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

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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 ..........................................................................

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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

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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 ..........................................................................

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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

7.2 Effect of Filler Size .......................................................

315

7.3 Effect of Filler Concentration .......................................

317

7.4 Effect of Filler Size Distribution ....................................

321

7.5 Effect of Filler Agglomerates .......................................

321

7.6 Effect of Filler Surface Treatment ................................

323

7.7 Effect of Polymer Matrix ..............................................

330

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

332

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

344

8.2 Effect of Filler Size .......................................................

344

8.3 Effect of Filler Concentration .......................................

345

8.4 Effect of Filler Size Distribution ....................................

350

8.5 Effect of Filler Agglomerates .......................................

356

8.6 Effect of Filler Surface Treatment ................................

360

8.7 Effect of Polymer Matrix ..............................................

372

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x

9.

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

387

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

390

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

396

9.2 Effect of Filler Size .......................................................

400

9.3 Effect of Filler Concentration .......................................

402

9.4 Effect of Filler Surface Treatment ................................

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

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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)

<2->

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+^I) 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, 7^APdie = 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 <£3 term. However, the values of the coefficients of higher order terms are less accurate since they must include more complicated interactions than it is theoretically possible to treat. Based on the work of Eyring et al. [96] who suggested that the term OC3^3 in equation (4.5) should be of the exponential form, Thomas [75] arrived at the following expression r\r = I + 2.5(1) +10.052 + 0.00273 exp(16.6<£)

(4.6)

Thus a closed form expression is obtained which fits experimental data as well as the power series form with three or four adjustable coefficients. The validity of expression (4.6) is for a filler concentration varying between 0.15 and 0.60. Another power series type of expression is given by Ford [77], who added higher order terms to his expression (4.4) for dilute suspensions, making use of the data of Vand [83], to give - = 1 - 2.50 + 11<£5 - 11.57

(4.7)

*ir

He suggested that the inclusion of the term 5 takes into account the onset of the inhibition of particle rotation, and the term ^7 takes into account the onset of particle interlocking. Mooney [97] proposed that at very high concentrations an Arrhenius type equation (with the addition of crowding factor (1 — ac)) of the following form could be used: *-«*&]

where ac has a value between 1.35 and 1.91. Mooney's relationship has been known to be in good agreement with experimental data ([66], [98]). The empirical expression (4.3) as suggested by Simha [74] can also be used for concentrated suspensions if the relationship for A() given by him is not approximated and all the higher order terms in are retained. Simha's analysis was based on the idea that the neighbors of each sphere in the suspension can be replaced by a rigid spherical enclosure, and that the finite size of the particles effectively shields the central particle from interaction with any other particle than the nearest neighbors as the concentration is increased. For concentrations up to m, experimental data are seen to validate the following empirical expression obtained by Eilers [99] * r = [ l + n 1^ J L (1 - 0/0m)J

Note that for -> 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 to obtain reliable estimates. Though equation (4.10) predicts the correct experimental trends at large values of $ -> 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 - <£/Apr2

(4.12b)

The adjustable parameter Ap is now considered to give some measure of the thickness of the immobilized polymer adsorbed on the filler surface and thereby indicates the affinity of the polymer for the filler [108]. B. Ellipsoidal particles The theory of the viscosity of dilute ellipsoidal suspensions without Brownian motion was developed by Jeffrey [109]. He observed that Einstein's equation (4.1) could be used to estimate the viscosity by appropriately defining the Einstein constant aE. Jeffrey [109] has tabulated the values of aE for prolate and oblate spheroids. He found that aE depended on the ratio of the semi-axes of the ellipsoid of rotation. For a prolate spheroid, Guth [110,111] developed the following relationship based on Jeffrey's theory [109] when the axis ratio re ^> 1. Thus, >7r = l + a rl 0 + ar22

(4.13)

where «"=<2tafe>) + 2

<4

"a)

-01& Pokrovskii [112] demonstrated theoretically that concentrated suspensions of solid ellipsoidal bodies in a Newtonian fluid give rise to a viscoelastic behavior. He showed that for such suspensions it is possible to use the concept of transverse viscosity which expresses the effect of normal stresses and found that the transverse viscosity increases with velocity gradient.

C. Rod-shaped particles Suspensions of rod-shaped particles have received greater attention than ellipsoidal particles due to the obvious increasing demand for fiber-reinforced plastics. Exact theories exist for the viscosity of very dilute suspensions of rod-shaped particles. However, this is not the case for higher concentrations. The reason is evident from the fact that differences in viscosities of suspensions of different shaped particles are small at low concentrations. With an increase in the concentration, particle rotation causes the frequency of contacts between adjacent particles to increase, resulting in an increase in the viscosity (Figure 4.1). For rod-shaped particles, the above effect is accentuated with an increase in the aspect ratio as can be seen from the work of

GLASS RODS GLASS PLATES QUARTZ GRAINS GLASS SPHERES EINSTEIN'S EQUATION

Figure 4.1 Effect of concentration on the relative viscosity of different shaped particles in water at a shear rate of 327.7 sec~1. (Reprinted from Ref. 81 with kind permission from The Institution of Chemical Engineers, Rugby, UK.)

Maschmeyer and Hill [113] who showed that the presence of only a few percent of relative long fibers in the particle distribution doubled or tripled the suspension viscosity. From the work of Clarke [81] shown in Figure. 4.1, it can be conclusively seen that the viscosity of a suspension is higher for the same concentration of the particles for larger departures in shapes from spheres. In the case of dilute suspensions of rod-shaped particles, too, Einstein's equation (4.1) can be used with the appropriately defined constant <XE. Burgers [114] has established the following relationship for straight rigid rod-shaped suspensions under shear

^^l-1.8)] ^^

(415)

where ra is the aspect ratio (/P/D) and 6 and \l/ are the spherical coordinates giving the orientation of the rod (see Figure 4.2). Burgers [114] found that rod-shaped particles (and also elongated ellipsoidal particles) cause an increase in the Einstein's coefficient with increasing aspect ratio (ra) as shown in Figure 4.3, for randomly oriented particles such as would occur at very low rates of shear. At high shear rates, of course, orientation of the rods would result in a decrease in the effective value of the Einstein coefficient.

Figure 4.2 Coordinate system for a rigid rod in a field of simple shear.

Figure 4.3 Einstein's coefficient as a function of the aspect ratio of randomly oriented rod-shaped particles,

For dilute suspensions in rotational motion, Mason [115] gave the following expression for critical concentration $cr below which the interaction between the rods can be neglected. Thus, 0 c r -¥ ^a

(4.16)

Blankeney [116] observed that below this critical concentration the relative viscosity of a rod-shaped particle suspension increases slowly and linearly with $. A slight non-linear region then follows after which the relative viscosity increases rapidly. Experimentally, it was found that the value of (/>cr depended .upon the nature of the solid phase. For example, Clarke [81] found that for glass rods cr = 0.15 and for PMMA spheres 0cr = 0.40. For concentrated suspensions of rod-like particles (i.e. for <j> > 0cr),

the following relationship was suggested by Simha [117] and later verified by Blankeney [116] >7r = l + aE + a(aE(/>)2

(4.17)

For randomly oriented rods with purely hydrodynamic interactions between them, Simha [117] determined the value of a to be equal to 0.73. For concentrated fiber suspensions in tube flow, Brodnyan [118] suggested the following equation:

,, =expp+ o^.-i)"Vj

(418)

but the experimental data of Brodnyan [118] themselves were not found to fit for high concentration of suspension. Taking into account the particle interaction and the degree of flocculation, Ziegel [119] proposed the following equation:

3

-( f^>--)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)

<-«fi?T®r

« _ Js1^nS f U \

Io

/

\

(4.23,

r/

Parkinson et al [126] arrived at a Mooney type equation for the relative viscosity of a dispersion of polydisperse spheres containing / size fractions as /

\

/ 2.5^1 \

/ 2.502 \

/ 2.50f \

"4E) -Mi^Mi=Iy-"(T=W

,,^,

<4 24)

'

where 015 0 2 ... 0f are the volume concentrations of each size fraction, and the values of /C1, fc2... fcz are derived from an empirical equation of the following general form Ic1 = 1.079 + exp(0.01008/Dp/) + exp(0.0029/D*,)

(4.25)

The above equation (4.24) is not suitable for a continuous distribution of particle diameters and the equation given by Mooney [97] has to be then resorted to: rjr = exp (2.50 T

Af"'' ) Ja1 1-0T 1 ^d(T7./

V

(4.26)

where 0 is the total volume concentration of spheres, a;l the crowding factor, Pf the particle size frequency function and alf O2 the upper and lower limits of the dimensionless radius, respectively. The two main difficulties in the use of equation (4.26) are that the nature of oc;j is known only in special cases and hence the distribution function to accurately describe the particle size distributions of different grades of the same filler is almost impossible to find. Hsieh [127] suggested a quadratic model of the following canonical form for an M-component system of particle size distribution. /M

M

\

Vr = exp ( Y, Bixt + £ V<*;) V i=l

Ki^j

(4-27>

'

where x{ is the weight proportion of the zth component in the blend of the total number of M components. The coefficients B1 and B^ were determined experimentally and for a tetramodal size distribution, Hsieh [127] has given the derived expression from equation (4.27) which was seen to agree well with the experiments for four basic aluminum trihydroxides with distinctively different particle sizes in an unsaturated polyester resin.

4.2.3 EFFECT OF THE NATURE OF THE PARTICLE SURFACE

Moreland [124] obtained viscosity data of suspensions of coal in mineral oil during which he concluded that slurry viscosities for irregular particles were greater than for spherical particles. A similar result was obtained by Clarke [81], who showed that suspensions of sharp-edged particles have a much larger viscosity than those of round-edged ones. Roughened particles are likely to increase the viscosity of suspensions in two ways: (a) primarily by harsh frictional contact and (b) secondarily, by trapping layers of inert liquid on its surface thereby causing an effective increase in the concentration as suggested by Ward and Whitmore [128]. 4.2.4

EFFECT OF THE VELOCITY GRADIENT

In the flow of suspensions, velocity gradient is known to cause two major effects, (a) migration of the particles towards the centre of the axis in a tube flow situation and (b) rotation of the particles. A. Wall effect and the central tube effect In pipe flow of a suspension, the presence of a wall and a velocity gradient causes the particles to migrate towards the tube axis and correspondingly decrease the concentration of the solid phase near the wall. This effect known as the wall effect has been observed by Goldsmith and Mason [129] for suspensions of different shaped particles in Poiseuille flow. A consequence of the wall effect is the central tube effect wherein the migration of the particles towards the tube axis causes an increase in the solid phase concentration within a narrow region of about 0.6.R1 (where R( is the internal radius of the tube). The wall effect and the central tube effect appear simultaneously and after a certain time from the start of the flow due to the velocity gradient being set up. Segre and Silberberg [130,131] have shown that these effects manifest themselves only at small concentrations of the solid phase. These effects have been taken into account during the estimation of the suspension viscosity of PMMA spheres in a ternary mixture of water, glycerol and 1.3butanediol by Segre and Silberberg [132]. A detrimental effect in the measurement of properties of a suspension, other than the viscosity, could result due to the presence of the wall and the central tube effect. Care should be taken during property measurements with probes through tube walls as the measured quantities could easily be the properties of the suspending medium rather than of the suspension. The migration of the particles towards the tube axis due to the

velocity gradient also leads to a characteristic slippage at the wall surface (Morrison and Harper [133]). A direct consequence of the slip is the increased flow rate through the tube compared to that which would occur if slip were absent. The value of shear rate calculated from the famous Mooney-Rabinowitsch equation will, therefore, be erroneous and a correction ought to be made as given by Jastrzebski [134]. B. Particle rotation At high concentrations of the solid phase, the existence of a velocity gradient results in rotation of the particles, followed by an increase in the viscosity of the suspension. Concentrated suspensions are known to exhibit non-Newtonian behavior and hence viscosity would change as the shear rate (velocity gradient) changes. It is generally assumed that the viscosity dependence on shear rate is due to some structural changes in the suspensions such as the breaking up of agglomerates during shear. For example, Chapman and Lee [135] as well as Goel [136] found that, at low shear rates (y < lsec"1) talcfilled polypropylene was more viscous than the base propropylene, but such was not the case at higher shear rates (7 > lsec"1) wherein break up of the network-like structure took place, resulting in the viscosity of the talc-filled systems equivalent to that of propropylene alone. Similar behavior was observed by Ferraro [137] for other systems. Concentrated suspensions often exhibit a yield stress below which no shear deformation takes place and after which the suspension could behave as a Newtonian, pseudoplastic or dilatant fluid. Shear dependent theories for such flows have been considered by Krieger and Dougherty [138] as well as Gillespie [139]. 4.2.5 EFFECT OF FLOCCULATION

The degree of flocculation of dispersed particles strongly influences the rheology of suspensions ([14O]). The viscosity of the suspension is known to increase sharply with flocculation, probably due to the relative immobilization of a fraction of the suspended particles trapped in the agglomerates. In shearing flow, both shear-thinning and shearthickening behavior have been observed. Govier and Winning [141] found concentrated clay suspensions to be shear-thinning. Heywood and Richardson [45] found that flocculated kaolin suspensions were shear-thinning and could be characterized by the power-law model at shear rates below 200OSeC"1. With increasing solid concentration, the flow behavior index n was seen to fall progressively while the consistency index K was seen to increase exponentially. At shear rates

above 2000 sec l, the behavior would be characterized by the Bingham plastic model with the yield stress and the plastic viscosity increasing monotonously with increasing solid concentration. Dilatant behavior is observed in certain flocculated systems with extremely high concentration of suspensions. Umeya and Kanno [142] found that concentrated suspensions of titanium dioxide in water stabilized with sodium pyrophosphate showed dilatancy. The effect of shear history on the dilatant behavior is shown in Figure 4.5, and is seen to be pronounced at high shear rates. A suspension deflocculated by shear does not recover its structure immediately after the removal of shear since the thermal energy of the particles is insufficient to overcome the energy barrier for the flocculation. As the degree of flocculation is decreased by applying high shear rates, the onset of dilatant flow occurs, the dependence of apparent viscosity on shear rate and the increase in the apparent viscosity decreases. Umeya and Kanno [142] also found that dilatant behavior of flocculated systems did

UNTTS

Water Ctaatase, 4 »0.4 *)

Shear Siren prrooudy appSed

Figure 4.5 The effect of shear histories on the flow properties of titanium dioxide (TiO2) water 27.7% suspensions stabilized with 5mg/g TiO2 of Na4P2O7-IOH2O. (Reprinted from Ref. 142 with kind permission from Society of Rheology, USA.)

depend upon the size of the particles as the degree of dispersion of larger particles was better than that of smaller ones. Hudson et al. [31] studied the time dependent effect of suspensions of pigment particles. Experimental work was undertaken to relate steady, transient and time dependent oscillatory flow in a unified manner. Hudson et al. [31] also developed a theory to explain the suspension behavior in terms of the structure formed as a result of flocculation. The viscosity rjf of a homogeneous suspension of floccules containing NJ floccules was given by In^ = f(0 f ) + vIn(N{) >7o

(4.28)

where r]0 was the viscosity of the suspending medium, f was a function of the volume fraction ($f) of the floccules in the suspension and v as a constant. A comparison of the experimental data with the theory enabled values of the model constants to be calculated. 4.2.6 EFFECT OF THE SUSPENDING MEDIUM

The nature of the suspending medium, whether Newtonian, viscous non-Newtonian, viscoelastic or viscoplastic affects the viscosity of the suspension. This has been the subject of research of a number of workers. Nicodemo et al. [143] studied the shear rate dependence of the viscosity of suspensions in non-Newtonian liquids and compared it with that of suspensions in Newtonian liquids. The relative viscosity, when the suspending medium was Newtonian, decreased with increasing shear rate toward an asymptotic value which was a function of the filler content. The concentration dependence could be correlated by equations given in section 4.2.1. When the suspending medium was non-Newtonian, Nicodemo et al. [143] found that the dependence of relative viscosity on shear rate was similar and could be correlated by a modified form of the equation of Krieger and Dougherty [138]. However, as the relative viscosity at high shear rates did not fit the existing equation, Nicodemo and Nicolais [144] searched for a mechanistic explanation. It was found that two different mechanisms in the range of low and high shear rates, respectively, existed for suspensions in polymeric solutions. When the suspending medium could be described by the power-law model, Kremesec and Slattery [145] derived the following expression for the viscosity of a dilute suspension of spheres If8 = (U-O0^)Ky"-1

(4.29)

The coefficient a0 was defined by the rate of energy dissipation within

the neighborhood of a typical sphere in the suspension. For n = \, a0 = 2.5 could be easily derived as in Einstein's equation (4.1). However, for n^\, the equations of motion, being nonlinear, are difficult to solve and hence only the upper and lower bounds for a0 as a function of n were obtained and compared with the available experimental data of Highgate and Warlow [146] to find reasonable agreement. 4.2.7

EFFECT OF ADSORBED POLYMERS

Polymers are often added as processing additives to suspensions used in ceramic forming operations [19]. Under certain conditions, the polymers adsorb on ceramic particles and alter the interparticle forces which control the state of dispersion and rheological properties of the suspension. An important criterion for achieving good particulate dispersion through steric stabilization is to ensure good coverage of the particle surfaces with the adsorbed polymer [147,148]. With low adsorption densities, the steric repulsive forces are likely to be too weak to prevent close approach of particles, thereby leading to flocculation due to van der Waals attractive forces as well as bridging flocculation due to polymer segments from one particle surface attaching themselves to the available surface site of another particle. Sacks et al. [19] studied the amount of polymer adsorbed and the state of particulate dispersion in aqueous silica suspensions containing different concentrations of poly (vinyl alcohol) (PVA). An adsorption isotherm for suspensions of 20 vol% silica at pH = 3.7, wherein the zeta potential for the chosen silica is zero, is shown in Figure 4.6. It can be seen that initial additions of PVA are almost completely adsorbed on the silica particles and very little polymer remains in solution. When overall PVA concentration is increased, the particle surface tends to become saturated with adsorbed polymer and then higher amounts of residual polymer are left in solution. The initial plateau region of the adsorption isotherm is associated with approximately monolayer coverage of the particle surface. The second plateau is probably an indication of either the development of denser packing of polymer molecules in the adsorbed monolayer or the development of multilayer adsorption. Figure 4.7(a) shows the relative viscosity versus shear rate behavior for suspensions at pH = 3.7 with varying polymer adsorption in the range of O to 1.1 mg adsorbed PVA/g silica. For the sake of comparison, the relative viscosity versus shear rate plot is also shown in the same figure for an electrostatically stabilized suspension prepared at pH = 7.0 with no PVA. At pH = 7.0, wherein the zeta potential is approximately — 55 mV, the silica suspension is extremely well

ADSORBED AMOUNT (mg PVA/g Silica)

20 vol% SHIcf

RESIDUAL POLYMER CONCENTRATION
dispersed judging from its low relative viscosity and Newtonian behavior over the entire shear rate range. In contrast, at pH = 3.7 when the zeta potential is approximately OmV, the silica suspension with no PVA is flocculated, adjudging from the high viscosity and highly shearthinning behavior. In Figure 4.7(a), it can be seen that initial additions of PVA result in higher viscosity and greater shear-thinning characteristics, thus indicating the occurrence of bridging flocculation at the low PVA adsorption densities. With additional PVA adsorption, the trend observed in Figure 4.7(a) is reversed as can be seen from Figure 4.7(b). Beyond the adsorbed amount of 1.1 mg adsorbed PVA/g silica, the bridging flocculation gives way to steric stabilization. Low suspension viscosity, approximately Newtonian behavior over the entire range of shear rate and absence of any yield stress are observed when the adsorption of PVA is such as to ensure monolayer coverage of particles. It is known that maximum flocculation occurs at approximately half coverage of the particles [44,149]. In the adsorbed PVA/silica suspension case, too, the maximum flocculation did occur at approximately half coverage of the particles as evidenced from the maximum value of the yield stress at that adsorption level as shown in

RELATIVE VISCOSITY

Amount Atf»0ib«d

RELATIVE VISCOSITY

SHEAR RATf (•"')

Amount Adaorfe**

SHEAU RATE (•'') Figure 4.7 Variation of relative viscosity with shear rate for 20 vol% silica suspensions prepared at pH = 3.7 with varying concentrations of poly(vinyl alcohol). The relative viscosity vs. shear rate plot for an electrostatically stabilized suspension (pH = 7.0) with no poly(vinyl alcohol) is also shown. (Reprinted from Ref. 19 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.)

Figure 4.8. It should be noted that the amount of adsorbed polymer depends not only on the overall suspension polymer-particle, polymerliquid and particle-liquid interactions. The adsorption densities are known to depend on the amount of isolated or free surface hydroxyl groups on the particles [150,151]. The effect of the adsorbed polymer in achieving different levels of dispersion also depends on the 'goodness' of the solvent for that polymer. For example, when the solvent is good, it is known [147,148] that steric stabilization is promoted because the loops and tails of the

YIELD STRESS (Pa)

ABSORBEb AMOUNT (mg PVA/g Silica) Figure 4.8 Variation of yield stress with adsorption density of poly(vinyl alcohol) on silica for suspensions prepared at pH = 3.7 with 20 vol% silica and varying concentration of poly(vinyl alcohol). Yield stress values were determined from shear stress vs. shear rate curves. (Reprinted from Ref. 19 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.)

polymer which extend out from the particle surface are well solvated. When the solvent is not good for the polymer, then bridging flocculation is likely to occur. The effect of solvent quality on particulate dispersion was investigated [19] using alumina (Al2O3)/poly(vinyl butyral) (PVB) suspensions prepared in methanol (MEOH), which is a poor solvent, and in a 3:1 volume ratio mixture of methyl isobutyl ketone (MIBK) with MEOH, which is a better solvent. Figure 4.9(a) and (b) shows the relative viscosity versus shear rate behavior in these two solvents for varying PVB concentration from O to 2.0vol%. In either of the solvents, highly shear-thinning behavior is observed when no PVB is added. These suspensions show poor stability against flocculation due to relatively low zeta potentials (< 25 mV) and therefore, low electrostatic repulsive forces [152]. The effect of PVB additions on the dispersion behavior can be seen from Figure 4.9 to be dependent on the suspension liquid composition. In MEOH suspension, though addition of PVB reduces the viscosity of the suspension, the system is shear-thinning indicating that the suspensions remain flocculated in poorer solvents for the added

RELATIVE VISCOSITY

CCMAMIC: 3O *«(% Al/>, 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% A^Os, 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 V<*X AJjO3 UOlHD : 3:1, MAK MEOH

SHEAR RATE U'1)

SHEAR STRESS (Pat

CtRAMtC : 45 «* % AI2O3 LIOiMD: 3:1, MBK/MCOH

SHCAM RATl U-1I Figure 4.10 Variation of shear stress vs. shear rate for 3:1 methyl isobutyl ketone/ methanol suspensions prepared using (a) 30 vol% alumina and (b) 45 vol% alumina with indicated poly(vinyl butyral) concentrations. (Reprinted from Ref. 19 with kind permission from The American Ceramic Society Inc., Westerville, Ohio, USA.)

PVB. In both cases, approximately Newtonian behavior is observed and the viscosities are relatively low, indicating that the suspensions are well dispersed. However, when the filler loading is 45 vol% Al2O3, the 0.25 vol% PVB suspension shows a yield stress and an initial region of highly shear-thinning behavior, indicating that the suspension is highly flocculated at low shear rates (Figure 4.10(a)). This happens because complete monolayer coverage of the Al2O3 particle does not occur at this concentration of polymer. In contrast, the system is far better

dispersed when the 0.5 vol% PVB suspension is used, due to the higher adsorption density. 4.2.8 EFFECT OF CHEMICAL ADDITIVES

Small additions of various common 'processing aids' such as surfactants, dispersants, coupling agents and lubricants do affect the rheological properties of suspensions. The effect of various surface modifiers on the viscous behavior and dispersion of alumina in water has been studied by Dow [155]. The suspensions were prepared by first adding 1.25g of surface modifier to 125 cc distilled (DI) water and then adding alumina powder after lhr. There were visually observed differences between the four surface modifiers studied. The zircoaluminate CAVCO MOD APG and the silane Z-6020 were completely dissolved to form clear solutions. On the other hand, the silane Z-6076 and titanate LICA 12 were found to form emulsions which showed small droplets suspended in the water. This was explained as due to the difference in the solubility of the different surface modifiers. On account of this, the alumina powders were not coated homogeneously by the later surface modifiers with poorer solubility [155]. Figure 4.11 shows the plots of shear stress versus shear rate for the suspensions prepared with alumina powder and different surface modifying agents. The suspension with zircoaluminate is well dispersed as indicated by the Newtonian flow behavior and low suspension viscosity which was determined to be about 2.8 centipoise. The good dispersion is also indicated from a comparison with the electrostatically stabilized alumina suspension prepared at pH = 4.0 with no chemical additives as shown in Figure 4.12. Suspensions containing silane Z-6076 and titanate LICA 12, on the other hand, showed large yield stresses, higher viscosities and highly thixotropic flow behavior, indicating that the suspension is highly flocculated. The suspension prepared with silane Z-6020 is also not as well dispersed as the suspension with the zircoaluminate but the relatively low yield stress, lower viscosity and absence of thixotropy indicates a much improved dispersion as compared with the silane Z6076 and titanate LICA 12. 4.2.9 EFFECT OF PHYSICAL AND CHEMICAL PROCESSES

Physical processes (e.g. crystallization) and chemical processes (e.g. polymerization) are known to modify the viscosity of a suspension with time. For a crystallization process, for example, Mistry and Warburton [156] have shown that if the solid phase concentration is small such that

SHEAR STRESS (Pa)

SHEAR STRESS (Pa)

in Water

Silane

Silane

Titanate UCA 12

SHEAR STRESS (Pa)

SHEAR STRESS (Pa)

in Water

Zircoaluminaie CAVCO UOD APG

SHEAR RATE (1/s)

Figure 4.11 Variation of shear stress vs. shear rate for suspension prepared with 20 vol% alumina in distilled water along with (1 wt% of filler) various surface modifiers as indicated. (Reprinted from Ref. 155.) Einstein's equation (4.1) holds and the crystals in the disperse phase increase with time, the relative viscosity can be estimated from the following expression. i?r(0 = l+a E (0o+*oO

(4.30)

where ^0 is the initial volume of the crystals, X0 the rate constant for volume increase of the crystals and t the time. Similarly, for concentrated suspensions, expressions like (4.6), (4.7) or (4.8) could be

SHEAR STRESS (Pa)

in Water

Zircoaluminate CAVCO MOD APG

DI Water

SHEAR RATE (1/s) Figure 4.12 Variation of shear stress vs. shear rate for suspensions prepared with 20vol% alumina in pH = 4 distilled water and with (1 wt% of filler) zircoaluminate. (Reprinted from Ref. 155.) used as first approximation with appropriately being replaced by (/>0 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
(4.32)

Smoluchowsky [161] and Booth [162] both accounted for the increase in viscosity as due to the electrical double layer round a charged particle in an electrolyte and gave expressions for ev which included the specific conductivity of the electrolyte, the dielectric constant of the suspending medium, the electrokinetic potential of the particles with respect to the electrolyte and the radius of the particle (which was to be large in comparison with the thickness of the double layer for the validity of the expression). Experimental verification by Chan and Goring [163] of the expressions for ev provided by Smoluchowsky [161] and Booth [162] gives confidence for their use. B. Second electroviscous effect The second electroviscous effect is due to the electrostatic repulsion between particles approaching each other and is directly proportional to the square of the particle concentration. The essential feature about this effect is that it occurs at high concentrations of the suspensions (unlike the first electrostatic effects) and when there is an overlap of the double layer. The additional dissipative effects that appear as a result of the repulsion bring about an increase in the viscosity. Chan et al. [164] showed that an expression of the form (4.17) can be used to account for the second electroviscous effect but the coefficient a would then strongly depend on the distance between the centers of the particles and consequently, on the particle concentration. The second electroviscous effect is, at times, known to give rise to non-Newtonian behavior of a suspension as observed by Harmsen et al. [165]. C. Third electroviscous effect The third electroviscous effect is due to the change of shape of suspending particles when their electrical free energy is modified by ionization and the presence of neutral salts. If a polymer molecule can undergo ionization. e.g. by reaction with a base or by reaction with some other ion-producing substance, electrostatic repulsion between the like charges introduced on the polymer chain modifies the partial molecular free energy of the polymer in the solution. With polymeric

electrolytes, non-Newtonian viscosity behavior is not uncommon, particularly at low salt concentrations or high polyion concentrations and the effect is known to increase with degree of ionization.

