, 79, 105 Kalousek, G.L. 113, 131, 136, 168 Kalyon, D.M. 189, 193, 201, 238, 240, 338, 357, 390, 394, 395, 409 Kamal, M.R.41, 50, 243, 307, 312, 336, 338, 393, 395, 413

Kambe, H. 138, 170 Kanno, T. 152, 173 Kasajima, M. 103, 110 Kataoka, T. 41, 48-50, 115, 132, 142, 143, 171, 223, 241, 243, 246, 248, 249, 255, 257, 304-6, 308, 312, 333-5, 338, 346, 390-2, 395, 410-2 Katz, H.S. 10, 44 Kawasaki, H. 41, 49, 243, 273, 305, 312, 334, 338, 391, 395, 411 Kearsley, E. 72, 85, 106 Kendall, K. 190, 238 Khadilkar, C.S. 41, 51, 136, 154, 157, 168, 173, 232, 236, 237, 241, 243, 266, 307, 312, 336, 338, 393, 395, 413 Killman, E. 155, 173 Kim, D.H. 243, 303 Kim, K.U. 115, 124, 132 Kim, K.Y. 43, 53, 113, 132 Kim, W.S. 208, 240 Kim, Y.W. 395, 398, 402, 403, 414 King, K.D. 136, 168 King, R.G. 115, 132 Kirkwood, R.B. 136, 167 Kishimoto, J. 257, 308 Kitano, T. 41, 48-50, 115, 132, 142, 143, 171, 223, 241, 243, 246, 248, 249, 255, 257, 304-6, 308, 312, 333-5, 338, 346, 390-2, 395, 410-2 Kitchner, J.A. 156, 173 Kizior, I.E. 165, 174 Klein, I. 194, 196, 238 Knappe, W. 121, 134 Knudsen, J.G. 139, 171 Knutsson, B.A. 41, 49, 243, 292, 305, 312, 313, 315, 331, 334, 338, 391, 395, 411 Kohan, M.I. 292, 311 Kolarik, J. 286, 310 Kondo, A. 124, 134 Kondu, AJ. 76, 107 Koran, A. 136, 168 Korn, M. 155, 173 Kosinski, L.E. 113, 132, 222, 241 Kossen, N.W.F. 136, 168 Kraus, G. 18, 45, 104, 111 Kremesec, VJ. 153, 173

Krieger, FM. 151, 153, 162, 173, 262, 264, 309 Krotova, N.A. 186, 238 Krumbock, E. 121, 134 Kruyt, H.R. 140, 171 Kubat, J. 397, 414 Kumins, C.A 38, 47 Kunio, T. 194, 239 Kuno, H. 264, 309 Kurgaev, E.F. 138, 170 Kwei, T.K. 38, 47 Lacey, P.M.C. 179, 180, 237 Lakdawala, K. 282, 310 Lamb, P. 74, 75, 106 Lamoreaux, R. H. 103, 110 Landel, R.F. 14, 45, 99, 109, 153, 173, 400, 414 Laskaris, A. 22, 46 Lau, H.C. 121 Laude, R.F. 36, 46 Lee, B.L. 115, 118, 132 Lee, D.I. 264, 265, 309 Lee. KJ. 243, 303 Lee, M.C.H. 236, 241 Lee, T.S. 41, 47, 115, 132, 151, 172, 243, 246, 303, 312, 332, 338, 390, 395, 407, 409 Lee, W.K. 98, 109, 166, 174, 397, 414 Lee, W.M. 41, 49, 243, 273, 305, 312, 334, 338, 391, 395, 411 Lefebvre, J. 136, 168 Leitlands, V. V. 130, 135 Lem, K.W. -41, 49, 243, 306, 312, 335, 338, 392, 395, 412 Lenk, R.S. 41, 47 Lens, W. 140, 171 Leonard, M.H. 14, 45 Leonov, A.I. 121, 134 Lerner, I. 37, 46 Lewellyn, M.M. 136, 168 Lewis, H.D. 266, 267, 309 Liaw, T.F. 41, 49, 243, 273, 305, 312, 334, 338, 391, 395, 411 Lightfoot, E.N. 136, 168 Lin, Y.H. 121, 134 Linnert, E. 41, 49, 243, 273-5, 305, 312, 334, 338, 391, 395, 411 Lipatov, Yu. S. 286, 310 Lippe, RJ. 113, 131

Litovitz, T. A. 103, 110 Lobe, V.M. 41, 48, 115, 132, 243, 246, 304, 312, 313, 317, 333, 338, 390, 395, 402, 410 Lockett, FJ. 396, 397, 414 Lodge, A.S. 43, 52, 66, 72, 77, 104, 107 Loh, J. 136, 168 Loppe, J.P.A. 41, 49, 143, 172, 243, 305, 312, 334, 338, 391, 395, 411 Lornsten, J.M. 98, 109, 130, 135 Loshach, S. 104, 111 Luebbers, R.H. 148, 172 Luo, H.L. 41, 50, 243, 273, 306, 312, 335, 338, 361, 392, 395, 412 Lyons, J. W., 43, 53, 113, 132 Macdonald, LF. 85, 108 Macedo, P.B. 103, 110 MacGarry, FJ. 37, 46 Mack, W.A. 189, 193, 238 Macosko, C.W. 130, 135 Maddock, B.H. 194, 239 Maheshri, J.C. 208, 240 Maine, F. W. 16, 45 Malkin, A. Y. 14, 41, 45, 47, 117, 133, 243, 307, 312, 319, 337, 338, 394, 395, 413 Mallouk, R.S. 196, 239 Mangels, J.A. 41, 51, 52, 266, 267, 309 Manley, R.StJ. 140, 171 Mannell, W.R. 136, 168 Mark, H. 140, 171 Markovitz, H. 43, 52, 66, 92, 104, 109 Maron, S.H. 255, 308 Marrucci, G. 67, 79, 97, 105, 109, 142, 171, 395, 414 Marsden, J.G. 35, 36, 46 Martelli, F. 204, 240 Maschmeyer, R.O. 145, 172 Mason, S.G. 140, 146, 150, 171, 172 Masuda, T. 86, 108 Matijevic, E. 262, 308 Matsumoto, S. 149, 172 Matthews, G. 175, 237 Maxwell, B. 103, 110, 118, 133 McCabe, C.C. 14, 45 McGeary, R.K. 139, 171 McGrath, J.E. 8, 44 McHaIe, E.T. 265, 309 Mclntire, L.V. 120, 133

