Filled Polymers: Science and Industrial Applications

Filled Polymers Science and Industrial Applications

Filled Polymers Science and Industrial Applications Jean L. Leblanc

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4398-0042-3 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

Contents Preface.......................................................................................................................xi Author Bio...............................................................................................................xv

  1. Introduction....................................................................................................1 1.1. Scope of the Book...................................................................................1 1.2. Filled Polymers vs. Polymer Nanocomposites...................................3 References........................................................................................................8   2. Types of Fillers............................................................................................ 11   3. Concept of Reinforcement......................................................................... 15 Reference........................................................................................................ 19   4. Typical Fillers for Polymers...................................................................... 21 4.1 Carbon Black......................................................................................... 21 4.1.1 Usages of Carbon Blacks.......................................................... 21 4.1.2 Carbon Black Fabrication Processes....................................... 21 4.1.3 Structural Aspects and Characterization of Carbon Blacks....................................................................... 24 4.1.4 Carbon Black Aggregates as Mass Fractal Objects..............30 4.1.5 Surface Energy Aspects of Carbon Black..............................44 4.2 White Fillers.......................................................................................... 49 4.2.1 A Few Typical White Fillers.................................................... 49 4.2.1.1 Silicates......................................................................... 49 4.2.1.2 Natural Silica............................................................... 52 4.2.1.3 Synthetic Silica............................................................ 53 4.2.1.4 Carbonates...................................................................54 4.2.1.5 Miscellaneous Mineral Fillers................................... 56 4.2.2. Silica Fabrication Processes..................................................... 56 4.2.2.1 Fumed Silica................................................................ 56 4.2.2.2 Precipitated Silica....................................................... 58 4.2.3 Characterization and Structural Aspects of Synthetic Silica.......................................................................... 62 4.2.4 Surface Energy Aspects of Silica............................................ 68 4.3 Short Synthetic Fibers.......................................................................... 69 4.4 Short Fibers of Natural Origin........................................................... 72 References...................................................................................................... 79

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Appendix 4.................................................................................................... 82 A4.1 Carbon Black Data............................................................................ 82 A4.1.1 Source of Data for Table 4.5............................................... 82 A4.1.2 Relationships between Carbon Black Characterization Data........................................................84 A4.2  Medalia’s Floc Simulation for Carbon Black Aggregate.............85 A4.3  Medalia’s Aggregate Morphology Approach............................... 86 A4.4  Carbon Black: Number of Particles/Aggregate............................ 89   5. Polymers and Carbon Black...................................................................... 91 5.1 Elastomers and Carbon Black (CB).................................................... 91 5.1.1 Generalities................................................................................ 91 5.1.2 Effects of Carbon Black on Rheological Properties............. 95 5.1.3 Concept of Bound Rubber (BdR).......................................... 108 5.1.4 Bound Rubber at the Origin of Singular Flow Properties of Rubber Compounds.......................... ............... 112 5.1.5 Factors Affecting Bound Rubber.......................................... 114 5.1.6 Viscosity and Carbon Black Level........................................ 121 5.1.7 Effect of Carbon Black on Mechanical Properties.............. 125 5.1.8 Effect of Carbon Black on Dynamic Properties.................. 140 5.1.8.1 Variation of Dynamic Moduli with Strain Amplitude (at Constant Frequency and Temperature)............................................................. 141 5.1.8.2 Variation of tan δ with Strain Amplitude and Temperature (at Constant Frequency)...................142 5.1.8.3 Variation of Dynamic Moduli with Temperature (at Constant Frequency and Strain Amplitude)..................................................... 142 5.1.8.4 Effect of Carbon Black Type on G′ and tan δ.................................................................... 144 5.1.8.5 Effect of Carbon Black Dispersion on Dynamic Properties................................................. 146 5.1.9 Origin of Rubber Reinforcement by Carbon Black............................................................................ 148 5.1.10 Dynamic Stress Softening Effect.......................................... 151 5.1.10.1 Physical Considerations........................................... 151 5.1.10.2 Modeling Dynamic Stress Softening as a “Filler Network” Effect............................................ 152 5.1.10.3 Modeling Dynamic Stress Softening as a “Filler–Polymer Network” Effect........................... 168 5.2 Thermoplastics and Carbon Black................................................... 172 5.2.1 Generalities.............................................................................. 172 5.2.2 Effect of Carbon Black on Rheological Properties of Thermoplastics........................................................................ 173

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5.2.3 Effect of Carbon Black on Electrical Conductivity of Thermoplastics........................................................................ 175 References.................................................................................................... 179 Appendix 5.................................................................................................. 185 A5.1 Network Junction Theory.............................................................. 185 A5.1.1 Developing the Model...................................................... 185 A5.1.2 Typical Calculations with the Network Junction Model.................................................................. 188 A5.1.3 Strain Amplification Factor from the Network Junction Theory................................................................. 190 A5.1.3.1 Modeling the Elastic Behavior of a Rubber Layer between Two Rigid Spheres................................................... 190 A5.1.3.2 Experimental Results vs. Calculated Data................................................ 191 A5.1.3.3 Comparing the Theoretical Model with the Approximate Fitted Equation.. .. ............ 192 A5.1.3.4 Strain Amplification Factor............................ 193 A5.1.4 Comparing the Network Junction Strain Amplification Factor with Experimental Data............. 194 A5.2 Kraus Deagglomeration–Reagglomeration Model for Dynamic Strain Softening............................................................. 196 A5.2.1 Soft Spheres Interactions................................................. 196 A5.2.2 Modeling G′ vs. γ0............................................................. 197 A5.2.3 Modeling G″ vs. γ0............................................................ 198 A5.2.4 Modeling tan δ vs. γ0........................................................ 200 A5.2.5 Complex Modulus G* vs. γ0............................................. 202 A5.2.6 A Few Mathematical Aspects of the Kraus Model...................................................................... 204 A5.2.7 Fitting Model to Experimental Data.............................. 206 A5.2.7.1 Modeling G′ vs. Strain.................................... 207 A5.2.7.2 Modeling G″ vs. Strain.................................... 209 A5.3 Ulmer Modification of the Kraus Model for Dynamic Strain Softening: Fitting the Model.............................................. 212 A5.3.1 Modeling G′ vs. Strain (same as Kraus)......................... 213 A5.3.2 Modeling G′′ vs. Strain..................................................... 215 A5.4 Aggregates Flocculation/Entanglement Model (Cluster–Cluster Aggregation Model, Klüppel et al.)............... 218 A5.4.1 Mechanically Effective Solid Fraction of Aggregate...................................................................... 219 A5.4.2 Modulus as Function of Filler Volume Fraction........... 220 A5.4.3 Strain Dependence of Storage Modulus........................ 221 A5.5 Lion et al. Model for Dynamic Strain Softening........................222 A5.5.1 Fractional Linear Solid Model.........................................222

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A5.5.2 Modeling the Dynamic Strain Softening Effect...........223 A5.5.3 A Few Mathematical Aspects of the Model.................. 226 A5.6 Maier and Göritz Model for Dynamic Strain Softening........... 227 A5.6.1 Developing the Model...................................................... 227 A5.6.2 A Few Mathematical Aspects of the Model.................. 229 A5.6.3 Fitting the Model to Experimental Data........................ 230 A5.6.3.1 Modeling G′ vs. Strain.................................... 231 A5.6.3.2 Modeling G″ vs. Strain.................................... 232   6. Polymers and White Fillers..................................................................... 235 6.1 Elastomers and White Fillers........................................................... 235 6.1.1 Elastomers and Silica.............................................................. 235 6.1.1.1 Generalities................................................................ 235 6.1.1.2 Surface Chemistry of Silica..................................... 236 6.1.1.3 Comparing Carbon Black and (Untreated) Silica in Diene Elastomers....................................... 237 6.1.1.4 Silanisation of Silica and Reinforcement of Diene Elastomers...................................................... 239 6.1.1.5 Silica and Polydimethylsiloxane............................. 246 6.1.2 Elastomers and Clays (Kaolins)............................................ 257 6.1.3 Elastomers and Talc................................................................ 260 6.2 Thermoplastics and White Fillers.................................................... 262 6.2.1 Generalities.............................................................................. 262 6.2.2 Typical White Filler Effects and the Concept of Maximum Volume Fraction.................................................. 266 6.2.3 Thermoplastics and Calcium Carbonates........................... 280 6.2.4 Thermoplastics and Talc........................................................ 291 6.2.5 Thermoplastics and Mica...................................................... 297 6.2.6 Thermoplastics and Clay(s)...................................................300 References.................................................................................................... 302 Appendix 6..................................................................................................308 A6.1 Adsorption Kinetics of Silica on Silicone Polymers...................308 A6.1.1 Effect of Polymer Molecular Weight..............................308 A6.1.2 Effect of Silica Weight Fraction....................................... 310 A6.2 Modeling the Shear Viscosity Function of Filled Polymer Systems............................................................................. 312 A6.3 Models for the Rheology of Suspensions of Rigid Particles, Involving the Maximum Packing Fraction Φm........................... 315 A6.4 Assessing the Capabilities of Model for the Shear Viscosity Function of Filled Polymers......................................... 319 A6.4.1 Effect of Filler Fraction..................................................... 320 A6.4.2 Effect of Characteristic Time λ0...................................... 320 A6.4.3 Effect of Yasuda Exponent a............................................ 321 A6.4.4 Effect of Yield Stress σc................................................... 321

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A6.4.5 Fitting Experimental Data for Filled Polymer Systems.............................................................. 322 A6.4.6 Observations on Experimental Data............................ 323 A6.4.7 Extracting and Arranging Shear Viscosity Data.................................................................. 324 A6.4.8 Fitting the Virgin Polystyrene Data with the Carreau–Yasuda Model.................................................. 324 A6.4.9 Fitting the Filled Polystyrene Shear Viscosity Data................................................................................... 326 A6.4.10 Assembling and Analyzing all Results........................ 332 A6.5 Expanding the Krieger–Dougherty Relationship...................... 335   7. Polymers and Short Fibers...................................................................... 339 7.1 Generalities......................................................................................... 339 7.2 Micromechanic Models for Short Fibers-Filled Polymer Composites..........................................................................................344 7.2.1 Minimum Fiber Length.........................................................344 7.2.2 Halpin–Tsai Equations...........................................................345 7.2.3 Mori–Tanaka’s Averaging Hypothesis and Derived Models...................................................................................... 351 7.2.4 Shear Lag Models.................................................................... 353 7.3 Thermoplastics and Short Glass Fibers........................................... 358 7.4 Typical Rheological Aspect of Short Fiber-Filled Thermoplastic Melts.......................................................................... 368 7.5 Thermoplastics and Short Fibers of Natural Origin..................... 370 7.6 Elastomers and Short Fibers............................................................. 375 References.................................................................................................... 383 Appendix 7.................................................................................................. 389 A7.1 Short Fiber-Reinforced Composites: Minimum Fiber Aspect Ratio..................................................................................... 389 A7.1.1 Effect of Volume Fraction on Effective Fiber Length...................................................................... 389 A7.1.2 Effect of Matrix Modulus on Effective Fiber Length...................................................................... 390 A7.1.3 Effect of Fiber-to-Matrix Modulus Ratio on Effective Fiber Length/Diameter Ratio......................... 391 A7.2 Halpin–Tsai Equations for Short Fibers Filled Systems: Numerical Illustration.................................................................... 391 A7.2.1 Longitudinal (Tensile) Modulus E11............................... 392 A7.2.2 Transversal (Tensile) Modulus E22. ................................ 393 A7.2.3 Shear Modulus G12............................................................ 393 A7.2.4 Modulus for Random Fiber Orientation........................ 394 A7.2.5 Fiber Orientation as an Adjustable Parameter. ......................................................................................394

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A7.2.6 Average Orientation Parameters from Halpin–Tsai Equations for Short Fibers Filled Systems.................................................................... 394 A7.2.6.1 Longitudinal (Tensile) Modulus E11.............. 395 A7.2.6.2 Transversal (Tensile) Modulus E22. ............... 396 A7.2.6.3 Orientation Parameter X................................. 396 A7.3 Nielsen Modification of Halpin–Tsai Equations with Respect to the Maximum Packing Fraction: Numerical Illustration........................................................................................ 396 A7.3.1 Maximum Packing Functions......................................... 397 A7.3.2 Longitudinal (Tensile) Modulus E11............................... 398 A7.3.3 Transverse (Tensile) Modulus Ey.................................... 398 A7.3.4 Shear Modulus G.............................................................. 398 A7.4 Mori–Tanaka’s Average Stress Concept: Tandon–Weng Expressions for Randomly Distributed Ellipsoidal (Fiber-Like) Particles: Numerical Illustration............................. 399 A7.4.1 Eshelby’s Tensor (Depending on Matrix Poisson’s Ratio and Fibers Aspect Ratio Only).............................. 399 A7.4.2 Materials’ Constants (i.e., Not Depending on Fiber Volume Fraction)...............................................................400 A7.4.3 Materials and Volume Fraction Depending Constants............................................................................ 401 A7.4.4 Calculating the Longitudinal (Tensile) Modulus E11. ...................................................... 402 A7.4.5 Calculating the Transverse (Tensile) Modulus E22....... 402 A7.4.6 Calculating the (In-Plane) Shear Modulus G12............. 403 A7.4.7 Calculating the (Out-Plane) Shear Modulus G23..........404 A7.4.8 Comparing with Experimental Data.............................404 A7.4.9 Tandon–Weng Expressions for Randomly Distributed Spherical Particles: Numerical illustration...................................................... 406 A7.4.9.1 Eshelby’s Tensor (Depending on Matrix Poisson’s Ratio Only)....................................... 406 A7.4.9.2 Materials’ Constants (i.e., Not Depending on Filler Volume Fraction)......... 406 A7.4.9.3 Materials and Volume Fraction Depending Constants..................................... 407 A7.4.9.4 Calculating the Tensile Modulus E...............408 A7.4.9.5 Calculating the Shear Modulus G.................408 A7.5 Shear Lag Model: Numerical illustration.................................... 409 Index........................................................................................................... 411

Preface This book is an outgrowth of a course I have taught for several years to master and doctorate students in polymer science and engineering at the Université Pierre et Marie Curie (Paris, France). It is also based on around 30 years of interest, research and engineering activities in the fascinating field of so-called complex polymer systems, i.e., heterogeneous polymer based materials with strong interactions between phases. Obviously, rubber compounds and filled thermoplastics belong to such systems. If one considers that, worldwide, around 40% of all thermoplastics and 90% of elastomers are used as more or less complicated formulations with so-called fillers, it ­follows that approximately 100 million tons/year of polymers are indeed “filled systems.” Quite a number of highly sophisticated applications of polymers would simply be impossible without the enhancement of some of their properties imparted by the addition of fine mineral particles or by short fibers, of synthetic or natural origin. The idea that, if a single available material cannot fulfill a set of desired properties, then a mixture or a compound of that material with another one might be satisfactory is likely as old as mankind. Adobe, likely the oldest building material, is made by blending sand, clay, water and some kind of fibrous material like straw or sticks, then molding the mixture into bricks and drying in the sun. It is surely one of the oldest examples of reinforcement of a “plastic” material, moist clay, with natural fibers that was already in use in the Late Bronze Age, nearly everywhere in the Middle East, North Africa, South Europe and southwestern North America. In a sense, the basic principle of reinforcement, i.e., to have a stiffer dispersed material to support the load transmitted by a softer matrix, is already in the adobe brick. Therefore, the “discovery” of natural rubber reinforcement by fine powdered materials, namely carbon black, in the dawn of the twentieth century surely proceeded from the same idea. At first, mixing rubber and carbon black was pragmatic engineering, it gave a better and useful set of properties, and the technique could be somewhat mastered, thanks to side developments, such as the internal mixer. The very reasons for the reinforcing effect remained unclear for a long time and the question only started to be seriously considered by the mid ­t wentieth century. Today, some light has been shed on certain aspects of polymer reinforcement, as will be reviewed through the book. But the story is surely not complete because any progress in the field is strongly connected with either the availability of appropriate experimental and observation techniques or theoretical views about polymer–filler interactions, or (and most likely) both. xi

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One of the starting points of my deep interest for filled polymers is the simple observation that, whilst having different chemical natures, a number of filled polymers, either thermoplastics or vulcanizable rubbers, exhibit common singular properties. This aspect will be thoroughly documented throughout the book but a few basic observations are worth highlighting here. Let us consider for instance the flow properties of systems that are as (chemically) different as a compound of high cis-1,4 polybutadiene with a sufficient level of carbon black and a mixture of polyamide 66 with short glass fibers. They share the same progressive disappearance with increasing filler content of the low strain (or rate) linear viscoelastic behavior. Regarding the mechanical properties, the effect of either fine precipitated calcium carbonate particles or short glass fibers on the tensile and flexural moduli of polypropylene are qualitatively similar but by no means corresponding to mere hydrodynamic effects. So, many filled polymer systems are similar in certain aspects and different in others. Understanding why is likely to be the source of promising scientific and engineering developments. The possibilities offered by combining one (or several) polymer(s) with one (or several) foreign stiffer component(s) are infinite and the just emerging nanocomposites science is an expected development of the science and technology of filled polymers, once the basic relationships between reinforcement and particle size had been established. For reasons that are given in Chapter 1, nanofillers have been excluded from the topics covered by the book, whose objectives are to survey quite a complex field but by no means offer the whole story. As stated above, teaching the subject is the origin of the book. In my experience, nothing must be left in the shadow when teaching a complex subject and all theories and equations found in the literature must be carefully checked and weighed, particularly if engineering applications are foreseen. I am not a theoretician but an experimentalist with an avid interest for any fundamental approach that might help me to understand what I am measuring. Therefore, whilst theoretical considerations that lead to proposals such as “property X is proportional to (or a function of) parameter Y,” i.e., X∝ Y or X∝ F(Y), may be acceptable in term of (scientific) common sense, they are of very little use for the engineer (and less so for the student) if the coefficient of proportionality (or the function) is not explicitly given. This is the reason why all equations displayed in the book have been carefully tested, using (commercial) calculation software. When one loads theoretical equations with parameters expressed in the appropriate units, then either the unit system is inconsistent and the software gives no results because the unit equation is considered, or the right units are used and the results of the theory can be weighed, at least in terms of “magnitude order.” If the results have the right order of magnitude, then the theoretical considerations are likely acceptable. If not… Such an exercise is always useful and I am grateful to my editor for having accepted, as appendices, a selection of calculation worksheets (obviously inactive in a printed book) that offer numerical illustrations of

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several of the theoretical considerations discussed in the book. Readers who are familiar with the calculation software I use will have no difficulties in implementing these appendices in their own work. As a last word, it is worth noting that writing a science book on an active field is (by essence) a never ending task since new interesting contributions are published every day. But working with an editor forces the scientistwriter to accept a deadline, in other words to make choices, to develop more certain subjects and drop other ones, and eventually to bring an end point, not final but temporary as always in science and industrial applications. Jean L. Leblanc Bois-Seigneur-Isaac

Author Bio Born in 1946, Jean L. Leblanc studied ­physico-chemistry at the University of Liège, Belgium, with a special emphasis on polymer science and received his PhD in 1976, with a thesis on the rheological properties on SBS bloc copolymers. He then joined Monsanto Company where, from 1976 to 1987, he held various positions in the rubber chemicals, the AcrylonitrileButadiene-Styrene plastics (ABS), and the santoprene• divisions. He left Monsanto in 1987 to join the italian company Montedison as manager, technical assistance and applied research, then moved to the position of manager applied research when Enichem took over Montedison in 1989. In 1988, he became fellow of the Plastics and Rubber Institute (U.K.) and in 1993 he qualified as European Chemist (EurChem). In 1993, he was elected Professeur des Universités in France and joined the Université Pierre et Marie Curie (Paris, France), as head of the then newly developed polymer rheology and processing laboratory, in collaboration with the French Rubber Institute. He is still in this position today and, since 1997, also teaches polymer rheology and processing at the Free University of Brussels (Belgium), as a visiting professor. He has written two books and more than 100 papers.