4.3 EXTENSIONAL FLOW Addition of fillers during the spinning of synthetic textile fibers and the phenomenon of drag reduction with fibers inculcated an interest in the study of the rheological properties of suspensions in extensional flows. The first of such studies was that of Bachelor [166,167], who provided expressions for extensional viscosity of suspensions with long slender particles subjected to extension. The analysis of Batchelor [166] resulted in the following expression for suspensions which are so dilute that the velocity field surrounding one particle remains unaffected by neighboring particles. Thus, . L

=3

1+

2 ^ |"0.64 + ln2ra

1.659

+

+

/

1

YIl

^ ^| 9i^LT5Ti^ ^^ Hai^)jl

^

.

00 (433)

YIQ denotes the viscosity of the suspending medium, $ the volume fraction of the particles having an aspect ratio ra. It is obvious from equation (4.33) that particles with large aspect ratios would give large values of the extensional viscosity even at small concentrations of the particles. Batchelor [167] considered the effect of hydrodynamic interaction of parallel elongated particles in pure extension on the bulk stress. The governing equations were solved under the assumption that the effect on one particle by all the others could be replaced by a cylindrical boundary condition (cell model) and the geometry and concentration of the particles satisfied the following inequality. ra > d > I

(4.34)

where d = Hf/Df (Hf being the average distance between the fibers and Df the diameter of the fiber). The resulting expression for extensional viscosity was as follows:

i+

*•=^ [ yy

(435)

Equation (4.35) predicts the extensional viscosity only for high concentrations and does not give results akin to equation (4.33) at lower concentrations. However, at an intermediate concentration of the suspended particles, both theories are known to predict nearly identical results and hence in this region the extensional viscosity can be estimated by interpolation.

An expression similar to the above was obtained by Insarova [168] for the extensional viscosity of uniformly distributed rigid rods subjected to axially symmetric extension as follows:

*-^°H(*-£*-i)]

(436)

Insarova [168] showed by calculation that for a particle with an aspect ratio ra = 50, the viscosity is increased by a factor of 4 for 0 = 0.01 and by 80 for (/> = 0.1. Equation (4.36) has not been tested with experimental data but the equations of Batchelor [166,167] have been the subject of experimental verification. With the then available data of Weinberger [169], Batchelor [166,167] compared the bulk stress in a suspension of glass-fiber rods (with (/> = 0.013) in each of the liquids, 'Indopol' and 'Silicone' but found that the extensional viscosity was much higher than that predicted by the close-particle theory. Later, Weinberger and Goddard [170] noted that suspensions used in Weinberger's work neither satisfied the dilutesuspension nor the close-particle description, but fell somewhere between the two. Proper interpolation was shown to predict that the tensile stress was 8.4 times that of the suspending medium from Batchelor's theory and was found to be in good agreement with the values of 9 to 10 found by Weinberger and Goddard [170] through experiment. Mewis and Metzner [171] studied the extensional flow of fiber suspensions (with 0.001 < < 0.01) and large aspect ratios (280 < ra < 1260). Experimental data showed that the extensional viscosities were independent of strain rate and up to 260 times larger than that of the suspending medium. The effect of particle concentration was as predicted by equation (4.35) proposed by Batchelor [167]. On the other hand, the effect of particle geometry was in close agreement with the theory only for lower values of ra like 282 and 586, but for ra = 1259 the experimental data were found to be about 30% lower than that predicted by Batchelor [167]. Kizior and Seyer [172] experimentally determined stress levels in extensional flow of suspensions with fibers having an aspect ratio of 340 and volume fraction of 0.001. It was found that the experimental value of the stress was higher than that predicted by Batchelor's theory. However, the dependence of stress on concentration and aspect ratio was well predicted by the theoretical expressions of Batchelor [167]. From the above experimental efforts, it can be concluded that sufficient evidence has been generated to verify the propriety of equations (4.33) and (4.35) given by Batchelor [166,167]. Equation (4.33) for dilute suspensions, equation (4.35) for close-particle distribution and

an interpolation between the two could predict the extensional flow behavior of suspensions of slender particles over a large range of particle concentration. However, the results of Batchelor [166,167] are valid only when the suspending medium is Newtonian. The entire body of literature which analyses the extensional flow of slender particles in non-Newtonian fluids goes to the credit of Goddard [173-175]. Goddard [173] derived a formula to describe the stress field for dilute suspensions of oriented slender fibers in a non-Newtonian fluid. The treatment was quite rudimentary but brought out an important result that particle-stress effect was considerably smaller in a shear-thinning non-Newtonian fluid compared to the Newtonian case, possibly due to tensile stiffening in the fluid itself. Qualitative agreement with the experimental data of Charrier and Rieger [176] was observed. A more sophisticated analysis of the same problem was presented by Goddard [174] using a general quasi-steady state rheological model for the suspending medium, with the assumption that in the near-field of the suspended particles the flow is shear-dominated and extends asymptotically to the extension-dominated flow in the far field. The conclusions of this analysis were the same as those arrived at earlier in Goddard [173]. The complex theoretical treatment of Goddard [173,174] are not included here. Results are available for the simplified case of the suspending medium being a power-law fluid in Goddard [175]. Goddard [175] carried out a qualitative comparison with the rheological data of Chan et al. [164] for simple shear and simple extension of polymer melts containing chopped glass fibers. It was seen that the agreement with the power-law theory was not satisfactory and even worse for the Newtonian case. Goddard concluded that the disagreement was not due to failure of the theory, but possibly because of lack of fiber alignment during experiments. Nicodemo et al. [177] were the first to show, experimentally, the spectacular reductions (at times by an order of magnitude) in the extensional viscosities of polymer solutions containing spherical particles (namely, glass microbeads) in contrast to solutions containing rod-shaped particles (namely, fibers), which are known to increase the extensional viscosities. The extensional viscosity of fiber suspension depends to a large extent on the orientation of the fiber and this has been the subject of study of a number of investigators (e.g. Taksermann-Krozer and Ziabicki [178], Bell [179], Goettler [180,181], Nicodemo et al. [182], Takano [183], Lee and George [184]). In polymer processing, as most of the flow takes place through converging dies, the study of the rheological properties in extensional flow and the orientation of the

polymer molecules and fiber are extremely important in order to control the quality of the end products. Despite the importance of extensional viscosity studies in filledpolymer processing it is unfortunate that the effects of various factors like size distribution of the fillers, nature of the surface, flocculation, etc. as in the case of shear viscosity (section 4.2), has not been studied at all. Only the effect of the shape of the filler on the extensional viscosity has been brought out by Nicodemo et al. [177]. A lot more research in this area is thus warranted. The main reason for the dearth of information in this area is probably due to the absence of a cheap and simple apparatus for the measurement of extensional viscosity. Most of the workers have had to design their own apparatus for determining extensional viscosities depending on the system to be measured and the facilities available.

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30. Heertjes, P.M., Smits, C.I. and Vervoorn, P.M.M. (1979) Some aspects of adsorption processes occurring on titanium dioxide particles, Prog. Organic Coatings, 7,141-66. 31. Hudson, N.E., Bayliss, M.D. and Ferguson, J. (1978) The rheology of pigment dispersions, Rheol Acta, 17, 274-87. 32. Saarnak, A. (1979) Characterisation of the interaction between pigments and binder solution using rheological measurements, /. Oil Col Chem. Assoc., 62, 455-60. 33. Dulik, D., Bernier, R. and Brauer, G.M. (1981) Effect of diluent monomer on the physical properties of bis-GMA-based composites, /. Dent. Res., 60, 983-9. 34. Vermilyea, S., Powers, J.M. and Craig, R.G. (1977) Rotational viscometry of a zinc phosphate and zinc polyacrylate cement, /. Dent. Res., 56, 762-7. 35. Vermilyea, S.G., Powers, J.M. and Koran, A. (1978) The rheological properties of fluid denture-base resins, /. Dent. Res., 57, 227-31. 36. Vermilyea, S.G., De Simon, L.B. and Huget, E.F. (1978) The rheological properties of endodontic sealers, Oral Surg., 46, 711-16. 37. Jacobsen, P.H., Whiting, R. and Richardson, P.C.A. (1977) Viscosity of setting anterior restorative materials, Brit. Dent. /., 143, 393-6. 38. Whiting, R. and Jacobsen, P.H. (1979) The evaluation of non-Newtonian viscosity using a modified parallel-plate plastometer, /. Mat. Sd., 14, 307-11. 39. Baker, F.S., Carter, R.E. and Warren, R.C. (1980) The rheological assessment of propellants, Paper presented at the 8th International Congress on Rheology, Naples, 1-5 Sept. 40. Chen, Y.R. and Hashimoto, A.G. (1976) Rheological properties of aerated poultry waste slurries, Trans. ASAE, 19, 128-33; (1976) Pipeline transport of livestock waste slurries, Trans. ASAE, 19, 898-902, 906. 41. Hashimoto, A.G. and Chen, Y.R. (1976) Rheology of livestock waste slurries, Trans. ASAE, 19, 930-4. 42. Brownell, W.E., Caso, G.B. and Braun, D.B. (1978) An additive to increase plastic strength, Amer. Cer. Soc. Bull, 57, 513-16. 43. Farooqui, S.I. and Richardson, J.F. (1980) Rheological behaviour of kaolin suspensions in water and water-glycerol mixtures, Trans. Inst. Chem. Engrs, 58,116-124. 44. Heath, D. and Tadros, Th.F. (1983) Influence of pH, electrolyte, and poly(vinyl alcohol) addition on the rheological characteristics of aqueous dispersions of sodium montmorillonite, /. Colloid Inter/. ScI, 93, 307-19; (1983) Influence of pH, electrolyte, and poly(vinyl alcohol) addition on the rheological behaviour of aqueous silica (aerosol) dispersions, /. Colloid Inter/. ScL, 93, 320-28. 45. Heywood, N.I. and Richardson, J.F. (1978) Rheological behavior of flocculated and dispersed aqueous kaolin suspensions in pipe flow, /. Rheol, 22, 599-613. 46. James, A.E. and Williams, D.J.A. (1982) Flocculation and rheology of kaolinite/quartz suspensions, Rheol Acta, 21,176-83. 47. Verreet, G. and Berlament, J. (1988) Rheology and non-Newtonian behavior of sea and estuarine mud, in Encyclopedia of fluid Mechanics (ed. N.P. Cheremisinoff), Gulf Publishing, Houston, TX, Vol. 7, pp. 135-49. 48. Cheremisinoff, N.P. (ed.) (1986) Encyclopedia of Fluid Mechanics, Vol. 5: Slurry Flow Technology, Gulf Publishing, Houston, TX.

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76. Simha, R. and Somcynsky, T. (1965) /. Colloid Sd., 20, 278-81. 77. Ford, T.F. (1960) /. Phys. Chem., 64,1168-74. 78. Cengel, J.A., Faruqui, A.A., Finnigan, J.W., Wright, C.H. and Knudsen, J.G. (1962) AIChE /., 8, 335-9. 79. McGeary, R.K. (1961) /. Amer. Ceram. Soc., 44, 513-22. 80. De Brujin, H. (1951) Disc. Faraday Soc., 11, 86. 81. Clarke, B. (1967) Trans. Instn Chem. Engrs, 45, 251-6. 82. Guth, E. and Simha, R. (1936) Kolloid Z., 74, 266. 83. Vand, V. (1948) Viscosity of solutions and suspensions. I, /. Phys. Chem. Colloid Chem., 52, 277-99; (1948) II, /. Phys. Chem. Colloid Chem., 52, 300-14. 84. Saito, N. (1950) /. Phys. Soc. Japan, 5, 4. 85. Manley, R. St. J. and Mason, S. G. (1955) Can. J. Chem., 33,163. 86. Harbard, E.H. (1956) Chem. Ind.f 22, 491. 87. Alfrey, T., Jr. (1952) Mekhanicheskie Svoistva Viskopolimerov, Izd. Innostranoi Literaturi, Moscow, p. 479. 88. Arrhenius, S. (1887) Z. Phys. Chem., 1, 285. 89. Fikentscher, H. and Mark, H. (1930) Kolloid-Z. 49,135. 90. Bungenberg de Jong, H.G., Kruyt, H.R. and Lens, W. (1932) Kolloid-Beihefte, 36, 429. 91. Papkov, S. (1935) Kunststoffe, 25, 253. 92. Hauwink, R. (1937) Kolloid-Z., 79,138. 93. Brede, H.L. and De Boojs, J. (1937) Kolloid-Z., 79, 31. 94. Mullins, L. (1968) Kauchuk i Rezina, 7,10. 95. Feldman, D. and Boiesan, V. (1971) Mat. Plastics, 8, 536. 96. Eyring, H., Henderson, D., Stover. BJ. and Eyring. E.M. (1964) Statistical Mechanics and Dynamics, Wiley, New York, p. 460. 97. Mooney, M. (1951) The viscosity of a concentrated suspension of spherical particles, /. Colloid Sd., 6,162-70. 98. Severs, E.T. (1963) Rheology of Polymers, Reinhold, New York, p. 65. 99. Eilers, H. (1941) Die viskositat von emulsionen hochviskoser stoffe als funktion der konzentration, Kolloid Zh., 97, 313. 100. Frankel, N.A. and Acrivos, A. (1967) On the viscosity of a concentrated suspension of solid spheres, Chem. Engg ScL, 22, 847-53. 101. Marrucci, G. and Derm, M.M. (1985) On the viscosity of a concentrated suspension of solid spheres, Rheol Acta, 24, 317-20. 102. Chong, J.S., Christiansen, E.B. and Baer, A.D. (1971) Rheology of concentrated suspensions, /. Appl. Polym. Sd., 15, 2007-21. 103. Kataoka, T., Kitano, T., Sasahara, H. and Nishijima, K. (1978) Viscosity of particle filled polymer melts, Rheol. Acta, 17, 149-55. 104. Kitano, T., Kataoka, T. and Shirata, T. (1981) An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers, Rheol. Acta, 20, 207-9. 105. Poslinski, A.J., Ryan, M.E., Gupta, R.K., Seshadri, S.G. and Frechette, FJ. (1988) Rheological behavior of filled polymeric systems. I. Yield stress and shear-thinning effects, /. Rheol, 32, 703-35. 106. Poslinski, AJ., Ryan, M.E., Gupta, R.K., Seshadri, S.G. and Frechette, FJ. (1988) Rheological behavior of filled polymeric systems. II. The effect of a bimodal size distribution of particulates, /. Rheol., 32, 751-71.

107. Wildemuth, C.R. and Williams, M.C. (1984) Viscosity of suspensions modeled with a shear-dependent maximum packing fraction, Rheol. Acta, 23, 627-35. 108. Stamhuis, J.E. and Loppe, J.P.A. (1982) Rheological determination of polymer-filler affinity, Rheol Acta, 21,103-5. 109. Jeffrey, J.B. (1922) The motion of ellipsoidal particles immersed in a viscous fluid, Proc. Roy. Soc., A102,161-79. 110. Guth, E. (1936) Kollaid-Z., 74,147. 111. Guth, E. (1938) Phys. Rev., 53, 926A. 112. Pokrovskii, V.N. (1967) KoIl(M Zh., 29, 576. 113. Maschmeyer, R.O. and Hill, C.T. (1975) Trans. Soc. Rheol., 21,183-94. 114. Burgers, J.M. (1938) Second Report on Viscosity and Plasticity, Nordemann, New York, p. 113. 115. Mason, S.G. (1950) Pulp Paper Mag. Can., 51, 83. 116. Blankeney, W.R. (1966) /. Colloid ScL, 22, 324-30. 117. Simha, R. (1949) /. Res. Nat. Bur. Stand., 42, 409. 118. Brodnyan, J. (1959) Trans. Soc. Rheol, 3, 61-8. 119. Ziegel. K. (1970) /. Colloid Inter/. ScI, 34,185-96. 120. Hashin, Z. (1969) in Contributions to Mechanics (ed. D. Abir), Pergamon Press, Oxford. 121. Nielsen, L. (1970) /. Appl Phys., 41, 4626-7. 122. Ward, S.G. and Whitmore, R.L. (1950) Studies of the viscosity and sedimentation of suspensions. Part I: The viscosity of suspensions of spherical particles, Br. J. Appl. Phys., 1, 286. 123. Ting, A.P. and Luebbers, R.H. (1957) AIChE /., 3,111-16. 124. Moreland, C. (1963) Viscosity of suspensions of coal in mineral oil, Can. J. Chem. Engg, 41, 24. 125. Shaheen, E.I. (1971) Rheological study of viscosities and pipeline flow of concentrated slurries, Powder Tech., 5, 245. 126. Parkinson, C., Matsumoto, S. and Sherman, P. (1970) /. Colloid Inter/. ScL, 33,150-60. 127. Hsieh, H.P. (1978) Polym. Engg ScL, 18, 928-31. 128. Ward, S.G. and Whitmore, R.L. (1950) Br. J. Appl. Phys., 1, 325-8. 129. Goldsmith, H.L. and Mason, S.G. (1965) Proc. 4th Int. Congr. Rheol., Interscience, New York, p. 65. 130. Segre, G. and Silberberg, A. (1961) Radial particle displacements in Poiseuille flow of suspensions, Nature, 189, 209-10. 131. Segre, G. and Silberberg, A. (1962) Behaviour of macroscopic rigid spheres in Poiseuille flow, Parts 1 & 2, /. Fluid Mech., 14, 115-35,136-57. 132. Segre, G. and Silberberg, A. (1962) Non-Newtonian behavior of dilute suspensions of macroscopic spheres in a capillary viscometer, /. Colloid ScL, 18, 312. 133. Morrison, S.R. and Harper, J.C. (1965) Ind. Eng. Chem. Fundamentals, 4,176. 134. Jastrzebski, Z. (1967) Entrance effects and wall effects in an extrusion rheometer during the flow of concentrated suspensions, Ind. Eng. Chem. Fundamentals, 6, 445-54. 135. Chapman, P.M. and Lee, T.S. (1970) Effect of talc filler on the melt rheology of polypropylene, SPE /., 26, 37-40. 136. Goel, D.C. (1980) Effect of polymer addition on the rheological properties of talc-filled polypropylene, Polym. Engg ScL, 20,198-201.

137. Ferraro, C.F. (1968) SPE /., 24, 74-8 (April). 138. Krieger, LM. and Dougherty, TJ. (1959) A mechanism to non-Newtonian flow in suspensions of rigid spheres, Trans. Soc. RheoL, 3,137-52. 139. Gillespie, T. (1966) /. Colloid ScL, 22, 554-62. 140. Fischer, E.K. (1970) Colloidal Dispersion, Wiley, New York, p. 154. 141. Govier, G.W. and Winning, M.D. (1948) AIChE Meeting, Montreal, Quebec. 142. Umeya, K. and Kanno, T. (1979) Effect of flocculation on the dilatant flow for aqueous suspensions of titanium dioxides, /. RheoL, 23,123-40. 143. Nicodemo, L., Nicolais, L. and Landel, R.F. (1974) Shear rate dependent viscosity of suspensions in Newtonian and non-Newtonian liquids, Polym. Engg ScL, 29, 729. 144. Nicodemo, L. and Nicolais, L. (1974) /. Appl. Polym. ScL, 18, 2809-18. 145. Kremesec, VJ. and Slattery, J.C. (1977) Trans. Soc. RheoL, 21, 469-91. 146. Highgate, DJ. and Whorlow, R.W. (1970) RheoL Acta, 9, 569-76. 147. Napper, D.H. (1977) Steric stabilization, /. Colloid Inter/. ScL, 58, 390-407. 148. Tadros, Th.F. (1982) Polymer adsorption and dispersion stability, in The Effect of Polymers on Dispersion Properties (ed. Th.F. Tadros), Academic Press, New York, pp. 1-37. 149. Eisenlauer, J., Killman, E. and Korn, M. (1980) Stability of colloidal silica (aerosil) hydrosols. II. Influence of the pH value and the adsorption of polyethylene glycols, /. Colloid Inter/. ScL, 74,120-35. 150. Tadros, Th.F. (1978) Adsorption of polyvinyl alcohol on silica at various pH values and its effect on the flocculation of the dispersion, /. Colloid Inter/. ScL, 64, 36-47. 151. Rubio, J. and Kitchner, J.A. (1976) The mechanism of adsorption of poly(ethylene oxide) flocculant on silica, /. Colloid Inter/. ScL, 57,132-42. 152. Sacks, M.D. and Khadilkar, C.S. (1983) Milling and suspension behavior of Al2O3 in methanol and methyl isobutyl ketone, /. Am. Ceram. Soc., 66, 488-94. 153. Sacks, M.D. (1984) Properties of silicon suspensions and cast bodies, Am. Ceram. Soc. Bull., 63,1510-15. 154. Morgan, RJ. (1968) A study of the phenomena of rheological dilatancy in an aqueous pigment suspension, Trans. Soc. RheoL, 12, 511-33. 155. Dow, J.H. (1992) PhD Thesis, University of Florida, Gainesville. 156. Mistry, D.B. and Warburton, B. (1971) RheoL Acta, 10,106-12. 157. Fryling, C.F. (1963) /. Colloid ScL 18, 713-32. 158. Krieger, LM. and Eguiluz, M. (1976) The second electroviscous effect in polymer lattices, Trans. Soc. RheoL, 20, 29. 159. Conway, B.E. and Dobry-Duclaux, A. (1960) in Rheology (ed. F.R. Eirich), Academic Press, New York, Vol. 3, pp. 83-120. 160. Dobry, A. (1950) /. Chim. Phys., 50, 507. 161. Smoluchowsky, M. (1916) Kolloid-Z., 18,190. 162. Booth, F. (1950) The electroviscous effect for suspensions of solid spherical particles. Proc. Roy. Soc., A203, 533-51. 163. Chan, F.S. and Goring, D.A. (1966) /. Colloid ScL, 22, 371-7. 164. Chan, Y., White, J.L. and Oyanagi, Y. (1977) Polym. ScL Eng. Report No. 96, University of Tennessee, Knoxville. May. 165. Harmsen, GJ., Schooten, J.V. and Overbeek, J.Th.G. (1953) /. Colloid. ScL, 8, 64-71.

166. Batchelor, G.K. (1970) Slender-body theory of particles of arbitrary crosssection in Stokes flow, /. Fluid Mech., 44, 419. 167. Batchelor, G.K. (1971) The stress generated in a non-dilute suspension of elongated particles by pure straining motion, /. fluid Mech., 46, 813-29. 168. Insarova, N.I. (1970) Inzh.-Fiz. Zh., 18, 347. 169. Weinberger, C.B. (1970) Extensional flow behavior of non-Newtonian fluids, PhD dissertation, University of Michigan. 170. Weinberger, C.B. and Goddard, J.D. (1974) Extensional flow behaviour of polymer solutions and particle suspensions in a spinning motion, Int. J. Multiphase Flow, 1, 465-86. 171. Mewis, J. and Metzner, A.B. (1974) The rheological properties of suspensions of fibres in Newtonian fluids subjected to extensional deformations, /. Fluid Mech., 62, 593-600. 172. Kizior, T.E. and Seyer, F.A. (1974) Axial stress in elongational flow of fiber suspension, Trans. Soc. Rheol., 18, 271-85. 173. Goddard, J.D. (1976) Tensile stress contribution of flow-oriented slender particles in non-Newtonian fluids, /. Non-Newtonian Fluid Mech., 1,1-17. 174. Goddard, J.D. (1976) The stress field of slender particles oriented by a nonNewtonian extensional flow, /. Fluid Mech., 78,177-206. 175. Goddard, J.D. (1978) Tensile behavior of power-law fluids containing oriented slender fibers, /. Rheol., 22, 615-22. 176. Charrier, J.M. and Rieger, J.M. (1974) Fibre Sd. Tech., 7,161. 177. Nicodema, L., De Cindio, B. and Nicolais, L. (1975) Elongational viscosity of microbead suspensions, Polym. Eng. ScL, 15, 679-83. 178. Takserman-Krozer, R. and Ziabicki, A. (1963) Behaviour of polymer solutions in a velocity field with parallel gradient. I. Orientation of rigid ellipsoids in a dilute solution. /. Polym. ScL, Al, 491-506. 179. Bell, J.P. (1969) Flow orientation of short fibre composites, /. Comp. Mat., 3, 244-53. 180. Goettler, L.A. (1970) Flow orientation of short fibers in transfer molding, 25th Annual Tech. Conf., SPI, Paper 14-A. 181. Goettler, L.A. (1974) Molding of oriented short-fiber composites, Report No. HPC-70-130, Monsanto/Washington University Association. 182. Nicodemo, L., Nicolais, L. and Acierno, D. (1973) Orientation of short fibers in polymeric materials resulting from converging flow, Ing. Ch. Ital., 9, 113-16. 183. Takano, M. (1973) Flow orientation of short fibers in rectangular channels, ARPA Report, HPC 73-165. 184. Lee, W.K. and George, H.H. (1978) Flow visualization of fiber suspensions, Polym. Engg ScL, 18,146-56.

Preparation of. filled

jr-

polymer systems

O

In context with the preparation of filled polymer systems, there are three terms, namely, compounding, blending and mixing, which are often synonymously or interchangeably used and though various researchers have defined these terms, one is at times faced with the dilemma of terminology [I]. In the present case, definitions of the terms are given as applicable to the subject matter and hence exclude any other connotations of the terms. Compounding is the term used for those cases wherein polymers are softened, melted and intermingled with solid fillers and other liquid additives to form filled polymer systems. Blending is defined as a process in which two or more components or ingredients are physically intermingled without engendering any significant change in the physical state of the components. The components are, normally, polymers to form polymer blends, and hence in the present context, will not be used to describe the intermingling of fillers with polymers. The word mixing is applied to both the processes of compounding and blending, and describes the process of intimate intermingling of polymers with fillers/additives or two polymers without any specific restrictions. It covers a broad spectrum of dispersion of various ingredients to form a homogeneous mixture on some definable small scale. 5.1 GOODNESS OF MIXING The important aspect in mixing is to evaluate the quality of mixtures [2] or the goodness of mixing [3]. The most straightforward method of characterizing the quality of the mixtures is to measure to what extent the desired properties have been attained. Industrial quality control follows that route whenever feasible. However, this requires a detailed

description of the structure of the mixture and to theoretically establish this, the spatial position of each ultimate minor particle in the matrix must be determined. It is often not possible to predict the exact path of individual ultimate particles during the mixing process because most compounding equipments achieve the mixed state randomly, though, of course, there are some in which mixing progresses at least in a partly ordered fashion. In the random mixed state there are more possible arrangements of the minor component than in the completely unmixed state. In the latter state the probability is unity and in the former case it is very large equal to 0

mixed

_ (N; + Nj)! ~ N;!Ni!

(5 1)

'

N; and N'2 are the number of ultimate particles of the major and minor component respectively. A very great number of possible random distributions have to be considered [4] to be the most uniform distribution which can be achieved with common random type compounding equipments as shown in Figure 5.1. From the inspection of Figures 5.1(d) and 5.1(e), it can be seen that in the case of filled polymer systems, this type of regular random state would undoubtedly be preferable to any other random state. In the case of small ultimate particles which cluster together to form large agglomerates (blobs), the clusters have to be broken up into ultimate particles (aggregates) and uniformly distributed by the dispersive mixing action. The random distribution of a component 'A' in a component 'B' is achieved if the probability of finding an ultimate particle of 'A' is the same at all points in the mixture and is equal to the volume fraction a of the component 'A' in the mixture. In the completely unmixed state as shown in Figure 5.1 (a), a sample of volume Vs will have a concentration of component A of either 1 or O. The probability to find XA = 1 is 'a' and to find XA = O is b, as shown in Figure 5.2 on the left-hand side. The mean is given by M=*

(5.2a)

°2 = d>

(5.2b)

and the variance as

The probability distribution for finding a particular level of concentration in a sample after drawing a great number of samples can be calculated [4] from the following expression

(a) THE COMPLETELY UNMIXED STATE.

O «1

COMPONENT: A,B VOLUME FRACTION: a,b ULTIMATE PARTICLE SIZE: a,0 Figure 5.1 Number of possible random distributions of fillers in a polymer matrix which can be achieved with common random type compounding equipments. (Reprinted from Ref. 4 with kind permission from John Wiley & Sons, Inc., New York, USA.)