McKelvey, J.M. 180, 194, 196, 237, 239 Mead, W. T. 398, 414 Meares, P. 1, 43 Medalia, A.L 14, 45, 319, 337 Meissner, J. 55, 96, 100, 104, 109, 113, 115, 128, 130, 132, 135 Meister, BJ. 91, 108 Mendelson, R.A. 100, 109 Menges, G. 41, 48, 243, 292, 304, 312, 333, 338, 390, 395, 410 Mennig, G. 41, 50, 121, 133, 243, 306, 312, 335, 338, 392, 395, 412 Mertz, E.H. 86, 108 Metz, B. 136, 168 Metzner, A.B. 71, 79, 88, 90, 98, 105, 107-9, 136, 165, 170, 174, 332, 337 Mewis, J. 136, 165, 170, 174, 332, 337 Middleman, S. 41, 47, 67, 79, 103, 105, 110, 196, 239 Mijovic, J.41, 50, 243, 273, 306, 312, 335, 338, 361, 392, 395, 412 Milewski, J. V. 10, 44, 268, 310 Millman, R.S. 17, 45 Mills, NJ. 41, 47, 117, 133, 243, 246, 303, 312, 332, 338, 345, 390, 395, 409 Minagawa, N. 41, 47, 115, 132, 243, 246, 277, 304, 312, 315, 317, 319, 333, 338, 390, 395, 410 Minoshima, W. 86, 108, 316, 337 Missavage, RJ. 136, 168 Mistry, D.B. 160, 173 Mitsumatsu, F. 41, 51, 243, 266, 307, 312, 336, 338, 393, 395, 413 Mizoguchi, M. 194, 239 Modlen, G.F. 397, 414 Mohr, W.D. 175, 181, 183, 196, 237 Mokube, V. 264, 309 Molau, G.E. 8, 44 Monte, SJ. 41, 48, 243, 273, 275-8, 292, 304, 310, 312, 333, 338, 390, 395, 410 Mooney, M. 118, 133, 141, 149, 171 Morawetz, H. 1, 44 Moreland, C. 148, 150, 172 Morgan, RJ. 158, 173 Morley, J.G. 17, 45 Morrison, S.R. 151, 172

Mount III, E.M. 194, 239 Muchmore, C.B. 136, 168 Mueller, N. 14, 45 Mujumdar, A.N. 88, 90, 108 Mullins, L. 14, 38, 45, 47, 140, 171 Munro, J.M. 136, 168 Munstedt, H. 117, 130, 133, 135 Murase, I. 41, 51, 243, 266, 307, 312, 336, 338, 393, 395, 413 Murty, K.N. 397, 414 Mutel, A.T. 41, 50, 243, 307, 312, 336, 338, 393, 395, 413 Mutsuddy, B.C. 41, 50, 52, 243, 266, 306, 312, 335, 338, 392, 395, 412 Nadim, A. 136, 170 Nadkarni, V.M. 11, 12, 41, 44, 50, 236, 237, 241, 243, 251, 252, 266, 273, 276-8, 284, 287, 290, 292, 299, 306, 307, 310-2, 330, 335, 336, 338, 384, 392-5, 412 Nagatsuka, Y. 41, 50, 243, 306, 312, 335, 338, 392, 395, 412 Nagaya, K. 41, 51, 243, 266, 307, 312, 336, 338, 393, 395, 413 Nakajima, N. 72, 105, 398, 414 Nakatsuka, T. 41, 49, 243, 273, 305, 312, 334, 338, 391, 395, 411 Napper, D.H. 154, 156, 173 Nazem, F. 41, 47, 243, 304, 312, 333, 338, 390, 395, 402, 409 Nevin, A. 117, 133 Newman, S. 123, 124, 134, 243, 307, 312, 336, 338, 394, 395, 397, 413, 414 Newnham, R.E. 41, 52 Nguyen, Q.D. 136, 170 Nichols, R. 213, 215, 240 Nicodemo, L. 153, 166, 167, 173, 174, 395, 397, 414 Nicolais, L. 36, 46, 153, 166, 167, 173, 174, 286, 310, 395, 397, 414 Nielsen, L.E. 14, 41, 45, 47, 51, 137, 147, 170, 172, 243, 257, 269, 307, 308, 312, 336, 338, 393-5, 413 Nikolayeva, N.E. 14, 45, 117, 133, 319, 337 Nishijima, K. 41, 48, 115, 132, 142, 143, 171, 243, 246, 304, 312, 333, 338, 390, 395, 410