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1 Introduction

1.1  Scope of the Book This book deals with the properties of filled polymers, i.e. mixtures of ­macromolecular materials with finely divided substances, with respect to established scientific aspects and industrial developments. So-called (polymer) composites, that consist of long fibers impregnated with resins, such as glass fibers reinforced polyesters or carbon fibers reinforced epoxy resins, are not within the subject of this book. Filled polymers discussed hereafter are heterogeneous systems such that, during processing operations, the polymer and the dispersed filler flow together. In other words, filled ­polymers are macroscopically coherent masses that exhibit interesting physical, mechanical, and/or rheological properties, often peculiar, but always resulting from interactions taking place between a matrix (the polymer) and a dispersed phase (the filler). It follows obviously that filled polymers have to be prepared through mixing operations, generally complex and requiring appropriate machines, in such a manner that a thorough dispersion of filler particles is achieved. Why does one prepare filled polymers? There are many reasons, all of them related to engineering needs. Generally one mixes fillers into polymers in order to modify properties of the latter, either physical properties, such as density or conductivity, or mechanical properties, for instance modulus, stiffness, etc., or rheological properties, i.e., viscosity or viscoelasticity. Occasionally, fillers are also used for economical reasons, as cheap additives that reduce material costs in polymer applications. Table 1.1 gives the relative volume costs of a few common mineral fillers in comparison with several polymers, using polypropylene (PP) as a reference. Clearly, only grinded calcium carbonate and finely divided clays can be considered as “economical” fillers; in all other cases, specific property improvements are sought when mixing the filler and the polymer. A few numbers allow underlining the economical importance of filled polymers. According to recently published market research reports (2007), the worldwide consumption of fillers is more than 50 million tons with a global value of approximately €25 billion. Many application areas are concerned, 1

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Filled Polymers

Table 1.1 Relative Cost of Mineral Fillers and Polymers Type of Filler or Polymer Grinded calcium carbonate Grinded clays Polyvinyl chloride Carbon black Polypropylene Talc Polyethylene Calcined clays Wollastonite (not treated) Natural rubber Ethylene-propylene rubber Treated calcined clays Styrene-butadiene rubber Silica Precipitated CaCO3 Polyamides

Relative Weight Cost (Polypropylene = 1.0) 0.3–0.6 0.4–0.7 0.7 0.7–1.2 1 1.1–1.4 1.1 1.5–1.7 1.6 1.6 1.6–1.9 1.7–1.9 1.7 1.7–1.9 1.9 3.0–6.0

Note: Table assembled using prices and quotations on the European market during the first sem­ester of 2008.

such as paper, plastics, rubber, paints, and adhesives. Fillers, either synthetic or of natural origin are produced by more than 700 companies all over the globe. In Western Europe, 17 millions tons of thermoplastics were consumed in 2005 with a significant part in association with 1.7 millions tons of mineral fillers. Polyvinyl chloride (PVC) and polyolefins (polyethylene PE, PP) are the main markets for mineral fillers, with calcium carbonate CaCO3 accounting for more that 80% of the consumption (in volume). In rubber materials, more that 90% of the applications concern “compounds”, i.e. quite complex formulations in which fillers are used at around 50% weight (some 30% volume). The Western Europe consumption of rubbers was 3.79 millions tons in 2006 (1.28 MioT natural rubber; 2.51 MioT synthetic elastomers) and some 2.25 millions tons carbon black were used in the interim. Preparing and using filled polymers is consequently a well established practice in the polymer field, particularly in the rubber industry where the first use of carbon black as a reinforcing filler can be traced back to the early twentieth century. There are consequently a number of pragmatic engineering aspects associated with the preparation, the development and the applications of filled polymers, not all yet fully understood, despite considerable progresses over the last 50 years. As usual, scientific investigations on filled polymer systems started later than empirical engineering (trial-and-error)

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and it is only the recent development of advanced investigation means that really boosted research and development work in this area, obviously connected with the contemporary physico-chemistry research on interfaces and interphases. Polymers, either elastomers or thermoplastics, offer a great variety of chemical natures, as well as the fillers, but curiously common effects and properties are (at least qualitatively) observed whatever is the chemistry of the polymer matrix and of the filler particles. This striking observation is the very origin of this book that intends to offer a survey of a quite complex field, with the objectives to highlight what most filler–polymer systems have in common, how proposed theories and models suit observations and, eventually what are the specificities of certain filled polymers.

1.2  Filled Polymers vs. Polymer Nanocomposites A filled polymer system is thus a polymer in which a sufficient quantity (volume) of a small size foreign rigid (or at least less flexible) material, e.g., powdered minerals, short glass fibers, etc., has been well dispersed in order to improve certain key properties of engineering importance, for instance modulus, stiffness, or viscosity. The reinforcing effect of carbon black in rubber is known for one century (1907, Silvertown, UK) and the mastering and understanding of its scientific aspect has led to the development of many high engineering performance products, for instance the automobile, truck, or aircraft tires. Starting in 1984, a series of patents obtained by Toyota1 described the use of organoclay additives for plastics as well as various plastic structures that could replace traditional components (e.g., ­aluminium parts) in automotive applications. Typically U.S. patent No. 4,810,734 described a production process for a composite material by firstly treating a layered smectite mineral having a cation exchange capacity (e.g., a phyllosilicate) with a swelling agent having both an onium ion and a functional group capable of reacting with a polymer and secondly forming a complex with a molten polymer. U.S. patent No. 4,889,885 described a composite material made with at least one resin selected from the group consisting of a vinyl-based polymer, a thermosetting resin and a rubber, and a layered bentonite uniformly dispersed in the resin. The layered silicate has a layer thickness of about 0.7–1.2 nm and an interlayer distance of at least about 3 nm, and at least one polymer ­macromolecule has to be connected to the layered silicate. Such patents prompted, over the last 20 years, a kind of cult research area for so-called polymer nanocomposites, whose origin of reinforcement is on the order of nanometers, but with the capability to deeply affect the final macroscopic properties of the resulting material. In certain cases, such materials exhibit properties not present in the pure polymer resin, whilst keeping the processibility, the other mechanical

4

Filled Polymers

polymer properties and the specific weight. Several types of polymeric nanocomposites can in principle be obtained with different particle nanosize, nature and shape: clay/polymer, carbon nanotubes, and metal/polymer nanocomposites. Let us consider the case of clay/polymer nanocomposites. The key aspect is obviously the successful formation of suitable clay/polymer nanostructures, essentially through an intercalation process. In the case of hydrophilic ­polymers (typically polyamides) and silicate layers, pretreatment is not necessary; but most polymers are hydrophobic and are not compatible with ­hydrophilic clays. Complicated and expensive pretreatments are thus required. For instance organophilic clays can be obtained from normally hydrophilic clay by using amino acids, organic ammonium salts, or tetra organic phosphonium solutions, to name a few reported techniques. Established methods are: solution induced intercalation, in situ polymerization, and melt processing. Solution induced intercalation consists of solubilizing the polymer in an organic solvent, then dispersing the clay in the solution and subsequently either evaporating the solvent or precipitating the polymer. Such a technique is obviously expensive, raises a number of environmental, health, and safety problems (common to all organic solution techniques), and in fact leads to poor clay dispersion. In situ polimerization consists of dispersing clay layers into a matrix before polimerization, i.e., mixing the silicate layers with the monomer, in conjunction with the polymerization initiator and/or the catalyst. This technique is obviously limited to polymers whose ­monomers are liquids, and therefore excludes most of the general purpose (GP) resins, namely polyolefins. In the melt processing technique the silicates layers, previously surface treated with an organo-modifier, are directly dispersed into the molten polymer, using the appropriate equipment and procedure. A priori, this technique would be the preferred route with most GP polymers, providing mixing/dispersion problems are mastered. In theory, extraordinary improvements of material properties are expected with polymer nanocomposites but, in reality, the overall balance of usage properties (i.e., mechanical, hardness, wear resistance, to name a few) in the best clay/polymer nanocomposites are much lower than in conventional fiber reinforced composites, or even in certain traditional filled compositions. It is indeed only in the low filler range, typically 4–5 wt%, and providing the dispersion of nanoparticles is nearly ideal, that nanocomposites show better mechanical performances, but at the cost of major difficulties in mass fabrication. At higher loading, the surface area of the silicate-filler increases, which leads to insufficient polymer molecules adsorbed on the clay surface. One may consider that polymer nanocomposites combine two concepts: composites (i.e., heterogeneous systems) and nanometer-size ­materials; the hope that manufacturing composites polymer material could eventually be achieved with a tight control at molecular level (i.e., the nanometer range) surely justifies fundamental research in this area, even if large scale industrial applications are not yet in sight. Certain thermoplastics, filled with nanometer-size

Introduction

5

materials, have indeed different properties than systems filled with conventional mineral materials. Some of the properties of nanocomposites, such as increased tensile strength, are routinely achieved by using higher conventional filler loading, but of course at the expense of increased weight and sometimes with unwanted changes in surface aspects, i.e., gloss with certain polymers. Obviously other typical properties of certain polymer nanocomposites such as clarity or improved barrier properties cannot be duplicated by filled resins at any loading. One may indeed consider that polymer nanostructured materials represent a radical alternative to the conventional filled polymers and polymer blends, because the utility of inorganic nanoparticles as additives to enhance polymer performance has been well established at laboratory level. The incorporation of low volume (1–5 wt%) of highly anisotropic nanoparticles, such as layered silicates or carbon nanotubes, results in the enhancement of certain properties with respect to the neat polymer that are comparable with what is achieved by conventional loadings (15–40 wt%) of traditional fillers.2 In principle the lower loadings would facilitate processing and reduce component weight, and in addition, certain value added properties not normally possible with traditional fillers are also observed, such as higher stiffness, reduced permeability, optical clarity, and electrical conductivity. But the chemical and processing operations to disrupt the low-dimensional crystallites and to achieve uniform distribution of the nanoelement (layered silicate and single wall carbon nanotube, respectively) continue to be a challenge. Most commercial interest in nanocomposites has so far focused on thermoplastics, essentially because certain polymer nanocomposites allow the substitution of more expensive engineering resins with less expensive commodity polymer nanocomposites, to yield overall cost savings. But such favorable cases are rare and restricted to very specific applications. A recent study by a market research company claims that, by 2010, nanocomposites demand will grow to nearly 150,000 tons, and will rise to over 3 million tons with a value approaching $15 billion by 2020.3 So far however the market for these new materials has not developed as expected and if, indeed, exfoliated (or surface treated) nanoclays are commercially available,4 their uses seem restricted to very specific cases. Packaging and parts for motor vehicles are nevertheless expected to be key markets for nanoclay and nanotube composites. With respect to the improved barrier, strength and conductive properties that they can offer, polymer nanocomposites should somewhat penetrate certain food, beverage, and pharmaceutical packaging applications, as well as specific parts for electronics. In motor vehicles, automotive manufacturers are expected to consider polymer nanocomposites either as replacement for higher-priced materials, or to increase the production speed of parts and to reduce motor vehicle weight by lightening a number of exterior, interior, and underhood applications. The future will weigh such expectations. Over the last decades, a considerable number of research papers have been published whose main subject is so-called polymer nanocomposites,5 i.e.,

6

Filled Polymers

mixtures or preparations involving macromolecular materials and small particles with dimensions in the nanometer range, with however a great deal of confusion in the author’s opinion. Indeed, a careful reading of published papers reveals that for certain authors, nanoparticles are entities with (equivalent) diameters up to a few tens nanometers, whilst others title their works with the heading nanocomposites but consider mixtures with particles in the micron range. It is also worth underlining that nanoparticles technology implies that individual representatives particles (i.e., spheres, platelets, etc.) are ideally dispersed in the polymer matrix, without agglomeration or flocculation. This aspect of polymer nanocomposites appears thus in sharp contrast with conventional filled polymer technology where elementary particles must be suitably clustered in complex tri-dimensional structures called “aggregates” to yield reinforcing properties. As will be extensively described in this book, this is the key aspect of the reinforcement of rubber with carbon black and high structure silica. In many published papers this ideal dispersion of nanoparticles is neither documented nor granted by the preparation (mixing) process, and therefore the reference to polymer nanocomposites is dubious. Despite the lack—so far—of significant industrial applications, polymer nanocomposites seem to be a fashion subject for fundamental research, with sometimes an unfortunate lack of reference to earlier works on more classical filled polymer systems, namely filled rubber materials, surely the oldest class of complex polymer materials of industrial importance. There are a number of recent books, reviews, and treatises on so-called polymer nanocomposites6–8 and elastomer nanocomposites.9,10 The present book is definitely not addressing the same subject, but rather so-called “filled polymer systems” that are nowadays used yearly in quantities of hundred thousands to million tons worldwide. In order to avoid confusion it is thus necessary to clearly define what are filled polymer systems, the very subject of the present book, in contrast with polymer naonocomposites. It is clear that industrial use is not a sufficient criterion to distinguish both classes of materials. Whilst mainly concerned with rubber reinforcement, Hamed offered recently quite a clear and wellsupported proposal to distinguish filled polymer systems, with respect to the smallest size d of the dispersed phase.11 The characteristic smallest dimension d depends of course of the actual geometry of the particles, the diameter for spheres and rods, the thickness for plates and scales. There are a number of available materials whose characteristic particle dimension is in the 1–100 nm range and therefore the prefix nano is ambiguously used in the literature. We will consequently somewhat follow the Hamed’s proposal: when the characteristic dimension d of the dispersed phase is between 1 and 10 nm, then one is dealing with nanocomposites, when 100 nm > d > 10 nm, then mesocomposites are involved, with d above 100, composite materials are referred with the prefix micro, and the prefix macro when very gross  (d > 104  nm) rigid “entities” are dispersed in a polymer. Further to this basic characterization, Hamed considers that the dispersed entities can be structured, either a priori

7

Introduction

by their nature or through their manufacturing process, or as a result of the kinetics and thermodynamics of phase separation that may occur during the preparation of the complex polymer system. The proposal is further elaborated in Table 1.2., with typical examples of concerned materials. With respect to Table 1.2, all filled polymer systems discussed in this book are either meso or microcomposites, and most of them have a considerable industrial importance. The proposal by Hamed is based on well sounded arguments on the mechanical properties of filled rubbers and is further reinforced by very recent observations on the likely origin of the unusual properties of (true) nanocomposites. Indeed as demonstrated by a number of authors, so-called anomalous rheological and mechanical properties of polymer nanocomposite systems are observed when the characteristic dimensions of (ideally) dispersed particles are in the 1–10 nm range, in fact commensurable with some typical dimensions of polymer dynamics, namely the reptational tube diameter (a few nanometers), as considered when modeling the behavior of entangled polymers. In fact polymer nanocomposites are distinguished by the convergence of length scales corresponding to the radius of gyration of the polymer chains, a dimension of the nanoparticle and the mean distance between the nanoparticles.12 It was therefore hypothesized that, when nanoparticles have such small dimensions, they have the capability to participate in the local polymer dynamics.13 Filled polymer systems of industrial importance, e.g., filled rubber compounds, filled thermoplastics are thus meso or microcomposites, ­possibly with a structuration (of the dispersed phase) at the nano or meso scale. Whilst no sizeable commercial application yet exist for nanocomposites rubbers or thermoplastics (to the author’s knowledge), considerable research has been made since 1984 with so-called ex-foliated layered silicate “nano-clays.” Exfoliation means that individual clay sheets, of around 1 nm thickness, have been separated and adequately dispersed in the matrix. Some reinforcement has indeed been demonstrated with such exfoliated nanoparticles but, generally with very specific rubber systems and/or at a cost of preparation that is hardly compatible with reasonable chances of commercialization.

Table 1.2 Classification of Filled/Composite Polymer Systems Designation

Characteristic Dimension (nm)

Nanofiller/particle composite

1–10

Mesofiller/particle composite

10–100

Microfiller/particle composite

100–10,000

Macrofiller/particle composite

 > 104

Example Polyamide/exfoliated montmorillonite Rubber compounds with highly reinforcing carbon blacks Polypropylene/grinded calcium carbonate Polymer concrete

8

Filled Polymers

It can further be commented that the level of reinforcement obtained in such systems is not even comparable with what is practically achieved with conventionally filled mesocomposite polymers, namely rubbers. No amorphous vulcanized rubber reinforced only with exfoliated clay has been reported to have a tensile strength in the 30 MPa range, as currently obtained with conventional carbon black filled compounds. One can nevertheless expect that, owing to their special geometries (plates or scales), properly dispersed exfoliated clays might enhance certain properties, such as gas impermeability, through barrier effects, or thermal or electrical conductivity, through appropriate orientation effects, and therefore find niche markets.

References







1. U.S. Patents: 4,472,538 (Composite material composed of clay mineral and organic high polymer and method for producing the same, September 18, 1984); 4,739,007 (Composite material and process for manufacturing same, April 19, 1988); 4,810,734 (Process for producing composite material, March 7, 1989); 4,889,885 (Composite material containing a layered silicate, December 26, 1989); 5,091,462 (Thermoplastic resin composition, February 25, 1992). 2. Q. Yuan, R.D.K Misra. Polymer nanocomposites: current understanding and issues. Mater. Sci. Technol., 22 (7), 742–755, 2006. 3. Nanocomposites. The Freedonia Group, Inc., Cleveland, OH, 2006. 4. For example, Nanomer® nanoclays from AMCOL Intern. Corp., Arlington Heights, IL; Cloisite® and Nanofil® from Southern Clay Products, Inc., Gonzales, TX; Bentone® from Elementis plc, Hightstown, NJ. 5. See for instance the following recent reviews: S.S. Ray, M. Okamoto. Polymer/ layered silicate nanocomposites: a review from preparation to processing. Prog. Polym. Sci., 28 (11), 1539–1641, 2003; H. Fischer. Polymer nanocomposites: from fundamental research to specific applications. Mater. Sci. Eng. C, 23 (6–8), 763– 772, 2003; Wang, Z.-X. Guo, S. Fu, W. Wu, D. Zhu. Polymers containing fullerene or carbon nanotube structures. Prog. Polym. Sci., 29 (11), 1079–1141, 2004; J. Jordan, K.I. Jacob, R. Tannenbaum, M.A. Sharaf, I. Jasiuk. Experimental trends in polymer nanocomposites—a review. Mater. Sci. Eng. A, 393 (1–2), 1–11, 2005. 6. P.M. Ajayan, L.S. Schadler, P.V. Braun. Nanocomposite Science and Technology. Wiley, New York, NY, 2003. ISBN: 9783527303595. 7. Y.-W. Mai, Z.-Z. Yu Ed. Polymer Nanocomposites. CRC Press, Baton Roca, FL, USA; 2006. ISBN 9780849392979; a review by an international team of authors with 13 papers on layered silicates/polymer compositions and eight papers on nanotubes, nanoparticles and inorganic-organic hybrid systems. 8. J.H. Koo. Polymer Nanocomposites. McGraw-Hill Prof., New York, NY, 2006. ISBN 13: 978-0071458214.