STAUSTtCAL DEFINITION OF THE STATE OF MIXTURE.

THE COMPLETELY UNMIXED STATE. P(X*)

VOLUME OF CONCENTRATION OF COMPONENT A MEAN: M = a VARIANCE: 0*= a.b

SAMPLE StZE V5

THE COMPLETELY MIXED STATE. P(X*)

VOLUME OF'CONCENTRATION MEAN:^ = a OFCOMPONENTA VARIANCE:**= a.b^

Figure 5.2 Variation of the probability of filler distribution with its concentration in the polymer matrix. (Reprinted from Ref. 4 with kind permission from John Wiley & Sons, Inc., New York, USA.)

eXp

P(XA)=

/ 1(XA-^u) 2 N (~2 V

7(2^)

^

<W

where the mean is given by /* = a

(5.3b)

*2 = ^

(5.3c)

and the variance as

Again it can be seen that for a sample of the size of an ultimate particle, the variance and the probability are the same as for the completely unmixed state. On the other end of the spectrum is a sample of very large size, possibly a whole batch, in which case the measured concentration will always indicate that the sample is well mixed. Obviously, in order to make a statement about the quality of mix, series of varying sample sizes have to be analyzed. During each series N" samples of the same size are collected. Using equation (5.3c), the variance for the completely mixed state for each series can be calculated introducing the sample size for that serve. The actual mean concentration XA and the actual variance S2 can be calculated [4] from the measured values XA. as follows (Table 5.1) —

1

N

"

X

X A = xpE A« v

I=I

S2

2 = IM^XX-XA) V X

(5.4)

(5.5)

f=l

If the mixture has been properly sampled then XA should not be significantly different from a. The normal proportion test can be used to determine if any observed difference is significant. The test is applied by calculating the quantity

Z-^-M*

,5.6)

Numerous indices to quantify the extent of mixing have been proposed and these are reviewed by Fan and co-workers [5,6]. In general these are based on standard deviation S or the variance S2 of the composition of spot samples taken from the mixture. The simplest such index [7] is given by S _ (j Tn1=- or Tn1=(5.7) O o where a is the standard deviation of composition for a random

Table 5.1 Mean: XA = ^7]T X A/ ; Variance: S2 == ^L-JT(X*, = XA)2 A/". . . number of samples Significance of XA Significance of S2 _ (XA - a)-v/A/" chi-square test Z ~ S Mixing index Mixing index

Perfectly mixed

Completely unmixed

E 2

;

K «

c

4^4 <7 0 -t7

In^-InS 2 In al - In
1

o

Q

o-a>H)

or2, = a. b = a(1 - a) the variance of the completely unmixed state Source: Ref. 4 (reprinted with kind permission from John Wiley & Sons, Inc., New York, USA).

homogeneous mixture defined as a mixture in which the probability of finding a particle of a given component is the same at all the locations in the mixture and a is equal to ^/[a(l — a)/N'] for a binary number of mixture where a is the proportion of one component and N' is the number of particles in the sample [7]. The number of particles in the sample are often difficult to estimate and in that case, the value of N' may be taken as VJOL. The sampling of mixtures and treatment of results has been extensively reviewed by Weidenbaum [8] as well as by Hold [4] and may be referred to for better understanding. Different ways of characterizing the degree of mix by a mixing index based upon a2 and S2 have also been proposed [9]. They are summarized in Table 5.1. The chi-square test is the statistical test of significance used in comparing S2 with a2. The chi-square test can be used to determine whether the difference between S2 and a2 is significant indicating a deviation from a random mix or a deviation by chance [1O]. If the mixing index is plotted vs. the size of the sample, the size of the domains with nonuniform

composition and the degree of nonuniformity can be established from the graph. Two properties which are useful in adjudging the goodness of mixing are the scale and intensity of segregation. Scale of segregation is a measure of the average separation between regimes comprising the same component and may be correlated to the average striation thickness, which is the average distance between like interfaces in a mixture [U]. On other hand, intensity of segregation is a measure of concentration [U]. Danckwerts [3] defines the scale of segregation SL as the integral of the coefficient of correlation between concentrations at two points separated by a distance SL= f RQjdr Jo

(5.8)

The definition of the coefficient of correlation is given by the following equation N " _ _ £(XAj. -X A )(X;-X A )

(5 9) ^ =ATS2 ' where X A/ , XA. are the concentrations at two points at a distance r. The variance S2 is calculated from the concentrations at all points using equation (5.5). The coefficient of correlation R(r) ranges from 1 to —1. A value of 1 indicates that both points in each couple have the same concentration. The value —1 indicates that the concentration in one point is pure minor and in the other is pure major. The intensity of segregation is defined as follows:

I = -0 2 O

(5.10)

It expresses the ratio of the measured variance to the variance of a completely segregated system. The intensity of segregation as defined in equation (5.10) reflects to some extent 'gross uniformity' on a scale of examination reduced to the scale at which texture or local structure is being examined. Danckwerts [3] thus introduced a different method of characterizing texture and local structure. In Figure 5.3, the sample on the left-hand side has a large scale of segregation and unless the intensity of segregation, i.e. the difference in concentration between these areas is very low, the sample is poorly mixed. The specimen on the right-hand side shows a small scale of segregation: in this case a higher intensity of segregation can still be tolerated without classifying the state of mix as poor. This provides a good insight into the importance of the scale of

SCALE OF SEGREGATION:

INTENSITY OF SEGREGATION:

GROSS UNIFORMITY

SCALE OF EXAMINATION TEXTURE

LOCAL STRUCTURE Figure 5.3 Definition of the scale and intensity of segregation along with the scale of examination. (Reprinted from Ref. 4 with kind permission from John Wiley & Sons, Inc., New York, USA.)

examination in relation to the interpretation of the results of the examination. Large scale examination renders information about the gross uniformity of a mixture. With decreasing scale, information is gained about the texture and finally the local structure of a mixture. Gross uniformity is important in most mixtures. However, the degree of fineness in the texture that is needed is, of course, dependent upon the individual application requirements of the mixed material. In order to evaluate the texture, the size of the analyzed samples has to be reduced to the level which permits characterization of the texture significant to a specific application. On the other hand, the local structure has to be examined at the scale of the ultimate particles. Naturally, the local structure would depend upon the efficiency of the mixing action and also upon the compatibility of the materials. In summary, the process of mixing involves the breaking down of the individual components into smaller elements and then dispersion of the

elements of one component in the space occupied by the other [12], thereby reducing compositional non-uniformity. In the case of the filled systems, the process includes the breaking down of agglomerates, their separation and segregation, until a final random distribution is achieved of each component in the system. The purpose of mixing is to attain an acceptable degree of homogeneity or uniformity of composition, assessed through the appropriate scale of scrutiny, which is defined [11,13,14] as the minimum size of the segregation region that would cause the mixture to be imperfect for the intended purpose. It is thus reasonable to define the perfect admixture as the state in which no variations in composition or morphology are observed at the relevant closeness of examination. 5.2 MIXING MECHANISMS

Having discussed the various methods of characterizing the quality (uniformity) of a mixture, the next step is to find functional relationships between these quantitative measures of the quality of the mixture and the mixing process, the mixer geometry, operating conditions, properties and initial conditions of the components. Theoretically each ultimate minor particle has to be followed throughout the mixing process to establish the final spatial distribution and evaluate it by applying these measures of quality. During the movement of the particles in the mixing action, three basic types of motion can be encountered as shown in Figure 5.4: molecular diffusion; eddy diffusion (turbulent flow); convection. In a viscous polymer system, the influence of molecular and eddy diffusion is generally negligible, leaving forced convective flow as the main mixing process [9]. If convection causes (a) the movement of the melt elements and the solid fillers from one spatial location to another such that the interfacial area between them increases or (b) that solid filler particles are distributed throughout the polymer matrix without necessarily increasing interfacial area, then distributive mixing is said to have occurred. Distributive or simple mixing is induced by plug or bulk convection. Bulk convection can be random or ordered. Ordered bulk convection rearranges constant volumina in a well defined way, and the progress of mixing action can be predicted mathematically. In random bulk convection, chunks of material are randomly redistributed, and modeling the mixing action is for most cases not possible. Distributive mixing is basically 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. Distributive mixing is influenced by the strain

CONVECTION PLUG (BULK) CONVECTION

LAMINAR CONVECTION

RANDOM ORDERED BULK BULK CONVECTION CONVECTION IMPROVES: GROSS UNIFORMITY TEXTURE LOCAL STRUCTURE

EXTENSIVE MIXtNG(STRAIN) LAMINAR, MIXING INTENSIVE, DISPERSIVE MIXING (STRESS)

MOLECULAR DIFFUSION

DISTRIBUTIVE SIMPLE MIXING

EDDY DiFFUStON NONEXISTENT

Figure 5.4 Various types of mixing mechanisms. (Reprinted from Ref. 4 with kind permission from John Wiley & Sons, Inc., New York, USA.)

imposed on the mixture. However, in the case of a viscoelastic polymer system containing agglomerated fine filler particles which show yield stress characteristics, the application of strain is necessary, but insufficient, to achieve mixing. The strain rate and hence shear stress imposed on the material determine the extent of mixing and this is known as dispersive mixing [9]. This type of mixing is accomplished by laminar convection through various types of flow: shear, extensional and squeezing. However, shear plays the major role during the mixing process and the main action occurs along streamlines. Dispersive mixing is basically defined as an operation which reduces the agglomerate size of the minor constituent to its ultimate particle size. During compounding of polymer melts with filler, it is the effect of the mixing action on the solid filler particle that achieves importance. Available research results for predicting the effect of a specific flow field on agglomerates indicates that there are a number of possible motions corresponding to the initial conditions of the velocity field and the shape of the particles. The motions are periodic and the resultant of

all the forces acting on the surface and their moments equal to zero. The periodic motion of the particles is necessary to maintain the forces and their moments in equilibrium. The forces create a stress distribution throughout the agglomerate which will result in the rupture of the agglomerate along a surface or surfaces where the stress exceeds the local cohesive forces between the ultimate particles. The breakdown of agglomerates in shear flow is described quantitatively by Tadmor and Gogos [9]. It has been shown that the maximum forces tending to divide a dumbbell shaped agglomerate in a fluid of viscosity rj is given by F max = ^yT1T2

(5.11)

where T1 and r2 are the radii of the two particles and y is the rate of change of shear experienced by the fluid, and occurs when the dumbbell axis is at an angle of 45° to the direction of shear. Thus improved mixing is achieved at higher shear stresses and, clearly, small filler particles are more difficult to separate than large filler particles, because the magnitude of the attracting forces increases with decreasing particle diameter. High shear stresses may be applied to nonreinforcing filler particles to achieve better dispersion. However, mixing of a polymer with a reinforcing fiber presents a different problem. In this case it is of importance not to break the fiber because only if the bond strength of the fiber with the matrix on both sides of a point of high stress is equal or greater than the strength of the fiber, is it possible to utilize the fiber optimally. Hence, stress levels will have to be maintained reasonably low to prevent drastic fiber breakage. Another point of importance that must be borne in mind is that dispersive mixing is laminar or streamline mixing. The streamlines never cross during the laminar flow of the mixing action. Hence initial nonuniformities of the distribution of the minor component across the streamlines of the mixing flow can only be eliminated if laminar mixing is combined with plug convection. Viewing mixing as distributive or dispersive is clearly an oversimplification when applied to practical mixing problems since both actions will occur simultaneously. Homogeneity will have been achieved when both the dispersive elements and the distributive elements of the mixing have achieved a fluctuation in average composition below a certain fixed and acceptable level. However, before either distributive mixing or dispersive mixing can effectively take place, there is an initial step in which the originally separate ingredients form a coherent mass. This process is known as incorporation or wetting. Filler particles, especially those of submicron size, are available in an aggregated form. At times, this is done on purpose by the suppliers

because agglomerates are easier to handle, are dustless, and are therefore less hazardous to health. However, due to a variety of forces such as van der Waals, electrostatic, liquid bridges, etc. [15] it is often the strong filler bonds that tend to hold finely divided particles together. In the preparation of filled polymer system, difficulties arise in dispersing the fillers in the polymer matrix due to the basic incompatibility between filler and polymer. Polymers, being organic in nature, have surface tensions that are quite different from those of most inorganic fillers. They have, thus, a natural tendency to resist wetting of the filler during compounding. Hence, during preparation of filled systems, there is certainly a possibility for increased agglomeration in this non-wetting environment, as well as an enhanced probability of forming microvoids around particle clusters. In summary, compounding of fillers with polymers basically involves four steps - the initial incorporation, then wetting, followed by deagglomeration and finally distribution of the fillers within the polymer matrix. This is achieved through the use of a variety of compounding techniques that are discussed below. 5.3 COMPOUNDING TECHNIQUES There are a number of types of compounding equipment that can potentially carry out the compounding of fillers with polymers. Compounding has traditionally been carried out on two-roll mills, internal mixers and nowadays single and twin screw extruders. The extruders are being increasingly adopted as a means of giving a consistent product in a readily usable form, i.e. pellet or sheet. The several techniques available to produce filled polymer materials vary greatly in their methods and results. Extensive experience with pre-compounded glass-reinforced polymers reveals that the physical properties of the product vary significantly (Table 5.2) with various compounding techniques [16-19]. What one should look for in a compounding machine and what one should consider when setting up a compounding operation are very important to understand. 5.3.1 SELECTION CRITERIA

When selecting a compounding equipment, careful consideration should be given to the size of the unit required. The full cost at the maximum daily or hourly production rate must be established. It should be borne in mind that use of several small equipment units rather than one large unit will usually increase labor costs and initial investment. Bearing in mind process economics, equipment availability and preference, the decision as to what compounding technique and what

Table 5.2 Effect of compounding on mechanical properties of fiber glass-filled polypropylene Compounding technique3

wt % glass fiber Type of fibers Tensile strength (psi) Flex, modulus (m psi) Izod impact ft Ib/in notched Heat defl. temp. C 264 psi (0F)

Single- Continuous Twin-screw Twin-screw Twin-screw Twin-screw mixer compounder compounder compounder compounder screw 1 extruder 4 2 3 25 23 25 25 25 25 1/8 in chopped glass

1/4 in chopped glass

Roving

Roving

Roving

1/8 in chopped glass

6100

4700

4900

5800

8000

8000

580

460

600

550

550

550

1.45

0.7

0.9

1.1

1.2

1.3

264

153

203

184

268

266

% of fibers smaller than 0.5mm Remarks

29 Glass was fed into the feed section

Screw with Moderate very strong screw after sections addition of after glass addition of glass

Mild screw after addition of glass (No. 14)

Source: Ref. 16 (reprinted with kind permission from American Chemical Society,' Washington DC, USA).

compounding equipment to utilize often centers around the question of how much shear is required. Compounding of non-reinforcing fillers usually requires the highest degree of dispersion. Therefore, equipment should be able to generate high shear stresses to separate the agglomerates, particularly since these fillers usually have very small particle sizes. In compounding reinforcing fillers, the opposite approach is taken as low shear compounding must be used to prevent damage to the fillers. The main consideration is to wet the filler uniformly, devolatilize and discharge. Each material formulation presents a distinct problem. It is important to know how sensitive the compound and/or compounding ingredients are to temperature. The need to control or change the temperature and pressure during compounding should be considered. Purchase of light-

duty equipment should not be overlooked if the material will be dispersed at a high temperature. Also, thought should be given to the materials which will come in contact with the final product. When dealing with glass fibers, abrasion or corrosion of the compounding equipment plays a substantial role in economics. Although a product may not be contaminated by plain steel, a polished stainless steel unit is easier to clean. Labour that is saved may more than justify the additional cost of a stainless steel unit. Other constructions such as the use of high manganese steel may also be considered. The flexibility of the unit is also worth a good look. Can the compounding equipment be readily cleaned to change from one product to another and adapt to future production demands? Thus, the simplicity, accuracy and adaptability of the compounding process are important. The equipment for compounding fillers and polymers must fulfil the following requirements: (a) provide steady-state running conditions, (b) give reproducibility of processing conditions, (c) have ease of cleaning and (d) show versatility to adapt to new formulations. The equipment has to be chosen with care to ensure that correct product properties are achieved while no undesirable effects are introduced. Proper selection of compounding equipment and optimizing equipment parameters can substantially increase the mechanical properties of the final product. To achieve optimum material quality, the equipment for compounding fillers into a polymer must be capable of performing some of the following process tasks: 1. Incorporation and homogenization of fillers without exceeding degradation temperatures; 2. Generation of sufficiently high internal shear stresses to facilitate good dispersion of non-reinforcing fillers; 3. Provision of uniform shear stress to each filler particle at any heat history; 4. Provision of short and uniform residence time distribution to minimize heat history; 5. Provision of precise temperature control over the process to ensure narrow temperature distribution throughout the process and at discharge, again to regulate and minimize heat history. Using the above discussed factors as the criteria for analysis, the processor can select either a separate or in-line system, whichever is most efficient for his needs. Each has its merits and each its limitations. A batch system may be considered more flexible since it need not be tied in with other in-line equipment, but, on the other hand, batch to batch variations, for whatever reason, may not be acceptable in the final product. A continuous system should give a consistent product, but to

be fully effective it must have the back-up of reliable feeding and takeoff equipment. Some of the factors which need to be considered in the choice of machinery have been dealt with; the various categories of machine available and the important features of each category [20-32] are discussed next. 5.3.2 BATCH MIXERS

A. Open mills The mixing mill predates all the other mixing devices [21]. Originally many mixing operations were carried out on two- or three-roll open mills [2O]. As shown in Figure 5.5, it consists of two counter-rotating differential speed rolls with an adjustable nip and imposes intense shear stresses on the material as it passes the nip. The major component, i.e. the polymer, is first melted by the introduction of heat to the rolls and from the shearing action between the rolls. Once the polymer is melted, fillers and other additives are introduced. Extensive mixing takes place at the entrance of the nip due to extensional flow. Intensive (dispersive) mixing is accomplished in the nip. To achieve homogenization some means of lateral mixing of the banding action must be included. There is little transverse mixing and thus constant operator attention is needed to displace the strip transversely. Folding of the sheet by the operator prior to reintroduction to the milling action takes care of a random plug flow distributive mixing. This method allows very good temperature control of the product since the rolls can be cooled or heated and they present a very large surface area to the polymer system. The region of high shear is very

NIP

"BANDING' ACTION

Figure 5.5 Schematic diagrams of a two-roll open mill showing the banding action and nip. (Reprinted from Ref. 20.)

small avoiding large temperature build-up. The degree of dispersion obtained with such a system can be very high. The two-roll mill is a mixer ideally suited to the processing of high viscosity materials and was used by Birchall et al. [33] to produce very high solids content moldable cement pastes. Although still available for use, this type of system would not be considered for anything but very specialist operations. Obvious disadvantages arise from the open nature of the operation, difficulty in obtaining an initial melt and initial incorporation of pigments and fillers, giving long cycles. The system is inevitably dirty and it is difficult to achieve a uniform product. B. Internal mixers The internal mixer too is among the oldest members of the mixer family. The cross section of such a mixer is shown in Figure 5.6. It has adopted the open mill mixing principle but has a completely enclosed mixing chamber in which two rotors are mounted. The rotors are sealed at each end to avoid leakage of material from the chamber. The internal mixer is generally a very high powered machine with drives up to 300OkW for a machine with 600 litre mixing chamber [2O]. Maximum capacity is about lOOOlb/batch, and depending upon the material to be mixed, production rates can be as high as 40 batches/hr. Three different mixing actions take place in the internal mixer [21]: intensive or dispersive mixing around the tips of the rotor blade; extensive mixing due to simple shear flow and extensional flow between the chamber walls and the cylindrical portions of the rotors and in the entrance region to the narrow gaps between rotor tip; and distributive mixing due to random plug convection, as a result of the interaction between the two rotor tips. The rotors can be fully intermeshing in which case they must obviously rotate at identical speeds, or non-intermeshing when, as on a two-roll mill, friction ratios (generally 1.3 to 1) can be applied. The speed of the rotors can be changed and even varied during the mixing cycle. It can be lowered for a certain period of time to cool the material down and then increased again to restart the mixing. The residence time of the material in the mixer, for instance, can be changed by changing the cycle time. Cycle times as low as 2-3 minutes are claimed and with use of refrigerated cooling still shorter times may be made possible. The amount of material put into the chamber can be varied to change the intensity of mixing and the cycle can be interrupted to add material at times which are most favorable to obtain optimum results. An internal mixer is never operated completely full of material, though pressure is maintained in the mixing chamber by means of a piston or

Figure 5.6 Cross-sectional view of an internal mixer. (Reprinted from Ref. 21 with kind permission from John Wiley & Sons, Inc., New York, USA.)

ram. It is found that by increasing ram pressures certain polymers can be completely massed even without heating to rotors or chamber. One of the earlier shortcomings of the internal mixer system when used for polymers was that of the single mass form of discharge, which was difficult to handle. Nowadays effective dump extruders are employed which can accommodate up to three batches from the mixer to give a continuous product through some form of pelletizing system. These dump extruders can be fitted with screens for elimination of agglomerates or foreign matter, and venting ports for removal of volatiles. One of the problems inherent in an internal mixer is that of batch to batch variation in the product. However, accurate automatic weighing

Table 5.3 Advantages and disadvantages of the internal mixer Advantages Versatility of operations Generally good temperature control Robust machines Relatively simple to operate Short mixing times Disadvantages High capital cost High and uneven power load Batch to batch variations dependent on accurate feeding and process control Rapid temperature rise needs good control Source: Ref. 20.

and dosing devices minimize changes in formulation, while automatic cycle control operating on parameters such as temperature, time or power input can reduce mixing variations. In fact, the internal mixer can be incorporated into a continuous flow of production by complete automation and buffering of its pulsating output. The advantages and disadvantages of the internal mixer are summarized in Table 5.3. The three most common internal mixers available are those of Parrel Bridge (Banbury), Frances Shaw (Intermix), and Guix. The internal mixer is certainly a versatile equipment and its operation can easily be adapted to a wide variety of mixing problems. Prime applications are for rubber and PVC compounding. These equipments are also being used for processing polyolefins, ABS, and polystyrene, along with thermosets such as melamines and urea. In the tire industry which faces the most difficult mixing problems, this type of mixer is still practically the only accepted mixing device. The tire industry has accepted the fact that the batch type mixer will not be replaced by a continuous mixing device in the near future, especially since the radial ply tire came into being, because this tire requires very tough types of rubber. The same is not the case as far as the plastics industry is concerned. In fact, predictions made several years ago that the batch type mixer will disappear and be replaced by continuous mixing equipment has come true in the filled polymers industry. Some of the continuous mixing devices are discussed below. 5.3.3 CONTINUOUS COMPOUNDERS

To meet the present demands for higher product quality and uniformity, combined with higher volume requirements, continuous

compounding equipments are required [17]. In continuous compounding systems, variation between batches is eliminated and the possibilities of human error are minimized. Because of economics and the large volume requirements of filled polymers, continuous systems are usually preferred. Any efficient processing system will allow the compounder to operate economically at high volume, and continuous compounding systems, being more efficient than batch systems, have capacities of 3000kg/hr and higher [16]. The two essential elements in a successful continuous system are absolute control over residence time and residence time distribution Continuous compounding and mixing [23,24] is accomplished by a variety of machines ranging from single-screw to twin screw extruders with variations in design and additional equipment units to achieve a range of end products. The type of compounder or mixer chosen depends upon the specifications and properties required of the end product, volume requirements, overall economics, and whether the compounding is done in-house or by a custom compounder. The basic functions of any compounding extruder are to melt the polymer and to introduce a closely controlled amount of shear energy into the melt to properly disperse and distribute additives or fillers. A. Conventional single screw extruders The simplest form of continuous compounding equipment is the single screw extruder [20,25]. The single screw extruder has been used for many years in polymer forming operations. The advantages and disadvantages of the single screw extruder are given in Table 5.4. There is much to be said for the mechanical simplicity of such machines, where motor drive can be transmitted through a straightforward gear box designed to accept the high thrust forces generated. Table 5.4 Advantages and disadvantages of the single screw extruder Advantages Mechanically fairly simple Relatively easy to operate Capable of high pressure generation Disadvantages Lack of positive conveying characteristics Limited compounding and homogenizing capabilities Large machines with long L/D Source: Ref. 20.