Pinder K.L. 113, 131 Pinto, G. 196, 240 Pipkin, A.C. 97, 109 Pisipati, R. 331, 337 Platzer, N.A.J. 8, 44 Plotnikova, E.P. 14, 45, 117, 133, 319, 337 Plueddemann, E.P. 35, 36, 38, 46, 47, 273, 277, 279, 310 Pokrovskii, V.N. 138, 143, 170, 172 Pollett, W.F.O 115, 132 Oda, K. 86, 108, 316, 337 Porter, R.S. 100, 101, 103, 104, 109-11, Ojama, T. 103, 110 126, 134, 299, 311, 398, 414 Okubo, S. 74, 106 Poslinski, AJ. 41, 51, 83, 107, 143, 171, Onogi, S. 86, 108 243, 250, 255, 257, 264, 269, 307, Opsahl, D.G. 136, 167 308, 312, 317, 319, 321, 336-8, Orenski, P.I. 35, 46 345-8, 353, 355, 393, 395, 413 Ostwald, W. 80, 107 Powell, R.L. 41, 50, 243, 306, 312, 335, Oswald, G.E. 136, 168 338, 393, 395, 412 Otabe, S. 41, 51, 243, 266, 307, 312, 336, Powers, J.M. 136, 169 338, 393, 395, 413 Pritchard, J.H. 299, 311 Ouchiyama, N. 268, 310 Prokunin, A.N. 130, 135 Overbeek, J. 163, 173 Oyanagi, Y. 41, 48, 75, 106, 163, 166, Rabinowitsch, B. 121, 133 173, 243, 248, 249, 255, 292, 304, Radushkevich, B.V. 97, 109, 128, 135 312, 315, 333, 338, 390, 391, 395, Raible, T. 130, 135 405, 410 Rajaiah, J. 247, 308 Rajora, P. 74, 106 Padget, J.C. 292, 311 Ramamurthy, A. V. 121, 134 Palmgren, H. 183, 238 Pao, Y.-H. 91, 108 Ramney, M. W. 35, 46 Papkov, S. 140, 171 Rauwendaal, CJ. 189, 200, 202, 214, Parish, M. V. 272, 310 238, 240 Reddy, K.R. 73, 106 Park, C.S. 243, 303 Park, HJ. 243, 303 Reiner, M. 54, 80, 104, 138, 141, 170, Parkinson, C. 149, 172 401, 414 Patel, R.D. 41, 49, 243, 273, 277, 278, Revankar, V.V.S. 262, 263, 309 305, 312, 334, 338, 391, 395, 411 Rhi-Sausi, J. 130, 135 Richardson, CJ. 98, 109 Paul, D.R. 75, 106, 243, 307, 312, 336, Richardson, J.F. 136, 151, 169 338, 394, 395, 413 Richardson, P.C.A. 113, 131, 136, 169 Payne, A.R. 38, 47 Pearson, J.R.A. 43, 52, 72, 106, 200, 240 Rideal, G.R. 292, 311 Rieger, J.M. 166, 174 Penwell, R.C. 103, 110 Riseborough, B. E. 16, 45 Petrie, CJ.S. 43, 52, 65, 73, 104, 106, Rodriguez, F. 41, 50, 243, 306, 312, 335, 130, 135 338, 392, 395, 412 Pett, R.A. 266, 268, 309 Roff, WJ. 1, 44 Peyser, P. 286, 310 Rogers, B.A. 136, 168 Philippoff, W. 100, 109, 113, 122, 124, Rogers, M.G. 126, 134 131, 134 Romanov, A. 384, 394 Pickthall, D. 36, 46 Rosen, M.R. 67, 79, 105 Pierce, P.E. 255, 308

Nishimura, T. 41, 48, 243, 246, 248, 249, 255, 257, 304, 308, 312, 333, 338, 346, 390, 391, 395, 410 Nissan, A.H. 71, 105 Nitanda, H.41, 51, 243, 307, 312, 336, 338, 393, 395, 413 Noll, W. 43, 52, 66, 67, 79, 104 Nomura, A. 194, 239 Noshay, A. 8, 44

Rosevear, J. 41, 49, 243, 305, 312, 334, 338, 391, 395, 411 Roteman, J. 38, 47 Rubio, J. 156, 173 Ruckenstein, E. 74, 106 Rudd, J.F. 100, 110 Rudraiah, N. 67, 79, 105 Rumpf, H. 272, 310 Runt, J.P. 41, 52 Rush, O.W. 30-33, 46, 243, 273, 304, 312, 333, 338, 390, 395, 410 Russel, W.B. 136, 170 Russell, RJ. 71, 105 Rutgers, LR. 138, 170 Ryan, M.E. 41, 51, 83, 107, 143, 171, 243, 250, 255, 257, 264, 269, 307, 308, 312, 317, 319, 321, 336-8, 345-8, 353, 355, 393, 395, 413