Introduction





9

9. S.D. Sadhu, M. Maiti, A.K. Bhowmick. Elastomer-clay nanocomposites. Chapter  2, 23–56. In Current Topics in Elastomer Research, A.K. Bhowmick Ed. CRC Press, Taylor & Francis Group, Boca Raton, FL, 23–562008. ISBN-13: 978-08493-7317-6. 10. M. Maiti, M. Bhattacharya, A.K. Bhowmick. Elastomer nonocomposites. Rubb. Chem. Technol., 81, 384–469, 2008. 11. G.R. Hamed. Rubber reinforcement and its classification. Rubb. Chem. Technol., 80, 533–544, 2007. 12. R. Krishnamoorti , R.A.Vaia. Polymer nanocomposites. J. Polym. Sci. Part B. Polym. Phys., 45 (24), 3252–3256, 2007. 13. M.E. Mackay. Anomalous rheology of polymer-nanoparticle suspensions. XVth International Congress on Rheology, Monterey, CA, August 3–8, 2008. Paper KL.11.

2 Types of Fillers In polymer technology, there are essentially two major classes of fillers, either extracted or fabricated. Minerals such as talc and clays (Al2O3, 2SiO2, 2H2O) are extracted, grinded, and possibly treated and therefore belong to the first class. Calcite (CaCO3) belongs to both classes, as it can be either extracted and grinded or obtained through a chemical process that involves precipitation. Carbon blacks result from the incomplete combustion of hydrocarbon feedstock, and are consequently fabricated fillers, as well as synthetic silica that are obtained through more or less complex chemical operations. Short fibers made either of glass, or of carbon, are fabricated products, and we arbitrarily include cellulose fibers also in the second class, because quite complex treatments are required before they can be used as a polymer reinforcing material. Moreover, many types of natural fibre have been considered for use in polymers as reinforcing materials including flax, hemp, jute, straw, wood flour, rice husks, sisal, raffia, green coconut, banana, and pineapple leaf fibre to name a few, but technical problems such as moisture absorption and low impact strength have sometimes restricted their development. Wood flour nowadays used to prepare so-called wood– polymer composites (WPC), which represents a growing market over the last decades,* can also be considered as a fabricated filler with respect to its preparation mode. Fillers for polymers exhibit in fact a stunning variety of chemical natures, particle sizes and shapes. Essentially three basic shapes can be distinguished: either spheres, or plaques (disks, lamellas) or rods (needles, fibers), as illustrated in Figure 2.1. Such basic shapes can be further combined to result in quite complex geometrical objects to which specific (reinforcing) properties can be associated. Carbon black aggregates offer typical examples of complex tri-dimensional structures whose shape specifically affects the reinforcing properties, as will be discussed hereafter. Most fillers, either extracted or fabricated, have a mineral origin, with the notable exception of carbon blacks that result from the thermal degradation of hydrocarbons. There are also a * In North America the WPC market amounts today to around 300,000 tons/year, essentially for building and garden applications, particularly decking and associated products. Estimated over $600 Mio in 2002, the USA and Canada segment is nowadays worth over $2 billion and worldwide estimates are in the $3 billion range. Market growth is slower in West Europe with a consumption of around 140,000 tons in 2002, over 200,000 tons in 2005 and estimated to reach some 270,000 tons in 2010 (source: A. Eder. WPCs – an updated worldwide market overview including a short glance at final consumers. 3rd Wood Fibre Polymer Composites Symposium, Bordeaux, France, March 21–27, 2007).

11

12

Filled Polymers

Spheres

Scales, flakes lamellas

Cylinders, rods, needles, fibers

Partial fusion elementary particles => aggregates Complex tri-dimensional object => structural effect of the filler

Figure 2.1 Fillers basic shapes and structure.

number of filler materials that have a vegetal origin, for instance wood flour, sisal, coco, or jute fibers. It is tempting to consider a classification scheme for polymer fillers but no overall system is available and the analysis of existing proposals reveal that their validity and interest strongly reflect the application considered. We will nevertheless consider a few logical possibilities, which underline certain specific aspects of the common property considered. Considerations based on the refractive index allow to draw a clear distinction between a filler and a pigment, whilst if certain fillers can be used to modify the color of a polymer (e.g., carbon black in polyolefin), not all pigmenting materials have reinforcing capabilities. Let us consider various materials and their respective refractive indices (Table 2.1). The refractive index of vacuum is (by definition) equal to 1, and most polymers exhibit indices around 1.5. One would consider that any given material has no capability to modify the color of another one if the respective refractive indices of both materials do not differ by more than 0.2. It follows that materials with refractive indices either above 1.3 or below 1.7 have practically neither clearing nor darkening effects on polymers. Consequently, a mineral whose refractive index is above 1.7 can potentially be used as a pigment (but can also have reinforcing capabilities), whilst materials whose refractive index is below 1.7 would be essentially considered as fillers.* A logical and broader approach would associate the origin, the production process and the reinforcing capabilities (Figure 2.2). In this manner, essentially four types of filler are considered: organic fillers of natural origin (liege, wood flour, vegetal fibers), organic fillers obtained by chemical * One notes however that such a classification makes no sense for “dark” fillers, such as carbon blacks, which do not refract light.

13

Types of Fillers

Table 2.1 Distinguishing Between Filler and Pigment with Respect to Refractive Index Material

Refractive Index

Vacuum Water Chalk Polymers Silica BaSO4 ZnO ZrO2 ZnS Diamond TiO2, Anatase TiO2, Rutile

1.00 1.33 1.35 1.50 1.55 1.64 2.08 2.17 2.37 2.42 2.55 2.75

Filler Organic

Inorganic

Natural

Synthetic

Natural

Synthetic

- Inactive - Semiactive - Active

- Inactive - Semiactive - Active

- Inactive - Semiactive - Active

- Inactive - Semiactive - Active

Liege Wood flour Fibres (jute, sisal,...)

Synthetic resins Minerals Cellulose derivatives (CaCO3, talc, clays,...)

Carbon blacks Silicas (fumed, precipitated) Metal oxides (TiO2, ZnO,...) Metal salts (BaSO4,...)

Figure 2.2 Classifying fillers with respect to fabrication process and reinforcing activity.

synthesis (synthetic resins, cellulose derivatives), mineral fillers of natural origin (essentially all extracted fillers) and mineral fillers obtained through chemical processes in the broad sense (carbon blacks, fumed and precipitated silica). Furthermore, for each type, one might distinguish materials as active, semiactive, or inert filler, depending how they boost, improve or do not affect certain mechanical properties of interest, for instance stiffness, tensile or flexural strength, and abrasion resistance, to name a few. Another approach, maybe less subjective, consists of paying attention to particle size because, as illustrated in Figure 2.3, there is a clear relationship between this characteristic and the reinforcing capabilities. Essentially

14

Filled Polymers

105

104

103

Degradative fillers Dilution fillers Semireinforcing fillers

102 Reinforcing fillers 101 Particle size (nm)

Grinded CaCO3 mica, talc

Clays Precipitated CaCO3 TiO2, ZnO Si aluminates Ca silicates Hydrated silica Anhydrous silica Carbon blacks

Figure 2.3 Classifying fillers with respect to particle sizes.

no reinforcement is obtained when particles are larger than 103 nanometer (nm) and too large particles deteriorate mechanical properties of polymer materials. The wide range of particle sizes (and structures) offered by the manufacturing of carbon blacks and synthetic silica clearly reflect in the  semireinforcing and reinforcing character of these fillers. The general relationship between reinforcing capabilities and particle size suggests obviously that a poorly dispersed mineral, whatever its ultimate particle size, is likely to deteriorate ultimate mechanical properties, for instance by reducing the elongation at break of vulcanized rubbers and thermoplastics. Indeed, large and badly dispersed particles are fracture initiation sites.

3 Concept of Reinforcement Whilst they can be added to polymers for other purposes, it is mainly for their reinforcing capabilities that certain fillers offer the largest interest. When compared to polymers, any mineral exhibits mechanical properties, such as modulus, stiffness, hardness, that are several order of magnitudes larger. Therefore, one may reasonably expect that mixing the latter with the former will result in a heterogeneous mixture that exhibits macroscopic mechanical properties, at least intermediate between those of the polymer and those of the filler. Reinforcement of elastomers by carbon black, discovered in 1907 in Silvertown, UK, is likely the most significant example of this effect, that really permitted the development of the emerging tire technology, strongly connected of course with the automotive industry. Essential in rubber technology, the concept of reinforcement is however very complex, even if relatively easy to capture at first sight. Indeed, when a filler is added to a polymer, practically all properties are affected, some in a positive manner, others negatively with respect to a given application. There has been much debate about which particular property should be considered as the most expressive in terms of reinforcement. In this respect, it is worth quoting here the opinion expressed by G. Kraus:1 A precise definition of the term «reinforcement» is difficult because it depends somewhat on the experimental conditions and the intended effects of the filler addition…it appears preferable to regard reinforcement broadly as the modification of the viscoelastic and failure properties of a rubber by a filler to produce one or more favorable results…

The reinforcing capabilities of a filler must consequently be appreciated with respect to a balance of properties, whose choice depends on the application considered. Let us consider the general trends exhibited by a rubber compound in which increasing quantities of active (e.g., carbon black) or inert (e.g., finely divided clay) have been added. As shown in Figure 3.1, certain properties will only either increase or decrease, for instance viscosity, hardness, but other ones will pass through extremes in the case of the reinforcing filler. This immediately suggests that there will be optimum loadings, for a given filler, in a given polymer, for a specific application. To establish the optimum filler level is therefore the most important task for the compounder, further complicated by the obvious requirement that the compound must remain processible at reasonable 15

Inert

Filler level (phr)

Inert

Active

Tensile strength

Filler level (phr)

Active

500

1000

50

60

70

80

90

Inert

Inert

Filler level (phr)

Active

Elongation at break

Filler level (phr)

Active

Hardness

Figure 3.1 Relative variation of rubber compound properties as imparted by active (reinforcing) or inert filler.

5

10

15

20

20

40

60

80

100

ML(1+4) at 100°C

MPa

Shore A %

400

500

10

20

30

40

50

% 100

200

300

mm3

(Mooney) viscosity

Active

Filler level (phr)

Inert

Abrasion

Filler level (phr)

Inert

Active

Compression set

16 Filled Polymers

17

Concept of Reinforcement

energy and labor costs; sometimes the excessive viscosity increase imparted by very active fillers, either limits their practical level in certain elastomers or requires additional modification in formulation, for instance higher  levels of processing oils, or plasticizers, which generally have a penalizing effects on certain mechanical properties of the vulcanized part. In general, the reinforcing activity of a filler depends on at least four criteria: • • • •

The particle size (always smaller than 100 nm) The structure (i.e., the spatial organization) The specific area The surface (chemical) activity.

The structure of the filler material refers to the fact that, during their manufacturing process, reinforcing fillers develop very complex tri-dimensional shapes, which are called aggregates in the case of carbon black. Aggregate structure appears thus as one of the most important aspect of reinforcement and is obviously related with the specific area. The quantification of structure and the measure of specific area are somewhat related, essentially because the adsorption of molecules of known size is used to assess both characteristics. The well-known BET (Brunauer, Emmet, Teller) method is used to measure the adsorption isotherm of nitrogen (N2) absorbed by powdery fillers, whilst the aggregate complexity is assessed by evaluating the maximum quantity  of larger molecules (for instance di-butylphthalate DBP, or cetyltriethylammonium bromide CTAB) than can be adsorbed on the external surface. As might be expected, there is a (loose) correlation between the socalled BET surface and the activity (or reinforcing capability) of a filler:

BET < 10 m2/g: inert filler BET = 10–60 m2/g: semiactive filler BET > 60 m2/g: active filler BET > 100 m2/g: very active filler

In fact, relationships between the reinforcing abilities and the characteristics of the filler are very complicated and, in general, one has to consider more than one criterion to make valid comparisons, useful for a given filler in a given polymer, for a given application. It is worth underlining that the concept of reinforcement has been more debated in the field of rubber science and technology than in the field of thermoplastics. The fact that, without suitable reinforcement, most elastomers exhibit so low mechanical properties that no interesting applications are possible is surely a reason. Another one is that most general purpose thermoplastics have known their tremendous development in the second

18

Filled Polymers

half of the twentieth century, in parallel with the expansion of petrochemistry, and have found immediately interesting applications “as such,” nearly without additives except a few protective chemicals. Polyethylene and polypropylene for instance are used to fabricate sheets and films by essentially exploiting their capabilities as semicrystalline polymers. No filler is needed to obtain the high mechanical properties that develop when crystalline structures are properly established and oriented. Polystyrene, ABS and other styrenics exhibit properties directly used in a number of applications, without the need of reinforcing fillers. Of course, in their usages, most thermoplastics must also meet a balance of properties but, except maybe polyvinyl chloride (PVC), the right material for a given application is obtained by controlling the macromolecular size and structure, essentially through a suitable adaptation of the polymerization process. The key role played by polymerization catalysts in the developments of polyolefins clearly supports this point. PVC is an exception because when suitably compounded with stabilisers, plasticizers, and other ingredients, a whole range of products can be obtained, essentially by changing the glass transition temperature of the material. It is quite symptomatic that the socalled “plastograph,” a small laboratory mixer, was specifically developed in the 1950s as a convenient tool to document the “plasticization” of PVC. The addition of fillers to thermoplastics polymers is thus quite a recent practice, around three decades old, whilst filled rubber compounds are used for more than a century. There is nevertheless another important, more technical reason for the different meaning of reinforcement in the rubber and plastics fields. In most of their applications, thermoplastics are used within the limits of their elastic behavior, generally below 10% strain. Indeed, once the yield strength limit is exceeded, permanent deformation occurs. It follows that most applications of thermoplastics are first concerned by the elastic behavior of the material; the viscoelastic character plays a secondary role, namely in what the long term variation of modulus is concerned through the creep phenomenon for instance. The situation is totally different with rubber materials, whose performance are controlled by their viscoelastic character, in a strain range that is substantially larger than for thermoplastics. For instance, with rubber materials, the tensile (Young) modulus is far less significant than the 100 or 200% modulus in most applications. It follows that the role played by fillers in “reinforcing” rubbers and thermoplastics is substantially different, as well as the balance of properties, as will be largely underlined throughout the book. A direct consequence is that the modeling of the filler’s effects in thermoplastics and in elastomers, whilst sometimes based on a similar theoretical background, is generally substantially differing in the supporting reasoning and therefore in the applicability. It is one of the objectives of this book to identify both the similitudes and the differences in those theoretical approaches, with respect to the class of polymer matrix considered.

Concept of Reinforcement

19

Reference

1. G. Kraus. Reinforcement of elastomers by carbon black. Adv. Polym. Sci., 8, 155–231, 1971.

4 Typical Fillers for Polymers

4.1  Carbon Black 4.1.1 Usages of Carbon Blacks Essentially, carbon black is the soot that results from the incomplete combustion of hydrocarbon materials, i.e., gas and oils. This definition does not pay tribute however to the high degree of development and control in use today in most industrial processes. The uses and the basic production principles of carbon black are lost in antiquity, but the development of controlled fabrication processes dates back to the previous century, resulting nowadays in highly sophisticated technologies with the capability to produce very fine and structurally complex materials, in accordance with the most recent standards of quality. As we will briefly see below, the term “carbon blacks” covers a very broad range of filler materials, with numerous applications, as outlined in Table 4.1. Except elastomer reinforcement, printing inks and several uses in the electrical industry, most application concern relatively low fraction of carbon black, typically below 5% volume. 4.1.2  Carbon Black Fabrication Processes Fabrication processes of carbon blacks all share the same principle: controlled heat decomposition of hydrocarbon products. Such processes are essentially chemical, either thermo-oxidative or mere thermal decomposition, as described in Table 4.2. Amongst the thermo-oxidative processes, the furnace black one is the most recent and nowadays the most important. As illustrated in Figure 4.1, the liquid combustible (either oil or gas) is sprayed in a flame of natural gas and hot air. Black smoke is produced that is a mixture of gas and carbon particles, initially nearly spherical elementary particles that partially fuse together to produce complex tri-dimensional objects called aggregates. Carbon black aggregates are quenched through water spraying that stops the pyrolysis and aggregation processes and cools down smoke, which is then filtered to recover solid particles. Unburned gas is treated and recycled in the process. 21

22

Filled Polymers

Table 4.1 Important Uses of Carbon Blacks Domain

Application

Elastomers Printing inks Enduction Thermoplastics

Fibers Paper Building Electrical industry

Reinforcing filler in tires and mechanical rubber goods Tinting, rheology modifier Black and gray tinting, color enhancement Black and gray tinting, color enhancement, anti-UV protection of polyolefins, high voltage cable shielding, application in semiconductors, static electricity dissipation Tinting Black and gray tinting, photograph protective paper Cement and concrete tinting Electrodes, dry batteries and cells

Table 4.2 Fabrication Methods of Carbon Blacks Chemical Process Thermo-oxidative decomposition

Method

Raw Material (Feedstock)

Furnace black Gas black (Degussa process) Lamp black

Thermal decomposition

Thermal black Acetylene black

Aromatic oils from coal tar or petrol distillates, natural gas Coal tar distillates, natural gas Aromatic oils from coal tar or petrol distillates Natural gas (or oils) Acetylene

Thermo-oxidative process : furnace black Air

Liquid feedstock atomized and sprayed into the flame

Smoke treatment for carbon black recovery

Oil

Gas

Flame from combustion of gas combined with preheated air

Smoke gas Water quench (stops pyrolysis)

Figure 4.1 Carbon black manufacturing process for furnace black.

23

Typical Fillers for Polymers

Thermo-oxidative process : lamp black Refractory bricks

Smoke treatment for carbon black recovery

Cooling water Flame under regulated air admission (=>partial combustion of feedstock) Air

Air

Oil (feedstock)

Figure 4.2 Carbon black manufacturing process for lamp black.