In the single screw extruder, a series of unit operations combine to transform the solid polymer in pellet or granular form at room temperature to a molten mass at elevated temperatures and further drive the mass at sufficiently high pressures into downstream shaping devices. The sequence of events is accomplished by different sections of the screw, namely, the feed, melting and metering zones. The polymer pellets or granules (virgin or sometimes in combination with regrind of scrap) are normally fed to the extruder throat through a conical hopper. The hopper geometry, the height of the fill and the physical properties of the particulate solids dictate the rate of delivery of solids to the feed zones of the extruders as well as the pressure under the hopper. The feeding through the hopper relies on the influence of gravity as well as the interparticle and particle/wall friction forces for the rate at which the material is picked up at the extruder feed throat. At times there are aberrations such as arching, piping and funneling flow in the hopper because of which the feed is improper and uncontrolled. Feeding by hopper is at times referred to as 'flood feeding7 as against 'starve feeding' which occurs when material is metered into the extruder feed throat by special feeding devices at controlled rates. Single screw extruders [9,10,34] are pumps that convey the material by a combination of pressure and drag flows. The operation of a single screw extruder relies on the frictional forces between the polymer and the barrel to push the polymer forward. When the ratio of polymer/ barrel friction vs. polymer/screw friction is low, then the polymer will adhere to the screw and simply rotate with it, resulting in no forward motion. If the ratio is high enough, then forward motion will occur. The lack of positive displacement characteristics means that the machines are far from self wiping. The section which is downstream of the feed zone is the melting zone. When frictional forces are present, their energy becomes apparent as heat, resulting in increased product temperatures. This effect assists the melting process, which is enhanced by pressure generation in the screw. At times, very large temperature differentials are generated between melt and unmelt. In an extreme, the region of unmelt may be surrounded by low viscosity melt and effectively insulated from the thermal or shear energy of the screw. The actual melting process in a single screw extruder is fairly well understood [34-47], and a good understanding of the extruder action too has been developed due to theoretical and computer modeling [48-51]. It is now widely accepted that, in a cold feed extruder, melting occurs at the barrel and screw surfaces and a melt pool accumulates on the leading edge of the extruder screw fight (see Figure 5.7). Melting of the solid bed takes place slowly over a length equivalent to several

BAKEtEL SDlPACE

FUMOP MOLTEN POLYMER

SOIJD MELT INTERFACE

SOLED

KLIOHT

CIRCULATORY FLOWOF PREVIOUSLY MELTED POLYMER

SCREW

Figure 5.7 Melting process in the conventional single screw extruder. (Reprinted from Ref. 29 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

screw diameters and the rate of melting is dependent on heat transfer across the relatively small interfacial area with the melt. Eventually the solid bed breaks up and complete melting is finally achieved. This gradual process often results in very long screws and length to diameter ratios as high as 30:1 are not uncommon. Single screw sizes may range from 0.06m to 0.2m in diameter with lengths of 20 to 42 times length to diameter ratio. The fact that the process is gradual does mean that different regions of polymer within a flight experience different thermal histories. Difference in heat history often can lead to non-uniformity of the product. The conventional single screw extruder is designed mainly to melt material and to pump it against pressure [21]. The flow rate Q with which the material is transported is expressed by the following equation Q = ON-P^

(5.12)

The extruder is also capable of mixing because of the shear action which moves the material along a spiral path through the screw channel. This happens in the metering zone which is downstream of the melting zone. However, the degree of mixing in the conventional single screw extruder is somewhat limited owing to the fact that the

movement of the material through the extruder and the mixing action are derived from the same shearing action. In the conventional single screw extruder, fluid flows through a channel of approximately rectangular cross section with either variable or constant depth. The two sides of the rectangle are the leading and trailing surfaces of the flight, the bottom corresponds to the screw root, and the top is the inside surface of the barrel. Fluid mechanical analyses are based on a frame of reference fixed to the channel. For shallow channels, curvature effects can be neglected and the channel visualized as being uncoiled and laid flat with barrel as an infinite flat plate moving diagonally across the channel at an angle defined by the angle of the helix. Unlike the more complex extruders,the helical flow behavior in single screw machines is well characterized [52] and well documented in books [34,53]. Fluid is conveyed forward along the channel as a result of the drag flow induced by the axial component of the relative motion between the barrel and the screw, while a pressure flow builds up in the reverse direction due to the resistance offered by the die at the outlet. The axial flow largely determines the extruder throughput, but it is the transverse flow, generated by the transverse component of barrel rotation, that is primarily responsible for mixing. An element of fluid traces a helical path as it undergoes deformation in its passage through the channel; the helix angle at any point being determined by the local velocity field. Simplified velocity profiles based on the work of Carley et al [54], which neglects end effects at the flights, provide the basis for most theoretical analyses of mixing in extruders. Although most materials to be mixed in extruders exhibit non-Newtonian behavior, Newtonian behavior is frequently assumed for simplicity. Extensions to account for purely viscous non-Newtonian effects are straightforward [34,55] and few significant qualitative differences in mixing behavior are observed. Mohr et al. [2] were the first to theoretically investigate the mechanism of mixing in extruders; they assumed an inverse linear relation between the striation thickness and strain by analogy to large unidirectional shear. McKelvey [10] used an average strain as a measure of mixing. Pinto and Tadmor [56] extended this approach to account for the distribution of strains experienced by different fluid elements. Bigg and Middleman [55] obtained similar relations for power law fluids, and Erwin [57] presented modifications to account for the effect of mixing sections. In the mixing aspect of the extrusion process, velocity distributions and, subsequently, stress distributions are the most important parameters [18]. The stress distribution inside the screw channel determines the degree of mixing which can be achieved (Figure 5.8). The mixing action in the extruder channel occurs only on the portion of

VELOCITY DISTRIBUTION OF DOWN CHANNEL FLOW

STRESS DISTRIBUTION OF DOWN CHANNEL FLOW

VELOCITY DISTRIBUTION OF CROSS FLOW

STRESS DISTRIBUTION OF CROSS FLOW

Figure 5.8 Velocity and stress distributions in single screw extruders. (Reprinted from Ref. 18 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

the material which passes over the flights as it alone is exposed to the shear action. The velocity distribution in the extruder channel in the down channel and the cross channel direction result in the spiral path of the material. The material does not back flow even at higher back pressures, but rather moves back and forth in a spiral fashion over a relatively short distance. Due to the spiral motion, the residence time in the extruder is reasonably uniform but the mixing action is very limited. By observing the velocity and stress profile of down channel and cross flow, it can be seen that the zero shear point exists inside the channel. In this area, very little or no mixing will occur. This has been confirmed by actual experiments. Further, the lack of effective reorientation of the material elements at the flights causes the bulk of the interfacial area to be oriented parallel to the channel bottom in a relatively short downstream distance,

regardless of the initial orientation. Consequently, the mixing achieved in single screw extruders is generally poor [58]. Efforts to improve the mixing efficiency of the single screw extruder have, therefore, never ceased. B. Modified single screw extruders When mixing is required then variations to the basic machine are usually considered. This involves installing mixing devices which would shift the zero shear stress point to various locations in the screw channel and hopefully achieve an adequate mix, if the equipment has a sufficient processing length. Most of these devices are designed to break down the smooth flow of material through the extruder, thereby mixing the melt with unmelt to give a more uniform distribution of polymer. (Figure 5.9). The devices being used include specialized screws for use with mixing flights or pins along with variations in pitch and screw form designed to break down the normal melt pattern or smearing sections where the polymer stream is forced over narrow gaps in the screw. There are a number of single-screw systems available with improved mixing devices such as the EVK, the fluted screw, the Maillefer type and others, which improve the mixing efficiency of the flow by redistribution of streamlines and reorientation of the intermaterial area at the expense of only a small increase in power requirements. In the simplest mixing device consisting of mixing pins, each mixing pin divides the helical flow into two streams, which are again combined into one upon passing the pin. This leads to growth of the interfacial area upon the division and combination of flow and introduces further velocity gradients, thereby increasing the total strain on the material. In the patented EVK screw design, material distributing elements are installed over the whole length of the screw which ensure an homogeneous temperature distribution. The screw length and shearing clearance requirement, however, vary depending on the material to be processed and the type of filler used. Wherever the same material formulation is run continuously without any major changes, the system performs quite well. In the fluted screw design, the barrier-type restrictions basically introduce narrow, albeit short, gaps through which all the melt elements are forced to pass at high shear stresses. This brings in some level of dispersing action besides reorienting the interfacial area, redistributing the streamlines and increasing the imposed total strain. These barriers may take different forms. A simple form is the dams or series of dams of almost the full channel height welded across the channel at selected locations. The potential problems with the fluted

Parallel interrupted mixing flights

Mixing pins

Ring barrier type

Union carbide fluted mixing section barrier type

Undercut spiral barrier type Figure 5.9 Mixing section designs in single screw extruders. (Reprinted from Ref. 25 with kind permission from The Institution of Chemical Engineers, Rugby, UK.)

screw are the hold-up of the material and dead spots [59,6O]. These can be minimized by proper selection of the helix angle, tapering the inlet channel to zero depth and by reversing the depth profile in the outlet channel. In the Maillefer type screw design, there is a spiral barrier or secondary screw flight extending over the entire length of the modified section of the screw. The mass and force balances across the Maillefer type screw are available [61] and can be used to estimate the increase in power requirements due to this modification. Additional barriers or flights may be introduced to split the flow into two or more streams, which in the limit arrives at a series of longitudinal grooves or flutes spread around the circumference of a cylinder. Static mixers, i.e. those which are motionless and do not contain any moving parts, may be incorporated into single screw extruders between the screw and the die in order to introduce distributive mixing. A review of the various types of static mixers, their mixing characteristics and performance is available [62]. They consist of a string of alternating right and left handed helical elements fixed in a tubular housing which helps in splitting and recombination of streams, resulting in a predicted increase in the number of striations. The energy for mixing is derived from the pressure loss incurred as the melt flows through the static mixer. On account of the substantial pressure drop introduced by a battery of these mixing elements, they have restricted use in filled polymer melt systems especially those with higher loading and high viscosities. Many different modifications of the geometry of the single screw extruder have been contrived (all of which are not described herein) but even then, the achievable level of dispersion of fillers is limited. Improvement in filler dispersion can be achieved only by powerful shear forces. If the speed is increased to intensify mixing, the output increases at the same time, whereby back pressure control is lost but, more importantly, the energy input to the polymer due to the viscous shear forces rises even more rapidly and can lead to excessive melt temperatures. It is this heat build-up which eventually limits screw speeds to even as low as 100-150 rpm. In summary, the limited compounding and homogenizing ability of single screw extruders usually restricts their application to straightforward melting and extrusion operations but use of mixing devices can extend their range. Their favorable economics makes them attractive for low volume compounding jobs especially for filled systems of mainly commodity plastics [63]. For tougher applications, such as compounding fillers with engineering polymers at low as well as high loadings, single screw kneaders and twin screw extruders are more ideally suited.

C. Single screw kneaders This form of continuous compounding equipment may appear as a variation of a single screw extruder. But truly it is not and hence not included in the category of modified single screw extruders, because of the radically different mixing action in it [27,64]. In this single screw compounding equipment, various types of protrusion, grooves or profiles are built into the barrel with corresponding elements along the flights of the screw in the particular section known as the 'kneading section' as shown in Figure 5.10. This induces dispersive mixing as the material passes repeatedly through the high shear stress zones in the narrow gaps formed by the interrupted screw flights and the kneading teeth on the barrel (Figure 5.11). The mixing action is unique in that the rotational motion gets superimposed by reciprocating motion, thereby giving a kneading effect to the melt. This results in a good dispersion of fillers in the polymer system even at relatively low shear rates, thus allowing good processing control for temperature sensitive materials. The kneading action is illustrated in Figure 5.12. It can be seen that the kneading teeth wipe off all the corresponding flanks of a flight during one full turn, and the material thus gets subjected to high shear stresses in the small gap between the flight flank and the barrel teeth, thereby giving good dispersive mixing. As a consequence of the good mixing action such extruders are generally only one half or even onethird of the length of conventional single screw units. Further, since the unit does not generate high pressures, when pelletized product is required, the kneader is generally coupled to a single screw extruder to pump the polymer up to die pressures. COOLED SOLIDS CONVEYIbTG SECTION

ENEADINGTEEIH FEED SCBEW ENEAD]NQ SCREW

SCREW FLIGHTS

ZONES

Figure 5.10 Schematic diagram of a single screw kneader. (Reprinted from Ref. 23 with kind permission from Gulf Publishing Co., Houston, Texas, USA.)

ISPLTT BARREL 2HEATJNGJACKET 3ENEADttJGTOOTH 4 SCBEW FLIOET SNAKROWQAP 6 CLAMPING BAR

Figure 5.11 Cross-sectional view of a single screw kneader. (Reprinted from Ref. 20.)

Typical examples of the single screw kneader are the Buss Kokneader and the Baker-Perkins Ko-kneader.

D. Twin screw extruders Twin screw extruders evolved as a result of the desire to overcome some of the limitations of single screw extruders [65]. They are the most expensive mixers per kilogram of output. The advantages of twin screw extruders are listed in Table 5.5. The term 'twin screw extruder' is basically a generic term but to treat all twin screw machines in one general category is truly not correct. Various types of twin screw extruders are commercially available and these differ widely in their operating principles and function [18] as can be seen from Figure 5.13. They differ in their abilities for distributive and dispersive mixing and hence in their application areas. A possible classification of twin screw extruders [30] is shown in Table 5.6 along with commercial examples for each category. However, it should be borne in mind that this listing is by no means exhaustive. The main distinction is made between intermeshing and nonintermeshing twin screw extruders (Figure

STARTING POSITION

FIRST QUARTER TDlN (90°)

FULL TURN COMPLETED

Figure 5.12 Kneading action in a single screw kneader. (Reprinted from Ref. 23 with kind permission from Gulf Publishing Co., Houston, Texas, USA.) Table 5.5 Advantages and disadvantages of the twin screw extruder Advantages Positive displacement characteristic gives throughputs independent of the nature of the product being processed Earlier and more complete melting allows machines to be shorter and gives better mixing Self-wiping characteristic avoids any product hang-up in the machines Disadvantages High capital cost Mechanically more complex with difficulties of fitting adequate thrust bearings to two closely positioned shafts Source: Ref. 20.

PARTIALLY INTERMESHING

COUNTER-ROTATING

LENOTHWISEAND CROSSWISECLOSED LENGTHWISE OPEN THEORETICALLY AND NOT POSSIBLE CROSSWiSE CLOSED THEORETICALLY POSSIBLE LENGTHWISE AND CROSSWISE OPEN 5_ BUTPRACTICALLY NOT REALIZED LENGTHWISE OPEN AND CROSSWISE CLOSED

CO-ROTATING THEORETICALLY NOT POSSIBLE "SB" DISKS SCREWS

FULLY INTERMESHING

SYSTEM

NOT INTERMESHING

NOT INTERMESHING

INTERMESHtNG

SCREW ENGAGEMENT

THEORETICALLY NOTPOSSIBLE

LENGTHWISE AND CROSSWISE OPEN

LENGTHWISEAND CROSSWISEOPEN

Figure 5.13 Types of twin screw extruders. (Reprinted from Ref. 18 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

5.14(a)), each of which may be divided on the basis of the direction of the rotation of the screw into co-rotating or contra-rotating (Figure 5.14(b)), also termed counter-rotating. The benefits and the drawbacks of each system have been discussed in many publications. However, arguments in favor of one system more often seem to be based on a personal preference or bias than on an objective engineering analysis. The plain fact that the systems have coexisted for a number of years must be sufficient evidence that these systems fulfil a practical need. In some applications, the co-rotating extruders may be better suited; in other cases, counter-rotating extruders may be preferred. For many filling operations, however, twin screw co-rotating systems have the widest application because of their high degree of flexibility. Instead of arguing against or in favor of a particular system, it is more useful to determine both the advantages and the disadvantages of the systems and to identify the applications for which each system would be best suited. The first known twin screw device was an intermeshing co-rotating extruder invented in 1939, and patented in 1949 [66]. Developments in twin screw extruders have come a long way since then. The mechanisms of different types of twin screw extruders shown in Figure 5.13 make a distinction between axially open machines, where there is a

Table 5.6 Classification of twin screw extruders

Corotating extruders

Low speed extruders -LMP Colombo —Windsor —Bamberger High speed extruders -Werner and Pfleiderer —Baker Perkins -Berstorff Conical extruders —Kraus Maffei —Cincinnati Milacron -AGM

lntermeshing extruders

Counterrotating extruders Twin screw extruders

Counterrotating extruders

—Cylindrical extruders —Leistritz —Kraus Maffei —Maillefer —Reifenhauser -Mapre —Kesterman —Troester, etc. Equal screw length -Bausano -Japan Steel Works Unequal screw length —Welding Engineers

Nonintermeshing extruders Corotating extruders Source: Ref. 30 (reprinted with kind permission from Society of Plastics Engineers Inc., Connecticut, USA).

No commercial examples known

intermeshing

non-intermeshing

co-rotating

co-rotating

contra-rotating

contra-rotating

Figure 5.14 Diagram showing the difference between (a) intermeshing and nonintermeshing as well as (b) co-rotating and contra-rotating or counter-rotating types of twin screw extruders. (Reprinted from Ref. 20.)

continuous passage between the inlet and outlet, and the axially closed machines, where the passage is interrupted at regular intervals. Considerable variations of screw design are possible to balance pumping and mixing characteristics [67]. The right-handed, forward, regular flighted elements shown in Figure 5.13 are only one of many types of elements employed in twin screw extruders. The other types may include left-handed, reverse, regular flighted elements, forward and reverse kneading discs, shearing discs, etc. Most twin screw extruders can be assembled using the building block principle, which involves the use of screw elements of different lengths/pitches and special kneading elements of various widths available as blocks that are interchangeable. This makes it possible to design the processing section exactly as required for obtaining optimum processing conditions because the screw elements and kneading elements blocks can be varied to tailor-make the screw configuration for the shear intensity required by a specific material. If properly designed,

twin screw extruders provide maximum process control, especially with respect to shear and stock temperature. Also, twin screw compounders allow the removal of large quantities of volatiles. The diversity among twin screw extruders is so large that a comprehensive discussion of all the various types would be an enormous undertaking. A description of some of the important characteristics of three basic types of twin screw extruders is given below. Nonintermeshing co-rotating twin screw extruders are not produced commercially and hence are not treated in this section. (i)

Intermeshing co-rotating screw extruders

These have the flights of one screw protruding into the channel of the other screw, thereby providing positive displacement pumping [68] when both screws rotate in the same direction. The degree of intermeshing can range from almost fully intermeshing to almost nonintermeshing with a corresponding range in the degree of positive conveying characteristics. The design of intermeshing screws includes variations in lengthwise and crosswise open and closed positions, which directly affect the conveying conditions, the mixing action, and the pressure buildup capacity of the screw system. Co-rotating twin screw extruders are commonly produced only as fully intermeshing and operate on the building block principle. Depending on the requirements of the compounding task at hand, proper screw elements are selected and assembled along with the right choice of kneading elements blocks. Commonly employed elements include right-handed and left-handed (forward and reverse) regular flighted elements, forward and reverse kneading discs, as well as mixing and shearing pins. For regular flighted elements, fully intermeshing corotating screws are open lengthwise, but are closed crosswise, except for leakage over the flights. When the twin screws are intermeshing, the extruder can be considered to be self-wiping, that is, all the surfaces within the processing chamber are wiped of polymer. This characteristic is important when heat sensitive materials are being processed. Selfwiping is advantageous to the operational economics as the screws require much less downtime for cleaning. Efficient self-wiping ensures no material hang-up in the machine and can be demonstrated by a short residence time distribution for the material being processed (Figure 5.15). In intermeshing co-rotating twin screw extruders, self-wiping characteristics can be improved through the use of kneading elements within the screw configuration which can allow varying degrees of mixing energy to be developed in the machine. Such elements are

INDICATOR

GOOD SELP WIPING

CONCENTRATION

POORSELF WIPING

TIME

Figure 5.15 Residence time distribution for self-wiping twin screw extruders. (Reprinted from Ref. 25 with kind permission from The Institution of Chemical Engineers, Rugby, UK.)

particularly useful in achieving good dispersion of fillers in the polymers. The co-rotation of the screw helps in the homogenization and dispersion because material is transferred back and forth between the screws. Co-rotating twin screw extruders allow for the possibility of adjusting the amount of axial mixing. Mathematical modeling can be found in the works of Erdmenger [69], Armstroff [70], Kim et al [71] and Booy [72,73]. Mixing studies were done by Todd [74], Wyman [75], Maheshri and Wyman [76,77]. Experimental data have been obtained by Werner [78], Jewmenow and Kim [79], and Kim et al [71,80,81]. In co-rotating machines, the flow pattern of polymer system is such that the material is actually conveyed in a figure 8-shaped pattern. Observing the velocity profile and the shear stress distribution for corotating machines as shown in Figure 5.16, it is seen that zero shear stress point can be influenced by operating variables, such as by changing the throughput and/or screw speed as well as by changing geometry. It is imperative that uniform stress distribution is maintained even with increased throughput. This is possible in corotating machines, because the degree of uniform dispersion is directly related to stress/strain distributions, and the most uneven stress distribution occurs at closed discharge which is meaningless in actual operation. In co-rotating machines, the proper selection and arrangement of individual screw sections and kneading elements [82] is customary in

Figure 5.16 Shear stress distribution in a co-rotating twin screw extruder. (Reprinted from Ref. 18 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

Figure 5.17 Screw design for (a) co-rotating (b) counter-rotating twin screw extruders. (Reprinted from Ref. 18 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

order to further improve the uniformity of stress distribution (Figures 5.17 and 5.18). Depending upon the design of the screw, number of tips or flights, the material forms a number of continuous tapes which spiral around both screws. In a three flighted screw, the material moves in five parallel channels which are continuous from one end of the extruder to the other, similar to a multi-flighted single screw. By selecting different numbers of flights, the intensity of the shearing action can be changed. Special shearing sections can be arranged between screw sections to perform intensive dispersive mixing. Co-rotating screws usually achieve compression by some means of restriction, either in the barrel or on the screw themselves. Overlapping disc or reverse flight screw threads can achieve the desired effect. Transfer of product from one screw to the other ensures rapid homogenization of melt with unmelt resulting in earlier and more complete melting in a relatively short distance. As a consequence of this rapid melting, co-rotating twin screw extruders tend to be much shorter than comparable single screw units. The residence time in the co-rotating twin screw extruder is in general more narrow than in the single screw extruder. Because of that uniformity, it is possible with a twin screw mixer, provided it is properly designed, to achieve the desired degree of mix with less

Figure 5.18 Screw design with kneading elements for co-rotating twin screw extruders. (Reprinted from Ref. 18 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

mechanical energy input and, therefore, less heat buildup than with other mixers. By starve feeding of a twin screw mixer, the amount of mechanical work input to the material can be varied. The materials to be mixed have to enter the twin screw mixer continuously in their proper proportions because the twin screw mixer, owing to the fact that the material passes through in almost a plug flow fashion, does not intermix in the axial direction. Since the material in this extruder is transported by positive displacement and not by frictional and viscous forces, the melt temperatures do not rise and, in fact, generally heat has to be applied externally. The intermeshing co-rotating extruders can be further subdivided into low and high speed machines as can be seen from Table 5.6. These low and high speed extruders are very much different in design, operational characteristics, and applications. The low speed extruders have a closely fitting flight and channel profile. Therefore, they have a high degree of positive conveying characteristics. However, because of the small mechanical clearances, they have to run at low screw speeds (generally 10-20 rpm) to avoid buildup of high local pressures that cause machine wear. These machines are used primarily in profile extrusion applications. In high speed co-rotating extruders, the design is such that there is considerable opportunity for the material to leak back from one screw channel into a channel of the other screw. Thus, these machines have a low degree of positive conveying characteristics. Because of the openness of the channels, material is easily transferred from one screw to another and pressure generation in the intermeshing region is less pronounced. This allows extruders to run at high screw speeds, as high as 500 rpm. These machines are primarily used in compounding operations, where use is made of the high shear rates and frequent reorientation of the material in the extruder. (ii) Intermeshing counter-rotating twin screw extruder This acts as a positive displacement pump due to the intermeshing of the screws as discussed earlier. The conveying capacity of the machine is virtually independent of the nature of the material being processed, especially when the screws are counter-rotating. However, it should be noted that full intermeshing is necessary but not sufficient condition for positive conveying. In some geometries, there is very little sealing of the screw channels, even when the screws are fully intermeshing. Positive conveying requires that the screw channels are closed off so that the material contained in the various channel sections is fully occluded. Any amount of back leakage into

upstream channel sections will adversely affect the positive conveying behavior. The output in volume per unit time is given by the difference between the theoretical flow rate and the sum of the leakage flows described in Figure 5.19 Q = QIH - QL = 2mNVc/60 - (Qr + Qc + Q4 + Q8)

(5.13)

where m is the number of thread starts per screw, Vc is the volume^ of the C-shaped channel between the flanks of successive flights and N is the rotation rate. Qr is the leakage between screw flight and barrel wall, Qc is the leakage between the screw flight and the other screw, Qt is the leakage between the flanks of screw flights and Q8 is the leakage between flanks perpendicular to the plane through the screw axis. The leakage paths have a strong influence on residence time in the extruder. The intermeshing point for counter-rotating screws acts as a 'calender nip7 forcing material through it. Only a small proportion of the material passes through the nip between the screws, while the remainder is carried along axially in closed chambers. Since the relative speeds of

TRANSPORT DIRECTION

Figure 5.19 Leakage flow paths in an intermeshing counter-rotating twin screw extruder. (Reprinted from Ref. 65.)

the two screws must be identical, high 'calender nip' shear forces are not generated. Material is forced through the nip but then out against the sides of the barrel. This effect can aggravate sticking to the barrel walls. The passage of material between the 'calender nip' generates normal stresses which tend to force the screws apart leading to potentially higher barrel wear and limitations on screw speeds for counter-rotating screw extruders. This is one of the reasons why counter-rotating extruders generally operate at lower screw speeds than co-rotating units, typically at less than 200 rpm [83]. Due to the lower relative velocities of the screws at the meshing point, the shear stresses in the counter-rotating extruders would be lower. In an analysis of the velocity and stress distribution in counterrotating twin screw machines, it is seen that there is actually no improvement regarding velocity profile and zero shear stress point location, in comparison to single screw machines. When superposition of the shear stress distribution of down channel and cross flow (for a given coordinate system) is done for counter-rotating machines, a characteristic minimum inside the screw channel is observed in the area of y/h = 0.38 (Figures 5.20 and 5.21). Shearing takes place in each individual spiral chamber between two contacts of the faces of flights with the surface of the root of the screw. The residence time distribution is therefore very narrow which is important for processing of heat

COUNTER-ROTATING MACHINES: SUM OF STRESSES OF DOWN CHANNEL AND CROSS FLOW COMPONENTS Figure 5.20 Sum of stresses of down channel and cross flow components for counter-rotating twin screw extruders. (Reprinted from Ref. 18 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

DOWN CHANNEL FLOW

CROSS FLOW

Figure 5.21 Schematic flow orientation in counter-rotating twin screw extruders. (Reprinted from Ref. 18 with kind permission from Society of Plastics Engineers Inc.,

Connecticut, USA.)

sensitive materials. The mixing efficiency can be increased by the introduction of special mixing sections. Since closely intermeshing counter-rotating twin screw units convey material by positive displacement in a series of closed C-shaped segments, compression is generally achieved, as in single screw extruders, by continuously decreasing the pitch or the flight depth. Thus the melting process can be an extended one and takes place gradually over a length equivalent to several diameters. Intermeshing counter-rotating twin extruders are available with conical as well as cylindrical screws. Conical twin screw extruders provide more space for the installation of thrust bearings while the cylindrical ones have the advantage of being amenable to be built by the building block method [3O]. The intermeshing counter-rotating twin screw extruders are normally used when narrow residence times for compounding are required, for example, PVC profile extrusion and color-concentrate preparations [84]. (iii)

Non-intermeshing counter-rotating twin screw extruders

These have the two screws rotating at the same frequency in different directions with the respective screw flights placed either facing each other or in a staggered fashion. Since the flights of one screw do not protrude into the channel of the other screw, the kinematics of the melt flow are quite similar to those in a single screw extruder except for the difference caused by the interaction of the flow field between the two screws. The meeting of the streams at the interface between the two screws increases the efficiency of extensive mixing and hence nonintermeshing counter-rotating twin screw extruders perform better than single screw extruders when called upon to do the same job. In fact it has been shown experimentally that the leakage flow at the nip can

penetrate and form re-oriented regions across the channel depth [85]. If the flights of the screw are staggered, this mode gives rise to even better extensive mixing, but, of course, at the cost of decrease in pumping efficiency. The disadvantage of the excessive leakage flow is that the residence time distribution in non-intermeshing counterrotating twin screw extruders is broader than that in single screw extruders [86]. Forward flighted, reverse flighted as well as cylindrical screw elements have been used [87] in non-intermeshing counter rotating twin screw extruders. The reverse flighted elements are often introduced with the idea of inducing extensive mixing in the longitudinal direction while the cylindrical elements are for sealing purpose when liquid additions are to be done in small quantities and hence located near to the liquid addition ports. Non-intermeshing screws are always open both lengthwise and crosswise. Non-intermeshing counter-rotating twin screw extruders are generally used in applications where dispersive mixing is not a necessity. The applications include polymer blending, preparation of hot melt adhesives and pigment concentrates. They can be used in the preparation of filled systems such as incorporation of glass fiber reinforcements in engineering polymers because the non-intermeshing screws will certainly protect the glass fiber from damage. E. Continuous internal mixers This form of mixing equipment is basically the conversion of the popular Banbury mixer of Parrel Corporation from the batch to continuous mode of operation [88]. In fact, the cross-section of the continuous internal mixer in the direction perpendicular to the axis of rotation of the rotors resembles that of the batch type Banbury mixer as shown in Figure 5.6. The design of the continuous internal mixer in the direction of the axis of rotation of the rotors can be seen from Figure 5.22. The twin rotors do not intermesh and rotate at slightly different speeds. Their special design ensures a high degree of longitudinal mixing and the discharge is adjusted to control the amount of material in the mixing chamber. The design of the continuous internal mixers follows, in most cases, the philosophy of controlling the forward action of the material through the mixer and the mixing action independently. Thus, the mixer basically involves a separate conveying section which propels the ingredients forward and a further section which tends to move the ingredients in the opposite direction for intensive mixing. In the intensive mixing section, the material is dispersed by passage through small gaps found between the rotors and the chamber walls, coupled

MIXING CHAMBER

FEED

ROTQR

DISCHARGE ORIFICE GATE

MATERIAL CNSCHARGtNG

Figure 5.22 Schematic diagram of the continuous internal mixer in the direction of the axis of rotation of the rotors. (Reprinted from Ref. 21 with kind permission from John Wiley & Sons, Inc., New York, USA.)

with kneading action between the rotors and roll-over of the material itself. There exists a very sensitive equilibrium condition under which the material is being moved forward against low resistance in the discharge orifice. If the resistance at the outlet is changed, the degree to which the mixer body is filled changes until the forward blades are able to overcome that resistance. If there is more material in the mixer, the actual residence time is longer and, if the throughput is kept constant, the mechanical work done on the material increases. Another way of changing the input of mechanical work to the material is by change of rotor speed without changing the throughput which is possible because the throughput is determined by the feed rate into the mixer. The materials have also to be fed continuously in the proper proportion because the mixing action in the axial direction is not very effective. The distribution of residence time and shear deformation in the continuous internal mixer with non-intermeshing mixing blades is broader than for the twin screw extruders of any type. Since this type of mixer cannot generate any appreciable discharge pressure necessary to extrude the material, it is often combined with an extruder for pelletizing. The continuous internal mixer is employed in applications involving polyolefin homogenization, PVC compounding as well as in the preparation of low filler loaded polyolefin composites. F. Co-rotating disk extruders The co-rotating disk extruder introduces the concept of two moving drag inducing surfaces [89,9O]. This is in contrast to all the earlier discussed continuous compounders wherein there was one moving boundary (the screw) along with one stationary surface in the channel (the barrel). This particular equipment known as the Disk pack processor manufactured by Parrel Corporation, consists of a single rotor with toroidal processing chambers fitted in a circular housing. The processing chambers may be connected in parallel, in series or in combination, and are assembled on the building block principle. In the co-rotating disk extruder, the material is transported by the action of two moving walls, the sides of the disks, which can form a parallel-faced or a wedge-shaped chamber as shown in Figure 5.23. An analysis of the velocity distribution in the parallel-faced chamber with a restriction to flow at the outlet shows two flow patterns [21]. A circular flow in planes perpendicular to the axis of rotation and another circular flow in cylindrical surfaces around the axis of rotation. The superposition of both flows is conducive to extensive mixing. To achieve intensive mixing, mixing blocks with narrow clearances between the surfaces of the blocks and rotating walls are introduced as

CHANNEL BLOCK WEDGE-SHAPED RADIAL-THICKNESS PROFILE

PARALLEL-FACED RADIAL-THICKNESS PROFILE

INLET OUTLET

SHAFT

SHAFT

DISK

DISK

BARREL

BARREL.