Scheffe, R.S. 265, 309 Scheiffele, G.W. 41, 51, 136, 154, 157, 168, 232, 236, 237, 241, 243, 266, 307, 312, 336, 338, 393, 395, 413 Schenkel, G. 207, 240 Schmidt, L.R. 41, 47, 123, 124, 134, 243, 304, 312, 333, 338, 345, 390, 395, 410 Schmitz, A.O. 125, 134 Schooten, J.V. 163, 173 Schott, H. 100, 110, 120, 133 Schowalter, W.R. 43, 52, 67, 79, 105, 121, 134 Schrader, M.E. 37, 46 Schramm, G. 67, 105, 113, 132 Schreiber, H.P. 72, 74, 105, 106 Schubert, H. 272, 310 Schultz, J.M. 1, 43 Schurtz, J.F. 41, 52 Saarnak, A. 136, 169 Schwartz, H.S. 20, 45 Sabia, R. 100, 110 Schwartz, R.T. 20, 45 Sabsai, O.Yu 14. 45, 117, 133, 319, 337 Scott, J.R. 1, 44 Sacks, M.D. 41, 51, 136, 154, 157, 158, Scott Blair, G.W. 43, 52 168, 173, 229, 232, 236, 237, 241, Sebastian, D.H. 194, 239 243, 266, 307, 312, 336, 338, 349, Segre, G. 150, 172 357, 370, 371, 393, 395, 413 Sellers, J.W. 18, 45 Saechtling, H. 1, 44 Sergeeva, L. M. 286, 310 Saini, D.R. 11, 12, 41, 44, 47, 50, 82, 87, Seshadri, S.G. 41, 51, 83, 107, 143, 171, 90, 92, 93, 100-2, 108-10, 115, 126, 243, 250, 255, 257, 264, 268, 269, 132, 134, 236, 237, 241-3, 251, 252, 307, 308, 310, 312, 317, 319, 321, 257, 258, 266, 273, 276-8, 284, 287, 336-8, 345-8, 353, 355, 393, 395, 290, 292, 300, 301, 306-8, 310-2, 413 330, 335, 336, 338, 340-3, 372, 373, Severs, E.T. 41, 47, 141, 171 Seyer, F.A. 165, 174 375, 381, 383-5, 392-5, 412, 413 Saito, N. 140, 171 Shaheen, EJ. 148, 172 Sakai, T. 41, 48, 243, 248, 249, 255, 257, Sharma, Y.N. 41, 49, 243, 273, 277, 278, 304, 308, 312, 333, 338, 346, 391, 305, 312, 334, 338, 391, 395, 411 395, 410 Shaw, H.M. 41, 51, 243, 266, 307, 312, Sakamoto, K. 118, 133 336, 338, 393, 395, 413 Salovey, R. 282, 310 Shenoy, A. V. 11, 12, 41, 44, 47, 50, 51, Sanchez, LC. 103, 110 81, 82, 86, 87, 90, 92, 93, 100-2, Sandford, C. 41, 48, 243, 273, 276-8, 107-10, 115, 126, 132, 134, 136, 292, 304, 312, 315, 333, 338, 390, 154, 157, 168, 229, 232, 236, 237, 395, 407, 410 241-3, 251, 252, 257, 258, 266, 273, 276-8, 284, 287, 290, 292, 299, Sarmiento, G. 113, 131 Sasahara, M. 41, 48, 115, 132, 142, 143, 300-2, 306-8, 310-2, 330, 335-8, 171, 243, 246, 248, 249, 255, 304, 340-3, 349, 357, 370-3, 375, 381, 312, 333, 338, 390, 391, 395, 410 383-5, 392-5, 412, 413 Shenoy, U. V. 81, 82, 86, 87, 107, 301, Saxton, R.L. 175, 196, 198, 237, 240 302, 311 Schaart, B. 214, 240

Sherman, P: 136, 149, 168, 172 Shete, P. 41, 49, 243, 273, 276, 277, 305, 312, 313, 319, 323, 334, 338, 391, 395, 411 Sheu, R.S. 41, 51, 136, 154, 157, 168, 232, 236, 237, 241, 243, 266, 307, 312, 336, 338, 393, 395, 413 Shida, M. 72, 105 Shinya, S. 194, 239 Shirata, T. 41, 49, 142, 143, 171, 243, 305, 312, 334, 338, 391, 395, 411 Silberberg, A. 150, 172 Simha, R. 138^1, 147, 170-2 Simon, R.H.M. 100, 110 Siskovic, N. 115, 124, 132 Skatschkow, W. W. 208, 240 Skelland, A.H.P. 67, 79, 105 Slattery, J.C. 153, 173 Smilga, V.P. 186, 238 Smith, J.H. 136, 167 Smits, C.T. 136, 169 Smoluchowsky, M. 163, 173 Snyder, J.W. 14, 45 Sobajima, A. 41, 51, 243, 266, 307, 312, 336, 338, 393, 395, 413 Somcynsky, T. 138, 139, 171 Soni P.L. 103, 110 Soppet, F.E. 266-8, 309 Southern, J.H. 75, 103, 106, 110 Spencer, R.S. 74, 106, 120, 133 Spriggs, T.W. 91, 92, 97, 108, 109 Spruiell, J.E. 86, 103, 108, 111, 316, 337 Stade, K. 189, 201, 238 Stamhuis, J.E. 41, 49, 143, 172, 243, 305, 312, 334, 338, 391, 395, 411 Steingiser, S. 104, 111 Stephenson, S.E. 130, 135 Sterman, S. 35, 46 Stevenson, J.F. 128, 135 Stober, W. 10, 44, 262, 263, 309 Stover, BJ. 141, 171 Studebarker, M. L. 14, 45 Suetsugu, Y. 41, 50, 243, 246, 273, 306, 312, 335, 338, 392, 395, 412 Sugerman, G.T. 41, 48, 243, 273, 275-8, 292, 304, 310, 312, 333, 338, 390, 395, 410 Sundstrom, D.M. 194, 239 Suzuki, S. 41, 51, 243, 307, 312, 336, 338, 393, 395, 413