After filtration, carbon black aggregates are first packed into agglomerates then into pellets (of millimeter dimensions) in order to produce roughly spherical grains that are easy to handle. The process has several advantages: first it is a totally sealed, so that full respect of environment is obtained second a precise control of elementary particle size (from 10 to 100 nm) and of aggregate structures is achievable.1 The lamp black process is likely the oldest industrial process and has consequently been the object of numerous engineering variants. Figure 4.2 describes the principle of a typical modern plant. The partial combustion of a feedstock (oil generally) in an atmosphere purposely poor in oxygen produces smoke, which is cooled down and filtered to recover carbon black particles that are subsequently flocculated. The control of the pyrolytic process is loose and results in a large distribution of elementary particle sizes (from 60 to 200 nm). This fabrication process tends to be abandoned today in favor of the much cleaner and more versatile furnace one. Degussa (now EVONIK), in Germany, developed the so-called gas black process in the 1930s. Initially coal tar was used, quite a common feedstock at that time, when carbochemistry was very important in a country that had limited access to petrol. Today, any kind of hydrocarbon feedstock may be used in the process. As shown in Figure 4.3, a carrying gas is flown over preheated oil, then the oil–rich gas feeds a burner. Smoke is in part captured on the wall of water-cooled rotating cylinders and removed with scrapers, and in part recovered through filtration. Very fine elementary

24

Filled Polymers

Thermo-oxidative process : gas black Off gas

Cooling water

Carrier gas Oil (feedstock) Heating gas

Rotating drum

Knife Air

Burner Carbon black

Figure 4.3 Carbon black manufacturing process for gas black.

particles are obtained, in the 10–30 nm range, which aggregate in a controllable manner. The thermal black process is discontinuous and consists essentially of “cooking” a mixture of natural (i.e., hydrocarbons) and inert (N2) gas in two reactors in tandem, where cycles of heating then decomposition are achieved (Figure 4.4). One of the reactors is heated for five to eight minutes by burning natural gas in presence of air, whilst the other, previously heated, is loaded with pure natural gas that thermally decomposes. When pyrolysis is complete, flushing the reactor and conveying the smoke to filtration equipment allows the recovery of carbon black particles. Depending on the ratio natural/inert gas, various ranges of large and gross particles are obtained, either from 120 to 200 nm or from 300 to 500 nm. Acetylene black is produced by pyrolysing acetylene at high temperature; essentially hydrogen and carbon are obtained. Very pure carbon black particles are obtained in the 30–40 nm range. All the processes described above yield a very wide range of carbon blacks, differing in a number of properties, as described in Table 4.3. 4.1.3  Structural Aspects and Characterization of Carbon Blacks In terms of consumption, the most significant usage of carbon blacks is rubber reinforcement. This effect was discovered in the early years of the twentieth century and played an essential role in the development of tire technology and consequently in the automotive industry. As we shall see later, there are still some unknown aspects of carbon black reinforcement

25

Typical Fillers for Polymers

Thermal process H2 Gas

Air

Chimney

Carbon black

Ovens

Natural gas Heating cycle

Decomposition cycle

Figure 4.4 Carbon black manufacturing process for thermal black.

Table 4.3 Carbon Black Production: Properties vs. Process Thermo-Oxidative Decomposition Property Specific area, N2 Adsorption, I2 Particle size DBP absorption Oil absorption Volatiles pH

Unit m2/g mg/g nm ml/100g g/100g %

Thermal Decomposition

Lamp Black

Gas Black

Furnace Black

Thermal Black

Acetylene Black

16–24 23–33 110–120 100–120 250–400 1–2.5 6–9

90–500 n.a. 10–30 n.a. 220–1100 4–24 4–6

15–450 14–50 10–80 40–200 200–500 0.6–6 6–10

6–15 6–15 120–500 37–43 65–90 0.5–10 7–9

Around 65 Ar. 100 32–42 150–200 400–500 0.5–2 5–8

but a basic consideration is surely the capability of carbon black to exhibit various levels of spatial organization, which are schematically outlined in Figure 4.5. Elementary particles of approximately spherical shape appear in the early stages of the pyrolysis process, when soot is being formed and assembled together through partial fusion to give complex tri-dimensional objects, called aggregates. During quenching, aggregates entangle into agglomerates, which are eventually pelletized into granules of millimeter dimensions. Aggregates are very difficult, if not impossible (in rubber mixing conditions) to break and consequently are likely to be the ultimate particle size when

26

Filled Polymers

Elementary particle (colloidal black)

10–90 nm

Partial fusion

Occuring in soot formation and quenching Entanglement

Aggregate

100–300 nm

Agglomerate

Carbon blacks Occuring during wet filtering and powder drying Compaction

Pellet

Specific area

104 –106 nm

2–4 mm

“Structure” Reinforcing character 10–8m

10–7m

Scaling

10–5m

10–3m

Figure 4.5 Spatial organization and reinforcing character of carbon blacks.

dispersion is optimal. They play an essential role in rubber reinforcement whilst residual agglomerates are considered as failure initiation sites in filled compounds. There is a consensus nowadays to consider aggregates as the ultimate carbon black form, the only relevant one when rubber (polymer) reinforcement is concerned. Carbon blacks exhibit various characteristics whose importance depends on the application considered. The specific area (m2/g) is obviously the most basic information for a given filler: the smaller the elementary particles, the higher the specific area of aggregates with equivalent spherical volume. The specific gravity of carbon black is in the 1.82–1.89 g/cm3 range. Note that the common value of 1.86 g/cm3 will be used in all the illustrative calculations made in this book. The elementary analysis of carbon blacks is roughly as follows (in wt%):

Carbon: 95.0–99.5 Hydrogen: 0.2–1.3 Oxygen: 0.2–0.5 Nitrogen: 0.0–0.7 Sulfur: 0.1–1.0 Ashes: below 1.0

27

Typical Fillers for Polymers

In addition, toluene extraction reveals traces of organic materials, essentially poly-aromatic hydrocarbons (below 0.5 wt%). A number of oxygenated chemical groups have been found on carbon black surface, such as carbonyls, carboxyls, pyrones, phenols, quinone, lactol, etc., but in minute quantities and all are removed by heating at 950°C in an oxygen free atmosphere. A number of standard characterization methods for carbon blacks (and other fillers) are listed in Table 4.4; most of them are described as International Organization for Standardization (ISO), American Society for Testing and Materials (ASTM), or Deutsches Institute für Normung (German Institute for Standards) (DIN) methods. One can distinguish three groups of methods with respect to the information sought for reinforcement purposes: specific area, structure and chemical analysis. In addition, there are methods for characterizing the final product, i.e., the carbon black in pellet form, the only one readily handled in polymer technology (essentially for health and safety reasons). The specific surface area is assessed either through iodine I2 adsorption (result is given in mg of I2 per g of carbon black), or through nitrogen N2 adsorption (result in m2/g of carbon black), or through cetyltrimethylammonium bromide CTAB adsorption (result in m2/g of carbon black) or through the tint strength (an indirect measure of the specific area). As expected, all Table 4.4 Standard Characterization Methods for Fillers Method

ISO

ASTM

DIN

Specific area Iodine I2 adsorption Nitrogen N2 adsorption CTAB adsorption Tint strength

1304 4652 6810 5435

D-1510 D-3037/4820 D-3765 D-3265

53582 66132

Structure DBP absorption Compressed DBP absorption Oil absorption*

4656 6894 787/5

D-2414 D-3493

53601 787/5

1125 787/2 787/18

D-1506 D-1509 D-1514

787/9

D-1512

1306

D-1513 D-3313 D-1511

Chemical analysis Volatiles Ashes Moisture* Sieving residue* Toluene extraction pH* Final product properties Bulk specific gravity Pellet hardness Pellet sizes distribution * DIN-ISO methods.

53552 53586 787/2 787/18 53553 787/9 53600

28

Filled Polymers

of these methods give similar but not equivalent information and there is no consensus regarding their respective advantages/disadvantages. Iodine adsorption for instance is sensitive to surface chemistry and the presence of polyaromatic hydrocarbons, but the method is considered correct for furnace and lamp blacks. The N2 adsorption is the BET method whose principle is based on the shape of adsorption isotherms.2 When a monolayer of material is adsorbed on a very uniform surface, a knee occurs in the isotherm before reaching another plateau that corresponds to the adsorption of a second layer. Ordinary surfaces are energetically quite heterogeneous as far as the adsorption energy in the first layer is concerned: however it is possible to work with carbon blacks because particles exhibit (at least locally) graphitelike facets that are quite homogeneous. Nevertheless the nitrogen molecule is small enough to reach pores and other small cavities within aggregates, and therefore results are sometimes obtained in excess with the area readily in contact with the polymer matrix. CTAB is a larger molecule than nitrogen and consequently only the “external” specific area is probed, leading to results better correlated with aggregates’ size. All methods for structure assessment are indirect and essentially consist of measuring the absorbed amount of a suitable chemical, for instance dibutylphthalate DBP. Results are expressed in ml or cm3 (of DBP) absorbed per 100 g of filler. The method consists of adding dropwise DBP to a known quantity of carbon black, which is malaxed in a calibrated laboratory mixer. Mixing torque is recorded and as long as the liquid just fills the “voids” between aggregates, the torque trace remains essentially flat. As soon as the whole external surface of aggregates is “wetted,” a coherent mass is obtained and a significant torque rise is observed. The more complex the structure, the higher the amount of absorbed DBP. As such, the method does not distinguish between aggregates and agglomerates, and this limitation is somehow overcome with the so-called compressed (or crushed) DPA absorption method. Before loading the mixer cavity the carbon black is compressed four times under a pressure of 24 MPa. Only the permanent structure, i.e., the aggregates, is expected to survive the crushing step. The size of the elementary particles and the structure of aggregates are the most important parameters in the ability of a given carbon black to reinforce a polymer. And both parameters essentially depend on the fabrication process. In time, a number of manufacturers went on the market with products, essentially described with respect to their manufacturing process and (expected) reinforcing character, essentially with respect to tire applications. For years, carbon blacks were described through acronyms, such as HAF, i.e., high abrasion furnace: a furnace black imparting a high resistance to abrasion to rubber (tread band) compounds, or ISAF-LS, i.e., intermediate superabrasion furnace-low structure: a furnace black of low structure offering an excellent resistance to abrasion, to name a few. Only carbon black experts familiar with rubber reinforcement aspects—and aware of the (sometimes) subtle differences between carbon blacks, described by similar

29

Typical Fillers for Polymers

acronyms but produced by different companies—were at ease with such a description. The essential role played by carbon blacks in rubber reinforcement prompted the American Society for Testing Materials (ASTM) to propose a standard classification and nomenclature, described in ASTM D1765 that became widely accepted, thanks to its simplicity. Figure 4.6 illustrates the principle of this nomenclature that invites all carbon black manufacturers to class their materials using a four character system: one letter (either N or S) and three numbers. The letter N means normal curing, meaning that the carbon black does not interfere (too much) with vulcanization chemistry; S means slow curing and concerns carbon blacks prepared with feedstock leaving chemical residues that affect the vulcanization process. S grade carbon blacks tend to disappear nowadays. The first number refers to the size of the elementary particles, at least in the 1986 version of the standard, because a recent proposal (in 1996) was made to assign the first digit a value (between 0 and 9) with respect to the specific area, as measured through N2 adsorption. The two last digits refer to the aggregate structure and are assigned to the carbon black by the manufacturer, with respect to various evaluation techniques, presently not standardized however. As a matter of fact, a very large diversity of carbon black grades has been and is still produced, essentially because there is an excessively large number of process variables, not all perfectly monitored despite encouraging progresses, as well as a great variety in the feedstock used. Despite modern Refers to aggregate structure N x

yz Refers to the size of elementary particles

Change in defining x

D1765–86

x

D1765–96

Typical average size of particles, nm

Average nitrogen N2 specific area, m2/g

1–10 11–19 20–25 26–30 31–39 40–48 49–60 61–100 101–200 201–500

> 150 121–150 100–120 70–99 50–69 40–49 33–39 21–32 11–20 0–10

0 1 2 3 4 5 6 7 8 9 Figure 4.6 ASTM classification of carbon blacks.

30

Filled Polymers

trends in standardization, this inevitably results in a large diversity of products. The present ASTM classification schema obviously offers a number of advantages but only the first digit (X in Figure 4.6) can be considered as solid information; in other words, for a given carbon black grade, its ASTM classification guarantees only the size range of the elementary particles. The structure that, in the opinion of the author, is likely the most significant aspect is “described” by the two last digits (yz). This description is however depending on the set of methods used by the manufacturer. To document this aspect, Table 4.5 was filled by compiling and averaging out typical test data for carbon black, as found in the trade or scientific literature, with respect to their ASTM designation. Sources of data, as well as a few interesting relationships between the quoted quantities are given in Appendix 4.1. Practical experience, essentially with respect to applications in tire technology, allows to somewhat distribute available carbon blacks in three main categories: • Highly reinforcing, so called “tread” blacks: series N100–N300 • Semireinforcing, so called “carcass” blacks: series N300–N600 • Weakly reinforcing: series N600, N700 The words “tread” and “carcass” refer to tire applications and with respect to wearing resistance, it is clear that tread band compounds need highly reinforcing blacks. As a rule of thumb, the higher the reinforcing capabilities of a carbon black, the more difficult its dispersion in the rubber, and consequently the more complex the mixing procedure. Conversely, low reinforcing blacks can be added to rubber formulations in very large quantities. An attractive manner to consider carbon black grades consists of plotting a parameter related to elementary particle size, for instance CTAB adsorption, vs. a parameter related to aggregate structure, for instance DBP absorption (Figure 4.7). Roughly speaking the reinforcing character increases as one moves along the increasing left to right diagonal. As can be seen in Figure 4.7, not all combinations of both parameters are available, and the largest variety of grades is found in the N300 series. 4.1.4  Carbon Black Aggregates as Mass Fractal Objects The central role plaid by aggregates in the reinforcing capabilities of carbon black is nowadays well established but recent progress in particles observation techniques, as well as fundamental studies on the physics of elementary particle aggregation through ballistic processes occurring during soot formation and quenching, shed new light on the particular nature of such complex objects. New concepts such as “fractal objects,” obviously inspired by the breakthrough work of B. Mandelbrot,3 were considered in describing carbon black aggregates.

1.02

1.20

1.30

1.25

N330

N343

N347

0.76

N326

N339

1.37

1.23

N242

N299

0.92

1.15

N220

1.24

0.73

N219

N231

0.85

N210

N234

1.09

1.27

1.33

N121

N125

1.13

N115

N134

1.14

N110

0.05

–

0.04

0.03

0.12

0.01

0.07

0.02

0.00

0.02

0.00

–

–

0.07

0.01

0.00

0.02

Std. dev.

ASTM D2414

Method:

Mean value

(dm³/kg)

Unit:

ASTM Nr

DBP Absorption

Data:

 

 

0.97

1.04

0.99

0.87

0.68

1.05

 

1.01

0.85

0.98

 

1.02

0.95

1.10

0.97

0.98

Mean value

ASTM D3493

(dm³/kg)

0.03

–

0.04

0.02

0.02

0.00

–

0.03

0.02

0.02

–

–

0.08

0.03

0.01

0.04

Std. dev.

Comp. DBP ab. 24M4

 

 

87.14

–

92.10

79.45

82.57

106.50

128.75

123.26

117.00

116.75

112.67

152.00

 

126.00

143.00

137.97

Mean value

3.50

–

4.32

4.28

8.22

1.80

9.91

1.97

0.00

4.19

6.35

–

5.29

–

4.82

Std. dev.

ASTM D3037

(m²/g)

N2 Adsorption

 

 

89.29

92.00

90.15

81.50

83.59

108.93

119.00

121.28

121.00

119.93

117.00

142.00

121.00

121.00

156.50

145.60

Mean value

2.75

–

1.22

1.36

2.69

1.62

–

1.86

–

2.24

–

–

–

0.00

4.95

2.00

Std. dev.

ASTM D1510

(mg/g)

I2 Adsorption

 

 

  125.45

85.40

91.31

80.54

82.82

105.90

–

118.63

108.50

111.56

134.00

126.00

118.50

128.00

(continued)

3.10

5.91

2.62

4.52

2.69

–

1.86

0.71

3.50

–

–

3.54

–

3.77

Std. dev.

ASTM D3765

(m²/g)

CTAB Adsorption

Mean value

Carbon Blacks—ASTM Designation vs. Characterization Data, as Compiled from Trade and Scientific Literature

Table 4.5

Typical Fillers for Polymers 31

0.35

0.38

N880

N990

0.04

0.08

0.02

0.00

0.04

0.04

0.02

0.17

0.12

0.04

0.14

0.11

0.08

0.01

0.03

–

0.02

 

0.37

 

0.63

0.59

 

0.81

0.58

0.86

0.73

0.83

0.84

0.83

1.18

0.90

1.13

1.14

0.96

Mean value

ASTM D3493

(dm³/kg)

0.02

0.02

0.01

0.05

0.03

0.02

0.03

0.02

0.04

0.01

0.06

0.10

0.01

–

0.02

Std. dev.

Comp. DBP ab. 24M4

 

9.09

12.25

29.64

 

25.00

33.88

28.47

38.12

34.85

35.75

41.12

42.70

248.67

93.48

84.30

88.05

71.43

Mean value

0.93

2.87

2.48

2.00

3.06

3.49

2.04

3.09

3.18

2.82

1.57

18.48

11.01

2.05

5.59

2.17

Std. dev.

ASTM D3037

(m²/g)

N2 Adsorption

Note: In the column for standard deviation – means that only one source of data was available.

0.65

0.74

N772

N774

1.17

0.73

N765

N770

1.29

0.66

N683

N762

1.24

0.94

N650

N660

1.03

1.17

N539

N550

1.15

1.86

1.55

N358

N375

1.57

N356

N472

1.21

N351

Std. dev.

ASTM D2414

Method:

Mean value

(dm³/kg)

Unit:

ASTM Nr

DBP Absorption

Data:

Table 4.5  (Continued)

 

9.40

 

29.05

30.00

28.00

32.70

28.20

34.54

35.73

35.00

42.58

42.75

250.00

88.72

84.00

82.90

68.22

Mean value

1.22

–

0.10

0.00

–

2.40

1.85

2.75

2.02

1.41

1.11

0.50

–

3.45

–

–

1.09

Std. dev.

ASTM D1510

(mg/g)

I2 Adsorption

 

9.70

 

30.37

33.00

 

36.17

29.75

40.24

37.52

36.00

41.32

41.33

147.50

91.72

88.00

87.50

73.23

Mean value

0.99

2.28

–

3.96

5.03

2.12

3.04

–

1.21

0.65

3.54

10.29

–

–

0.52

Std. dev.

ASTM D3765

(m²/g)

CTAB Adsorption

32 Filled Polymers

33

Typical Fillers for Polymers

140.0 N110

CTAB adsorption (m2/g)

120.0 100.0

60.0

r cte

ra N330

ha

N326 ng c ci for n i Re

80.0

N299 N339 N347

20.0

N990 0.20

0.40

N774 N762

N660

0.60 0.80 1.00 DBP absorption (dm3/kg)

N356

N351

N550

40.0

0.0 0.00

N234

N220

1.20

N683

1.40

1.60

Figure 4.7 Carbon blacks reinforcing capabilities with respect to parameters related to elementary particle size and aggregate structure.

Carbon black aggregates can indeed be viewed as mass fractal objects whose description results from the so-called “fractal scaling law”: two parts of a fractal object, a larger one of size DL and a smaller one of size DS, are ­statistically equivalent if the latter is enlarged by a factor DL/DS. Applied to the case of an (carbon black) aggregate of overall size D made of Np aggregated ­elementary particles of size d, such a law leads to the following equality (see Figure 4.8a): F



 D Np = α    d

(4.1)

where α is a prefactor, also called front factor and F is the so-called massfractal dimension of the aggregate, which depends on the conditions for the aggregation process. The mass fractal F describes how the mass of an object varies with its size. This concept was first applied by Kaye4 and Flook5 to the determination of the perimeter type fractal of carbon black aggregates, then was used by Bourrat et al.6 Ehrburger-Dolle and Tence7 for the structural characterization of a few carbon blacks, and by Herd et al.8 who reported an extensive study comparing the utility of fractal and Euclidean geometries in characterizing quite a large series of 19 carbon black grades, with DBP adsorption ranging from 35 to 174 cm3/100 g. By measuring the perimeter and the mass fractal values for various carbon blacks in the dry state, these authors found that

Size d

Size D

Np particles

(b)

Size d

Geometrical distance R

Fractal path L

(c)

Size d

Length L

Figure 4.8 Fractal geometry description of aggregates (a) basic dimensions of an aggregate, (b) concept of fractal path, (c) Chain-like aggregate.