PROCESSCHAMBER Figure 5.23 Sectional view of co-rotating disk extruder without mixing blocks. (Reprinted from Ref. 21 with kind permission from John Wiley & Sons, Inc., New York, USA.)

shown in Figure 5.24. The extensive mixing action described above takes place in the material accumulated in front of the mixing blocks. Mixing pins are often necessary to impose high strains and reorient the melt. This is because, at the same pressurization capability, the shear rates with two walls moving are considerably smaller than with only one moving wall. Co-rotating disk extruders are useful in the preparation of color concentrates, polymer blending as well as in the compounding of fillers with polymers. 5.3.4 DUMP CRITERIA

From among the compounding techniques discussed earlier, the most common practice for incorporating fillers in polymers has been to make use of internal mixers such as the Banbury mixer for batch mixing and the twin screw mixer/extruder for continuous operation. In laboratory small-scale processes, the Brabender plasticorder (Brabender, Duisburg, Germany) or the Haake Rheocord (Haake Buchler Instruments, Inc., Saddle Brook, NJ) is often used for conducting the mixing operation. They have accessories for batch mixing as well as continuous operation, but the batch mixer is most popularly used for laboratory research purposes. It is most important to determine the dump criteria, i.e. the moment when the mixing is deemed complete.

CHANNBLBUOCK MUET

OUTLET SHAFT

MIXING BLOCK

BARREL DISK

CHANNEL BLOCK

INtCT OUTLET MIXING BLOCK DISK Figure 5.24 Sectional view of co-rotating disk extruder with mixing blocks. (Reprinted from Ref. 21 with kind permission from John Wiley & Sons, Inc., New York, USA.)

In the batch mixers, the completion of the mixing operation is normally adjudged by the observed constancy of the torque level in the mixer's motor drive unit after a certain length of time. A constant torque implies internal homogeneity of the mixture that is achieved within the system's capability limits. A typical torque-time curve generated during a mixing operation [91] of filler and polymer is shown in Figure 5.25. The major peak represents the moment when the addition of all the filler and polymer to the mixing bowl is complete and when the shearing action begins to disperse the filler into the polymer matrix. As the filler becomes better dispersed, the viscosity of the composite begins to decrease, as exemplified by the recorded torque, till it reaches its equilibrium value beyond which the mixedness would not improve or would improve only marginally due to the system constraints. In fact, it is not advisable to continue mixing once the equilibrium torque level is reached because it could only lead to thermomechanical degradation of the polymer, especially in highly

TEMPERATURE

(*C)

(M-KG) TORQUE

TIME (mfn.)

Figure 5.25 A typical torque-time curve generated in an internal mixer during compounding of fillers into polymers. (Reprinted from Ref. 91 with kind permission from Marcel Dekker, Inc., New York, USA.)

filled systems, where local shear rates between filler particles could be extremely high. Time and temperature have been the most commonly used criteria to determine when to terminate the mixing process. The aims are to guarantee the quality of the end product, avoid overmixing, and reduce variation between batches. There is an increasing body of evidence that a more precise and reproducible control of the mixing cycle can be obtained by following the energy inputs at various stages in the cycle. This is because the changes in power consumption are indicative of the wetting, deagglomeration and dispersion stages in the process. Mixing to a preset time does not allow for variations in metal temperature at the start of the mix, for cooling rate or for ingredient addition times. This can result in significant batch to batch variation. When mixing to a predetermined temperature the major limitation is the accuracy with which the batch temperature can be measured. The large heat-sink provided by the machine often makes temperature measurement inaccurate, though there are now available infrared probe thermocouples that are more accurate. Mixing to a predetermined power input into the batch overcomes these limitations and gives improved batch to batch consistency for mixes requiring longer mixing times. However, following the work or

energy input alone is not sufficient. In addition, one has to establish the effect of process variables on the shape of the power curve. In other words, a recording chart, which indicates both the instantaneous power and the integrated power or work done is required. Furthermore, these data should be additional to the established control criteria of time and temperature and not a completely separate alternative set of criteria. 5.4 COMPOUNDING/MIXING VARIABLES A number of variables affect the final quality of the mix. Some variables are more sensitive to the changes than others. It is important to identify these variables and understand their sensitivity so that the mixing can be carried out under optimum conditions. Variables affecting the compounding operation could be the machine variables or operating variables as shown in Table 5.7. It has been recognized that these variables influence the quality of the compounded filled systems, but the separate effects of these changes are often difficult to determine. This is mainly due to the strong interactions between most of these variables. Hence, in the discussion of these variables and their effects, some general statements may have to be made and clear-cut guidelines may at times be difficult to give. In all cases, the efficacy of the compounding action is adjudged by the level of dispersion achieved as exemplified by a study of the rheological characteristics of the compounded filled polymer system rather than through the product properties in the solid state. Capillary rheometry has been the most popular technique for obtaining the steady state rheological response, as it provides information at deformation rates high enough to be close to those encountered during polymer system processing. However, the high shear rates achievable in the capillary rheometer do not permit the rheological parameters to Table 5.7 Variables affecting the compounding operation Compounding variables

Machine variables

Mixer type

Rotor Mixing geometry time

Operating variables

Rotor speed

* Applicable particularly to internal mixers.

Ram Chamber pressure* loading*

Mixing Order of temperature ingredient addition Next Page

Steady shear viscous

r*

properties

D

There is extensive literature on the rheology of filled polymer systems [1-75] including reviews [41,42,49] and chapters in a number of books [76-85]. The bulk of the literature deals with rheology of systems in the filler loading range of 20-40% by volume. In the present context, this range will be considered as the low filler loading range and will be referred to thus whenever necessary. Aspects relating to this loading level have been effectively reviewed by Utracki and Fisa [41]. The filler loading range between 40 to about 60% by volume will be referred to here as the high filler loading range. A review of the rheology of highly filled polymer melt systems is also available [82]. The enhanced interest in the rheology of highly filled polymer systems is the intended use of the polymers as binders during ceramic processing [48,66-70,73-75] and for the preparation of functional composites [60-63]. Table 6.1 Various types of fillers used in Ref. 29 Fillers

Source

Carbon black Titanium dioxide Calcium carbonate Mica Glass beads Franklin fibers Aramid fibers Cellulose fibers Glass fibers

Continental Carbon du Pont Pfizer Hayden Mica Company Potters Industries, Inc. Certain Teed Corp. du Pont Westvaco Fiberfil

Specific Size, (im Aspect ratio gravity (particle or fiber diameter) 1.8 4.1 2.7 2.9 2.48 2.96 1.44 1.5 2.5

0.045 0.18 0.5 Flake, 25-50

10 0.5-2 12.2 12 12.7

-100 -60 -100 -10

Source: Ref. 29 (reprinted with kind permission from American Chemical Society, Washington DC, USA).

The rheology of filled polymer systems will be discussed in this and subsequent chapters under various headings, namely steady shear viscous properties, steady shear elastic properties, unsteady shear viscoelastic properties and extensional flow properties. The effects of filler type, size, size distribution, concentration, agglomerates, surface treatment as well as the effect of the polymer type will be elucidated wherever information is available. The material parameters that are involved in steady shear measurements are the steady shear viscosity and the normal stress difference. These will be treated in separate chapters as the information available on these two parameters is not balanced equally. The information on steady shear viscosity of filled polymer systems far outweighs that which is available on other parameters. Hence, the effect of a number of variables on the steady shear viscosity is discussed in utmost detail. 6.1

EFFECT OF FILLER TYPE

The effect of filler type on the steady shear viscosity of a filled polymer system has been brought out in the work of White et al. [29]. The polymer used was polystyrene of one specific grade, namely, Dow Styron 678U. The filler loading was fixed at 20vol%. Nine different types of filler were used as summarized in Table 6.1. The filler types included rigid fillers (such as glass fibers, mica, glass beads, calcium carbonate, carbon black, titanium dioxide and Franklin fibers) and flexible fillers (such as aramid fibers and cellulose fibers). They covered different types of shapes such as spherical (glass beads), particulate (calcium carbonate, carbon black, titanium dioxide) and fibrous (glass fibers, aramid fibers, cellulose fibers, Franklin fibers). Steady shear measurements were carried out using the Rheometrics Mechanical Spectrometer at a fixed temperature of 18O0C using a coneplate mode. Plots of steady shear viscosity rj as a function of shear rate y for various filled systems at a fixed loading of 20vol% are shown in Figures 6.1 and 6.2. The curves exhibit qualitative as well as quantitative differences. The filled systems containing glass fibers, aramid fibers, cellulose fibers, mica and glass beads exhibit qualitatively similar behavior with low shear rate Newtonian viscosities and a decreasing viscosity function at higher shear rates. On the other hand, the filled systems containing calcium carbonate, carbon black, titanium dioxide and Franklin fibers exhibit unbounded viscosity buildup at low shear rates. Generally, the viscosity levels in these systems are higher than those found in the former group of fillers. The extent of the steady shear viscosity increase is the lowest for 3-

FILLED POLYSTYSENE

DOTTS

FRANKLINFIBERS

Figure 6.1 Variation of steady shear viscosity with shear rate for filled-polystyrene melts at 20vol% of various types of fillers as indicated. (Reprinted from Ref. 29 with kind permission from American Chemical Society, Washington DC, USA.)

FILLED POLYSTYMNE

GLASS BEADS ARAKOD FIBERS CELLULOSE FIBERS MICA GLASS FIBERS PS

UNITS

Figure 6.2 Variation of steady shear viscosity with shear rate for filled-polystyrene melts at 20voi% of various types of fillers as indicated. (Reprinted from Ref. 29 with kind permission from American Chemical Society, Washington DC, USA.)

dimensional spherical fillers such as glass beads, higher for 2dimensional platelet fillers such as mica and highest for !-dimensional fibrous fillers such as glass fibers. When considering rigid and flexible fillers, the increase in the level of steady shear viscosity would be more for rigid fillers than for flexible fillers because they resist deformation to a greater extent. This is evident when the effect of glass fibers is compared with aramid and cellulose fibers filled PS systems in Figure 6.2. All three fibers have nearly the same fiber diameter (Table 6.1), yet the filled system with glass fiber shows the highest viscosity even when the aspect ratio of the glass fiber is the lowest. The existence of unbounded viscosity buildup at low shear rates as shown in Figure 6.1 is not an effect of the filler rigidity or shape, but of the filler size and will be discussed in the next sub-section. 6.2

EFFECT OF FILLER SIZE

Systems in Figure 6.1 exhibit the yield stress which is a manifestation of strong particle-particle interaction and has direct relation to the particle size. From Table 6.1, it can be seen that carbon black, titanium dioxide and calcium carbonate have a much smaller particle size than mica and glass beads. Similarly, from among the fibers, it is the Franklin fiber that has a much smaller diameter than the other three fiber types. It is only the smaller size fillers, specifically those having diameters below 0.5 jam, that have shown the yield stress. The smaller the particles, the higher is the yield value [27]. In fact, with decreasing particle size, the entire steady shear viscosity curve is pushed to a significantly higher level [19,27]. The existence of yield stress in the steady shear viscosity curve for filled polymer systems containing small size fillers has been noted by other researchers [2,3,7,11,12,52] as well. Yield stress is observed when the filler particles are extremely small as with carbon black [19] or finely divided mineral particles [2,3,11,12,52]. Apparently, the particles link temporarily into a network of finite strength. It must be borne in mind that there may be a difference between yield stress values obtained by extrapolating a flow curve to zero shear rate [7,11] and those obtained by stressing a sample at rest until it yields [12]. For filled systems with larger particles, the response to deformation is determined by hydrodynamic interaction and not by particle-particle interaction. These non-interacting particle systems would not show the yield stress as can be seen from Figure 6.2 wherein the smallest particle dimension exceeds 10 jim (Table 6.1). Tanaka and White [86] have developed a cell theory of the steady shear viscosity of a suspension of interacting spheres to account for the viscous behavior of filled polymer systems. Essentially, they compute the enhanced energy dissipation through the existence of interaction

energy between the particles, E1n^ and hydrodynamic viscous dissipation Evis. Specifically the ratio of the viscosity r\ of the suspension to that of the matrix rj° is i//f/° = (4it + Evis)/^

(6.1)

This energy-based method of calculation of viscosity Y\ is due to Einstein [87], who considered hydrodynamic dissipation in a very dilute suspension of non-interacting spheres. Tanaka and White [86] base their calculations on the Frankel and Acrivos [88] cell model of a concentrated suspension, but use a non-Newtonian (power law) matrix. The interaction energy is considered to consist of both van der WaalsLondon attractive forces and Coulombic interaction, i.e.

Efat = EV + EC

(6.2)

which are computed from the calculations of Hamaker [89] and Derjaguin [90] for interacting spheres. It is shown that Eint/Evis = B1 (0, n)/y,

Evis/^is = B2(0, n)

(6.3)

so that

n(4>* y) = TY/V + q*(, y)

(6.4)

where

TY=CA

Hf

1

I2Ce0^

^ L/^-ir-ir-

^'^W(^MVl

(6 5a)

-

(63b)

In equations (6.5a) and (6.5b), TY is the yield stress, n is the power law index, K is the consistency index, C is a positive constant representing the total number of nearest neighbours of each sphere, AH is Hamaker's constant, $m the maximum volume fraction, e0 the dielectric constant of the matrix, ke the thickness of the electrostatic interaction layer, ^0 the surface potential of the particulates, D the particle diameter and y the shear rate. Basically, equation (6.4) along with (6.5b) is identical to the empirical Herschel-Bulkley [91] model given by equation (2.45), and has been suggested also by Jarebski [92]. There are other theories which relate yield stress to volume fraction and particle size, and these are available in Rajaiah [93]. From equation (6.5a), it can be seen that the yield stress increases when volume fraction or surface potential or Hamaker's constant increases and when particle size decreases. The usefulness of equation (6.5a) is limited by the fact that most filler particles are neither monodisperse particles nor are values of Hamaker's constant, the

surface potential and thickness of the interaction layer readily available. If the electrostatic interaction is assumed to be negligible compared to van der Waals interaction, then TY may be expressed as follows

-^b^iT

If a value of (j) of 0.2 is taken, this predicts TY to be 101 ~ 102 pascal for a diameter of 1 jam, 103 ~ 104 pascal for 0.1 |im and 104 ~ 105 pascal for 0.01 ^m if Hamaker's constant is 10"12 - 10~13 erg. Tanaka and White [86] were able to show semi-quantitative agreement between calculated and experimental values of yield stress for typical fillers such as carbon black, titanium dioxide and calcium carbonate. Yield stresses are generally observed in dispersions with fine particle size having a high specific surface area and when the filler loading is high. Increasing filler concentration has a pronounced effect on the rheology of the filled systems. 6.3 EFFECT OF FILLER CONCENTRATION With increasing concentration of the 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 on the geometry and surface activity of the filler particles. For example, high aspect ratio fillers would begin to interact at much lower concentrations, while non-agglomerated large size spherical particles would not interact until about 20vol% [94]. The final concentration-dependent regime is that when a complete network formation occurs due to particle-to-particle contact, which would occur at concentrations of less than 1% for fibers and 40vol% for nonagglomerated, randomly dispersed spherical particles. This is exemplified by the sharp rise in the relative viscosity rjT at the concentration beyond which particle-to-particle contact occurs as can be seen from Figure 6.3(a). Various fillers were dispersed in different polymer matrices [21,22] and their relative viscosity vs. volume fraction was plotted as shown in Figure 6.3(a). It is seen that the relative viscosity of the filled polymer system adjudged at the same volume fraction varies with the properties of the filler such as shape, size, size-distribution, surface appearance, etc. It is quite obvious what the effect of the physical nature of the filler surface would be on the steady shear viscous properties of the filled polymer systems. The higher the surface roughness the greater the resistance to flow deformation and hence the viscosity of filled system

FITJJRD POLYEIHYLENE AND POLYPItOFYLENE

Figure 6.3(a) Variation of relative viscosity at constant shear stress with filler volume fraction 0 for various filled polymer systems. The capital letters indicate the filler and the numbers designate the average aspect ratio of the fiber. The dashed lines represent the curve through the experimental points. (Reprinted from Ref. 22 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

would be higher. This can be deduced intuitively. Experiments to verify this are not easy because it is very difficult to find two fillers with the same size and size distribution but different surface roughness. Some indication of the effect of roughness can be got from the data available in the works of Kataoka et al. [21] and Kitano et al. [22]. This is shown in Figure 6.3(b) which gives the relative viscosity of systems filled with precipitated calcium carbonate (CC) and natural calcium carbonate (NC). The surface of the precipitated calcium carbonate is much rougher than that of the natural calcium carbonate, and as a consequence, the viscosity of systems filled with CC is seen to be higher than those filled with NC Of course, this is not the isolated effect of roughness alone. The CC particles had a much smaller size and a much narrower size distribution than the NC particles and hence these factors would also contribute to the viscosity increase. In Figure 6.3(a), the values of r\r of materials filled with natural calcium carbonate powder NC are the lowest because they are basically

CAI£IOM CAKBONATS FELED POLYEIEYIENE;

Figure 6.3(b) Variation of relative viscosity as a function of the volume fraction for polyethylene melts filled with precipitated calcium carbonate (CC) and natural calcium carbonate (NC). (Reprinted using data from Refs 21 and 22 with kind permission from Steinkopff Verlag Darmstadt, Germany.)

particulate systems with an average of about lOjim particle size. The result of NC in polyethylene (PE) and in polypropylene (PP) can be approximated by a single curve, indicating that the matrix does not influence rjr as long as the filler surface is not treated with any specific treating agents. The talc particles (TK) being disk-shaped show a relative viscosity higher than those of natural calcium carbonate (NC) filled polymer systems. Further, it can be seen that the value of r\r for systems filled with glass fibers (GF) and carbon fibers (CF) increases rather sharply with increasing average aspect ratio. It should be noted that the relative viscosity shown in Figures 6.3(a) and (b) is the one determined at constant shear stress, rather than at constant shear rate, because it is independent of shear stress. This fact was reaffirmed by Polinski et al. [71] as shown in Figure 6.4. Where the data are plotted using relative viscosity at a constant shear rate then a set of curves with varying shear rate would be obtained as shown in Figure 6.5. The solid lines are the predictions of equation (6.7) given by Jarzebski [92] as

GLASS SPHERE ELLH) THEBMOPLASUC POLYMER UNITS PASCALS

CONE-A]MD-PLArE CAPElARY EXPERIMENTAL CURVES

Figure 6.4 Variation of relative viscosity at constant shear stress as a function of the shear stress for a suspension of 15 jam diameter glass spheres in a thermoplastic polymer. (Reprinted from Ref. 95 with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.)

M

>=1 k/*M"

<67)

Agreement between data and theory can be seen to be excellent. 0m is taken as 0.62 and the values of power law index n are chosen to be different during the calculation depending on the chosen shear rate. At shear rates below 1 s~l the polymer matrix and the filled system at most concentrations is near Newtonian while the value of n reduces to 0.75 at 10s""1 and finally to 0.5 at shear rates in excess of 1Os"1. Of course, the point of onset of shear thinning moves to lower shear rates with increasing filler concentrations. It is also seen in Figure 6.5 that the data at three different temperatures superimpose on a single curve for each shear rate considered. This is in contrast to the findings of Saini et al [63] who also prepared plots of relative viscosity with filler volume fraction for four different polymer matrices at three levels of loading as shown in

GLASS SPHERE FILLED THERMOFLASUC POLYMER

JARZEBSZI'S EQUAHOK AS AFUNCTIQN QFn

Figure 6.5 Variation of relative viscosity at constant shear rate with filler volume fraction. (Reprinted from Ref. 95 with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25,1055 KV Amsterdam, The Netherlands.)

Figures 6.6-6.9. The curves are distinctly different at the various temperatures in all cases and the change in viscosity with filler concentration is exponential, in particular, with a major rise occurring beyond a filler volume concentration of 0.25 to 0.30. Note that the plot in Figure 6.5 is on a semi-logarithmic scale while Figures 6.6-6.9 are not. Hence, it is possible that the differences in values are masked in Figure 6.5, thereby leading to a conclusion that the plot is unique and independent of temperature. A closer look at the plot does indicate this. The temperature sensitivity of the relative viscosity can be easily estimated [63] as follows. The viscosity of a polymer melt decreases with increasing temperature due to the greater free volume available for molecular motion at the higher temperature. In the case of filled systems, the free volume change is limited only to the polymer fraction of the composite.

BARIOM FERRJTE FILLED LOW DENSIIY POLYEIHYLlHE

Figure 6.6 Variation of the relative viscosity with volume fraction of ferrites for low density polyethylene at three different temperatures. (Reprinted from Ref. 63 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

BARIUM FERRTTE FILLED POLYURETHANE TEERMOPLASIIC ELASTOMER

Figure 6.7 Variation of the relative viscosity with volume fraction of ferrites for polyurethane thermoplastic elastomer at three different temperatures. (Reprinted from Ref. 63 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

BARIDM FERRIIE FILLED STYRENE • ISOPlENE - STYRENE BLOCK COPOLYMER

Figure 6.8 Variation of the relative viscosity with volume fraction of ferrites for styrene-isoprene-styrene block copolymer at three different temperatures. (Reprinted from Ref. 63 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

BAHIUM FERSITE FILLED POLYESIER ELASTOMER

Figure 6.9 Variation of the relative viscosity with volume fraction of ferrites for polyester elastomer at three different temperatures. (Reprinted from Ref. 63 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

It is therefore expected that the filled polymer melt viscosity would be less temperature sensitive than the unfilled melt viscosity. Referring to Figures 6.6 through 6.9 at a given filler loading, the relative viscosity increases with increasing temperature. This is because the drop in the unfilled melt viscosity with increasing temperature is more than the drop in the filled melt viscosity. The Arrhenius type equation, (2.6a), proposed by Eyring [96], is most commonly used to analyze the temperature dependence of melt viscosity. For unfilled polymer systems, the viscosity ^0 T at a temperature T can be related to the viscosity rjojR at a reference temperature by the following expression: %T = ^rRexp(^J

(6.8)

where E0 is the activation energy of the flow process for the unfilled system and R is the universal gas constant. For a filled system, with volume fraction (j> of the filler equation (6.8) can be written as ^.T = ^.rRexp^J

(6.9)

where ^ r and ^TR are the viscosities for a filler concentration of 0. Thus combining equations (6.8) and (6.9) gives ln^ = ln!VlR + ^- £ o) >?0,T

>?0,TR

(6>10)

^T

It is obvious that a plot of ^^,T/^O,T vs- V^ on a semi-logarithmic scale would result in a straight line from whose slope AE which is the difference in the activation energy of viscous flow due to presence of the filler can be calculated. The values of E0 (i.e. the activation energy for viscous flow of unfilled systems) and values of E0 (i.e. activation energy for viscous flow for filled systems) are shown in Table 6.2. It is seen that the E0 values are lower than E0 values in all cases, indicating that rj^T is less temperature sensitive as compared to r]QT which is obviously true because the fillers provide very little free volume change with temperature in relation to the matrix. In Figures 6.3, 6.5-6.9, the *-axis is not normalized though this is a common practice to be followed if a theoretical equation fit of the Maron-Pierce [97] type is sought as shown in Figure 6.10. The simplest one parameter equation which has been evaluated [21,22], and tested extensively [71,72] is given as

i/r = (i - 4>/m)~2

(6.ii)

Table 6.2 Values of activation energies calculated from equation (6.10) Filler amount E (kJ/mole) @ y = 100 sec'1 SIS block copolymer Polyurethane thermoplastic elastomer Polyester elastomer Low density polyethylene

0% E0

40% v/v E40

47% v/v E47

57% v/v E57

23.5 71.9 53.8 15.9

18.3 50.2 36.9 5.7

15.8 42.9 34.7 5.1

12.0 39.7 28.7 4.9

Source: Ref. 63 (reprinted with kind permission from Society of Plastics Engineers Inc., Connecticut, USA).

GLASS SPHERE FILED POLYMERS CHEVRON GRADE CHEVRON GRADE CHEVRON GRADE CHEVRON GRADE THERMOPLASTIC THERMOPLASTIC THERMOPLASTIC EQUATION (6.11)

Figure 6.10 Variation of average relative shear viscosity as a function of the reduced volume fraction of filler. (Reprinted from Ref. 71 with kind permission from Society of Rheology, USA.)

In Figure 6.10, the maximum packing parameter 0m was determined [71] experimentally by liquid displacement technique as 0.62. Several different values have been reported in the literature. From geometrical considerations, m ranges from 0.52 for simple cubic packing to 0.74 for rhombohedral packing. However, Kitano et al. [22] have suggested an average value of 0.68 for spherical particulates whose aspect ratio is close to one. Nielsen [78], on the other hand, noted that maximum packing usually ranges from 0.601 to 0.637 for random packing of spheres, and it is generally smaller for agglomerated and non-spherical fillers. Chong et al. [98] used a value of 0.605 for uniformly sized glass beads which is in agreement with the observation of Nielsen [78]. Another method of interpreting flow data of filled polymer systems is by following the route suggested by Shenoy and Saini [51]. Since the relative viscosity r\r at constant shear stress gives a uniqueness to the plot (Figure 6.3 vs. Figure 6.5), it is logical to choose a rheological parameter which is determined under constant shear stress conditions, and also one which is far from the zero shear rate region wherein filled systems generally show the yield stress. The melt flow index (MFI) becomes an ideal choice because it is determined at constant shear stress in the medium shear rate range. Using the inverse relationship between steady shear viscosity and MFI [99], Shenoy and Saini [51] write the modified form of Doolittle's equation [100] in the following form: In MFI(T, 0) = In A" - —1—

J(T9 0)

(6.12)

where MFI(T, 0) is the melt flow index of the filled polymer system at temperature T and containing 0 volume fraction of filler, A' is a constant. It is now assumed that the addition of filler alters the free volume state of the unfilled polymer. The free volume of the polymer is specified by /(T, O). Its melt flow index is written as MFI(T, O) and the following relationship derived from equation (6.12) would then hold: InMFI(T, O) = InA" - -L^

(6.13)

The altered free volume state of the filled polymer system is given, of course, by equation (6.12). As a first approximation, the free volume in the altered free volume state resulting from the addition of a filler can be considered to reduce the free volume of the reference medium and to be a linear function of the volume fraction (/> of the added entity similar to that given by Fujita and Kishimoto [101]: /(T, 0)=/(T, O)-/J(T^

(6.14)

where /J(T) represents the difference between the free volumes of the polymer and the filled system. Combining equation (6.12) to (6.14) and rearranging the terms gives ^=-2.303/(T,Q) +2'3°^0)l log *MFI

^

P(T)

(6.15)

where aMFl = MFI(T, 0)/MFI(T, 0) Based on the assumptions made during the derivation of equation (6.15), it is imperative to choose MFI(T, O) > 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\^i + ' D ^a/) Tl 1

'-'+*<*-'*>•

x

/pv

——1 , x \Ji f n

<"8>

EP--k~D.m

and

D11 = J]D1./.

(6.19)

Table 6.4 Effect of particle size distribution on maximum packing density of randomly dispersed spheres Ratio of particle diameter(s) of the size fractions

Cy1 Qf 11 VTCy 1 Cy11VTCy1JMQCy1 Cy1, 1 /7Cy1, 1 /49Cy1, 1 /343Cy1

Ratio of fractions (wt%)

100 84/15 75/14/11 72/14/10/3

Maximum packing density (vol%)

64 86 95 98

Source: Ref. 124 (reprinted with kind permission from American Chemical Society Inc., Westerville, Ohio, USA).