Swanborough, A. 189, 190, 193, 238 Sweeny, K.H. 265, 269, 309 Szalanczi, A. 397, 414 Szczesniak, A.S. 136, 168 Tadmor, Z. 59, 104, 180, 183-5, 194, 196, 217, 237-40 Tadros, Th.F. 136, 154-6, 169, 173 Tager, A. 1, 43 Takahashi, M. 41, 51, 86, 108, 243, 307, 312, 336, 338, 393, 395, 413 Takano, M. 166, 174 Takserman-Krozer, R. 166, 174 Tammela, V. 41, 48, 243, 259, 304, 312, 333, 338, 390, 395, 398, 410, 414 Tan, C.G. 10, 44, 262, 263, 309 Tanaka, H. 41, 48, 49, 243, 244, 246, 247, 248, 268, 292, 305, 308, 310, 312, 313, 315, 317, 323, 334, 337, 338, 344, 391, 395, 396, 400, 403, 404, 407, 411, 414 Tanford, C. 1, 43 Tanner, R.I. 43, 52, 67, 72, 73, 79, 97, 104, 105, 106, 109 Taylor, R. 118, 133 Taylor, N.H. 292, 311 Teutsch, E.G. 41, 50, 279, 292, 310 Theberge, J. E. 16, 45 Thiele, J. L. 400, 414 Thomas, D.G. 138-42, 170 Thomson, J.B. 16, 45 Thurgood, J.R. 136, 168 Thurston, G.B. 136,167 Ting, A.P. 148, 172 Tiu, C. 81, 107 Tobolsky, A. V. 8, 44, 117, 133 Todd, D.B. 189, 208, 228, 238, 240 Tolstukhina, F.S. 14, 41, 45 Tomkins, K.L. 41, 51, 243, 266, 307, 312, 336, 338, 393, 395, 413 Tordella, J.P. 73, 75, 106, 120, 133 Travers, A. 22, 46 Trela, W. 41, 52, 266, 267, 309 Tremayne, P. 41, 49, 243, 305, 312, 334, 338, 391, 395, 411 Trementozzi, Q.A. 123, 124, 134, 397, 414 Trottnow, R. 72, 106 Trouton, F.T. 93, 109 Truesdell, C. 66, 67, 79, 104

Tsao, I. 88, 90, 108 Tsutsui, M. 103, 110 TuR, P. 113, 131 Turcotte, G. 136, 168 Turetsky, S. B. 226, 241 Turnbull, D. 103, 110 Tusim, M.H. 194, 239 Uebler, E.A. 79, 107 Uhland, E. 74, 106, 121, 133 Uhlherr, P.H.T. 81, 107, 113, 131 Umeya, K. 152, 173 Usagi, R. 82, 107 Utracki, L. A. 41, 43, 49, 52, 103, 110, 136, 170, 243, 305, 312, 334, 338, 392, 395, 411 Van Buskirk, P. R. 226, 241 VanderWeghe, T. 41, 49, 243, 273, 276, 277, 305, 312, 313, 319, 323, 334, 338, 391, 395, 411 Van Doren, R.E. 136, 168 Van Suijdam, J.C. 136, 168 Van Wazer, J.R. 43, 53, 113, 132 Vand, V. 140, 171 Veal, CJ. 136, 168 Vermeulen, J.R. 194, 239 Vermilyea, S.G. 113, 132, 136, 169 Verreet, G. 136, 169 Vervoorn, P.M.M. 136, 169 Vinogradov, G.V. 14, 41, 45, 47, 97, 109, 117, 128, 133, 135, 243, 307, 312, 319, 337, 338, 394, 395, 413 Virbsom, L. 217, 240 Vlachopoulos, J. 194, 239 Volpe, A.A. 37, 47 Von Mises, R. 401, 414 Wagner, A.H. 395, 409 Wagner, M.H. 84, 85, 108, 343, 394 Wagnes, M.P. 18, 45 Wales, J.L.S. 117, 121, 124, 133, 134 Walk, C. 215, 240 Walker, J. 66, 104 Wall, D.R. 136, 168 Walters, K. 41, 43, 47, 52, 53, 67, 71, 79, 84, 105, 107, 113, 116-8, 132 Wang, R.H. 179, 237 Warburton, B. 160, 173 Ward, S.G. 148, 150, 172, 264, 309