(a)

34 Filled Polymers

35

Typical Fillers for Polymers

the mass fractal dimension F was in the 2.19–2.85 range, i.e., a mean value of 2.44±0.15. This indicates that carbon blacks have a moderately rough surface, since a smooth surface has a value of F = 2. Indeed, Göritz et al.9 used scanning tunneling microscopy (STM), atomic force microscopy (AFM) and small-angle x-ray scattering (SAXS) to study the surface structure of quite a broad range of carbon blacks, from N115 to N990. They well documented the surface topography of carbon blacks and demonstrated that graphitized (2700°C treatment) high structure blacks, e.g., N115 and N234, lost all surface roughness and exhibit typical flat huge local terraces. Moreover x-ray scattering experiments gave access to the surface fractal dimension which was found to vary systematically from 2.27 (N115) down to about 2.0 (N990). The smaller the primary particle diameter the higher the measured fractal dimension. Consequently surface roughness decreases with increasing primary particle diameter and is related to the reinforcing character of carbon black grades. One would therefore expect fractal dimensions of carbon blacks to be correlated with DBP absorption numbers in the dry state. Figure 4.9 is for instance drawn using mass fractal dimensions reported by Herd et al. and average DBP absorption data from Table 4.5. Mean DBPA data and their standard deviation were used in drawing the graph. If, indeed, there is a loose linear correlation between both characteristics of carbon black, one can hardly expect the measurement of mass fractal dimensions to become a valid replacement candidate for the well spread and much easier ASTM methods, particularly with respect to the complexity of the former. 2.0

DBPA, dm3/kg

1.5

1.0

0.5

0.0

2

2.2

2.4 2.6 Mass fractal dimension F

2.8

3

Figure 4.9 Mass fractal dimension vs. (mean) DPA absorption number of carbon black. (Mass fractal data from C.R. Herd, G.C. McDonald, R.E. Smith, W.M. Hess., Rubb. Chem. Technol., 66, 491–509, 1993. DBPA data from Table 4.5.)

36

Filled Polymers

Table 4.6 Surface Energy Components for Carbon Black Carbon Black Grade N110 N220 N234 N326 N330 N347 N550 N660 N762 N774 N880 N990

Specific Surface Area N2 (BET) [m²/g] 140.0 118.0 123.3* 83.2 76.5; 80.0 85.8 39.7; 43.2 39.4 32.5 ; 24,0 29.0 12.3 7.9; 10.3

Dispersive Component γ ds (at 150°C) [mJ/m²] 270.4 235.2 382.0* 186.5 196.9; 150.4 192.9 134.4; 173.4 124.7 126.4  ; 132.8 118.1 113.1 71.8; 78.7

Polar Component p sp I benzene ( γs ) (at 150°C) [mJ/m²] 120.0 103.9 93.0* 90.2 85.9; 80.2 87.9 75.0; 75.0 71.1 77.7 ; 74.0 63.8 63.9 56.6; 58.8

Source 29 29 55 29 29,30 29 29,31 29 29,31 30 31 29,31

sp measured at 180°C. * BET value from Table 4.5; γ ds and I benzene

Nevertheless, the fractal description of carbon black aggregates on one hand brings quite a convincing theoretical support for the former interpretation of DBP absorption results by Medalia and, on the other hand, provides the starting argument for several recent theoretical descriptions of certain nonlinear effects associated with the reinforcement of rubbers by carbon black. As largely illustrated by published transmission electron micrographs (see Herd et al.8,10 for instance), most carbon black aggregates exhibit a branching structure that can be considered in terms of fractal geometry. If L is the shortest connecting (fractal) path between any two arbitrarily chosen elementary particle of an aggregate (see Figure 4.8b), then this quantity is related to their geometrical distance R in the three dimensional (Euclidean) space through the following relationship: C



L  D = β   d d

(4.2)

where β is a prefactor of the order or unity and C, the so-called ­connectivity exponent, readily related to the branching structure of the aggregate. Indeed for a chain-like aggregate (i.e., without any branches; see Figure 4.8c), the fractal path is the length of the chain and consequently the connectivity exponent C equals the mass fractal dimension F and has the value of 2. It follows that, because they have many branches, most carbon black aggregates have connectivity exponent C significantly

37

Typical Fillers for Polymers

smaller than two. In fact, when performing their TEM/AIA (transmissionelectron-­m icroscopy/automated-image-analysis) study on a representative sampling of 19 different carbon black grades, Herd et al. 8,10 classified aggregates in four specific shapes: spheroidal, ellipsoidal, linear, and branched. Different aggregate shapes exist within a given grade of carbon black but it seems that the highest percentages of branched aggregates are found in the highly reinforcing carbon black grades. In weight percent, the branched aggregates quantity decreases as both the DBP absorption number and the surface area decrease. With respect to reinforcement, branched aggregates have the greatest influence on properties such as modulus, tear, and wear resistance, which are known to be connected with the effectiveness of the aggregate in “occluding” the polymer from deformation. The (mass) fractal description of carbon black suits obviously the effects of aggregate branching and, as we will see, is the background of advanced modeling approaches. The size D of an aggregate can be viewed as the diameter of its spherical envelope with respect the well-known void volume concept of aggregates introduced by Medalia.11,12 Whilst apparently not aware of the concept of “fractals,” it is quite clear that Medalia somewhat foresaw the fractal nature of carbon black aggregates when he wrote: “the effective volume of a carbon black aggregate composed of Np particles is proportional to Np raised to a power greater than unity, so that with increasing number of particles per aggregate, the aggregate becomes more open and more voluminous.” The effective volume of an aggregate cannot however be directly assessed with any precision and, therefore with respect to the at-the-time capabilities of electronic microphotography techniques, Medalia suggested to consider that the effective volume of an aggregate is that of a sphere of the same (mean) projected area as the aggregate (see Figure 4.10). If Des is the diam-

Equivalent sphere of diameter Des

Size D

seen by TEM

Size d

Np particles

Figure 4.10 Concept of equivalent sphere for a single aggregate.

Project area A of the aggregate

38

Filled Polymers

eter of the equivalent sphere and A is the measured projected area of the aggregate, it follows that:



Ves =

π Des3 π  4 A  =  6 6  π 

3/2

=

4 A 3/2 3 π

(4.3)

Obviously the volume of solid carbon within an aggregate is the product of the number of particles Np times the volume of an elementary particle (assumed to be spherical). It follows that the measured projected area (a two dimensional quantity) of an aggregate can be related to the project area of its calculated equivalent sphere (a three dimensional quantity) through a scaling law. Medalia et al. performed a so-called “floc simulation” to establish the following equality:13,14

 A Np =   Ap 

1/ε

or A = Ap N pε

(4.4)

where Np is the number of elementary particle of projected area Ap and ε a scaling exponent. From his floc simulation, Medalia reported a value of 0.87 for ε , which however seems to be an unfortunate printing mistake since, using his published data, it can be shown that the correct value is 0.847 (see Appendix 4.2). If d is the (average) diameter of the elementary particles of the aggregate, it follows from Equation 4.4:



A=

 4A π d2 ε N p or N p =  4  π d 2 

1/ε



(4.5)

Using ε = 0.847 , Equation 4.5 is rewritten as N p = ( 4 A/πd 2 )1.18 , with an exponent slightly different from the one reported by Medalia (i.e., 1.15), not only in his original publication but also in all his subsequent ones, and moreover blindly used by a number of other authors. It is worth noting that the exponent 1.18 is still far, but closer to the surface fractal exponent of ≈ 1.8 as derived later from colloid agglomeration modeling,15,16 and also confirmed by experimental results on carbon black filled EthylenePropylene-Diene Monomer rubber (EPDM) compounds.17 From Equations 4.3 through 4.5 it follows that the diameter of the equivalent sphere is given by:

Des = d N pε/2 or Des = d N p0.4235

(4.6)

39

Typical Fillers for Polymers

The exponent in Equation 4.6 is sufficiently different from the one reported by Medalia (i.e., 0.435) to bring large differences in calculated Des when either the diameter d of the elementary particle increases and/or when the number Np of elementary particles is large. However, the misprint in Medalia publications must not shade his merit in having foreseen the fractal nature of carbon black aggregates. In addition, Medalia has thoroughly elaborated practical formulas to convert an easily measured quantity (i.e., the DBP absorption number, in cm3/100 g filler) into the number of particles in an aggregate (see Appendix 4.3 for details and numerical illustrations). The solid volume Vs of an aggregate is nothing else that the volume of an elementary particle (of diameter d) times the number of particles, and by combining Equations 4.3 and 4.5, it follows: Vs = N p



π d3  4 A  = 6  π d 2 

1/ε

π d3 6

(4.7)

Using his floc simulation approach, Medalia has established the following practical relationship between the so-called “void ratio,” i.e., the ratio of the equivalent sphere Ves to the solid volume Vs, and the DBP absorption number, i.e.

CF

Ves (1 + vf ) g − 1 = DBPA ρ 0.0115 Vs C

(4.8)

where CF = 0.765: correction factor accounting for difference between the projected area of the equivalent sphere and the projected area of the aggregate (around 8.5% reduction in diameter) vf = 0.46: void fraction for randomly packed spheres C = 1.4: correction for partial fusion of primary particles in aggregate g = 0.94: anisometry correction factor for non-perfect alignment of aggregate’s main axis with projection plan ρ = filler specific gravity [carbon black: ρ = 1.86 g/cm3] DPBA = di-butylphtalate absorption (cm3/100 g filler) 0.0115: correction for DBPA end point (i.e., 1.15/100 , to take into account that at the end of the DBP absorption test, the sample contains around 15% air) By combining the equations for the volume of the equivalent sphere (Equation 4.3), for the projected area (Equation 4.5) and for the volume of solid carbon in the aggregate (Equation 4.7) with Equation 4.8, one gets immediately:

ε    ε + 2 − 1

N p

= ( 1 + DBPA ⋅ ρ ⋅ 0.0115 ) ⋅

C CF ⋅ g ⋅ ( 1 + vf )

40

Filled Polymers

or ε    ε + 2 − 1

N p



= ( 1 + DBPA ⋅ ρ ⋅ 0.0115 ) ⋅ 1.333

(4.9)

if one replaces the various correction factors by their values given above. The number of elementary particle in an aggregate can consequently be assessed from the DBPA number, the specific gravity ρ of carbon black (1.8 g/cm3) and the value assigned to ε , using: N p = [ 1.333 ⋅ ( 1 + DBPA ⋅ ρ ⋅ 0.0115 )] 2/( 3 ε − 2 )



(4.10)

Depending on the value used for ε , the result yielded by Equation 4.10 can be very different, as shown in Figure 4.11, in fact largely underestimated using ε = 0.87 , as published by Medalia. For instance, for a typical High Abrasion Furnance (HAF) grade, e.g., N330 (DBPA = 102 cm3/100 g), the correct value ε = 0.847 gives 211 particles/aggregate, whilst ε = 0.87 gives 114. The more reinforcing the carbon black, the larger the difference. It is interesting to compare the number of particles per aggregate as calculated with Equation 4.10 (Medalia’s; based on TEM analysis and “floc” 800 700

ε = 0.847

Number of particles

600 500 400 300 ε = 0.87 (Medalia)

200 100 0

0

0.5

1 DBPA, dm3/kg

1.5

2

Figure 4.11 Assessing the number of elementary particles in a carbon black aggregate; curves were calculated with Equation 4.10, ρ = 1.86 g/cm3, DBPA values in the range 38 (N990) to 157 (N356) cm3/100 g and the ε values given in the figure. (Data (◽) are from A.I. Medalia, F.A. Heckman, Carbon, 7, 567–582, 1969.)

41

Typical Fillers for Polymers

Number of particles/aggregate (Medalia)

800

600

400

200

0

0

200 400 600 Number of particles/aggregate (Fractal approach)

Figure 4.12 Comparing carbon black aggregates as described either through the TEM data analysis by Medalia and or the fractal approach.

simulation) with the estimation obtained using the mass fractal approach, i.e., Equation 4.1, and reported particles and aggregates dimensions (from Herd et al. for instance; see Appendix 4.4 for details). As shown in Figure 4.12, with respect to the equality line, an agreement is obtained only if the front factor α in Equation 4.1 is taken equal to around 11, i.e., more than 10 times the value guessed by some authors.18 Within the spherical aggregate envelope, one can distinguish the solid Vs and the void Vv volumes, whose relative importance is expressed in terms of volume fraction, i.e.: −1



V   Φ = 1+ v   Vs 

(4.11)

With respect to fractal geometry, this solid volume fraction is expressed in terms of basic dimensions of the aggregate as follows :

Φ=

N d3  d =α   D D3

3− F



(4.12)

where α is the so-called “front factor.” This fraction is the volume occupied by the (fractal) solid aggregate with respect to the overall volume occupied in the three dimensional space, and is obviously related to the aggregate surface accessible to polymer chains in a compound. When carbon black volume

42

Filled Polymers

fraction is large enough in a compound—in practice when the loading is above the so-called percolation level (i.e., Φ ≈ 0.12 − 0.13 ), all aggregates are expected to entangle (or at least to connect) and to form a secondary aggregated structure in the polymer matrix. As we have seen before, during the production process (precisely during the quenching), carbon black aggregates flocculate into agglomerates, that are further compacted by the final pelletizing step (see Figure 4.5). Carbon black pellets is the easier handling form of the filler, readily used in polymer compounding. Such pellets are very friable and no much (mixing) energy is needed to split them into agglomerates which correspond in fact to a close packing state of aggregates. Agglomerates have also a fractal nature with a mass fractal dimension of the order of three and a connectivity exponent of around one. Aggregates are recognized for decades as the filler structural state that plays the key role in rubber reinforcement, but some authors have recently argued that, when above a critical concentration threshold they are well dispersed in a polymer matrix, they form a kind of tenuous secondary structure, which helps in understanding certain aspects of the reinforcement of elastomers through a filler aggregates networking effect (Klüppel and Heinrich18), as we will see later in detail. Whatever is the packing state of aggregates into agglomerates, and the compaction degree of agglomerates into pellets, there is a certain degree of “voids” such that the solid fraction of carbon black pellets is given by:

Φ pellet = 1 + ρ

DBPA 100

(4.13)

The fraction Φ pellet is found nearly equal to the volume fraction Φ for the aggregate (Equations 4.11 and 4.12), which means of course that the front factor α is also close to one for carbon black pellets. The mean number of aggregates within the total volume R3 of a single agglomerate is defined as N aa = n R 3 with n the number density of the aggregate, i.e.

n=

Φ N d3

(4.14)

The number N aa is obviously related to the degree of interpenetration of aggregates in each other and, with respect to Equation 4.1 it follows:

N aa = α − 3/F Φ N (( 3/F ) − 1)

(4.15)

In relation with the void volume in carbon black aggregates, Medalia19 introduced the (debated) concept of “occluded rubber,” defined as the fraction of

Typical Fillers for Polymers

43

polymer that has penetrated the internal void space of filler aggregates and is thus shielded from deformation, at least partially. Medalia classified voids in a compacted carbon black in two categories: within and between aggregates, and he developed two relationships that permit to assess their relative importance from easily measured quantities, i.e.

Ratio

 1 + 0.02139 DBPA  =  ( 1 − Φ ) ×  − 1   1.46





voids volume within aggregate overall agglomeratee volume

Ratio

(4.16)

voids volume between aggregate overall agglomeratte volume

1 + 0.02139 DBPA   =  Φ − ( 1 − Φ ) ×  − 1   1.46 where DPBA is expressed in cm3/100 g, the factor 0.02139 is the product ρ × 0.0115 with ρ = 1.86 g/cm3 and the factor 1.46 is related with the Medalia’s assumption11 that, at the endpoint of an oil absorption test, the remaining void space between aggregates is 31.5% so that 1 + ( 31.5/100 − 31.5 ) = 1.45985. Such considerations about the fractal nature of carbon black are the background for recent theoretical developments on the very origin of the reinforcing effect of the filler and of certain aspects of the mechanical properties of rubber parts. During efficient mixing operations, carbon black pellets are expected to completely disappear and agglomerates to fully separate into their constitutive aggregates, the latter being ultimately evenly distributed in the rubber matrix. A “well dispersed” state can of course be considered in terms of an even statistical distribution in a given rubber volume, but in the opinion of the author, it is more interesting to define the ideal well dispersed state as the situation where all the reinforcing entities of the filler, in the occurrence the aggregates, have developed their maximum interaction potential with the rubber matrix. In other words, in the optimum dispersion state, the maximum available specific area of the aggregate is in contact with elastomer chains. It is now easy to understand that there will be a tremendous difference in carbon black effects on mechanical (and rheological) properties of rubber compounds, depending one is in the low concentration regime or above a critical concentration level. Below this critical concentration level, well mixed aggregates are sufficiently separated from each other and it is essentially the surrounding rubber matrix that support and transmit the stress. Above a critical filler level, obviously not much depending on the grade of carbon black, there are enough aggregates for a secondary carbon

44

Filled Polymers

black network to be formed through direct aggregate—aggregate interactions, with the resulting capability to support and transmit stress in the compound. As local aggregate density in the rubber matrix increases, a kind of aggregates flocculation occurs which, amongst other effects, can be considered as the very origin of phenomena such as dynamic stress softening. 4.1.5  Surface Energy Aspects of Carbon Black In addition to specific surface area and the fractal nature of carbon black as discussed above, it may be expected that rubber–filler interactions, which are the roots of reinforcement, somewhat depend upon the surface activity of the particles. The so-called surface activity is not however a clearly defined concept as many phenomena might be involved, from Van der Waals proximity forces (around 4 kJ/mole) to specific chemical interactions (e.g., hydrogen bonding, ≈ 20 kJ/mole; ionic bonds, ≈ 30 kJ/mole). Despite the considerable literature on the subject, there is so far no standard method to measure surface activity. Rubber grade carbon blacks contain small quantities of chemically combined hydrogen (0.2–1.0%), oxygen (0.1–4.0 %) and even sulfur (up to 1.0%) depending on the quality of the feedstock and the process. Over the years, a large variety of oxygen containing functional groups, most in minute quantities, has been detected in carbon blacks, for instance carboxyl and hydroxyl groups, phenol, lactones, quinones, ketones, aldehydes, hydroperoxydes, etc., (Figure 4.13). It must be noted however that a number of reported data that support the picture offered in Figure 4.13 have been obtained on lamp or gas blacks, obviously very sensitive to contamination by oxygen and other heteroatoms. Advanced and sophisticated analytical techniques performed on (modern) furnace blacks give quite a different picture. Indeed, Bertrand and

Ketone

O

O

O

HO C

Carboxyl

Pyrone O

O

C

O Lactone HO Hydroxyl

O

O Quinone

Figure 4.13 Chemical functions detected on (lamp and gas) carbon black surface.