Here ^n denotes the maximum packing of spheres of uniform size which may be taken to be the average value of 0.619 as suggested by Nielsen [78]. D3 is the average diameter of the different sizes of the particulates, D1 is the diameter of the fth component, and the abbreviation (D1 ~ D3) is defined as: (D1 - D3) = O

for D1 < D3

(6.2Oa)

and

(D, - D3) = D1 - D3

for D1 > D3

(6.2Ob)

In addition,/ is the number fraction of the zth component defined by:

*-%/£%

<«»

where v{ is the volume fraction of the zth component in the mixture. The number fraction may also be directly obtained if a size distribution of the solids, / vs. D1, is available as in Table 6.3. The maximum packing parameter can be calculated with equations (6.17)-(6.21) for any solids mixtures of spherical particulates, and equation (6.11) may then be used to estimate the value of the filled system viscosity at a fixed shear stress level. Poslinski et al. [72] undertook the study of the influence of a bimodal size distribution of glass spheres on the rheology of filled polymer systems. They used small and large glass spheres of sizes as given in Table 6.3. The average diameter of the two size ranges calculated by equation (6.19) was 15 and 78 (im, respectively, and therefore the particle size ratio was 0.192 based on these average values. Bimodal size distributions were prepared by mixing together various volume percents of the smaller 15 ^m spheres with the larger 78 jam spheres; specifically, <£s = 0,10,30,50,75 and 100%. Figure 6.19 shows the maximum packing data obtained for a bimodal solids mixture of 15 and 78 ^m glass spheres with s, the volume percent of smaller spheres, ranging from O to 100% of the total solids mixture. The experimental values of the maximum packing fraction were obtained by the sedimentation technique, and these indicated that m increased for a bimodal system depending on the amount of the smaller spheres. Also shown in Figure 6.19 is the value of the maximum packing calculated from the data of Chong et al. [98] for a bimodal distribution of spheres at (/> = 0.60 and Ss = 0.138 as well as Sweeny's data [115] at 4> = 0.55 and <5S = 0.048. The highest value of the maximum packing parameter, 0.752, was obtained by mixing approximately 15% of the 15jnm and 85% of the 78 jam glass spheres in the total solids volume. The error bars shown in Figure 6.19 represent the random error in the experimental measurements of the maximum packing.

GLASS SPHERE FILLED POLYBUTENE Eq.(6.17) USING DISTRIBUTION VALUES Eq.(6.17) USING AVERAGE VALUES DATAOF POSDNSiQ ET AL. [72 ] DATA OF CHONG ET AL. [ 98 ] DATAOF SWEENY

[115]

Figure 6.19 Variation of maximum packing as a function of the bimodal size distribution of the 15 and 78 jam glass spheres. (Reprinted from Ref. 72 with kind permission from Society of Rheology, USA.)

Various bimodal mixtures were compounded into a polybutene grade 24 polymer obtained from the Petrochemical Division of the Chevron Company, at total solids concentrations up to 60% by volume. The polybutene polymer was a Newtonian liquid at room temperature whose viscosity and density at 220C were 25 Pa sec and 898kg/m3, respectively. Due to the relatively high viscosity of the polybutene, it was found that the glass spheres suspended in the polymer remained dispersed for at least 24 h before complete settling occurred. Upon addition of glass spheres to the polybutene grade 24 matrix, the magnitude of the shear viscosity at 220C was observed to increase and it remained approximately constant for shear rates ranging from 0.1 to 100 sec'1. As a result, the relative viscosity data determined at the same shear stress were obtained by averaging data over the entire shear rate range of measurement. To see the effect of particle size and size distribution, the average relative viscosities of the various glass sphere suspensions are plotted against $s, the volume percent of the 15 j^m spheres, ranging from O to 100% of the total solids mixtures as shown in Figure 6.20. When the

GLASS SPHERE FILLED POLYBUIEHE TOTAL SPHERES

EQUATION (6JlJ) DATA OF OHONO ET AL. [98] DATAOF SWEENY 1115]

Figure 6.20 Variation of average relative shear viscosity as a function of 0S, the vol% of the 15 jam glass spheres in the total solids mixture suspended in a polybutene grade 24 matrix at 220C. (Reprinted from Ref. 72 with kind permission from Society of Rheology, USA.)

two sizes were mixed together, the relative viscosities were actually reduced, especially for volume fractions of total solids greater than 0.3. The lowest values of i\r were obtained for 0S = 10 to 30% of the smaller spheres, the same range in which the maximum packing parameter was observed to be highest. The 60% by volume loading level could not be achieved for a unimodal size distribution due to difficulty in obtaining a fully homogeneous compound; however, it was possible to reach this high concentration for 0S = 10, 30 and 50%. The solid lines in Figure 6.20 represent the predictions of equation (6.11) with the maximum packing fraction determined by equation (6.17) using the particle size distribution values of the bimodal components listed in Table 6.3. The agreement between theory and experiment is excellent. The optimum reduction in the relative viscosity is correctly predicted by equations (6.11) and (6.17) to be (j)s — 15%, which corresponds to the highest value of the maximum packing.

6.5

EFFECT OF FILLER AGGLOMERATES

The presence of agglomerates has a considerable effect on the maximum possible loading of the filler, as can be seen from Table 6.5. If the agglomerates are all broken down into individual particles, then 63.7 vol% of the filler can be incorporated into the polymer. However, if a dozen particles are clustered together in each agglomerate, then the maximum loading possible would be only 43vol%. At maximum filler loading, the viscosity of the filled system is close to a hundred to a thousand times greater than that of the polymer itself, as can be seen from Figure 6.10. Agglomerates occlude liquid in their interparticle voids, thereby increasing the relative viscosity value at any given solids loadings. Thus, the viscosity would be the same for 43vol% of agglomerated filler with a dozen particles in each agglomerate as it is for 63.7 vol% of non-agglomerated particles. For highly filled polymer systems, it is therefore important to reduce the degree of agglomeration to a minimum level, in order to decrease the system viscosity for easier processing and to increase the extent of filler loading. The adhesion forces between particles have been discussed by Rumpf and Schubert [131]. In addition to electrostatic and van der Waals forces, the presence of liquid and solid bridges contributes to high agglomerate strength. The dispersion of powders in liquids is influenced in part by the wetting characteristics of the liquid on the solid surface, and is related to the hydrogen-bonding capability of the liquid which can be expressed as its cohesive energy density [132]. The initial drying of powders and the heating associated with processing of a high molecular weight organic vehicle help to reduce the effect of Table 6.5 Effect of aggregration on maximum filler concentration 0m Number of particles in aggregated spheres

Maximum filler concentration 0m, (vol%)

1 3 5 8 12 18 32 57 122 250

63.7 60 53 47 43 39 37 35 34 32

Source: Ref. 76 (reprinted with kind permission from Marcel Dekker Inc., New York, USA).

liquid bridges. The presence of solid bridges when a small amount of additive has been incorporated in the powder via a salt by spray drying, such as magnesia in alumina or yttria in zirconia, gives rise to more serious problems. Such agglomerates can only be broken down by milling of the powder or by using mixing devices which impose high shear stresses. High shear mixing generates velocity gradients which are normally strong enough to break agglomerates in fillers of micron size or larger. In the case of submicron fillers, the forces holding the particles together are stronger than the shear force created by the velocity gradient. In such circumstances, even high shear mixing does not break the agglomerates or eliminate the microvoids within the filled system, and one has to take recourse to filler surface treatment [9,15,34,36,37,43-45,52,54,60] which is capable of altering the interparticle forces, resulting in deagglomeration, better dispersion, and hence lower viscosity with maximum possible loading. 6.6

EFFECT OF FILLER SURFACE TREATMENT

An approach to improving filler-polymer compatibility is the use of surface-modifiers on the filler through a pretreatment process. Surface treatment is meant to help the polymer wet the filler and disperse it. However, the extent of wetting and degree of adhesion are different for each polymer, filler, and surface modifier combination. Some modifiers that will effectively coat the filler surface may not interact with the polymer, whereas others that are compatible with the polymer may not adhere to the filler, and hence, will not effectively modify its surface. It is thus very difficult to predict the performance of the surface modifier a priori for any filler-polymer combination. It is, therefore, a common practice to select the appropriate surface modifier for a filler-polymer combination rather empirically. One could, of course, refer to the extensive information available in the form of books, reports and papers [133-135] on various types of surface modifiers listed in Table 1.5. Surface modifiers are generally bifunctional molecules with one end capable of adhering to the filler and the other end compatible with the polymer, and at times even capable of reacting with it. There have been extensive studies on the effect of surface treatment on the steady shear viscous properties of filled polymer systems [9,13,15,18,32,34,36,38, 39,52,53,6O]. There is enough evidence that the increased melt viscosity introduced by filler addition can be significantly reduced by the application of surface modifiers to the fillers. Some typical examples of viscosity reductions achieved through the addition of surface modifiers to

selected filled polymer systems are given in Table 6.6. The maximum reduction in viscosity is seen to be 73% in the 40phr high aspect ratio mica in unsaturated polyester. Reductions in viscosity can also be achieved through the use of wetting agents such as W-900 manufactured by Bye-Mallinckrodt as shown by Cope and Linnert [32]. With low-profile fillers like clay, silica and mica, viscosity reductions of 70, 80 and 90% respectively have been observed. Presence of low-profile filler in a 50% blend of wollastonite and polyester have been seen to give 35 to 41% lower viscosity. Cope and Linnert [32] observed an interesting synergistic effect when wetting agents and silanes were used in combination. A 60% quartz/epoxy

Table 6.6 Effect of surface modifiers on some selected filled polymer systems Polymer

Filler, loading (phr)

Unsaturated polyester Unsaturated polyester Unsaturated polyester Unsaturated polyester Unsaturated polyester Unsaturated polyester Unsaturated polyester Unsaturated polyester Unsaturated polyester

Unsaturated polyester Unsaturated polyester Unsaturated polyester Unsaturated polyester HOPE HOPE HOPE HOPE HOPE HOPE PP PP

Wollastonite, 150 Wollastonite, 150 Calcined kaolin clay, 100 Calcined kaolin clay, 100 Hydrous kaolin clay, 88 Hydrous kaolin clay, 88 Talc, 100 Talc, 100 Mica (high aspect ratio), 40 Mica (high aspect ratio), 40 Silica 10/*, 120 Silica 10/1, 120 Alumina trihydrate Alumina trihydrate Wollastonite, 43 Wollastonite, 43 Calcium carbonate, 150 Calcium carbonate, 150 Barium sulfate, 150 Barium sulfate, 150 Calcium carbonate, 50 Calcium carbonate, 50

PP PP

Calcium carbonate, 100 Calcium carbonate, 100

PP

Calcium carbonate, 100

PP

Calcium carbonate, 100

Unsaturated polyester

Surface Viscosity Melt flow Source modifier, (Pa sec) index Ref. amount (g/10min) (phf)

90 54 56 37 95 36 58 30 104

[13] [13] [13] [13] [13] [13] [13] [13] [13]

A-174,0.5

28

[13]

A-174,0.5

52 14 65 48

16.5 16.5 16.3 17.8 16.3 22.0 4.4 5.2

[13] [13] [13] [13] [13] [13] [13] [13] [13] [13] [30] [30]

3.6 4.7

[30] [30]

5.0

[30]

5.5

[30]

A-174,0.5 A-174, 0.5 A-174,0.5 A-174,0.5

A-174,0.5 TTS, 3 TTS, 3 TTS, 3

Stearic acid, 1.0 Stearic acid, 0.2 Stearic acid, 1.0 Stearic acid, 1.5

system produced a 60% viscosity reduction with a hydrophobic wetting agent and an 80% reduction with a silane/BYK additive combination. The general conclusions of Cope and Linnert [32] are that greater effects occur with increasing filler concentration and with those fillers that are most difficult to wet and disperse. Similarly, thermoplastic filled systems produce greater effects than those seen in thermosets because these resins do not wet fillers well. Cope and Linnert [32] have shown how wetting agents can be effectively used in low density sheet molding compound (SMC) for lowering viscosities. Observing the published data, it is seen that, in some cases, the effect on the steady shear viscosity is only marginal. There are a few select cases [9,18] wherein the viscosity is seen to reduce even below that of the virgin polymer (Figures 6.21(a) and (b)). Figure 6.21(b) taken from Monte and Sugerman [18] shows how the concentration of the titanate changes the flow behavior of 70% calcium carbonate filled polypropylene. The work of Han et al [15] also indicates that the melt viscosity decreases with the addition of TTS to CaCO3-PP and fiber glass-PP system. The extent of decrease depends on the type of the surface modifier and the type of the filler-polymer combination. It was found that the effect of TTS in decreasing the viscosity was more than in the case of ETDS-201; so also there was a profound effect on the viscosity of CaCO3-PP system than fiberglass-PP system. A possible

MICA FILLED POLYPROPYLENE

UNITS

40 % FILLER UNTKEAIED 40%FILLERTEtEATED WTTH Z-6032 SHANE UNFILLED

Figure 6.21 (a) Variation of steady shear viscosity with shear rate. (Reprinted from Ref. 9 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

CALCIDM CARBONATE FILLED POLYPROPYLENE UNITS

70% FILLER TREATED WTIH 0.5 w % KR TTS 70% FILLER TREATED WTIH 1.0 w % KR ITS UNFILLED

Figure 6.21 (b) Variation of steady shear viscosity with shear rate. (Reprinted from Ref. 18 with kind permission from Marcel Dekker Inc., New York, USA.)

explanation for this viscosity decrease has been suggested by Han et al. [15,34] to be due to the plasticizing effect of the organic groups in titanate coupling agents. It is more likely that the titanium oxide monomolecular layer formed by monoalkyl titanates and the silanol monomolecular layers formed by silanes on the filler surface, modify the surface energy so that the viscosity of the filled polymer system is reduced. The effect of viscosity reductions in these systems [15,34] was not as drastic as observed [9,18] in Figures 6.21(a) and (b). The normal expected behavior from the use of surface modifiers is one in which the viscosity reduction occurs to a different extent depending on the type of surface modifier, its amount and method of treatment. The maximum achievable reduction would most likely be very close to the flow behavior of the unfilled polymer at least at higher shear rates. This is well illustrated in the work of Saini et al. [60] for highly filled polymer systems. Saini et al. [60] used pretreated fillers in their study of the effect of surface modifier on the melt rheology of styrene-isoprene-styrene block copolymer. The platelet-shaped barium ferrite that was used as filler was not monosized, and had an average particle size of about 3|im. Two silanes: Z6075 and Z6076; and three titanates: KR38S, KR138S, and KRTTS (Table 1.5 for chemical description) were used as surface modifiers. The treatment method consisted of using a solution of 95 parts methanol and 5 parts of water by volume for the silanes and 100 parts of xylene for the titanates. Surface modifiers of an amount equal

to 1% by weight of barium ferrite were dispersed in the solvent. The slurry was formed by wetting the required quantity of barium ferrite with the prepared solution and stirring for about 30 minutes. It was then allowed to stand overnight and the solvent was removed in the oven at 1050C for silanes and 13O0C for titanates. The viscosity vs. shear rate data obtained by Saini et al. [60] for 57vol% of barium ferrite pretreated with different surface modifiers is shown in Figure 6.22(a). It can be seen that all titanates aid in reducing the viscosity of the filled polymer system while the silanes show a very marginal effect. Though there have been a number of articles in the literature [7,9,15,34,36,44,133-135] on the mechanism by which surface modifiers act, not all seem to show coherent agreement. Some have

BAMDM FERBITE FILLED STYIUENE-ISOPSENE-STYaENE UNITS

FERSITE UNTREATED (57 v% FILLER) FESSTTE TREATED WTTH SHANE - Z6076 FEESITE TREATED UTCTH SILANE - Z6075 FERBITE TREATED WITH TJTANATE - KR38S FERBITE TREATED WITH TTTANATE - KR138S FERBITE HEATED WTTH TTTANAIE' KR TTS UNFILLED

Figure 6.22(a) Variation of apparent viscosity with shear rate at 22O0C for unfilled, 57 vol% untreated ferrite filled, and 57vol% treated ferrite-filled SIS block copolymer systems. (Reprinted from Ref. 60 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

postulated [15,43] that there is a tendency to produce a plasticizing effect, due to the modification of the interfacial characteristics of the filled system. The titanates in the case of Saini et al. [60] could have acted as good dispersing agents, but not as coupling agents, and provided no chemical bridge between the polymer and the filler particles. The scanning electron micrographs shown by Saini et al. [60] give evidence of the fact that the dispersion level is better with the titanates than with the silanes. However, since no low steady shear data were presented, this conclusion cannot be ascertained with confidence. It is quite plausible that the titanates act as good dispersing agents and provide a plasticizing effect as well, thereby resulting in drastic reductions in viscosity of the filled polymer systems [6O]. The effect of surface modifier concentration was also studied by Saini et al. [60] as shown in Figure 6.22(b). It is seen that the optimum concentration is in the range of 0.6-0.8% by weight of filler. Their results confirm the range suggested by Monte and Sugerman [18] and Sharma et al. [36]. The exact optimum concentration would, of course, vary for each filler, polymer, and surface modifier combination, and may need to be determined each time. It must be emphasized here that the method of pretreatment also plays an important role in the final performance of the filled polymer system. This point has not been addressed by earlier investigators. The silanes, for example, are known to undergo hydrolysis/condensation

VISCOSTlT RAUO

%TTS

Figure 6.22(b) Variation of viscosity ratio at a shear rate = 100 sec"1 of treated to untreated 57vol% ferrite-filled SIS with different percentages of KR TTS for determining the optimum treatment level. (Reprinted from Ref. 60 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

reactions, and the extent of these reactions is a function of the nature of the solvent phase, silane concentration in solution, pH of the solution, and filler treatment time [133]. Prehydrolysis of the silane is known to be beneficial because the silanols formed in the reaction can then react with the filler surface. However, two silanols can undergo a parallel self-condensation reaction reducing the systems activity towards the filler. By adjusting the pretreatment variables in a favorable manner, it would thus be possible to make either of the possible reactions dominate. This would naturally result in differences in the efficacy of the filler surface modification. Further, it is known that the preferred orientation of a silane molecule at the inorganic filler surface is with silanol function attached to it and the organofunctional group extending away from it. Deposition of the silane from water or alcohol would naturally assist this favorable orientation, provided the pH is equal to or less than the isoelectric point of the inorganic surface. By altering the pH, it might be possible to deposit the silane in an inverted fashion as suggested by Plueddeman [133]. This would alter the rheological characteristics of the filled polymer system. It is not known to what extent the rheology of the filled system could be altered by manipulations in the treating method, and this may need systematic investigation. 6.7 EFFECT OF POLYMER MATRIX The effect of the polymer matrix on the steady shear viscosity of filled systems would depend on the chemical nature of the polymer as well as its unfilled viscous characteristics. The polymer melt itself could be Newtonian, moderately shear-thinning or predominantly shear-thinning. In the case of polymers like nylon, polyethylene terephthalate (PET) and polycarbonate (PC), the steady shear viscosity is near Newtonian up to even a shear rate of a little over 10Os"1 as can be seen from Figures 6.23-6.25. As an example of a moderately shear-thinning polymer, polyetherimide (PEI) could be considered [136] as shown in Figure 6.26; while, for a predominantly shear-thinning polymer, polypropylene (PP) is a reasonable example as shown in Figure 6.27. Note that in Figures 6.24 and 6.27, the plots are shown as shear stress vs. shear rate which is, of course, another method of representation of the same information. In all cases, it can be seen that the addition of fillers whether they are particulates like calcium carbonate (CaCO3) or glass fibers (GF), systematically moves the entire curves to a higher level. The viscosity increase is greater at lower shear rates whereas lower at higher shear rates. The extent of the effect would, of course, depend on the nature of the polymer and the filler type, size, size distribution, etc. Next Page

Steady shear elastic properties

-7 /

Despite the fact that the literature on the rheology of filled systems is extensive [1-85], the work related to the steady shear elastic properties of such systems is quite limited [5,19,27,29,31,34]. This is, of course, due to the experimental difficulties in these measurements. The usual signs of elastic behavior in filled systems are recognized by the presence of larger normal stress differences during relatively low shear measurements on a cone and plate rheogoniometer and by higher exit pressure as well as larger die swell values during high shear capillary or slit rheometry. As normal stress difference is mathematically connected with exit pressure and recoverable shear strain and, conceptually, with die swell ratio, knowledge of these would also provide the same information. The exit pressure is the small finite value pexit which is obtained when the pressure in the capillary or slit is plotted against downstream distance and the extrapolated pressure at the exit is non-zero. With appropriate assumptions, it has been shown that pexit can be related to primary normal-stress difference [77]. However, since the theory has been questioned [86] and a controversy arisen [87], it has not gained popularity as a useful measure of elasticity. The use of the recoverable shear strain to determine the effect of fillers on the elastic properties of polymer melts has been uncommon, whereas study of die swell ratio has been extensive, especially in the determination of the effect of carbon black on the extrudate swell of elastomers. Some special problems exist in normal stress measurement of filled polymer systems. Abnormal effects at low rates are observed due to the interactions between the measuring equipment and the yield stress property associated with the filled systems. As a result, a proper zero point for measurement cannot be obtained. Furthermore, the uncertainty is proportional to the level of the yield stress [19]. It is thus difficult to measure normal stresses accurately especially at low shear rates [27].

In low shear measurements, the difficulty lies in the gap setting of the cone and plate, firstly, due to the high residual stresses which do not relax easily and secondly, due to the filler particles which interfere on account of their individual particle size or the agglomerate size. In high shear measurements, the particle or agglomerate size again may interfere with the extrusion process on account of the obstruction that they may create at the die exit. Nevertheless, with a great deal of careful experimentation, useful elastic property data on filled polymer systems can be generated as has been shown by some researchers [5,19,27,29,31,34]. The generated data may appear confusing at the outset if looked at cursorily. The presence of fillers in the polymers seems to show decreases as well as increases in elasticity and the reports may appear contradictory at times. However, a careful look at the elastic properties data of filled polymer systems shows that it is the method of representation and certain categories of fillers which show unexpected trends. The effects of various parameters on the steady shear elastic properties are discussed in various subsections as was done for the steady shear viscous properties in the preceding chapter. However, all subcategories may not be covered and the extent of discussion is at times concise due to limited information. 7.1

EFFECT OF FILLER TYPE

The experimental studies of the influence of fillers on the rheological properties of polymer melts by White et al. [29] best illustrate the effect of filler type. The steady shear elastic data were generated in terms of the first normal stress difference using the cone and plate arrangement of the Rheometrics Mechanical Spectrometer at a fixed temperature of 18O0C. The filled systems were prepared with a fixed grade of polystyrene Dow Styron 678U and involved the use of nine different types of filler as summarized in Table 6.1. Various filler types were thus covered from rigid to flexible, as well as spherical, particulate to fibrous. The filler loading was fixed at 20vol% and hence the generated data bring out the exclusive effect of the filler type on the elastic properties. Figures 7.1 and 7.2 show the variation of primary normal stress difference with shear stress for the various filled systems studied [29]. It is seen that glass beads hardly affect the values of N1 while the presence of Franklin fibers, titanium dioxide (TiO2), calcium carbonate (CaCO3), carbon black (CB) and mica, respectively, show greater and greater reductions in N1. However, on the other hand, the addition of certain fibers like cellulose fiber, glass fiber and aramid fiber, respectively, increases N1 values to a larger and larger extent.

FILLED POLYSTYRENE UNTTS PASCALS

PS GLASS BEADS FRANZLINFIBER TiO2 CaCO3 CB MICA

Figure 7.1 Variation of primary normal stress difference with shear stress for filled polystyrene melts containing 20 volume fraction of various types of fillers as indicated. (Reprinted from Ref. 29 with kind permission from American Chemical Society, Washington DC1 USA.)

FILLED POLYSTYRENE UNITS PASCALS

ARAMID FIBER GLASS FIBER CELLULOSE FIBER PS

Figure 7.2 Variation of primary normal stress difference with shear stress for filled polystyrene melts containing 20 volume fraction of various types of fillers as indicated. (Reprinted from Ref. 29 with kind permission from American Chemical Society, Washington DC, USA.)

From Table 6.1 it can be concluded that fibrous fillers (aramid, glass and cellulose fiber) tend to increase the normal stress difference value at a fixed shear stress, particulate fillers (titanium dioxide, calcium carbonate and carbon black) tend to depress it while spherical fillers (glass beads) leave it nearly unchanged. The large increase in normal stresses in fiber-filled polymer systems is probably due to a hydrodynamic particle effect, associated with orientation in the direction of flow [29]. There is an indication that the increase in N1 is a function of the fiber modulus and aspect ratio. The higher the modulus and the longer the fibers, the larger the value of N1 as depicted in Figure 7.2. It has been noted elsewhere by Czarnecki and White [28] that mastication of fiber-filled systems breaks up the fibers into smaller lengths and also reduces the value of N1. Among the fibers studied in Table 6.1, the lone fiber which depresses the value of N1 is the Franklin fiber. Though the aspect ratio of the fiber is comparable to the other fibers, its diameter is much smaller. Thus, the observed lowering of the normal stress in this case is more of an effect due to size than its shape and hence is discussed in the next subsection. 7.2

EFFECT OF FILLER SIZE

Normal stress measurements for polymer melts filled with large particles [14,28,29,31] and those filled with small particles [7,15,27,29] are available. When plotted in terms of N1 vs. T12, the primary normal stress differences are consistently found to be reduced by the presence of smaller particles (for example, Figure 7.1). From Table 6.1, it can be seen that titanium dioxide, calcium carbonate and carbon black are small in size compared with the other fillers and show lower normal stress differences. From among the fibers, it is the Franklin fiber which has the smallest dimension, almost in the range of the three particulate fillers, and thus behaves in the manner similar to that of TiO2, CaCO3 and CB. However, the extent of normal stress difference reduction is less due to the effect of the high aspect ratio which tends to raise the value as explained in section 7.1. It should be noted that Figures 7.1 and 7.2 are plotted as N1 vs. T12 rather than N1 vs. y. Where the latter representation was used then the data would appear as shown in Figure 7.3(a). Only curves for particulate fillers - titanium dioxide and carbon black - are shown for illustration purposes. It can be seen that both these fillers appear to increase normal stress differences of the polymer melt at the same shear rate. Thus from Figure 7.3(a) it can be concluded that small particulate fillers like carbon black and titanium dioxide increase the elasticity of polymer melts. This is contrary to the trend in Figure 7.1 for the same filled systems using the same data but different representation wherein

FILLED POLYSTYRENE UNITS PASCALS

Figure 7.3(a) Variation of primary normal stress difference with shear rate for filled polystyrene melts containing 20 volume fraction of carbon black and titanium dioxide fillers. (Reprinted from Ref. 27 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

it was concluded that small participate fillers decrease the elasticity of polymer melts. It is important to establish, therefore, which of the representations is truly correct. In order to do this, it is best to deal with another useful parameter of elasticity, namely, the steady-state shear compliance /e which gives a measure of the stored energy or the elastic recovery of the system. When represented mathematically as /e = (T11 - T22)/2T12, the plot of J6 vs. y would give the correct trend based on a molecular interpretation. The usefulness of plotting normal stress N1 vs. shear stress T12 for unfilled polymer melts is known [77,88,89]. Such plots are independent of temperature and molecular weight (though not its distribution), and rheological behavior for such viscoelastic fluids is interpretable in terms of the steady state shear compliance Je. Addition of the carbon black reduces the N1 values at fixed T12 as already discussed with reference to Figure 7.1. The same effect would be observed in case /e values are plotted against shear rate. Another method of representation of elasticity is through the use of the characteristic time of the system. Figure 7.3(b) gives the plot of relaxation time vs. shear rate for high density polyethylene melts containing glass fibers. It is seen that the presence of glass fibers increases the relaxation time and gives the same interpretation as Figure 7.2. Thus, it can be concluded that the method

GLASS FIBEJl FTlLED POLYSTYRENE

UNITS

GLASS FmER GLASS FIBER

Figure 7.3(b) Variation of relaxation time with shear rate for polystyrene melt filled with glass fiber. (Reprinted from Ref. 14 with kind permission from John Wiley & Sons, Inc., New York, USA.)

of representation of N1 vs. T12 as given in Figures 7.1 and 7.2 is the correct one and should be followed. Particularly in the case of filled systems, the shear rate or velocity gradient may not be continuous at the interface between the filler and polymer, whereas shear stress is likely to be continuous at the interface assuming no slippage occurs between filler and polymer [81]. Hence, interpretation of normal stress difference through shear stress becomes more meaningful. In systems showing a yield stress, as the shear stress is decreased towards it, the normal stress difference approaches zero [27,90] as suggested by the developed phenomenological constitutive equations [19]. However, this response is not observable in Figures 7.1 and 7.2 because of lack of normal stress data at very low shear rates. The view that N1 is small relative to T12 because of the existence of a yield value is, of course, a reasonable one. 7.3 EFFECT OF FILLER CONCENTRATION In order to obtain data representation in the form of N1 (or ^1 = N1/^2) vs. volume fraction , attempts [7,71] have been made to obtain graphical similarity with the viscous counterpart as shown in Figure 6.10. Minagawa and White [7] showed that the ratio of N1 or ^1 of the

filled systems to that of the unfilled polymer matrix at the same shear rate was independent of the shear rate and an increasing function of . The function was not found to be unique and depended on the chosen filler-polymer combination. An empirical Taylor series expansion for \l/ltT at low volume fraction was shown to successfully describe the increase with volume fraction in the range of O < 0 < 0.25. Poslinski et al [71], on the other hand, found that all their glass sphere data could be correlated by the following form analogous to equation (6.11) for relative viscosity *-£tf=" -*/«*

™

The ^1 r experimental values were obtained by averaging the relative data over the entire range of shear rate measurement from 1 to 10Os"1. It can be seen from Figure 7.4 that the experimental data are fitted rather well by the above equation (7.1) over the entire volume fraction range up to as high as 0.6. In Figure 7.4, the value of 0m was taken as 0.62 as

GLASS SPHERE FILLED POLYMERS CHEVRON GRADE-18 CHEVRON GRADE-24 CHEVRON GRADE -32 CHEVRON GRADE -122 THERMOPLASTIC @ 130 0 C THERMOPLASTIC (§150° C THERMOPLASTIC @ 170 0C EQUATION a-1)

Figure 7.4 Variation of average relative primary normal stress coefficient 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.)

determined experimentally [71] by the liquid displacement technique. Despite the fact that theoretical interpretation of experimental data [7,71] works out well as represented, intuitively the predicted trend is reversed. The presence of the particulate fillers has been shown to decrease the elasticity of the polymer melt as demonstrated in Figure 7.1. This fact is also supported by exit pressure measurements [5,34]. With increasing filler concentration, it is thus logical to expect that normal stress difference would decrease. This has, in fact, been shown by Han [77,81] as given in Figure 7.5(a). It has also been shown [7] that extrudate swell decreases with increasing filler concentration as can be seen from the plot given in Figure 7.5(b) of extrudate swell ratio vs. filler concentration for high density polyethylene melts filled with titanium dioxide determined at a constant shear rate (10 sec"1) with a capillary having an L/D ratio of 28. Hopper [91], Gotten [92], Medalia [93], Vinogradov et al [94] and White and Crowder [6] all studied the effect on die swell ratio of carbon black-filled polymer systems. The unanimous observation was that, as the filler concentration was increased, the die swell ratio decreased showing a decrease in the elasticity. The effect of this was an improved extrusion characteristic and an increased value of the critical shear stress before melt fracture could occur. Plots of relaxation time vs. filler concentration as given in Figure 7.5(c) for various polymers filled with TiO2 particulate filler, support the fact that elasticity decreases with increasing concentration.