Warren, R.C. 113, 132, 136, 169 Wasiak, A. 103, 110 Waston, W.F. 18, 45 Watson, C.A. 179, 237 Watson III, J.G. 194, 239 Weidenbaum, S.S. 180, 237 Weill, A. 74, 106 Weinberger, C.B. 165, 174 Weiss, Y. 8, 44 Weissenberg, K. 71, 105, 115, 121, 132, 133 Werner, H. 208, 240 Westman, A.E.R. 266, 267, 309 Westover, R.F. 299, 311 Whalen, TJ. 266, 268, 309 Whelan, J.P. 41, 51, 292, 299, 311 White, J.L. 14, 41, 45, 47-50, 75, 76, 85, 86, 103, 106-8, 111, 115, 118, 120, 124, 130, 132-5, 163, 166, 173, 243, 244, 246-8, 273, 277, 292, 304-6, 308, 312, 313, 315-7, 319, 323, 331, 333-5, 337, 338, 344, 390-2, 395, 396, 400-5, 407, 410-2, 414 Whiting, R. 113, 131, 136, 169 Whitmore, R.L. 148, 150, 172, 264, 309 Whittaker, R.E. 38, 47 Whorlow, R. W. 43, 53, 113, 132, 154, 173 Wigotsky, V. 200, 240 Wildemuth, C.R. 143, 172 Wilkinson, W.L. 67, 79, 104 Willard, H. 136, 167 Willermet, P.A. 266, 268, 309 Williams, D.J.A. 136, 169 Williams, M.C. 136, 143, 168, 172 Williams, M.L. 99, 109 Williams, R.M. 41, 52, 266, 267, 309 Willmouth, P.M. 103, 110 Winning, M.D. 151, 173 Winter, H.H. 131, 135 Wissbrun, K.F. 41, 47, 98, 109, 299, 311 Withers, V.R. 136, 168 Wolf, R.F. 18, 45 Wong, W.M. 41, 49, 243, 273, 306, 312, 335, 338, 360, 361, 387, 388, 392, 394, 395, 412 Wood, R. 189, 238 Woodthorpe, J. 41, 51, 223, 224, 241, 243, 266, 307, 312, 336, 338, 393, 395, 413

Wright, B. 100, 109 Wright, C.H. 139, 171 Wu, S. 41, 48, 243, 292, 304, 312, 333, 338, 391, 395, 410 Wyman, C.E. 208, 240 Yamada, M. 299, 311 Yamashita, S. 41, 49, 243, 273, 305, 312, 334,338,391,395,411 Yanovsky, Yu.G. 41, 51, 243, 308, 312, 337, 338, 394, 395, 414 Yao, J. 215, 240 Yazici, R. 395, 409 Yoo, HJ. 41, 48, 243, 273, 276-8, 292, 304, 312, 315, 333, 338, 390, 395, 407, 410

Young, C.C. 194, 239 Youngblood, E.L. 136, 168 Zahorski, S. 43, 52 Zaikov, G.E. 41, 51, 243, 308, 312, 337, 338, 394, 395, 414 Zakharenko, N.V. 14, 41, 45, 285, 310 Zapas, L. 72, 85, 106 Zhao, G.Y. 262, 263, 309 Ziabkki, A. 103, 110, 166, 174 Ziegel, K.D. 147, 172, 384, 394 Ziemanski, L.P. 35, 46 Zingel, U. 41, 48, 243, 292, 304, 312, 333, 338, 390, 395, 410 Zisman, W.A. 19, 45