Typical Fillers for Polymers

45

Weng20 used time-of-flight secondary ion mass spectrometry (ToF-SIMS) and x-ray photoelectron spectroscopy (XPS) to characterize various furnace blacks, either commercial or experimental, quite representative of the reinforcing carbon blacks available today. They carefully interpreted the various spectra obtained before and after toluene extraction of carbon blacks and came to the conclusion that there are only C and H on carbon black surface, with nearly no oxygen. Even after heat temperature treatment (1000°C), hydrogen containing fragments were still detected in ToF-SIMS spectra. This strongly supports the view that carbon black surface is locally graphitic in nature, with broken graphitic plan edges supporting only C–H or maybe some pendant methyl groups. Carbon black surface chemistry therefore plays nearly no role in the exceptional reinforcing capabilities of this filler. It is moreover well established today that oxygen complexes at the surface of carbon black particles are not essential for reinforcement in most rubbers, with the notable exception of polar elastomers, e.g., butyl rubber. As indeed convincingly demonstrated by Gessler et al.21 some 30 years ago, oxygen functionality on carbon black is a requisite to high-order interactions only for butyl rubber. Indeed when the surface oxygen on channel black is removed by high temperature treatment under inert gas, these authors observed significant loss in reinforcement (damping properties) of butyl rubber. Conversely when furnace blacks (for butyl rubber) are activated by heating at 250–300°C in a stream of oxygen, a significant benefit is observed in terms of reinforcement. With nonpolar elastomers, i.e., most general purpose rubbers, including Natural Rubber (NR), Butadiene Rubber (BR), Styrene-Butadience Rubber (SBR), Ethylene-Propylene Rubber (EPR) and EPDM, the occurrence of chemical reactions with functional groups on carbon black surface is far less convincing. The heating of carbon black below 800°C does not result in graphitization of the filler, i.e., there are no significant changes in the crystallinity of the inner particles, but it does remove most of the chemisorbed surface oxygen. However, rubber–filler interactions remain essentially unaffected, as shown by Dannenberg’s experiments with SBR compounds.22,23 There is therefore a consensus today to consider that the strong rubber–carbon black interaction is not necessarily resulting at all from chemical reactions involving oxygen complexes at the surface of particles. However if the rubber is containing specific reactive groups, then quite logically it may be interesting to consider purposely surface oxidized carbon blacks to promote chemical interaction, as shown for instance by Manna et al.24 with epoxidized natural rubber (ENR). Indeed, with respect to a 60 phr (part per one hundred rubber) Intermediate Super Abrasion Furnace Carbon black (ISAF) (likely N220) filled ENR reference compound, a corresponding 60 phr oxidized ISAF/ENR compound with 4 phr silane coupling agent exhibits twice higher tensile and tear properties, largely below however with what can be readily obtained with conventional natural rubber and carbon blacks. It is thus well established today that carbon black surface chemistry plays a very minor role, is any, in the reinforcement of general purpose elastomers,

46

Filled Polymers

i.e., essentially diene rubbers (NR, BR, SBR) and EPDM, more than 90% of the overall rubber consumption. As we will see, the situation is completely different with other fillers, namely silica, for which the surface chemistry of particles plays the essential role. An attempt to quantify the role of filler surface chemistry is to consider the so-called “surface activity,” generally assessed through the surface energy γ s , which consists of two main components,25 i.e. γ s = γ ds + γ sp



(4.17)

where γ s and γ s are respectively the so-called dispersive and polar (or specific) components. Such properties are measured by inverse gas chromatography (IGC),26 a technique in which the filler to be characterized is used as the stationary phase and the injected solute is the so-called probe. In practice, the filler particles are carefully poured into a stainless steel column of appropriate diameter, typically a few mm. Suitable model chemicals, in dilute solution, are used as probes in order to quantify their interactions with the filler. When the probe is operated at infinite dilution, information is obtained concerning the adsorption of the solute on a solid surface, by use of the Henry’s law that considers the standard free energy of transferring one mole of vapor from a gas phase (at the standard pressure of 1 atmosphere, or 101 kPa) to a standard state on the surface.27 By injecting a series of homologous n-alkanes (e.g., pentane to decane) as probes, the dispersive component of the surface energy γ ds is obtained from the free energy of adsorption, by considering the slope of the measured standard free energy for adsorption vs. the number of carbon atoms for different n-alkanes. The specific (or polar) component is derived from the difference in the free energy of adsorption between a polar probe and a real or hypothetical n-alkane with the same surface area (see details elsewhere25,26,28–31). Table 4.6 gives typical data as reported in literature; as usual data for the same grade slightly differ between authors. Carbon blacks exhibit a high dispersive component, actually proportional to their specific surface area, and a relatively low polar component, not much differing whatever the grade. As we will see later, the reverse is seen with reinp forcing silica grades, which have a lower dispersive component, but a high γ s . With carbon blacks, the reinforcing effect is thus essentially achieved by means of strong filler–rubber interactions, and the polar component is reflected by a relatively weak carbon black network, at least providing that the filler volume fraction is below the so-called percolation level. Above that level, some authors have recently argued that branched aggregates readily entangle through complex topological interactions in such a manner that the carbon black network plays the key role in modulus enhancement, as we will see later in detail. New equilibrium gas adsorption techniques were recently used to analyze the surface energy distribution of carbon blacks.32–35 By deconvoluting the d

p

47

Typical Fillers for Polymers

energy distribution function of adsorbed ethylene into four Gaussian peaks, it can be concluded that there are four different energetic sites on the surface of the filler (Figure 4.14). The nature of these energetic sites is however a matter of interpretation that depends on the model considered for the surface of the filler. For instance, if one considers as with some authors36,37 that the surface of elementary carbon black particles consists of turbostatic graphitic crystallites beside areas of amorphous carbon, then one could assign such energetic sites to different features:

Site I: graphitic surface of a crystallite Site II: amorphous carbon zone. Site III: crystallites edges Site IV: slit shaped cavities (or boundaries between two crystallites) Amorphous carbon zones (II)

Distribution of surface energy f(Q), kJ/mol

Graphitic planes (I)

0.20

Crystallite edges (III)

Adsorption of C2H2 on at T = 233 K N220 carbon black

0.16

Overall energy distribution function

I

0.12

Fraction (%) of energetic sites III IV I II

0.08 II

0.04 0.00

Slit shaped cavities (IV)

0

III

N115 N220 N550 Graphite

69 84 93 94

13 7 6 0

15 7 1 4

3 2 <1 2

IV

10 20 30 Surface energy Q, kJ/mol

40

Figure 4.14 Types of energetic sites on the surface of carbon black with supporting data (From: A. Schröder. Charakterisierung verschiedener Rußtypen durch systematische statische Gasadsorption; Energetische Heterogenität und Fraktalität der Partikeloberfläche. PhD thesis, University of Hannover, Germany, 2000.)

48

Filled Polymers

The fraction of higher energetic sites (ie. II and III) is then found to decrease with the reinforcing capabilities of the black and to disappear almost completely through graphitization. It can therefore be hypothesized that these sites play an important role in filler–filler and filler–elastomer interaction. However, because of the relatively low energies of these sites, weak interaction forces (London, Van der Waals, etc.) between carbon black surface and unsaturated segments in the rubber backbone may prevail, with obviously a significant cumulative effect due to the large number of such sites. With respect with the macroscopic effects involved in reinforcement, the exact nature of the surface of carbon black particles might appear as detail knowledge. But the key information is the demonstration of the surface roughness of elementary carbon black particles, whether one considers a disordered array of graphitic crystallites or a step like surface resulting from a spiraling growth process. It is quite remarkable that nearly all carbon blacks have a unique surface roughness on atomic scales with a surface fractal dimension d f ≈ 2.6 , in agreement with the most recent theoretical models for physical concepts of surface growth that are nowadays found to be valid in many different fields in nature. One such theoretical approach is the so-called Toom model, i.e., a relatively simple probabilistic cellular automaton that appears to be “generic” for a variety of physical patterns.38 In the context of this book, such fundamental theoretical works might seems quite remote, but it is striking to see that through calculation with the appropriate equations, step-like surface patterns are built that are very similar to what is observed on AFM images of carbon black.39 It seems thus that there is a convergence of experimental and theoretical results that describe the rough surface of elementary carbon black particles as a pattern of overlapping scales, further reinforced by the recent demonstration of the role of fullerenes, particularly C60, as precursors for the chemical reactions involved in the formation and nucleation of carbon black in the furnace combustion process.40,41 Another indirect demonstration of the importance of the very surface of filler particles in the development of strong interactions with polymer materials is provided by the fact that, when chemically grafting short hydrocarbon chains on carbon blacks, the reinforcing capabilities of the filler are either lost or at least severely penalized. Vidal et al.42 showed, for instance, that the esterification (via carboxyl groups at the surface) of carbon blacks, using either methanol or hexadecanol, is associated with a strong decrease of the interactions the filler can develop with any molecule. The surface free energy is found to decrease whilst the surface energy homogeneity is increasing. As expected, when grafting filler particles, interparticle interactions are lower, an effect that might be beneficial for dispersion mechanisms; but polymer–filler interactions are also decreasing, which results in a severe drop in reinforcement.

Typical Fillers for Polymers

49

4.2  White Fillers 4.2.1 A Few Typical White Fillers So-called white fillers are materials of various nature and origin that are used in both elastomers and thermoplastic polymers. Most of them are mere natural products, i.e., found as such in nature and submitted to more or less complicated mechanical and physical processes, such as sorting, washing, grinding, sieving, etc. Others are elaborated natural products, i.e., slightly modified through relatively simple chemical treatments, and a few are essentially synthetic materials, i.e., not found as such in nature but obtained either as side products of chemical processes or through purposely chemical synthesis. Typical natural fillers are extracted from lodes (in quarries or mines), sorted and washed if necessary, nearly always grinded to the appropriate particle size (at least below 10 µm) and sieved with respect to the desired distribution of dimensions. Chemically, mineral white fillers are silicates, carbonates or oxides, with specific surface area ranging from 1 to 50 m2/g, therefore considered as inert (i.e., not reinforcing) to semiactive (slightly reinforcing) materials. We briefly describe hereafter the most common white minerals used as fillers for polymer materials. 4.2.1.1  Silicates Kaolinites (clays): general chemical formula: Al2O3.2SiO2.2H2O. Specific gravity: 2.6 g/cm3. Clays belong to the phyllosilicate class (old greek: ϕυλλον, plate), which means that, at microscopic level, they exhibit lamellar forms that are reduced to roughly hexagonal small plates by grinding (typical aspect ratio: 20:1). Depending on the mineralogical quality of the deposit, its geological environment and its accessibility, the extraction and treatment of clay can be more or less complicated but the main steps of the process are always the same: extraction (mining, excavation or high pressure water disintegration), grinding, separation from foreign minerals, e.g., sand, mica, etc., washing, sorting and screening, then thickening, filtering, drying and final grinding. One distinguishes the “hard clays” (30 m2/g specific surface area), which are semiactive fillers, and the “soft clays” (10 m2/g) considered as inert fillers. The product form is a fine powder with a moisture content of around 1%. Mohs hardness is low, typically 2.5, and pH is 5–7.5 (Note that detailed considerations on the Mohs hardness are given in Section 6.2.1). China clay or Kaolin (from the chinese “kao-ling,” hill) is a hydrated aluminium silicate crystalline mineral (kaolinite) formed over many millions of years by the hydrothermal decomposition of granite rocks. Hydrous kaolin is characterized by its fine particle size, plate like or lamellar particle shape and chemical inertness. There are many production sites in China and in

50

Filled Polymers

Europe, the most important deposit is in Cornwall, England (discovered in 1745 by W. Cookworthy and initially used to produce white chinaware). Montmorillonite is a very soft type of clay, named after Montmorillon, Vienne, France where the first deposit was discovered in 1847, but is found in many places worldwide. It is a hydrated Na, Ca, Al, Mg silicate hydroxide (Na,Ca)0,33 (Al,Mg)2 (Si4O10)(OH)2.nH20 whose particles are plate-shaped (average diameter: 1 µm) and made of silicate sheets spaced by 1–2 nm. Montmorillonite has many usages, for example as component of drilling mud, as soil additive (due to its high capability to absorb water), and as animal feedstuff component (anticaking agent) and became recently a popular component in many recent research works on so-called nanocomposites, thank to its capabilities to be relatively easily exfoliated by polymer chains. Exfoliated montmorillonite provides to certain polymer compositions interesting properties at very low loading, in the 2–5% volume fraction range, which however have not yet led to sizeable industrial applications, to the author’s knowledge. In connection with the on-going research efforts on so-called polymer nanocomposites, the surface modification of clay is worth to be briefly mentioned. Recently Liu43 published a comprehensive review on the advances in surface modification of natural clay minerals with polymers, including the modifying methods and mechanisms, characterizing and analytical techniques, and potential applications. Two main approaches are so far exploited: either physical adsorption or chemical grafting of functional polymers on the surfaces of the clay minerals. Physical adsorption, a process essentially driven by thermodynamic effects can somewhat modify the nature of the mineral surface and improve the local physical and chemical properties, without modification of the structure of the clay. However the forces between the adsorbed molecules and the clay mineral might be weak, and the process is essentially reversible, for instance nearly annihilated as temperature increases. Coated clays, with fatty acids or oleates for instance, belong to this physical adsorption approach and are readily used in the current industrial practice of filled polymers. Grafting of functional polymers to the surface of clay minerals induces stronger interactions, in principle irreversible and therefore less sensitive to temperature. Two strategies can be used: either one-step or two-steps grafting methods (Figure 4.15). The former consists in the condensation of functionalized polymers with reactive groups of clay particles. Relatively loose polymer brushes are obtained because chemisorption of the first fraction of chains hinders the diffusion of the following chains to the surface for further attachments. The two-step grafting technique allows higher grafted densities to be achieved, because a monolayer of polymerizable (macromonomer) or initiator (macroinitiator) molecules is first covalently attached to the clay surface, then further activation produces chains growth from the interface, which is limited by the diffusion of monomers to the active species only. Chemical grafting of clay minerals to produce so-called organo-clays are complicated and expensive processes however and there is no doubt that they offer a route to new materials with

51

Typical Fillers for Polymers

One-step grafting X X X

+

Y

X

Two-step grafting X X X

+

Y - A - A*

Z - A - A* Z - A - A* Z - A - A*

A

Z-A Z-A Z-A

Figure 4.15 Mechanisms for one-step and two-step grafting reactions on mineral surfaces.

predefined structure and performance that ultimately will allow to prepare truly advanced materials, e.g., organic/inorganic polymer nanocomposites or even biomimetic materials. There are not yet indication for significant use of such polymer-modified clay in material technology. Calcined kaolins (calcined clays): obtained by treating kaolinites at temperature higher than 1000°C in order to remove water and modify the structure. An anhydrous aluminium silicate is obtained, with increased whiteness and hardness, better electrical properties, and smoother size and shape of particles. Calcination occurs in three steps: at 700°C the dehydroxylation of the kaolin is complete forming a poorly crystalline metakaolin, at 980°C an amorphous defect spinel is formed which undergoes a recrystallisation in an amorphous glass at temperatures above 1100°C. Products obtained within this temperature range provide enhanced mechanical properties and ­chemical resistance in rubber compounds, especially when coated with silane. Calcined kaolin improves the thermal properties of agricultural films, gives better electrical properties in PVC cables and, suitably coated, may act as a functional filler in engineering thermoplastics. Silane coated grades provide higher stiffness, toughness and dimensional stability to polyamide moldings. Talc: hydrated magnesium silicate of general chemical formula: Mg8(OH)4. Si8O20 or 3MgO.SiO2.H2O. Specific gravity: 2.7–2.8 g/cm3. Another phyllosilicate with a lamellar structure and a very low hardness. Depending on the deposit, other minerals can be present, e.g., chlorite, dolomite, sometimes in the few percentage range. Particle sizes range from 2 (semiactive) to 15 µm (inert filler). Talc is a common filler for polypropylene with loading up to the 30% weight and is also used in the rubber industry as antitack agent for storing uncured rubber strips before further processing. The lamellar surface of talc is organophilic, which means that such a mineral has a natural affinity for organic substances, including of course most hydrocarbon polymers. Mica: a group of phyllosilicates exhibiting near perfect basal cleavage, owing to the hexagonal sheet-like arrangement of its atoms. Muscovite is potassium aluminum silicate hydroxide fluoride, KAl2(AlSi3O10)(F, OH)2, and

52

Filled Polymers

the most common of the mica group. It has a layered structure of aluminum silicate sheets weakly bonded together by layers of potassium ions, which allow a perfect cleavage in thin sheets of flakes, which are flexible and elastic. Color is white to colorless, with a vitreous to pearly luster. Mohs hardness is 2–2.5 and specific gravity is 2.75–2.90 g/cm3. Properly delaminated, milled and sorted, mica from muscovite is used in certain thermoplastics as a reinforcing filler. Biotite, another common phyllosilicate within the mica group, with the formula K(Mg, Fe)3AlSi3O10(F, OH)2, is less used as a filler for polymers, owing to its dark brown color. Because it is frequently associated with other minerals, scrap and flake mica is produced all over the world but the largest deposits are in India, China, and Brazil. 4.2.1.2  Natural Silica Quartz: nearly pure SiO2 is obtained by finely grinding quartz, the most abundant mineral on earth. Specific surface area (BET) is below 5 m2/g, which confines such a silica in the class of essentially inert fillers. The high hardness (7 on Mohs scale) severely limits its use as filler for thermoplastics owing to wearing problems on processing equipment. However, certain grades find uses in specific elastomer applications. Neuburg silica: a natural mixture of lamellar kaolinite and corpuscular quartz, that forms a loose structure. Specific gravity: 2.6 g/cm3. It is mined in the region of Neuburg (Germany), and treated by hydrocyclonisation to yield nearly spherical grains (1–5 µm diameter) that exhibit a grape structure (BET: 11 m2/g). Mohs hardness is up to seven for the quartz particles, but somewhat tampered by the lower harness (around 2.5) of kaolinite. Available from Hoffmann Minerals GmbH, Neuburg, Germany, under the tradename Sillitin®, typical grades have 65/30 to 78/17 %silica/%kaolinite ratio with around 5% other minerals. It is a very specific filler considered as inert to semiactive44 and is mainly used in the rubber industry. Wollastonite: a white calcium silicate (CaSiO3), with sometimes traces of Fe, Mg or Mn substituted for Ca, mined in certain regions (China, India, USA, Germany, Sweden), which has a typical acicular structure (i.e., packed microscopic needles), with diameter-to-length ratio ranging from 1/3, 1/5 to 1/15. Mohs hardness between 4.5 and 5; specific gravity: 2.87–3.09 g/cm3. Wollastonite is used as a filler for thermoplastics and is the essential white filler for fluoroelastomers (FKM). With respect to its unique structure, wollastonite could also be considered as a natural short mineral fiber. Micronized wollastonite grades are available with very high aspect ratio (L/D), which are especially designed for thermoplastic and thermoset applications where a high degree of reinforcement is needed. Such grades could in principle be used instead of glass fibers, providing the adequate surface treatments are made with respect to the properties of the polymer matrix. In thermoplastics compounding Polycarbonate (PA) 6 and 66, Polypropylene (PP), Polycarbonate (PC), Polyurethane (PU), wollastonite is expected to provide higher heat distortion temperature, better impact