CALCIUM CARBONATE FILLED POLYPROPYLENE UNITS

Figure 7.5(a) Variation of primary normal stress difference with shear stress for polypropylene at 20O0C filled with calcium carbonate. (Reprinted from Refs 77, 81 with kind permission from Academic Press Inc., New York, USA.)

TUAKIDM DIOXIDE FILLKD HIGH DENSITY POLYETHYLENE

Figure 7.5(b) Variation of extrudate swell ratio with volume fraction for high density polyethylene at 18O0C filled with titanium dioxide. (Reprinted from Ref. 7 with kind permission from John Wiley & Sons, Inc., New York, USA.)

XUAITDM DIOXEDB FILLED POLYMERS UNITS

Figure 7.5(c) Variation of relaxation time with volume fraction for polymer melts filled with titanium dioxide. Data at shear rate of 0.5sec~1. (Reprinted from Ref. 7 with kind permission from John Wiley & Sons, Inc., New York, USA.)

It is obvious that the mobility of the 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. Note that Figure 7.5(a) is a plot of N1 vs. T12 rather that y and correctly so. The trend of increasing elasticity with increasing filler concentration will be noticed only in fiber-filled systems as evidenced from Figure 7.3(b). Thus, it should be borne in mind that equation (7.1) may be used for data representation but not for data interpretation, as the predicted trend does not describe the actual situation. In fact, it might be better to define ^1 r as equal to N1(^, T12VN1(O, T12) and then seek a relationship with (j). 7.4 EFFECT OF FILLER SIZE DISTRIBUTION Poslinski et al. [72] undertook the study of the influence of a bimodal size distribution of glass spheres on the rheology of filled polymer systems as already discussed in section 6.4. The bimodal size distribution was prepared by mixing together various vol% of the smaller 15 |im spheres with the larger 78 jim spheres given in Table 6.3. Various bimodal mixtures were compounded into a polybutene grade 24 polymer obtained from Petrochemical Division of the Chevron Company, at total solids concentrations up to 60% by volume. To see the effect of particle size and size distribution on elasticity of the system, the average relative normal stress difference coefficients are plotted against (/>s, the volume percent of the 15|im spheres, ranging from O to 100% of the total solids mixture as shown in Figure 7.6. Similar to the relative viscosity case in Figure 6.20, the relative primary normal stress coefficient is also reduced when the two sizes of spheres are mixed together, and again the lowest values are obtained for 0S = 10 to 30% of the smaller spheres, which happens to be the same range when the maximum packing parameter is the highest. The solid lines in Figure 7.6 represent the predictions of equation (7.1) with maximum packing fraction determined by equation (6.17) using the particle size distribution values of the bimodal components listed in Table 6.3. The agreement between theory and experiment is quite adequate. However, as discussed in section 7.3, there is a problem with data interpretation because the curve of 60% by volume of total spheres shows the highest elasticity which is incorrect. 7.5 EFFECT OF FILLER AGGLOMERATES Agglomerates occlude liquid in their interparticle voids and thereby leave a less volume fraction of the liquid around it. This would create

GLASS SPHERE FILLED POLYBDTENE TQTALSPHERES

EQUATION CT. D

Figure 7.6 Variation of average relative primary normal stress coefficient as a function of 0S, the volume percent of the 15^m glass spheres in the total 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.)

an apparent situation of higher filler loading than is actually present. Hence, the effect of filler agglomerates would be similar to that of filler concentrations; or, in other words, with larger number of filler agglomerates, the system would behave Theologically in a manner similar to a system with a higher filler concentration than what actually exists. It can be thus expected that with increasing number of filler agglomerates when dealing with particulate fillers, the normal stress difference would be lower. 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. In an unagglomerated filled system, the extent

of normal stress difference lowering would be less if the particle size is larger. When an agglomerate is formed or present, it is as though the particle size of the filler has increased throughout the system. Thus, with increasing number of particles in the agglomerates, the extent of normal stress difference lowering decreases. On the other hand, because of the occluded liquid in the interparticle voids of the agglomerates, the extent of normal stress difference lowering increases. It is basically the net effect of these two opposing factors that determines exactly how much the normal stress difference would be lowered. In the case of fiber-filled systems, too, the effect of agglomerates would be to lower the normal stress difference. It has already been shown in section 7.1 that fiber-filled systems show increases in elasticity. The extent of the increase would be thus reduced if agglomerates are formed because the fibers that gather to make up the agglomerates are restrained and cannot orient during flow. There are no experimental data specifically to support the intuitive thoughts put forth in this section. The reason is that determination of normal stress difference in the presence of agglomerates is extremely difficult. The agglomerates interfere with the gap setting during cone and plate rheological measurements due to their increased size. Despite lack of actual data, the conclusions on the effect of agglomerates can be drawn by carefully understanding the effect in analogous situations as done here. 7.6

EFFECT OF FILLER SURFACE TREATMENT

One of the effective methods of reducing the number of filler agglomerates in a filled polymer system is through the use of surfacemodifiers such as those listed in Table 1.5. Surface modifiers are generally bifunctional molecules with one end capable of adhering to the filler and the other end compatible with the polymer, and at times even capable of reacting with it. Surface treatment basically helps the polymer to wet the filler better and disperse it, thereby reducing and preventing agglomeration because of promotion of filler-polymer contact as against filler-filler contact. Research work on the effect of surface treatment on the steady shear elastic properties of filled polymer systems is limited [27,34] but good enough to draw adequate conclusions. Figure 7.7 shows the effect of surface treatment on 30% calcium carbonate filled polystyrene [27]. It can be seen that the data are presented in both forms of representations. N1 vs. y (Figure 7.7(a)) and N1 vs. T12 (Figure 7.7(b)). In Figure 7.7(a), it appears that surface treatment reduces elasticity to a level even below that of the pure polymer. However, conclusions drawn from this type of representation

CALCIUM CARBONATE FILLED POLYSTYRENE UNITS PASCALS

UNTREATED TELEATED

Figure 7.7(a) Variation of primary normal stress difference with shear 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.)

CALCIDH CARBONATE FILLED POLYSTYUENE UNITS PASCALS

TKEKTED UNTREATED

Figure 7.7(b) Variation of primary normal stress difference with shear 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.)

would not be correct because the untreated 30% CaCO3 shows an increase in elasticity, which is against intuition and logic as discussed in section 7.2. Thus, the representation given in Figure 7.7(b) is the true one and shows that surface treatment increases the values of N1 as against that of the untreated system of a fixed filler loading at the same level of shear stress. Figure 7.8 gives the plot of first normal stress difference vs. shear stress for CaCO3 filled polypropylene. Similar to the effect observed in Figure 7.7(b), the use of the surface modifiers, namely, silanes Y9187 and AIlOO, shows an increase in the normal stress difference over that of the untreated system irrespective of the temperature of measurement. Note that Y9187 is an N-octyltriethoxysilane while AIlOO is a yaminopropyltriethoxysilane. The same effect is, however, not found to be the case when considering a different filler such as glass beads in the polypropylene.

CALCIUM CAEBONATE FILLED POLYPROPYLENE UNITS

PASCALS

Figure 7.8(a) Variation of primary normal stress difference with shear stress at 20O0C for calcium carbonate filled polypropylene treated with silane surface modifiers. (Reprinted from Ref. 34 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

CALCIUM CABBONATE FILLED POLYPROPYLENE

UOTTS PASCALS

Figure 7.8(b) Variation of primary normal stress difference with shear stress at 24O0C for calcium carbonate filled polypropylene treated with silane surface modifiers. (Reprinted from Ref. 34 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

Figure 7.9 shows that neither the silane Y9187 nor AIlOO is effective in influencing the normal stress difference of the untreated glass beadsPP systems at 20O0C. However, at 24O0C, the surface modifier Y9187 decreases the melt elasticity while AIlOO increases it. Note that the data in Figures 7.8 and 7.9 are all in the high shear rate range and obtained using the Han slit/capillary rheometer. When the surface modifier is changed from the silane to a titanate in the case of the CaCO3-PP system, as in Figure 7.10, the effects are qualitatively not different from those observed in Figure 7.7(a) or (b). It is seen that the normal stress difference of the titanate KR-TTS treated CaCO3-PP system is higher than that of the untreated CaCO3-PP system. In Figure 7.10, the low-shear data were obtained on the Weissenberg rheogoniometer and the high-shear data were got using the Han slit/capillary rheometer. When the filler is changed to glass fiber and the same titanate KRTTS is used as a surface modifier, it is seen that the untreated and

GLASS BEADS FILLED POLYPROPYLENE UNITS PASCALS

PP PP/GLASS BEADS PP/GLASS BEADS/Y9187 PP/GLASS BEADS/AIlOO

Figure 7.9(a) Variation of primary normal stress difference with shear stress at 20O0C for glass bead filled polypropylene treated with silane surface modifiers. (Reprinted from Ref. 34 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

GLASS BEADS FILLED POLYPROPYLENE UNITS PASCALS

P? PP/GLASS BEADS PP/GLASS BEADS/Y9187 PP/GLASS BEADS/Al 100

Figure 7.9(b) Variation of primary normal stress difference with shear stress at 24O0C for glass bead filled polypropylene treated with silane surface modifiers. (Reprinted from Ref. 34 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

CALCIUM CARBONATE FILLED POLYPROPYLENE UNTTS PASCALS

Figure 7.10 Variation of primary normal stress difference with shear stress for calcium carbonate filled polypropylene treated with titanate surface modifier TTS. Open symbols represent Weissenberg rheogoniometer data and closed symbols represents Han slit/capillary rheometer data. (Reprinted from Ref. 34 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.) treated filled polypropylene systems do not show any difference in the normal stress characteristics as shown in Figure 7.11. The normal stress difference data in Figures 7.7-7.11 all go to show that the effect of surface modifier is quite system specific. Hence,

GLASS FIBER FILLED POLYPROPYLENE UNITS PASCALS

PP

PP/GLASS FIBER/ITS PP/GLASS FIBER

Figure 7.11 Variation of primary normal stress difference with shear stress for fiber filled polypropylene with 50% untreated and treated filler using titanate surface modifier TTS. Open symbols represent Weissenberg rheogoniometer data and closed symbols represent Han slit/capillary rheometer data. (Reprinted from Ref. 81 with kind permission from Academic Press Inc., New York, USA.) extrapolation of information with regard to this effect can at times be dangerous. The efficiency of the surface modifiers will depend on the type of filler, type of polymer, amount of modifier and method of treatment as already discussed in section 6.6.

7.7

EFFECT OF POLYMER MATRIX

As already mentioned in section 6.7, the effect of polymer matrix on the rheological properties of filled polymer systems would depend on the chemical nature of the polymer as well as its unfilled rheological properties. In order to understand this effect, it would be essential to observe normal stress difference response using different polymer systems but with the same filler of a fixed size/size distribution and at a fixed level of loading. This information is not available from one source as in the steady shear viscous case [63]. Hence, this effect is exemplified by presenting data from different sources for different polymer systems but with the same fillers. Of course, the physical characteristics of the chosen filler are unlikely to be the same and so also, it is unlikely to find a particular filler loading as a common denominator in all cases. Nevertheless, the data would give some general idea of the effect of the polymer matrix.

GLASS FIBER FILLED NYLON

UNITS PASCALS

NYLON 6,6/33% GLASS FIBER NYLON 6,6

Figure 7.12 Variation of primary normal stress difference with shear stress for glass fiber filled nylon 6,6 with 33 vol% of fiber. (Reprinted from Ref. 95 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

Figures 7.12-7.14 show the effect of glass fibers at different loading levels on the normal stress difference behavior of nylon, polycarbonate (PC), and polystyrene (PS). Since the plots have been made as N1 vs. T12, they are independent of temperature. This is specifically clear from Figure 7.12 which shows a uniqueness of data at the three temperatures of 2750C, 2850C and 2950C [95]. In Figure 7.13 for polycarbonate [31], it is seen that N1 does not rise systematically with filler concentration at a fixed shear stress. The 30% loaded system coincides with the 10% loaded glass fiber filled polycarbonate instead of lying between the 20% and the 40%. No apparent explanation can be presented for such anomalous behavior. In the case of glass fiber filled polystyrene [28], the loading level is limited to 10% and 22% and hence whether a reversal of the type that was observed in Figure 7.13 for PC takes place cannot be established. Figure 7.11, which also gives the normal stress difference information

GLASS FIBER FTTIED POLYCARBONATE

MAKROLON GRADE

FILLER

UOTTS PASCALS

Figure 7.13 Variation of primary normal stress difference with shear stress for glass fiber filled polycarbonate of different grades of Makrolon. (Reprinted from Ref. 31 with kind permission from John Wiley & Sons, Inc., New York, USA.)

GLASS FIBER FILLED POLYSTYlEKE UOTTS PASCALS

PS/22% GLASS FIBER PS/10% GLASS FIBER PS

Figure 7.14 Variation of primary normal stress difference with shear stress for glass fiber filled polystyrene with different volume fraction. (Reprinted from Ref. 28 with kind permission from John Wiley & Sons, Inc., New York, USA.). on glass fiber filled systems [81] but using a different polymer, namely polypropylene (PP), presents an unusual behavior. It is seen that this is the lone case where a fiber filled system shows a decrease in elasticity of the polymer melt. The behavior is inexplicable considering the fact that even a Newtonian fluid containing fibers is known to show an increase in elasticity [96]. REFERENCES 1. Bruch, M., Holderle, M. and Friedrich, C. (1997) Rheological properties of polystyrene filled with hairy PMMA-particles, Paper 8-F presented at the 13th international meeting of the Polymer Processing Society (June 10-13). 2. Chapman, P.M. and Lee, T.S. (1970) Effect of talc filler on the melt rheology of polypropylene, SPE Journal 26, 37-^0. 3. Mills, NJ. (1971) The rheology of filled polymers, /. Appl Polym. ScI1 15, 2791-805.

4. Nazem, F. and Hill, C.T. (1974) Elongational and shear viscosities of beadfilled thermoplastic, Trans. Soc. RheoL, 18, 87-101. 5. Han, C.D. (1974) Rheological properties of calcium carbonate-filled polypropylene melts., /. Appl Polym. ScL, 18, 821-9. 6. White, J.L. and Crowder, J.W. (1974) The influence of carbon black on the extrusion characteristics and rheological properties of elastomers: polybutadiene and butadiene-styrene copolymer, /. Appl Polym. ScL, 18,1013-38. 7. Minagawa, N. and White, J.L. (1976) The influence of titanium dioxide on the rheological extrusion properties of polymer melts, /. Appl. Polym. ScL, 20, 501-23. 8. Faulkner, D.L. and Schmidt, L.R. (1977) Glass bead-filled polypropylene Part I: Rheological and mechanical properties, Polym. Engg ScL117, 657-64. 9. Boira, M.S. and Chaffey, C.E. (1977) Effects of coupling agents on the mechanical and rheological properties of mica-reinforced polypropylene, Polym. Engg ScL117, 715-18. 10. Bigg, D.M. (1977) Rheology and wire coating of high atomic number metal low density polyethylene composites, Polym. Engg ScL, 17, 745-50. 11. Kataoka, T., Kitano, T., Sasahara, M. and Nishijima, K. (1978) Viscosity of particle filled polymer melts, RheoL Acta, 17,149-55. 12. Kataoka, T., Kitano, T. and Nishimura, T. (1978) Utility of parallel-plate plastometer for rheological study of filled polymer melts, RheoL Acta, 17, 626-31. 13. Copeland, J.R. and Rush, O.W. (1978) Wollastonite: short fiber filler/ reinforcement, Plastic Compounding, 1, 26-36 (Nov/Dec). 14. Chan, Y., White, J.L. and Oyanagi, Y. (1978) Influence of glass fibers on the extrusion and injection molding characteristics of polyethylene and polystyrene melts, Polym. Engg ScL, 18, 268-72; (1978) A fundamental study of the rheological properties of glass fiber-reinforced polyethylene and polystyrene melts, /. RheoL, 22, 507-24. 15. Han, C.D., Sandford, C. and Yoo, HJ. (1978) Effects of titanate coupling agents on the rheological and mechanical properties of filled polyolefins, Polym. Engg ScL, 18, 849-54. 16. Menges, G., Geisbusch, P. and Zingel, U. (1979) Kunststoffe, 7, 485. 17. Arina, M., Honkanen, A. and Tammela, V. (1979) Mineral fillers in low density polyethylene films, Polym. Engg ScL, 19, 30-9. 18. Monte, SJ. and Sugerman, G. (1979) A new generation of age and waterresistant reinforced plastics, Polym. Plastics Tech. Engg, 12,115-35. 19. Lobe, V.M. and White, J.L. (1979) An experimental study of the influence of carbon black on the rheological properties of a polystyrene melt, Polym. Engg ScL, 19, 617-24. 20. Wu. S. (1979) Order-disorder transitions in the extrusion of fiber-filled polyethylene terephthalate) and blends, Polym. Engg ScL, 19, 638-50. 21. Kataoka, T., Kitano, T., Oyanagi, Y. and Sasahara, M. (1979) Viscous properties of calcium carbonate filled polymer melts, RheoL Acta, 18, 635-9. 22. Kitano, T., Kataoka, T., Nishimura, T. and Sakai, T. (1980) Relative viscosities of polymer melts filled with inorganic fillers, RheoL Acta, 19, 764-9. 23. Kitano, T., Nishimura, T., Kataoka, T. and Sakai, T. (1980) Correlation of dynamic and steady flow viscosities of filled polymer systems, RheoL Acta, 19, 671-3.

24. Crowson, R.J., Folkes, MJ. and Bright, P.P. (1980) Rheology of short glass fiber-reinforced thermoplastics and its applications to injection molding I. Fiber motion and viscosity measurement, Polym. Engg ScL, 20, 925-33. 25. Crowson, RJ. and Folkes, MJ. (1980) Rheology of short glass fiberreinforced thermoplastics and its application to injection molding II. The effect of material parameters, Polym. Engg ScL, 20, 934-40. 26. Goel, D.C. (1980) Effect of polymeric additives on the rheological properties of talc-filled polypropylene, Polym. Engg ScL, 20,198-201. 27. Tanaka, H. and White, J.L. (1980) Experimental Investigations of shear and elongational flow properties of polystyrene melts reinforced with calcium carbonate, titanium dioxide and carbon black, Polym. Engg ScL, 20, 949-56. 28. Czarnecki, I. and White, J.L. (1980) Shear flow rheological properties, fiber damage and mastication characteristics of aramid, glass and cellulose-fiber reinforced polystyrene melts, /. Appl Polym. ScL, 25, 1217-44. 29. White, J.L., Czarnecki, I. and Tanaka, H. (1980) Experimental studies of the influence of particle and fiber reinforcement on the rheological properties of polymer melts, Rubber Chem. Tech., 53, 823-35. 30. Hancock, M., Tremayne, P. and Rosevear, J. (1980) Fillers in polypropylene II. /. Polym. ScL, (Polym. Chem. Edn), 18, 3211-17. 31. Knutsson, B.A., White, J.L. and Abbas, K.A. (1981) Rheological and extrusion characteristics of glass-fiber reinforced polycarbonate, /. Appl. Polym. ScL, 26, 2347-62. 32. Cope, D.E. and Linnert, E. (1980) The lowdown on loading down resins using hydrophobic encapsulation, Plastic Engg, 37-9 Qune). 33. Kitano, T., Kataoka, T. and Shirata, T. (1981) An empirical equation of the relative viscosity of polymer melts filled with various inorganic fillers, Rheol. Acta, 20, 207-9. 34. Han, C.D., Van der Weghe, T., Shete, P. and Haw, J.R. (1981) Effect of coupling agents on the rheological properties, processing and mechanical properties of filled polypropylene, Polym. Engg ScL, 21, 196-204. 35. Stamhuis, J.F. and Loppe, J.P.A. (1982) Rheological determination of polymer-filler affinity, Rheol. Acta, 21,103-5. 36. Sharma, Y.N., Patel, R.D., Dhimmar, LH. and Bhardwaj, LS. (1982) Studies of the effect of titanate coupling agent on the performance of polypropylene-calcium carbonate composite, /. Appl. Polym. ScL, 27, 97-104. 37. Nakatsuka, T., Kawasaki, H., Itadani, K. and Yamashita, S. (1982) Phosphate coupling agents for calcium carbonate filler, /. Appl. Polym. ScL, 27, 259-69. 38. Lee, W.M., Abe, D.A., Chipalkatti, M.H. and Liaw, T.F. (1982) Rheological properties of particulate-filled linear low density polyethylenes, Proc. Ann. Conf. Reinf. Plast. Compos. Inst. Soc., Plast. Ind., 37 (12D), 7. 39. Juskey, V.P. and Chaffey, C.E. (1982) Rheology and tensile properties of polypropylene reinforced with glycerol-treated mica, Can. J. Chem. Engg, 60, 334-41. 40. Hinkelmann, B. (1982) Zur analytischen beschreibung des fullstoff-einflusses auf das fliessverhalten von kunststoffschmelzen, Rheol. Acta, 21, 491-3. 41. Utracki, L.A. and Fisa, B. (1982) Rheology of fiber or flake-filled plastics, Polym. Composites, 3,193-211. 42. White, J.L. (1982) Rheological behavior of highly filled/reinforced polymer melts, Plastics Compounding, 47-64 (Jan/Feb).

43. Bigg, D.M. (1982) Rheological analysis of highly loaded polymeric composites filled with non-agglomerating spherical filler particles, Polym. Engg ScL1 22, 512-18. 44. Bigg, D.M. (1982) Rheological behavior of highly filled polymer melts, Polym. Engg Sc/., 23, 206-10. 45. Althouse, L.M., Bigg, D.M. and Wong, W.M. (1983) Evaluating the effectiveness of filler surface treatments, Plastics Compounding, (March/ April). 46. Lem, K.W. and Han, C.D. (1983) Rheological behavior of concentrated suspensions of particulates in unsaturated polyester resin, /. Rheol., 27, 263-88. 47. Daley, L.R. and Rodriguez, F. (1983) Flow properties of ethylene-propylene terpolymer filled with silica modified by silane coupling agents, Ind. Eng. Chem. Prod. Res. Dev., 22, 695-8. 48. Mutsuddy, B.C. (1983) Influence of powder characteristics on the rheology of ceramic injection molding mixtures, Proc. Brit. Ceram. Soc., 33, 117-37. 49. Chaffey, C.E. (1983) Reinforced thermoplastics: through flow to use, Ann. Rev. Mater. Sa'., 13, 43-65. 50. Shenoy, A.V., Saini, D.R. and Nadkarni, V.M. (1983) Rheograms of filled polymer melts from melt-flow index, Polym. Composites, 4, 53-63. 51. Shenoy, A.V. and Saini, D.R. (1983) Interpretation of flow data for multicomponent polymeric systems, Colloid Polym. ScL, 261, 846-54. 52. Suetsugu, Y. and White, J.L. (1983) The influence of particle size and surface coating of calcium carbonate on the rheological properties of its suspension in molten polystyrene, /. Appl. Polym. ScL, 28,1481-501. 53. Luo, H.L., Han, C.D. and Mijovic, J. (1983) Effects of coupling agents in the rheological behavior and physical mechanical properties of filled nylon 6, /. Appl. Polym. ScL, 28, 3387-98. 54. Bigg, D.M. (1984) Complex rheology of highly filled thermoplastic melts, Proc. IX Intl. Congress on Rheology in Mexico, Adv. in Rheology, 3, 429-37. 55. Kitano, T., Kataoka, T. and Nagatsuka, Y. (1984) Shear flow rheological properties of vinylon and glass-fiber reinforced polyethylene melts, Rheol. Acta, 23, 20-30. 56. Kitano, T., Kataoka, T. and Nagatsuka, Y. (1984) Dynamic flow properties of vinylon fiber and glass fiber reinforced polyethylene melts, Rheol. Acta, 23, 408-16. 57. Suetsugu, Y. and White, J.L. (1984) A theory of thixotropic plastic viscoelastic fluids with a time-dependent yield surface and its comparison to transient and steady state experiments on small particle filled polymer melts, /. Non-Newtonian Fluid Mech., 14,121-40. 58. Hinkelmann, B. and Mennig, G. (1985) On the rheological behavior of filled polymer melts, Chem. Engg Comm., 36, 211-21. 59. Bretas, R.E.S. and Powell, R.L. (1985) Dynamic and transient rheological properties of glass-filled polymer melts, Rheol. Acta, 24, 69-74. 60. Saini, D.R., Shenoy, A.V. and Nadkarni, V.M. (1985) Effect of surface treatment on the rheological and mechanical properties of ferrite-filled polymeric systems, Polym. Engg ScL, 25, 807-11. 61. Saini, D.R. and Shenoy, A.V. (1986) Viscoelastic properties of highly loaded ferrite-filled polymeric systems, Polym. Engg ScL, 26, 441-5.