Index

Index terms

Links

A Activation energy

100

Alumina filled polyethylene

349 388

353

358

362

370

286

374

377

381 377

Amorphous polymer definition Antistatic agent

5 23

Aramid fiber filled polystyrene

245

Arrenhius-Eyring equation

99

Azidosilanes

26

314 28

B Bagley correction

121

Barium sulfate filled polyethylene

274

Barium ferrite filled polyester elastomer

253

polyethylene

253

polyurethane thermoplastic elastomer

253

styrene-isoprene-styrene block copolymer

254

277

286

374

381

384

385

387


469

470

Index terms Batch mixers

Links 189

Biaxial extension

64

extensional viscosity

65

Blending definition

175

Block copolymer

7

Branched polymer

3

C Calcium carbonate filled polyethylene

249

274

293

polypropylene

249

259

274

276

294

319

325

328

245 407

295

314

324

397

314

polystyrene

282

Carbon black filled polyethylene

287

polystyrene

245

260

283

295

316

397

401

404

Carbon fiber filled polyethylene Cauchy stress tensor

249 57

Cellulose fiber filled polystyrene

245

Central tube effect

150

Characteristic time

54

Chemical bonding theory

37

Chemical additives Chlorinated paraffins

314

160 28


471

Index terms

Links

Chrome complexes

25

Commodity plastics

6

Complex viscosity

60

Complex modulus

61

Compounding definition

175

process tasks

188

techniques

175

variables

221

chamber loadings

231

mixer type

223

mixing temperature

232

mixing time

225

order of ingredient addition

236

ram pressure

229

rotor geometry

224

rotor speed

229

Constitutive equation

57

Constrained recoil

62

Contact angle

19

Continuous compounders

20

192

Copolymer definition

7

Coupling agent characteristics

29

mechanism

35

Cox-Mertz method or rule

86

Creep compliance

62

Creep

61


472

Index terms

Links

Crystalline polymer definition

5

D Deborah number definition Deformable layer theory

54 37

Dilatant behavior

152

Dispersive mixing

184

Distributive mixing

183

Dolomite filled polyethylene

262

Draw resonance

73

Dump criteria

218

Dynamic loss modulus

60

storage modulus

60

viscosity

60

E Effect of filler agglomerates steady shear elastic

321

viscous

272

unsteady shear viscoelastic

356

Effect of filler concentration extensional

402

steady shear elastic

317


473

Index terms

Links

Effect of filler concentration (Continued) viscous

248


345

Effect of filler size extensional

400


315

viscous

246


344

Effect of filler size distribution steady shear elastic

321

viscous

262


350

Effect of filler surface treatment extensional

405


323

viscous

273


360

Effect of filler type extensional

396


313

viscous

244

unsteady shear


474

Index terms

Links

Effect of filler type (Continued) viscoelastic

344

Effect of matrix additives unsteady shear viscoelastic

387

Effect of polymer matrix steady shear elastic

330

viscous

279

unsteady shear viscoelastic Einstein equation

372 138

Elastic solid description

39

Elastic energy

122

54

Elastomers definition

2

examples

2

Electroviscous effect

163

Ellipsoidal particles

143

6

Engineering thermoplastics examples

7

Entry flow

75

Extension biaxial

64

planar

65

uniaxial

62

Extensional viscometer extrusion method

130


475

Index terms

Links

Extensional viscometer (Continued) filament stretching

128

Extensional viscosity biaxial

65

planar

65

uniaxial

65

Extensional rate biaxial

64

planar

65

uniaxial

63

Extensional flow

55

Extra stress tensor

57

Extrudate swell

71

62

65

164

F Filled polymer definition

11

examples

13

13

Filled polymer rheology definition

54

Filler cost-effectiveness

15

definition

9

ellipsoidal

10

11

fibrous

10

11

flexible

10

geometry

19

inorganic

11

organic

11

platelike

10

11


476

Index terms

Links

Filler (Continued) rigid

10

selection

13

spherical

10

surface

19

surface treatment

21

types

10

Filler-polymer interactions

16

Finishes

22

Flocculation

11

17

151

Flow curve ideal unfilled

40

filled

42

Flow curve

40

entry

75

extensional

55

shear

55

42 62

Fluids Newtonian

66

67

77

non-Newtonian

66

77

78

pseudoplastic

67

68

thixotropic

67

68

70

viscoelastic

67

77

78

245

314

123

327

78

Franklin fiber filled polystyrene

G Glass bead/sphere filled polypropylene

340


477

Index terms

Links

Glass bead/sphere filled (Continued) polystyrene

245

314

Glass transition temperature definition

5

Glass bead/sphere filled polybutene

256 355

270

318

322

styrene acrylonitrile

123

thermoplastic polymer

251

256

318

347

nylon

280

296

330

polycarbonate

261

281

298

polyether ether ketone

300

polyetherimide

281

299

polyethylene

249

408

polyethylene terephthalate

261

280

297

polypropylene

187

249

329

polystyrene

245

295

314

347

Glass fiber filled

Graft copolymer

331

317

7

H Homopolymers Hooke's law in shear

6 122

I Intensity of segregation

181

Interface definition

16

Interfacial energy

20


322

478

Index terms

Links

Interfacial (Continued) region Internal mixers

16 190

Interphase definition

16

L Liquid crystalline polymer

9

Loss tangent

60

Lubricant

23

M Matrix additives examples

43

Melt flow index

127

Melt flow indexer

126

Melt fracture

73

Melting temperature definition

5

Mica filled polyethylene

263

polypropylene

275

294

polystyrene

245

314

Mixing definition

175

goodness

175

index

179

mechanisms

183

temperature

232


479

Index terms

Links

Mixing devices in single-screw systems

198

dispersive

184

distributive

183

pins

198

sections

199

time

225

variables

221

Modulus complex

61

dynamic loss

60

dynamic storage

60

Molecular weight description

39

distribution

39

N Network polymer

3

4

behavior

77

78

description

66

generalized

67

Newtonian fluids

Non-Newtonian fluids behavior

77

78

description

66

67

Normal stress components

58

difference cone and plate definition

117 59


480

Index terms

Links

Normal stress (Continued) parallel-disk

118

Nylon glass fiber filled

280

296

330

270

318

322

261

281

298

331

barium fernte filled

254

286

374

377

unfilled

285

380

O Open mills

189

P Packing of filler

139

Particle size distribution

147

Particle surface effect

150

Planar extension

65


65

Plasticizer

23

Plate separation

77

Poly (vinyl chloride) fine particles filled

103

Polybutene glass bead/sphere filled

256

347

355 Polycarbonate glass fiber filled Polyester elastomer

Polyether ether ketone glass fiber filled

300


381

481

Index terms

Links

Polyetherimide glass fiber filled

281

299

alumina filled

349 388

353

358

362

370

barium ferrite filled

253

286

374

377

381

barium sulfate filled

274

286

calcium carbonate filled

249

274

293

359

carbon black filled

287

carbon fiber filled

249

dolomite filled

262

glass fiber filled

249

mica filled

263

quartz filled

258

silica filled

359

silicon filled

89

Polyethylene

steel sphere filled

340

talc filled

262

titanium dioxide filled

320

unfilled

285

zirconia filled

354

408 293

353

293

380

261

280

296

27

28

Polyethylene terephthalate glass fiber filled Polymeric esters Polymerization addition