Typical Fillers for Polymers

53

and scratch resistance and less shrinkage of injection molded parts, all benefits associated with its needle-shaped particle structure. Most wollastonite-filled compounds are used by the automotive sector (interior and exterior parts, underthe-hood components) and the electrical industry (insulation materials). There are some ultrafine wollastonite grades, either untreated or surface coated modified, which no longer have anisotropy related reinforcing performance; such grades are used in thermoplastic polyesters and plateable/paintable nylons for improved adhesion and reflectivity. Some specially treated wollastonite products exhibit conductive properties so that they could replace conductive carbon black grades or carbon fiber in thermoplastic olefins and engineering alloys. Miscellaneous siliceous fillers: there is a wide variety of mineral deposits that, either as such or suitably treated, may offer fine particles of interest, with various shape, structure and/or composition. For instance, perlite is a volcanic siliceous rock (around 75% SiO2 and 15% Al2O3), naturally occurring as hollow, near spherical micro-pearls, with a water content in the 3–5% range. When shock treated at high temperature (above 600°C), perlite expands and develops an internal rigid foamed structure. Suitably milled, grinded and sorted, such minerals exhibit low reinforcing properties, equivalent to N600 or N700 carbon blacks, but they could be useful in certain specialty elastomers, for instance fluoroelastomers. Their hardness might however give wear problems in polymer processing equipments. Another example is provided by so-called diatomaceous earths which are deposits of the skeletons/shells of microanimals. Microscope observation reveals a great variety of delicate and perforated structures. The typical chemical composition of diatomaceous earth is 86% SiO2, 5% Na2O, and some magnesium and iron oxides. The natural product can be calcined so that the particles fuse together to produce grades of controlled average particle size. Apart their well known usage as supports for catalyst in chemical engineering, such siliceous products could also find niche applications as either extenders of semireinforcing fillers for certain type of polymers. Except a few technical data in trade literature, nearly no scientific information is available (to the author’s knowledge) about the actual use of these minerals in polymer applications, but the chemical inertia of such “exotic” minerals makes them worth of consideration as filler materials for rubber or plastic parts with pharmaceutical and food applications. Generally such minerals have a high absorption potential for hydrocarbon products owing to their organophillic character. Therefore they could also be interesting alternatives to low molecular weight polymers as used in the encapsulation of (rubber) chemicals. 4.2.1.3  Synthetic Silica Synthetic silica (silicon dioxide) are prepared either by pyrogenation of silica tetrachloride or by precipitation from a solution of alkaline silicates through the action of acids or metal salts (see next section). Synthetic silica exhibit specific surface area in the 100–200 m2/g range and are therefore active fillers. Table 4.7 gives a comparison of some physical properties and the chemical

54

Filled Polymers

Table 4.7 Properties of Precipitated Silicic Acids (Silica) vs. Active Silicates Precipitated Silicic Acids Physical Data

Unit

Very Active

Active

Silicates Active

Specific area (BET) Moisture (2 h, 105°C)

m /g %

160 6

130 6

90 6

Fire loss (2 h, 1000°C) Specific gravity Apparent density

% g/cm3 kg/m3

4 2.0 250

4 2.0 250

7 2.2 250

Chemical Composition Silicic acid SiO2.H2O Aluminium oxide Al2O3 Magnesium oxide MgO Sodium sulfate Na2SO4 Sodium oxide Na2O Sulfites SO3

% % % % % %

98 0.5 – 1.5 – –

90 8.5 – 0.5 – –

81 8 2 – 8 1

2

composition of precipitated silicic acids (silica) vs. active silicates. Two major differences emerge from this comparison: first the active character is associated with the specific surface area, second the purer the silica, the better its reinforcing character. The usage of silica was recently boosted in the rubber industry by the development of the so-called “green tire” technology, pioneered by Michelin in the late 1980s. As we will see later, high structure silica impart interesting dynamic properties to tire tread compounds when compared to carbon black, namely a lower viscous dissipation and therefore a better efficiency in term of so-called “rolling resistance.” However a chemical treatment (using a bifunctional silane) is necessary to achieve a good balance of properties. 4.2.1.4  Carbonates Chalk, whiting, and limestone are natural calcium carbonate (CaCO3) that, finely grinded, is likely the most widely available and utilized mineral as thermoplastic additive and is sometimes used as an inert filler in rubber compounds. Specific gravity: 2.70–2.83 g/cm3 (depending on crystal form; calcite less dense than aragonite). Calcium carbonate ores are found in either sedimentary or metamorphic rocks. Sedimentary rocks result from sediment or from transported fragments deposited in water, e.g., limestone, formed from inorganic remains, such as shells and skeletons. Metamorphic rocks, e.g., as marble, slate, quartzite, formed when rock masses were subjected to high heat and pressure in geological time. Whilst the main element in calcium carbonate deposits is calcium, other elements are present, magnesium (Mg), iron (Fe), and manganese (Mn) essentially that affect whiteness, hardness and specific gravity.

55

Typical Fillers for Polymers

Available in a wide range of particle sizes (range 500–1000 nm), ground CaCO3 Ground Calcium Carbonate (GCC) is a low cost filler added to extend and cheapen the widest range of thermoplastic polymer systems. In contrast, synthetic calcium carbonate as obtained by precipitation, has generally a finer and more regular particle size and therefore exhibit a semi-active character. Ultrafine, suitably coated (for instance with calcium stearate) precipitated CaCO3 Precipitated Calcium Carbonate (PCC) can even be considered as a sophisticated (and of course more expensive) filler with the following characteristics: primary particle size (in the 20–70 nm range); narrow particle size distribution; regular and controlled crystalline shape (rhombohedral for calcite; orthorhombic for aragonite). PCC is prepared through a three steps process: first calcium oxide is obtained by calcining crude calcium carbonate at around 900°C, then water is added to give calcium hydroxide, and eventually carbon dioxide is passed through this solution to precipitate the desired calcium carbonate:

CaCO3→CaO + CO2↑



CaO + H2O→Ca(OH)2 Ca(OH)2 + CO2→CaCO3↓ + H2O

The fineness of the particles, as well as the crystal form is controlled by adjusting the temperature, the concentration of reactants and the timing of the operations. Because it is a synthetic material, PCC offers a range of technical capabilities that are beyond the ones of GCC fillers, particularly when ultrafine coated grades are considered. Indeed, the fabrication of synthetic calcium carbonate through the above chain of chemical reactions is so sufficiently versatile that morphology and crystalline size of PCC can be varied at will. With respect to potential uses in polymer technology, ultrafine coated PCC grades exhibit several important characteristics:

1. Primary particle size in the 20–70 nm range, 2. Regularity of shape (essentially rhombohedral), 3. Narrow particle size distribution, 4. High purity, and 5. Surface coating (generally with calcium stearate).

There are in fact a number of plastic applications where PCC cannot be challenged by any GCC grade. Calcium carbonate, is a very common mineral, with deposits nearly everywhere on the planet and more than 60 large companies are producing and processing it; but only a few companies have worldwide significance: Minerals Technologies Incorporated (MTI), by far the largest with around 55 operating units worldwide, Omya, the largest producer of ground calcium carbonate in the world, is the second, Imerys has eight plants, Solvay (Belgium) is an important producer and merchant of PCC in Europe, Okutama Kogy (Japan) produces both GCC and PCC and there are also several Chinese companies of growing importance.

56

Filled Polymers

Dolomite, named for the French mineralogist Deodat de Dolomieu, is a natural blend of calcium and magnesium carbonate CaCO3.MgCO3 that is commonly found in deposits of a sedimentary rock called dolostone. Two types of materials are called dolomite, a true chemically uniform calcium magnesium carbonate CaMg(CO3)2, and a dolomitic limestone, i.e., an irregular mixture of calcium and magnesium carbonates. Dolomite is harder and denser than the calcite form of CaCO3 (limestone), and is more chemically inert, namely more resistant to acid attack. When finely grinded, dolomite can be used as an inert filler in thermoplastics. 4.2.1.5  Miscellaneous Mineral Fillers Barium sulfate: BaSO4; specific gravity: 4.5 g/cm3; essentially used to increase the density of polymer systems. High purity synthetic barium sulfate, often referred as “blanc fixe” is used as filler and extender in a great variety of plastics. Blanc fixe is precipitated with well-controlled particle diameter, in the micron range, from pure and filtered barium salt and sodium sulfate solutions. It can then be used in producing translucent plastic sheets or as an additive, which improves mould release of injection-molded parts. Lithopone (ρ = 4.2 g/cm3) is a nontoxic white pigment made by coprecipitating BaSO4 and ZnS, which can be used for the same purpose, providing the zinc sulfide does not interfere with certain chemical aspects of the polymer material. Zinc oxide: ZnO; not considered as a filler for polymers but used in the rubber industry for its major role in the complex sequence of chemical reactions that are involved in bridging elastomer chains (so-called vulcanization). Typical loading is 5 phr, always associated with 2–3 phr of stearic acid that is reacting with ZnO to yield zinc stearate, readily soluble in the compound, at the vulcanization temperature. Magnesium oxide: MgO; used as chemical buffer of reaction products that have an acidic character. For instance, MgO is a necessary ingredient in the curing formulation of fluoroelastomers where it captures the fluorohydric acid (HF) resulting from chains bonding. Calcium oxide: CaO; used as a moisture absorber in certain polymer materials. Table 4.8 compares the properties of a few selected white minerals used as fillers in the rubber and thermoplastic industries. 4.2.2  Silica Fabrication Processes 4.2.2.1  Fumed Silica So-called fumed silica results from a pyrolitic process, as the product of the hydrolysis of silicon tetrachloride vapor in a flame of hydrogen and oxygen, according to the following set of chemical reactions:

2 H2 + O2→2 H2O SiCl4 + 2 H2O→SiO2↓ + 4 HCl↑

2.65 2.87–3.09 2.65–2.70 2.70–2.72 2.40–2.90 4.3–4.5

CaCO3 CaCO3.MgCO3

BaSO4

7–20

3–12 5–20a

n.a. 3–100a 100–240

10–30a 2–12a 19–140a

Surface Area (m2/g)

3

3 3.5–4.0

7.0 4.5–5.0 7.0

2.5 1 2.0–2.5

Mohs Hardness

0.44

78–82 80–90

69–72 62–73 44.7

93–102 80–85 100

Specific. Heatb Cp (at 298 K)(J/g.K)

1.3–1.5

3.5–5.0 4.9–5.5

6.5 n.a. 21

2.0–3.2 3.0–5.3 2.2–2.3

Thermal Conduc­tivityb (W/m.K)

a

Specific surface area data were compiled from various sources and may vary considerably, outside the quoted range, depending on the milling/grinding process. b Thermal data were obtained from various sources of geological information; the actual data of the mineral used as filler in the polymer industry might be slightly different.

Quartz (grinded) Wollastonite Silica (synthetic) Calcium carbonate (synthetic) Dolomite (grinded) Barium sulfate (blanc fixe)

Al2O3.2SiO2.2H2O 3MgO.SiO2.H2O KAl2(AlSi3O10)(F,OH)2 K(Mg,Fe)3AlSi3O10(F,OH)2 SiO2 CaSiO3 SiO2

Clay Kaolinites Talc Mica

2.60 2.70–2.80 2.75–2.90

Chemical Composition

Filler

Specific Gravity (g/cm3)

Typical White Minerals Used as Fillers for Polymers

Table 4.8

Typical Fillers for Polymers 57

58

Filled Polymers

Overall reaction:

oC SiCl4 + 2 H2 + O2 1800  → SiO2↓ + 4 HCl↑

During the combustion process, elementary spheres of molten silica are formed, whose diameters are the 7–30 nm range, depending of the process parameters. Silica grades with final surface areas ranging from 400 to 100 m2/g, respectively are produced. The molten silica spheres, so-called primary particles, collide and fuse with one another to give three dimensional, branched, chain-like objects (“string of pearls”). The largest dimensions of these chainlike objects are in the few tenths of a micron range. As the fused, aggregated primary spheres cool down below the fusion temperature of silica (approximately 1710°C) further collisions occur which lead to reversible mechanical entanglement or agglomeration. Further agglomeration also takes place when silica is collected from the fumes. The residual adsorbed HCl on fumed silica surface is then reduced to less than 200 ppm through calcination. Certain grades go through additional densification processes that raise the bulk density, for instance from the normal average value of 0.03 to more than 0.07 g/cm3 depending on the grade. The true density of fumed silica is 2.2 g/cm3. As expected, the purity of the starting materials, the completeness of the pyrolitic process and the care in handling the product condition the degree of chemical purity of fumed silica. Only traces of metallic contaminants, e.g., aluminum, bore, calcium, zinc, etc., in the ppm range are found in typical fumed silica grades. As no ionic impurities are present in significant amount, fumed silica has excellent dielectric properties. Fumed silica is a non-porous, amorphous (i.e., noncrystalline) material. 4.2.2.2  Precipitated Silica Basically silica can be obtained by a wet process from solutions of alkali silicate, preferably sodium silicate, from which amorphous silica is precipitated by adding acid. After is has been filtered, washed and dried, the product consists of 85–90% SiO2 and 10–15% water, either present in the material structure (silanol groups) and/or physically bound on the surface. By controlling the precipitation parameters, such as temperature, pH, electrolyte concentration, stirring rate and duration, silica with different surface areas can be obtained, in the 25–700 m2/g. Various grades of amorphous precipitated silica are obtained from a process that essentially uses sand as starting material (Figure 4.16). Sand and sodium carbonate are first blended then molten by fusion in a furnace at 1400°C. The resulting vitreous silicate is dissolved under pressure in hot water. The liquid silicate is diluted at an appropriate concentration in water before performing the precipitation stage. Silica particles precipitate when adding sulfuric acid (or solutions of metal salts for high performance grades), with sodium (metal) sulfate as byproduct which is removed by filtration, and washing. Silica gel formation is avoided by stirring at elevated

Dissolution tank

Furnace 1400° C

Sodium carbonate

Dilution tank

H2O

Liquid silicate

Figure 4.16 Manufacturing process of precipitated silica.

Vitreous silicate

Blender

Sand

Spraying and drying

Liquefaction

SiO2 + H2O

Washing

Filtration

Slurry

Precipitation Dry powder silica

Bags

H2SO4

Bin-bags

Trucks

Packaging and delivering

Storage silos

Pelletizing

Grinding

Micronizing

Typical Fillers for Polymers 59

60

Filled Polymers

temperatures. Silica suspended in water is then dried by spraying and the dry powder is further treated to give various grades with respect to application requirements. Key steps in the process are precipitation, filtration, and spraying/drying but the microscopic particle morphology is essentially determined by the ­precipitation stage, during which controlling the composition and the ratio of reactants, the reaction time, the temperature and the concentration monitors silica properties. When precipitating silica, the reaction mixture is maintained in the alkaline pH region, in order to control the aggregation of primary particles whose diameter is larger than 5 nm. The precipitation process affects silica characteristics such that structure (oil absorption and compressibility), chemical properties (pH, silanol group density and optical properties). Drying is of course a major cost component in manufacturing precipitated silica. The most common technique is spray drying but rotary drying is also used, which give rise to different particle shapes, degrees of agglomeration, and to a lesser extent porosity. The dry powder silica (which still contains 6–8% of hydrogen bonded water) can be further treated in a number of ways (grinding, milling, micronizing) in order to reduce the size of the clusters (agglomerates) formed during the drying process and to obtain specific particle size distributions. Final products are stored in silos under controlled atmospheric conditions, before packaging and delivering. Precipitated silica is a safe amorphous (noncrystalline) mineral, an inert, nontoxic, powder, and considered as nonhazardous with respect to manufacture, transportation and handling, in contrast with finely grinded natural silica (quartz) or silicates which can give pulmonary deceases (lung tumors or silicosis). Essentially, precipitated silica seems to consist in bulky three dimensional assemblies of primary particles, approximately spherical. One may consider that, as precipitation proceeds, elementary silica particles, whose surface is rich in hydroxyl groups, tend to adhere each other and strongly stick together through hydrogen bonding. Then, through packing of additional elementary particles, these first clusters grow in size higher than 4–5 nm before they further agglomerate in larger assemblies. Direct observation of such a cluster formation mechanism is hardly possible but is easily supported by basic thermodynamic considerations. Indeed, hydrogen bonding corresponds to an energy of around 20 kJ/mole and, during the precipiation, the thermal energy that could conflict with the packing of elementary particles can be estimated as RT ≈ 2.69 kJ/mole , would the precipitation occurs at 50°C. So thermal agitation cannot really hinder clustering. An additional argument is provided when considering the time it takes for a spherical particle of diameter d to diffuse in a medium of viscosity η at temperature T, i.e.

tdiff =

3 π η d3 16 kB T

 4 π  1/3    − 2  3 Φ  

(4.13)

where kB is the Boltzmann constant (1.3 × 10 –23 J/K) and Φ the volume fraction of particles. If one considers that the viscosity of the suspending

Typical Fillers for Polymers

61

medium (i.e., water + sulfuric acid) is around 2 × 10 –6 Pa.s, then one ­calculates tdiff ≈ 6.3 × 10−9 s at T = 50°C. Such a diffusion time is so short that clustering of precipitating elementary SiO2 particles can be considered as a nearly instantaneous process. Once in close contact, clustered elementary particles will then remain so because cumulated hydrogen bonding forces between particles is considerably larger than thermal energy. It follows that large clusters of elementary particles of silica is expected a common observation. Indeed, Medalia has published comparative electronic microscopy photographs of a typical assembly of carbon black particles, of fumed silica particles and of precipitated silica particles.45 There are both similarities and differences between carbon black and so-called high structure silica, either fumed or precipitated. At equal aggregate/cluster cross dimension, carbon black elementary particles are larger, and fumed silica has a more open structure than precipitated silica. In his paper, Medalia underlines the difficulties in obtaining such pictures: first the filler must be thoroughly dispersed in a liquid, then the suspension is disposed on the appropriate support and the liquid is evaporated, using special precautions to prevent flocculation. It seems obvious that such a technique can hardly prevent the occurrence of artefacts, if the particles stick together owing to hydrogen bonding. It follows that if indeed aggregates are likely the ultimate form of carbon blacks, no certainty can be established in the case of silica, except maybe fumed silica, because the process occurs at such high temperature that local fusion of elementary particles is plausible. The author doubts that it is the case for precipitated silica. It is worth underlining here that, if it is well established that the aggregate is the reinforcing object with carbon black, there is indeed some controversy today about the actual role of silica clusters. The above thermodynamic considerations would lead to the conclusion that, if only interparticle hydrogen bonds are involved in the clustering process, then silica clusters might be considerably more brittle structure than carbon black aggregates. Consequently the ideal (theoretical) state of dispersion of silica could be as well an homogenous distribution of elementary SiO2, in sharp constrast with carbon black for which it is well established that the reinforcing structure is the aggregate. As we will see later, with diene rubbers, reinforcement by silica is only achieved through a chemical modification of silica surface and the strong silica-rubber interactions that develop only during the vulcanization stage. Recent data suggest that in an optimal dispersion stage, silica clusters could be totally dislocated with spherical silica particles evenly distributed.46 Would it be the case, clustered silica particles, maintained together in the rubber matrix by hydrogen bonding (≈20 kJ/mole, to be compared with C–C, 347 kJ/mole and C = C, 613 kJ/mole), are indeed relatively weak structures that surely do not play the same role as carbon black aggregates. The true meaning of the terms “high structure silica,” as used by silica manufacturers would need therefore clarification.