62. Shenoy, A.V. and Saini, D.R. (1986) Quantitative estimation of matrix filler interactions in ferrite-filled styrene-isoprene-styrene block copolymer systems, Polym. Composites, 7, 96-100. 63. Saini, D.R., Shenoy, A.V. and Nadkarni, V.M. (1986) Melt rheology of highly loaded ferrite-filled polymer composites, Polym. Composites, 7, 193-200. 64. Shenoy, A.V. and Saini, D.R. (1986) Wollastonite reinforced polypropylene composites: dynamic and steady state melt flow behavior, /. Reinf. Plastics Comp., 5, 62-73. 65. Mutel, A.T. and Kamal, M.R. (1986) Characterization of the rheological behavior of fiber-filled polypropylene melts under steady and oscillatory shear using cone-and-plate and rotational parallel plate geometry, Polym. Composites, 7, 283-94. 66. Edirisinghe, MJ. and Evans, J.R.G. (1987) Rheology of ceramic injection molding formulations, Br. Ceram. Trans. /., 86,18-22. 67. Sacks, M.D., Khadilkar, C.S., Scheiffele, G.W., Shenoy, A.V., Dow, J.H. and Sheu, R.S. (1987) Dispersion and rheology in ceramic processing, Adv. in Ceramics, 24, 495-515. 68. Dow, J. H., Sacks, M.D. and Shenoy, A.V. (1988) Dispersion of ceramic particles in polymer melts, Ceram. Trans. (Ceram. Powder Sci. UA), 1, 380-8. 69. Hunt, K.N., Evans, J.R.G. and Woodthorpe, J. (1988) The influence of mixing route on the properties of ceramic injection moulding blends, Br. Ceram. Trans. /., 17-21. 70. Takahashi, M., Suzuki, S., Nitanda, H. and Arai, E. (1988) Mixing and flow characteristic in the alumina/thermoplastic resin system, /. Am. Ceram. Soc., 17,1093-9. 71. Poslinski, A.J., Ryan, M.E., Gupta, R.K., Seshadri, S.G. and Frechette, FJ. (1988) Rheological behavior of filled polymer systems I. Yield stress and shear-thinning effects, /. Rheol, 32, 703-35. 72. Poslinski, AJ., Ryan, M.E., Gupta, R.K., Seshadri, S.G. and Frechette, FJ. (1988) Rheological behavior of filled polymeric systems II. The effect of a bimodel size distribution of particulates, /. Rheol, 32, 751-71. 73. Ishigure, Y., Nagaya, K., Mitsumatsu, F., Otabe, S., Hayashi, K., Sobajima, A. and Murase, I. (1989) Relationship between the flow characteristics of highly filled alumina or zirconia-organic binder and the properties of sintered products in injection molding processing, Rep. Gifu Pref. Ind. Res. Tech. Center, 21, 51-70. 74. Dow, J.H., Sacks, M.D. and Shenoy, A.V. (1990) Dispersion of alumina particles in polyethylene melts, Ceram. Trans. (Ceram. Powder Sci. Ill), 12, 431-42. 75. Edirisinghe, MJ., Shaw, H.M. and Tomkins, K.L. (1992) Flow behavior of ceramic injection moulding suspensions, Ceramics Int., 18,193-200. 76. Nielsen, L.E. (1974) Mechanical Properties of Polymers and Composites, Marcel Dekker, New York, Vol. 2, Ch. 7, 379-86. 77. Han, C.D. (1976) Rheology in Polymer Processing, Academic Press, New York, 7,182-8. 78. Nielsen, L.E. (1977) Polymer Rheology, Marcel Dekker, New York, Ch. 9, 133-57. 79. Paul, D.R. and Newman, S. (1978) Polymer Blends, Academic Press, New York, 1, Ch. 7, 295-352.

80. Vinogradov, G.V. and Malkin, A.Y. (1980) Rheology of Polymers, Mir Publishers, Moscow, 380-402. 81. Han, C.D. (1981) Multiphase Flow in Polymer Processing, Academic Press, New York. 82. Shenoy, A.V. (1988) Rheology of highly filled polymer melt systems, in Encyclopedia of Fluid Mechanics, (ed. N.P. Cheremisinoff), Gulf Publishing, Houston, TX, 7, 667-701. 83. Yanovsky, Yu.G. and Zaikov, G.E. (1990) Rheological properties of filled polymers, in Encyclopedia of Fluid Mechanics, (ed. N.P. Cheremisinoff), Gulf Publishing, Houston, TX, 9, 243-76. 84. Carreau, PJ. (1992) Rheology of filled polymeric systems, in Transport Processes in Bubbles, Drops and Particles (eds R.P. Chhabra and D. Dekee), Hemisphere Publishing, New York, 165-90. 85. Advani, S.G. (ed.) (1994) Flow and Rheology in Polymer Composites Manufacturing, Elsevier Science BV. 86. Boger, D.V. and Derm, M.M. (1981) Capillary and slit methods of normal stress measurements, /. Non-Newtonian Fluid Mech., 6,163-85. 87. Han, C.D. (1982) Polymer News, 8,111-14. 88. Oda, K., White, J.L. and Clark, E.S. (1978) Correlation of normal stresses in polystyrene melts and its implications, Polym. Engg ScL, 18,15-28. 89. Minoshina, W., White, J.L. and Spruiell, J.E. (1980) Experimental investigation of the influence of molecular weight distribution on the rheological properties of polypropylene melts, Polym. Engg ScL, 20,1166-76. 90. White, J.L. and Tanaka, H. (1981) Comparison of a plastic-viscoelastic constitutive equation with rheological measurements on a polystyrene melt reinforced with small particles, /. Non-Newtonian Fluid Mech., 8,1-10. 91. Hopper, J.R. (1967) Effect of oil and black on SBR rheological properties, Rubber Chem. TechnoL, 40, 463-75. 92. Gotten, G.R. (1968) Rubber Age, 100, 51. 93. Medalia, A.T. (1970) Morphology of aggregates VI. Effective volume of aggregates of carbon black from electron microscopy; application to vehicle absorption and to die swell of filled rubber, /. Colloid Inter/. ScL, 32,115-31. 94. Vinogradov, G.V., Malkin, A.Ya., Plotnikova, E.P., Sabsai, O.Yu. and Nikolayeva, N.E. (1972) Rheological properties of carbon black filled polymers, Int. J. Polym. Mat., 2,1. 95. Pisipati, R. and Baird, D.G. (1981) Correlation of rheological properties of filled nylon melts with processing performance, SPE ANTEC, 27, 32-4. 96. Mewis, J. and Metzner, A.B. (1974) The rheological properties of suspensions of fibers in Newtonian fluids subjected to extensional deformations, /. Fluid Mech., 62, 593-600.

Unsteady shear

Q

viscoelastic properties

O

In the preceding two chapters, various effects on steady shear viscous and elastic properties of filled polymer systems were discussed. The present chapter focuses on the unsteady shear viscoelastic properties of these systems. The unsteady shear characteristics are mainly discussed with respect to small-amplitude oscillations, namely, dynamic rheological data. In some cases, the thixotropic sweep responses and the stress relaxation behavior are also included because they bring out the rheological characteristics in some situations in a much better manner. The extensive literature [1-85] on the rheology of filled polymer systems, however, contains quite limited information on the unsteady shear data [1,8,43-45,54,61,62,64,68,71,72,74,91,92]. The reason for this is because unsteady shear data were normally not used when dealing with low loading levels of fillers where the bulk of information is available. For highly filled systems, this is the only mode of obtaining reliable rheological data, but since the work on highly filled polymer systems is not extensive, the information on unsteady shear data is naturally limited. With higher filler loading, it becomes increasingly difficult to gap set in a cone and plate arrangement of the rheogoniometer; whereas through the use of the parallel-plate arrangements, it is possible to obtain rheological data at any level of loading. The use of steady shear in the case of highly filled systems is not recommended because the material trapped between the plates of the rheometer during data generation normally tends to hang outside the plate dimensions, thereby giving erroneous results. On the other hand, when dealing with unsteady shear data generation, small-amplitude dynamic oscillation keeps the material between the plates of the rheometer intact and hence gives much more reliable and reproducible data. Thus, dynamic data are often the preferred mode for viscoelastic information of highly filled polymer systems. Of course, it need not be restricted only to high filler loadings

as it would certainly provide reliable information even in low filler loadings as well. As a matter of fact, the most reliable rheological data on filled polymer systems can be obtained through the use of dynamic oscillatory measurements. The dynamic rate sweep and the dynamic strain sweep would be most suitable for assessing the internal structure of the system. During a strain sweep, a plot of storage modulus vs. percentage strain at low frequency would be the best indicator of the level of homogeneity in the system. A decrease in storage modulus with percentage strain would be an indicator of the occurrence of the structural breakdown in the system. During a frequency sweep, it is best to maintain the strain as low as possible within the system constraints, in order to be in the linear viscoelastic region of the material. The response of complex viscosity, storage modulus, loss modulus and tangent delta that is then obtained would give a measure of the dispersion of the filler in the matrix. A highly agglomerated system would show the existence of a yield at low frequency, the storage modulus would be high and vary minimally with frequency giving a more solid like response and the tangent delta would be lower. However, generation of reliable and consistent data in case of filled systems depends to a large extent on the preparation of the sample for the rheological test. Premolding samples under high pressure to a shape and size as would be used for the parallel-plate rheometer test, would ensure that the variation in the observed rheological response is related to the filled system characteristics and is not an experimental artifact. During generation of dynamic data, it is important to use a fixed low amplitude when collecting data for comparison on various systems. The effect of the amount of strain during dynamic data measurement on filled polymer systems has been brought out by Bigg [44]. Figure 8.1 shows the strong effect of strain on the complex viscosity and storage modulus for low density polyethylene filled with 50vol% of spherical stainless steel particles at 16O0C. It can be seen that the difference between the complex viscosity at 1(T1 rad/s with a 1% strain (rf = 106 Pa.sec) to that with a 25% strain (rf = 2 x 104 Pa.sec) is almost two orders of magnitude. Much greater differences are observed in the storage modulus response with increasing strain. When the unfilled low density polyethylene is tested using different levels of strains, it is seen from Figure 8.2 that the complex viscosity remains unaffected by strain and remains so at all temperatures. Thus, for unfilled polymers, it may be all right to determine dynamic data at any level of strain. However, for filled systems, because of the sensitivity of the dynamic response to the level of strain, it is always important to determine the dynamic viscoelastic properties at a strain that is low enough not to affect the material response. As the filler loading increases, the level of strain

STEEL SPHERE FILLED LOW DENSITY POLYETHYLENE UNITS

STRAIN

Figure 8.1(a) Effect of strain on the complex viscosity vs. frequency curves for NA-250 low density polyethylene with 50vol% of spherical stainless steel particles at 16O0C. (Reprinted from Ref. 44 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

below which the response is unaffected is reduced [44]. For loadings close to the maximum packing fraction, this strain level may be even below the minimum obtainable within the experimental constraints. In such cases, one could use a higher strain level, but keep this value fixed in all systems studied so that meaningful comparison of information can be done. This would not induce any errors if the systems studied have same filler loadings, but in systems containing different loadings levels, this point ought to be borne in mind when drawing conclusions from the generated data. In Figure 8.1, it is worth noting that at all levels of strains, a yield stress (indicated by a slope of -1 on the In 77* - lnco plot) is observed at frequencies below 1 rad/sec. This is because the filler loading level was high (50vol%) and the interparticle network dominated the rheological behavior of such highly filled polymer systems at all levels of strain [44]. A similar behavior was observed by Saini and Shenoy [61] when

STEEL SPHRRF. FILLED LOW DENSITY POLYETHYLENE UNITS PASCALS RAD/SEC STRAIN

Figure 8.1(b) Effect of strain on the storage modulus vs. frequency curves for NA-250 low density polyethylene with 50vol% of spherical stainless steel particles at 16O0C. (Reprinted from Ref. 44 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

dealing with high filler loading using a different type of filler, namely, barium ferrite. The theoretical equations for modeling this type of dynamic response would also be different from the conventional ones. Theoretical equations for modeling dynamic response of filled systems have been suggested for low [64] as well as high [61] levels of loading and have been discussed at length in sections 2.3.3 and 2.3.4. However, some important equations are recalled and presented here for convenience. Shenoy and Saini [64] suggested a form similar to that of Carreau [86] but modified it appropriately for describing the dynamic rheological behavior of low filled systems as: 11,I = IiJSI(I+ AV)^

(8.1)

where r\l = zero-frequency viscosity function, q* = complex viscosity, co = frequency, /I = time constant, N = power-law parameter.

STEEL SPHERE RLLED LOW DENSTY POLYETHYLENE UNITS

STRAIN

Figure 8.2 Complex viscosity vs. frequency curves for unfilled NA-250 low density polyethylene at various temperatures and different levels of strain. (Reprinted from Ref. 44 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

This equation is ideal for use when the complex viscosity curves show a plateau region in the low frequency range along with a decreasing trend in the higher frequency range (like Curve 1 in Figure 1.3). The model given by equation (8.1) thus naturally presumes no yield stress. On the other hand, when the filler loading increases, the particle network formed leads to the existence of yield stress in the low frequency region. In such cases of highly filled systems, Saini and Shenoy [61] proposed a modified form of the above equation as:

m

=^F^

<8 2)

where K — complex viscosity at frequency of 1 rad/s and at ri = n" ri = slope of the \q*\ vs. co curve in the region of 0.1 < CD < I n" = slope of the \rj*\ vs. co curve in the region of 1 < co < 102

'

Equation (8.2) was suitable for the systems studied by Saini and Shenoy [61] because the complex viscosity in all cases showed the existence of two distinct straight lines one in the low frequency region below CD equal to 1, and the other in the high frequency region above co equal to 1 (like Curve 3 in Figure 1.4). However, in cases where the complex viscosity vs. frequency curves show an initial yield stress followed by a plateau and then a frequency-thinning region, (like Curve 2 in Figure 1.4), neither of the two equations suggested above can be used. In such circumstances, it might be necessary to break up the curves into two or three regions and fit separate equations in each region. A correlation between complex viscosity rf and the storage modulus G7 was derived [61,64] by analogously following the method of Wagner [87,88] for relating the elastic response to the viscous response of the material (equation (2.61) in section 2.3.4). The expression obtained [61,64] was as follows: G'

1 dto'lfro)

^-~^~d^~

(8 3)

-

where m' is an adjustable parameter. Equations (8.1) and (8.3) can be combined to give: G_ 2Nr^Co 2 (N+1) co m for low filler loadings. Equations (8.2) and (8.3) can be combined to give: % = Ad - n'K'-d + «r-Ml + n (n' 7Jfy J CD m I (1 - n)(l + or)J

(8.5)

for high filler loadings. The loss modulus G" vs. frequency can be easily predicted from the above equations by using the definition: G" = VIOf G>)2 - G'2]

(8.6)

It was found [61,64] that the theoretical predictions given by equations (8.1)-(8.6) agreed well with the experimental determinations for each of the rheological parameters, namely, rf, G and G". All the model parameters with the exception of N varied with filler loading. The adjustable parameter m" itself varied with frequency in one case [61]. Since the model parameter can be correlated with filler compositions, reasonably good estimations of the various rheological characteristics can be made for different filler loadings through the use of equations (8.1)-(8.6). The effect of various parameters on the unsteady shear viscoelastic properties are discussed at varying lengths in the following subsections.

The conclusions that can be drawn from unsteady shear data are most of the times not very different from those that can be got from steady shear data. However, there are some cases, such as in the effect of agglomerates or filler surface treatment, where the unsteady shear data provide a better insight than the steady shear data. In such cases, the subsections are dealt with in detail. Whereas in other cases, the discussion is kept to a limited extent as the derived information is analogous to the steady shear case. 8.1

EFFECT OF FILLER TYPE

In Chapters 6 and 7, the effect of filler type on the steady shear rheological properties was elucidated through the response of filled polymer systems having a fixed polymer but with nine different types of filler at a single level of loading. This information was available from one data source [29]. However, the same is not the case with unsteady shear data. There is no single source wherein different fillers have been used at a fixed filler loading in a single polymer matrix. The reason is that where such data were to be generated, it would not lead to any new conclusions other than those which have already been drawn from the steady shear data. Complex viscosity vs. frequency data would show that viscosity increase would be the highest for fibrous fillers and lowest for spherical fillers whereas particulate and platelet fillers would lie in between. Further rigid fillers would show higher complex viscosity than flexible fillers at the same frequency if their dimensions were similar. When storage modulus data are generated over an entire frequency range at a fixed strain, the conventional method of representation would be a plot of G' vs. CD. In the case of G vs. CD plots are viewed for different types of filler, then it would be seen that all fillers increase storage modulus at any frequency - the extent of the increase being the highest for fibrous fillers while less for platelike or particulate fillers and least for spherical fillers. Non-interacting fillers such as spherical fillers would show variation of G with frequency whereas interacting fillers like fibrous fillers would show greater independence with frequency as they would depict more solid like behavior with higher storage of elastic energy. 8.2

EFFECT OF FILLER SIZE

The smaller the size of the filler, the greater the particle-particle interaction and this reflects greatly on the unsteady shear viscoelastic properties. In the case of complex viscosity vs. frequency curves, it would be natural to expect yield stress with decreasing particle size as

was discussed in section 6.2. With larger particles, since the response to deformation is determined by hydrodynamic interaction and not by particle-particle interaction, such filled systems do not show yield stress. In the case of storage modulus, the plot of G vs. co is comparable with the plot of N1 vs. y shown in the earlier chapter. All conclusions that could be drawn through the plot of N1 vs. y would hold good when G vs. a> are viewed. But then it must be borne in mind that the type of data representation (N1 vs. y or N1 vs. T12) does make a difference to the derived conclusions. In the case of storage modulus, the data representation is always in the form of the G vs. co curves and hence the conclusions should be the same as those derived from the N1 vs. y curves. Actual unsteady shear data on filled polymer systems using a single polymer matrix but different sized fillers at a fixed level of loading from one source are unfortunately not available for validation. 8.3 EFFECT OF FILLER CONCENTRATION Dynamic data in the form of complex/dynamic viscosity and storage/ loss moduli vs. volume fraction (j> are available [8,71]. Faulkner and Schmidt [8] obtained dynamic modulus data on glass bead filled polypropylene composites using about 10% strain over the frequency range of 0.01-10 rad/sec. The data for frequencies between 0.01O.lrad/sec were found to have larger scatter and great deviations from the higher frequency trends. Hence, the data range was restricted to 0.1-10 rad/sec and the relative storage and loss moduli data were determined in this range. One representative curve at 11 rad/sec is shown in Figure 8.3. It is seen that the glass beads enable the polymer to store more energy elastically and to dissipate more mechanical energy when compared at equal strains, with trends similar to those obtained by Mills [3] for glass bead polystyrene systems. Within the limited range of volume fraction of up to 0.26, it has been shown [8] that the loss moduli (or, in other words, the dynamic viscosity) increases at a greater rate than the storage moduli with increasing filler volume fraction. Faulkner and Schmidt [8] correlated their data as follows:

1

and

87

m-'+ *

< - °>

G"Qft» co) = I + 2 0 + 3.30 2 ___

(8.Tb)

The above equations clearly show how the viscous response dominates

GLASS BEAD FILLED POLYPROFYUNE LOSS MODULUS STORAGE MODULUS RAD^EC

RELATIVE MODCfLUS

Figure 8.3 Variation of relative storage and loss moduli with volume fraction of filler in glass bead filled polypropylene system at frequency co = 1 rad/sec and strain y = 10%. (Reprinted from Ref. 8 with kind permission from Society of Plastics Engineers Inc., Connecticut, USA.)

the elastic response with increasing filler concentration in the case of glass bead filled polypropylene. Poslinski et al. [71] showed that when average relative dynamic viscosities in higher frequency ranges are plotted as a function of the reduced volume fraction, /m, the data approximately fall on a single curve which is independent of the polymer matrix and temperature as shown in Figure 8.4. The value of <j>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 <1

15(70%)

^84/^16 PSD 3.2

Narrow Bimodal

1 .2-0.35 8.0-1 .5

3.4 3.4 5.3

Narrow Narrow Broad

Table 8.2 Maximum packaging fraction of various filled systems used in the study of the effect of filler particle size distribution (PSD) Type

PSD

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

Narrow Bimodal Narrow Narrow Bimodal Broad Broad Unspecified

0m 0.63 0.69 0.57 0.63 0.70 0.60 0.70 0.62

Surface treatment None None None W-910, Kemamide E W-910, Kemamide E None W-910, Kemamide E Kemamide E, W-900, none

Source: Ref. 54.

gives the maximum packing fraction of the filler that could be loaded in the polyethylene matrix under the same mixing conditions. As expected, it was found [54] that using a broad distribution or a bimodal dispersion improved the level of maximum packing fraction as compared to that using a narrow size distribution. In the case of random shaped alumina and zirconia, which normally exist in agglomerated form, matrix additives were needed to achieve the maximum packing fraction. In the case of unagglomerated spherical steel particles, a bimodal distribution alone could produce the desired effect, as the smaller particles could easily be segregated into the interstices of the larger particles. Of course, its non-random packing is not completely achievable in a random mixing process. Broadly distributed powders do have random packing arrangement at their maximum packing fraction. Bigg [54] has shown how the difference between the two packing arrangements can be detected by changes in the storage modulusfrequency response of the filled polymer melt during dynamic measurements in the low strain region. Figure 8.8 shows that, though the complex viscosity for bimodal distribution of stainless steel spheres and alumina particles is the same, their storage modulus responses are different. Bigg [54] attributed the initial drop in the modulus value at low frequency to the movement of the smaller particles during their flow into the interstitial spaces between the larger particles. The low level of oscillatory motion provides sufficient energy to initiate the movement and create a more efficient packing. This occurs favorably for bimodal distribution but fails for broad distribution of zirconia particles, as shown in Figure 8.9. Even the use of matrix additive does not show the effect observed for the bimodal distribution. Bigg [54] concluded that bimodally distributed powders had the potential of higher loadings than broadly distributed powder, and one cannot achieve the theoretical maximum fraction by the usual random mixing procedures.

FILUED LOW DENSITY POLYETHYLENE

UNITS PASCALS

STAINLESS STEEL ALUMINA

Figure 8.8 Variation of complex viscosity and storage modulus with frequency, showing the effect of bimodal distribution of stainless steel and alumina in low density polyethylene at 20O0C. (Reprinted from Ref. 54.)

The study of the influence of bimodal size distribution of fillers on the rheology of glass sphere filled polymer systems has also been done by Poslinski et al [72]. The bimodal size distributions were prepared by mixing together various volume percentages of the smaller 15 pm spheres with the larger 78 ^m spheres given in Table 6.3. Various bimodal mixtures were compounded into a polybutene grade 24 polymer obtained from Petrochemical Division of the Chevron Company, at total solids concentrations of up to 60% by volume. The effect of the particle size and size distribution on dynamic viscosity is shown in Figure 8.10 through a plot of average relative dynamic viscosity vs. s, the vol% of the 15 jim spheres, ranging from O to 100% of the total solids mixture. It was found that in the limit of low frequencies, the dynamic viscosities of the filled polymer systems [72] were equal to the shear viscosities at the same shear rate. However, the relative dynamic viscosity was found to be a decreasing function of the frequency approaching a constant value at the higher frequencies. Consequently, relative values were obtained by averaging the data in

ZIRCONIA. FILLED LOW DENSITY POLYEIHYLENE

UNITS PASCALS

WIHOTJT ADDITIVE WTTH KEMAMIDE ADDITIVE

Figure 8.9 Variation of complex viscosity and storage modulus with frequency for zirconia-filled polyethylene at 20O0C with an additive. (Reprinted from Ref. 54.) Table 8.3 Summary of important torque values for mixing polyethylene with alumina powders treated with different surface modifying agents Sample description No additive Silane Z-6020 Silane Z-6076 Zircoaluminate CAVCO MOD APG Titanate LICA-12

Peak torque (g.m)

Final torque (g.m)

370 305 200 425 170

70 60 60 15 50

Source: Ref. 92.

the 10 to 100 rad/sec frequency range. As shown in Figure 8.10, the experimental data at higher !frequencies indicate that the relative dynamic viscosities are also lowered when the two sizes of spheres are mixed together, however, the reduction is not as significant as observed for both rjr and ^1 r discussed in sections 6.4 and 7.4, respectively. The lowest values of relative dynamic viscosity are seen to be obtained for (/>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

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

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

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

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

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

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)

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

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)

Table E1 (continued) Symbol 1

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)

Table E1 (continued) Symbol 1

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

Table E1 (continued) Symbol 1

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)

Table E1 (continued) Symbol 1

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)

Table E1 (continued) Symbol 1

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)

Table E1 (continued) Symbol 1

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 <5S £ e £f s £B £c £P C r\ r\(y) rja ^A rjc ^E f/E(0 T/EB f7EP ^f /ymc v\mo

Biaxial extensional rate Critical extensional rate Planar extensional rate Capillary entrance end effect correction for viscous effect Steady shear viscosity Steady shear viscosity function Apparent viscosity of a fluid in Figure 4.5 Viscosity value read from intersecting asymptotes Viscosity of the filled composite material Uniaxial extensional viscosity Extensional viscosity as a function of time Biaxial extensional viscosity Planar extensional viscosity Viscosity of a homogeneous suspension of floccules Viscosity of the polymer matrix Viscosity of the unaffected polymer

Units 3

Equation 4

sec"1

(2.12), (2.13), (2.15), (2.16) (3.7)

sec"1 J

(3.13) (1.1)

J

(1.1)

J

(1.1) (2.14H2.17)

m2/sec sec'1 sec"1 sec"1 sec~1

(6.5) (5.15) (2.22)-(2.26), (2.66a), (2.66b), (3.19), (3.20) (2.27), (2.28) (2.29), (2.30) (3.12), (3.14)-(3.16)

Pa.sec Pa.sec Pa.sec

(2-9)

Pa.sec

(2.44b)

Pa.sec

(6.22)

Pa.sec Pa.sec

(2.31), (2.66a), (2.66b)

Pa.sec Pa.sec Pa.sec

(2.32) (2.33) (4.28)

Pa.sec Pa.sec

(6.22) (6.22)

Table F1 Continued Symbol 1

Description 2

Units 3

Equation 4

rjN

Viscosity of filled polymer system in the Newtonian plateau region Relative steady shear viscosity which is the ratio of viscosity of filled to unfilled polymer system Relative viscosity at fixed shear rate Relative viscosity of a mixture of two suspended particles Viscosity of a dilute suspension of spheres Trouton's viscosity in uniaxial stretching Volume viscosity Zero shear viscosity; also viscosity of the suspending medium

Pa.sec

(2.45b)

Pa.sec

(6.11)

rjr 0/r)f >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
Parameters of the Oldroyd model Characteristic time Time scale of deformation Function of 0 Mean of composition of a random homogeneous mixture Interaction parameter Kinematic viscosity Constant Polymer melt density Density of the suspending medium Standard deviation of composition of a random homogeneous mixture Variance of composition of a random homogeneous mixture Upper and lower limits of the dimensionless radius, respectively Average extensional stress Shear stress Wall shear stress Yield stress Special value of shear stress when steady shear viscosity is half the zero shear viscosity Amplitude of the sinusoidal variation of shear stress Shear stress components of the stress tensor Normal stress components of the stress tensor Primary normal stress difference Secondary normal stress difference Extra stress tensor Volume fraction of the suspended particles or fillers Weight fraction of filler in Figures 6.17(a),6.17(b) and 6.31

sec sec sec

(4.3) (5.2a), (5.3b) m2/sec kg/m3 kg/m3

(4.19) (5.15) (4.28) (4.22), (4.23) (5.7) (5.2b), (5.3c) (4.26)

Pa Pa Pa Pa Pa

(2.67a) (2.36), (2.39) (3.11), (3.12) (2.45), (6.5a), (6.6) (2.41) ,(6.32)

Pa

(2.14)-(2.16)

Pa

(2.2), (2.6), (2.9)

Pa

(2.6)-(2.8)

Pa Pa Pa

(2.10) (2.11) (2.3), (2.6), (2.34)

Table F1 Continued Symbol 1

Description 2

Units 3

Equation 4

00 01 ' 02> 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

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

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

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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

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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

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473

Index terms

Links

Effect of filler concentration (Continued) viscous

248

unsteady shear viscoelastic

345

Effect of filler size extensional

400

steady shear elastic

315

viscous

246

unsteady shear viscoelastic

344

Effect of filler size distribution steady shear elastic

321

viscous

262

unsteady shear viscoelastic

350

Effect of filler surface treatment extensional

405

steady shear elastic

323

viscous

273

unsteady shear viscoelastic

360

Effect of filler type extensional

396

steady shear elastic

313

viscous

244

unsteady shear

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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

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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

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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

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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

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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

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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

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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

extensional viscosity

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

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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

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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

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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

extensional viscosity

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

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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

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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

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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

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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

extensional viscosity

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

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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

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