1

condensation

1

Polypropylene calcium carbonate filled

249

259

274

276

294

319

325

328


282

482

Index terms

Links

Polypropylene (Continued) glass bead/sphere filled

123

327

346

glass fiber filled

187

249

294

329

mica filled

275

294

talc filled

249

294

unfilled

294

Polystyrene aramid fiber filled

245

314

calcium carbonate filled

245

295

314

324

397

283 401

295 404

314

314

317

332

295 401

314 406

316

407 carbon black filled

245 316

260 397

cellulose fiber filled

245

314

Franklin fiber filled

245

314

glass bead filled

245

314

glass fiber filled

245

295

mica filled

245

314

titanium dioxide filled

245 320

260 397

unfilled

403

Polyurethane thermoplastic elastomer barium ferrite filled Pressure hole error

253 77

Purely elastic solid definition

54

Purely viscous liquid definition

54


483

Index terms

Links

Q Quartz filled polyethylene

258

293

R Rabinowitsch-Weissenberg correction Random copolymer Residence time

121 7 227

Resitols

4

Resols

4

Restrained layer theory

37

Reversible hydrolyzable bond theory

38

38

Rheological models Carreau

81

complex viscosity

86

dynamic modulus

90

Ellis

80


93

general

81

Herschel-Bulkley

83

modified Carreau

86

modified Herschel-Bulkley

83

normal stress difference

84

power-law

80

steady shear viscosity

79

87

Rheology definition

39

Rheometer (see also Viscometer) capillary type

113

definition

112

114

118


484

Index terms

Links

Rheometer (Continued) rotational-type

113

types

114

Rheometry definition

12

Rod-shaped particles

144

Rotational viscometer

113

S Scale of segregation

181

Scale of scrutiny

183

Segregation intensity

181

scale

181

Self-cleaning time

228

Semi-crystalline polymer definition

5

Shear flow extensional

55

steady simple

55

unsteady simple

55

59

Shear stress capillary components

124 58

Shear rate capillary

124

cone and plate

116

definition Silanes

56 25


485

Index terms

Links

Silica filled polyethylene

359

Silicon filled polyethylene

89

Single screw extruders conventional

193

modified

198

Single screw kneaders Small-amplitude oscillatory flow

201 59

60

Specialty polymers examples Spherical particles Steady state compliance Steady shear viscosity unification Steady simple shear flow

7 137 86 287 55

Steel sphere filled polyethylene Stick-slip phenomenon Stress relaxation

340

353

121 61

Stress growth

61

normal

58

relaxation

61

shear

56

58

181

196

Striation thickness Styrene acrylonitrile glass bead filled

123


486

Index terms

Links

Styrene-isoprene-styrene block copolymer barium ferrite filled

254 381

277 384

unfilled

285

380

Surface wettability theory

286 387

374

37

Surface modifiers effects

160

examples

26

mechanism

35

27

silane treated alumina

362

barium ferrite

277

calcium carbonate

325

glass beads

327

mica

275

suggested

34

367

294

titanate treated alumina

362

barium ferrite

277

calcium carbonate

276

glass fibers

329

366 294

328

Surface treatment effect

30

method

24

25

Suspension concentrated

139

definition

136

dilute

137

rheology

136


377

487

Index terms

Links

T Talc filled polyethylene

262

polypropylene

249

294

251

256

Thermoplastic polymer glass/sphere filled

318

347

Thermoplastics definition

1

examples

2

Thermosets definition

2

examples

2

Time characteristics residence scale of deformation self-cleaning time Titanates

54 227 54 228 26

28

Titanium dioxide filled polyethylene polystyrene

320 245

260

295

314

320

397

401

406

Trouton viscosity

95

Tubeless siphon

78

Twin screw extruders classification

202 205

intermeshing co-rotating counter-rotating

207 21

non-intermeshing This page has been reformatted by Knovel to provide easier navigation.

316

488

Index terms

Links

Twin screw extruders (Continued) counter-rotating Two-roll mill

214 189

U Uebler effect

79

Uniaxial extension

62


65

Unification of steady shear viscosity Uniform copolymer

287 7

Unsaturated polyester clay filled

274

mica filled

274

silica filled

274

talc filled

274

wollastonite filled

274

Unsteady simple shear flow

55

V Viscoelastic fluid behavior

77

Viscoelastic effects

71

78

Viscoelasticity description

54

Viscometer (see also Rheometer) cone and plate

115

extensional

128

parallel-disc

117

Viscometric functions

59


489

Index terms

Links

Viscosity complex

60

definition

59

dynamic

60

extensional

65

function

59

shear

57

58

39

54

Viscous liquid description

W W-L-F equation

99

Wall slip (see Stick-slip phenomenon) Wall effect

150

Weissenberg effect

71

Wettability

19

Wetting

185

Wollastonite filled polyethylene

274

polypropylene

87

88

94

Z Zirconia filled polyethylene

354


Rheology Of Filled Polymer Systems

Rheology of Filled Polymer Systems

Particulate-Filled Polymer Composites,

Polymer and Composite Rheology

Rheology for Polymer Melt Processing (Rheology Series)

Rheology and Processing of Polymeric Materials: Volume 1: Polymer Rheology

Rheology for Polymer Melt Processing

Rheology and Processing of Polymeric Materials: Polymer Rheology

Polymer Characterization: Rheology, Laser Interferometry, Electrooptics

Polymer and Composite Rheology, Second Edition

Filled Elastomers Drug Delivery Systems

Handbook of Multiphase Polymer Systems

Multicomponent Polymer Systems

RHEOLOGY AND PROCESSING OF POLYMERIC MATERIALS Polymer Processing

Photonic Polymer Systems

Photonic Polymer Systems

Polymer-surfactant systems

Rheology of the Earth

Polymers, Polymer Blends, Polymer Composites And Filled Polymers: Synthesis, Properties, and Applications

Principles of polymer systems, Volume 1

Handbook of Elementary Rheology

Polymer Melt Rheology: A Guide for Industrial Practice

Polymer Characterization: Rheology, Laser Interferometry, Electrooptics (Advances in Polymer Science, Volume 230)

Handbook of elementary rheology

Rheology of Complex Fluids

Fluid Mechanics of Viscoelasticity (Rheology Series) (Rheology Series)

Rheology Fundamentals

Polymer Synthesis Polymer-Polymer Complexation

Computational Rheology

Rheology: An Historical Perspective (Rheology Series)

Filled With The Spirit

Rheology Of Filled Polymer Systems