62

Filled Polymers

Whilst there is no standard classification scheme (as for carbon blacks), there are three main types of synthetic precipitated silica: • Standard grades • Grades for cosmetic applications (e.g., toothpaste), animal feeding and human foodstuff, which require particular specifications • High performance grades for various industrial applications, namely elastomer reinforcement. 4.2.3  Characterization and Structural Aspects of Synthetic Silica In sharp contrast with carbon black, there is no standard classification for commercially available silica but elementary particle size and the morphology of the reinforcing object (that we call the “cluster” in the case of silica, keeping the term “aggregate” for carbon black) are recognized as two important parameters for reinforcement performance. However, surface characteristics play a larger role in the case of silica than for carbon black, thanks to the rich surface chemistry of silica particles. Depending on the manufacturer and the manufacturing process, i.e., essentially by pyrogenation or by precipitation, different characteristics are measured, not always with standard methods. Primary characteristics include (see Table 4.9 for typical ranges): • Specific surface area: using the well known BET method for nitrogen adsorption (ASTM D3037/4820). • Average cluster size (or morphology): DBP absorption is used (ASTM D2414); TEM can be used for a more precise (but also more tedious) measure of cluster and elementary particle sizes. • Porosity: ideally through mercury absorption, to be preferred to more common oil absorption techniques. • pH: silica are normally slightly acidic (5.5–7.0). • Purity: most synthetic silica contain more than 98% silicon dioxide, with minute impurities such as Fe2O3 and water soluble Na2O, plus some other elements depending on the process and or the starting material. SiO2 content is determined gravimetrically by fuming off with hydrofluoric acid. The sulphate and chloride contents are determined by potentiometric titration. Minutes impurities are assessed using analytical techniques such as atomic absorption or ion chromatography. • Specific gravity and tamped density: the specific gravity of silica is approximately 2 g/cm3 but it is the tamped density that is the first industrial concern with respect to material handling, health and safety of workers. The tamped density (also called bulk density) is calculated from the weight of a sample with respect to a given

63

Typical Fillers for Polymers

Table 4.9 Typical Property Ranges for Synthetic Silica Characteristic Particle size (μm) N2 specific surface area (m2/g) Moisture content at 105°C, 1 h (max%) Ignition loss at 10000C for 1 h (max%) Bulk density (g/cm3) pH 5% Aqueous suspension Water absorption (%) Oil absorption (%) Residue on 325 mesh, wet sieving (max%) Soluble salt (%max) R2O3 (max%) SiO2 (max%) Specific gravity (g/cm3) Refractive index

Precipitated Silica 10–12 110–240 5 10 0.08–0.40 6.5–7.2 175–250 180–270 0.3 0.5 0.3 89 1.95 1.46

Fumed Silica 0.007–0.3 100–400 0.5–2.0 5–6 0.05–0.2 3.7–4.8 – 250–320 0.5 – – 99.8 2.0 –

volume, and is reported in g/dm3 or in g/cm3. Final process treatments, such as grinding, milling, and micronisation allow to somewhat control the tamped density: • • • •

Between 0.26 and 0.36 g/cm3 for so-called micropearl silica Between 0.1 and 0.3 g/cm3 for powder silica Between 0.26 and 0.36 g/cm3 for milled silica Between 0.05 and 0.1 g/cm3 for micronised silica

• Moisture and ignition loss: depending on the manufacturing process, silica contain both physically and chemically bound water (obviously more for precipitated silica). The physically bound water, generally referred as “drying/moisture loss,” is released by heating to 105°C for two hours (usually in the range of 5–8 wt%) whereas dehydration of surface silanol groups occurs at high temperatures (between 200 and 1000°C). Bonded water in the form of silanol groups, generally referred as “ignition loss,” is removed after two hours of ignition time at 1000°C. The ignition loss is determined on samples that are previously dried for two hours at 105°C. With respect to their respective moisture content, precipitated and fumed silicas are also referred by some authors as hydrated and anhydrous silicas, respectively. In addition to the above primary physical characteristics, there is a trend (at least in the R&D literature) to characterize the surface activity of silica in terms of specific components of the adsorption energies of chemicals with

64

Filled Polymers

known polarities. IGC techniques are used (see Section 4.1.5) but the information is available only for a few commercial grades (see below) Manufacturers are practically the only source of characterization data for (commercial) silica grades but how the structure of silica affect the properties of polymer compounds remains poorly documented, essentially because few test results can be considered for analysis. Amongst the various reasons that can be offered for this situation, there is surely the fact that, compared with carbon black, there are no simple and reliable test methods for directly characterizing the structure of silica. In addition there is nearly no availability for (fumed or precipitated) silica with similar surface areas but different structures. Table 4.10 gives typical data as provided by several suppliers of silica. In their technical data sheets, nearly all manufacturers describe their products essentially with respect to the specific surface area assessed using the BET method, the percent weight loss after heating for 1–2 hours at 105°C, the pH of a 5% suspension in water and the product form. Most producers also specify the SiO2 content, generally above 98% for the precipitated silica and above 99.8 for the fumed silica; a few ones give the tapped density and typical particle dimensions. In 1976, with respect to the already established ASTM classification for carbon blacks; Wagner47 suggested a letter/number classification schema for silica with a letter, either “A” or “H” to reflect the production method, the former for fumed silicas because they are relatively “anhydrous,” the latter for precipitated silicas, which have around 5% bound and adsorbed water. Silicas prepared through other processes would receive the heading letter “C.” A first digit would refer to the elementary particle size group and two last digit would be arbitrarily assigned but with respect to the complexity of the structure. This very clever proposal by Wagner has unfortunately not (yet) been adopted, despite its obvious advantages. It must be noted however that, thanks to their very rich surface chemistry, silica are prone to relatively easy chemical modification, either using the vast possibilities of silane chemistry or more generally all the condensation reactions that organic chemistry can offer with hydroxyl groups. Quite a large number of so-called “modified” silicas (and silicates) are consequently available, not all however for polymer applications, but whose detailed description is outside the scope of the present book. Chemically modifying silica has a cost however, sometimes excessive when organic solvents have to be eliminated. Therefore, we will be pay attention to modified silica only in the (rare) case of real advantages in significant industrial applications. In rubber technology, when used in nonpolar elastomers, silica must be treated with a silane, but despite early attention paid for presilanated products, the massive use of silica (in tire technology) developed when it became practically feasible to perform in situ silanisation of silica, i.e., during the compounding operations, essentially in internal mixers. Such a development was a tremendous challenge over the last quarter of the twentieth century, nowadays globally achieved through a combination of sound engineering pragmatism and advanced scientific research, as we will describe in Chapter 6.

Ultrasil

Rubbersil

Vulkasil

Glassven (Venezuela)

Lanxess (Germany)

Product Name

1. Precipitated Silica Evonik [ex Degussa] (Germany)

Producer

Silica, Suppliers’ Data

Table 4.10

880 360 AS7 VN2 VN2 GR 7000 GR VN3 VN3 GR 7005 RS-50 RS-250 RS-200 RS-150 RS-120 C S S/KG N

Grade

68 ± 22 200 ± 10 180 ± 10 150 ± 10 120 ± 10 50 ± 10 175 ± 20 175 ± 20 125 ± 20

35 50 60 125 125 175 175 175 185

N2 Spec. Area (m2/g)

9.0 ± 0.7 6.2 ± 0.8 6.2 ± 0.8 6.9 ± 0.7

10.5 9 11.5 6.9 6.9 6.8 6.2 6.2 6 6.0–9.0 6.0–6.8 6.0–6.8 6.5–7.3 6.5–7.3

pH

5.5 ± 1.5 5.5 ± 1.5 5.5 ± 1.5 5.5 ± 1.5

5.5 5.5 5.5 5.5 5.5 6.5 5.5 5.5 5 n.a. n.a. n.a. n.a. n.a.

Heating Loss (%)

Powder Powder Powder Powder Granules Granules Powder Granules Spherical particles Powder Powder Powder Powder Powder Powder Powder Granules Powder (continued)

Product Form

Typical Fillers for Polymers 65

Cabot (USA)

2. Fumed silica

Rhodia (France)

Producer

Cab-O-sil

Zeosil

Product Name

Table 4.10  (Continued)

200

M-5P M-7D PTG MS-75D MS-55 255 ± 25

198 ± 2 200 255

100 ± 15 130 ± 15 155 ± 5 160 200 200 200

110 ± 15 110 ± 15 160 ± 20 170 ± 20 180 ± 20 220 ± 10

175

N2 spec. Area (m2/g)

L-90 LM-130 LM-150 LM-150D HP-60 M-5 M-5DP

175 GR 1115MP 115 GR 1165 MP 165 GR 195 GR 215 GR

Grade

pH

3.7–4.3 3.7–4.3 3.7–4.3 3.8–4.3 3.7–4.3

3.7–4.3 3.7–4.3 3.7–4.3 3.7–4.3 3.8–4.3 3.7–4.3

6.8 6–7 6–7 6–7 6–7 5.5–7 6–7

<1.0 <1.5 <1.5 <1.5 1.5

<0.5 <1.0 <1.5 <1.0

0.5 1

6 ~8 ~8 ~8 ~8 ~8 ~9

Heating Loss (%)

Product Form

Powder Densed powder Powder Densed powder Powder

Powder Powder Powder Densed powder Powder Powder Powder

Granules Micropearls Granules Micropearls Granules Granules Granules

66 Filled Polymers

Evonik [ex Degussa] (Germany)

Aerosil

H-5 HS-5 EH-5 130 130V 150 150V 200 300 300 SP 380 400 ± 20 130 ± 25 130 ± 25 150 ± 15 150 ± 15 200 ± 25 300 ± 30 300 ± 30 380 ± 30

300 325

3.7–4.3 3.7–4.3 3.7–4.3 3.7–4.7 3.7–4.7 3.7–4.7 3.7–4.7 3.7–4.7 3.7–4.7 3.7–4.7 3.7–4.7

<1.5 <1.5 <1.5 <1.5 <1 <1.5 <1.5 <1.5 <1.5 <1.5 <2 Powder Densed powder Powder Powder Densed powder Powder Densed powder Powder Powder Structure modified Powder

Typical Fillers for Polymers 67

68

Filled Polymers

4.2.4  Surface Energy Aspects of Silica The surface chemistry of silica is richer and more clearly defined than that of carbon black. As illustrated in Figure 4.17, silica surface is occupied by sizable quantities of siloxane and silanol groups, giving rise to hydrogen interaction with either “free” or “bound” water. Free water is easily removed by drying at 105–250°C, while bound water is only released at 900–1000°C and results in fact from the condensation of vicinal silanol groups. An important consequence of such an oxygen rich surface chemistry is that strong inter-particles interactions occur through hydrogen bonding. In nonpolar polymers, this give rise to a poor dispersibility and therefore special compounding procedures are needed. But suitable chemical promoters, e.g., silanes, ­easily form covalent bonding and on one hand make mixing easier and on the other hand participate in the reinforcing capabilities of the filler. Of course, polar polymers and notably certain specialty elastomers such as polydimethylsiloxanes, naturally interact with silica, the latter thanks to their similar chemistry. Such a chemistry reflects in the surface energy as it can be estimated through IGC, whose results can be conveniently expressed in terms of disp d persive γ s and polar γ s components. As we have seen, carbon blacks exhibit a high dispersive component, in the 100–400 mJ/m2 range (at 150°C), actually proportional to their specific surface area, and a relatively low polar component, nearly constant, whatever the grade. On the reverse, reinforcing silica grades have a lower dispersive component (in the 20–60 mJ/m2 at 150°C), but a higher γ sp . This explains why filler–filler interaction dominates in silica H

O

H

O

O

Si

Si

Si

O

O O O O

O

O

Vicinal silanols

OH Si

OH Si

OH OH Si

O

OH

O

OH

Si

O O O O O O O Free silanol Silanediol Silanetriol => hydrogen bonding (rare, if any) between particles

Si

Si O

O

O

Surface density : 2 to 8 OH per nm2 (depending on specific area and production process) Strong reactivity with water (moisture) : H H H O O +H O H H H H 2 H H H H +H2O H O O O O O O O -H2O -H2O Si Si Si Si Si Si Si Si Si O O O O O

O O O O O

Figure 4.17 Surface chemistry of silica.

O O O O OO O O

O

Siloxane

O Si

O O OO OO O O

Typical Fillers for Polymers

69

filled compounds, thus leading to the formation of silica networks and the associated difficulties in dispersing such a material in nonpolar polymers. Dispersive and polar components would therefore appear as a key information regarding the surface activity of silica, unfortunately available only through research literature on a few commercial grades, and with differences in tests conditions as used by the authors. Table 4.11 gives published data on commercial silica grades, as compiled in the literature. d Anhydrous, fumed, silicas have a higher γ s than hydrated, precipitated silicas and, moreover, the lower the specific surface area, the lower the dispersive component, irrespective of the production process. According to Wang et al.,25 the difference in the dispersive component between both types of synthetic silica might depend on the surface topology of their particles, and less on the concentration of hydroxyl groups. In contrast with carbon black, the primary silica particles (that cluster in “string of pearls” or other tri-dimensional complex structure) is the smallest dispersible unit and, therefore, the likely reinforcing entity. Strongly clustered elementary silica spherical particles can be separated only when exerting dispersive forces larger than the sum of all interparticle hydrogen bonds. Indeed, owing to their silanol rich surface, silica particles easily assemble through hydrogen bonding to form clusters, leading eventually to a relatively soft network, detrimental to reinforcement. Therefore one needs to chemically promote the rubber–filler interaction by using bi-functional chemicals and in the meantime to shield the surface of silica particles in order to decrease interparticle interactions. Such bi-functional promoters, on one side, react with silanol groups that densely cover the surface of silica particles and, on the other side, will undergo covalent bonding with rubber chains during vulcanization. When compared with carbon black, the reinforcement with silica proceeds thus from a totally different concept, with a strong role played by chemistry.

4.3  Short Synthetic Fibers There are many fibers of synthetic origin, most of them spun or extrusiondrawn semicrystalline polymers (e.g., polyolefins, polyamides, polyesters), which are not used as fillers for polymers. The main reason is that, for a material to play a potential reinforcement role as a filler for polymer, large differences in certain key properties must exist between the filler and the polymer matrix. It follows that only three types (or classes) of fibrous products can be considered as valid short synthetic fibers candidates for polymer reinforcement: glass fibers, carbon fibers and aramid fibers. Glass fibers of control diameter (in the 10 µm range) are produced by melt spinning techniques, essentially by pushing a molten glass of appropriate

p p p p f f f f f f f f f f f f f f f f f

Type

134 134 181 181 128 128 189 189 300 90 122 154 167 183 196 209 216 258 285 407 417

N2 (BET) Specific Area (m²/g) 21.3 22.9 27.5 34.3 27.3 30.7 40.9 44.3 55.0 35.0 42.0 59.0 58.5 58.0 56.0 55.5 59.0 80.0 81.0 86.0 94.5

Dispersive Component γ ds (mJ/m2) 73.1 64.0 79.6 71.9 48.1 45.4 57.4 55.1 51.0 127.0 120.0 144.0 142.0 147.0 149.0 144.0 149.0 150.0 154.0 167.0 157.0

Polar Component γ (mJ/m2) p s

* * * * * * * * * ** ** ** ** ** ** ** ** ** ** ** **

IGC Test Conditions

* At 150°C with heptane for dispersive component; benzene for polar component. ** At 120°C with an alkane for dispersive component; THF for polar component.   Ultrasil and Aerosil are Evonik trade marks for precipitated and fumed silica; Cab-O-Sil is Cabot trade mark for fumed silica.

®

Ultrasil VN2 Ultrasil® VN2 Ultrasil® VN3 Ultrasil® VN3 Aerosil® 130 Aerosil® 130 Aerosil® 200 Aerosil® 200 Aerosil® 300 Cab-O-Sil® L90 Cab-O-Sil® LM130 Cab-O-Sil® LM150D-T Cab-O-Sil® LM150 Cab-O-Sil® LM150D-B Cab-O-Sil® M7D Cab-O-Sil® M5 Cab-O-Sil® HP60 Cab-O-Sil® M75D Cab-O-Sil® HS5 Cab-O-Sil® S17D Cab-O-Sil® EH5

Silica grade

Surface Energy Components for Silica

Table 4.11

25 29 25 29 25 29 25 29 28 56 56 56 56 56 56 56 56 56 56 56 56

Source

70 Filled Polymers

Typical Fillers for Polymers

71

composition through a platinum crown with small holes. Continuous fibers with diameters in the 5–24 µm range are drawn, cooled down and chopped (cut) to the desired length. Depending on the composition, various type of glass fibers are produced: E-glass (i.e., electrical grade glass), S-glass (high stiffness), C-glass (chemical grade, for its best resistance to chemical attack), R-glass (high strength, special grades, mainly for aerospace glass-reinforced applications). E-glass is an alumino-borosilicate, low alkali composition (SiO2, 54%; Al2O3, 14%; CaO + MgO, 20–22%; B2O3, 10%; Na2O + K2O, < 2%), which gives excellent fiber forming capabilities. It is therefore a high ­production rate, low cost grade, almost exclusively used as reinforcing material. S-glass (SiO2, 65 wt%;.Al2O3, 25 wt%; MgO,10 wt%) is a higher stiffness grade, more difficult to produce than E-glass and therefore more expensive. It is used only when high mechanical properties are required. C-glass (SiO2, 64.6%; Al2O3 Fe2O3, 4.1% CaO, 13.4%; MgO, 3.3%; Na2O·K2O, 9.6%; B2O3 BaO, 5.0%) is also a fiber reinforcement grade, made and applied specifically for high chemical resistance. The most frequently used short glass fibers (SGF) for filling (thermoplastic) polymers are E-glass and S-glass grades. Such grades have typical diameters of 10 µm, with length in the 1–10 mm range; fiber aspect ratios are thus in the 100–1000 range. Carbon fibers combine very high tensile properties (strength and modulus) with low weight. As the high tensile strength is maintained up to extremely high temperatures, carbon fibers are ideal materials for lightweight structures in highly demanding applications, e.g., in aerospace. It can be considered that Thomas Edison, in the late nineteenth century, was the first to produce carbon fibers as filament for incandescent lamps. But the first commercial carbon fibers were not really produced until the early 1960s.48 The basic principle is first to manufacture a carbon rich filament, then to carbonize it at very high temperature (above 2000°C) under controlled atmosphere conditions, in order to eliminate noncarbon atoms. A number of different processing techniques as well as different precursor routes were developed, but the most important among them are polyacrylonitrile (PAN-fibers) and mesophase-pitch-precursor (MPP-fibers). Carbon fiber filaments have diameters in the 5–10 µm range but are produced as yarns of several thousand filaments. Short carbon fibers can be produced by chopping with fibers aspect ratio in the 100–1000 range. Aramid fibers, best known under the commercial trade name Kevlar®, are polypara-phenylene terephtalamide fibers first commercialized by Dupont in 1971, that combine high strength and flexibility with light weight. A similar product with roughly the same chemical structure was introduced by Akzo in 1978 under brand name Twaron® and is now manufactured by Teijin. Spun as long fibers, typically yarns of 700–1000 filaments of around 12 µm diameter, Kevlar was first used as a replacement for steel wires in racing tires, then was used to make tissues and fabric sheets that are nowadays used as such or as a component in multilayer composites. Short aramid fibers are produced either as pulp (a highly fibrillated form), or as floc (precision

72

Filled Polymers

cut short fibers less than 1 mm in length), or as staple (precision cut short fibers of 7–10 mm length), to be used as a fibrous filler for polymers. Short aramid filaments are thus available with fiber aspect ratios in the 80–800 range. Table 4.12 gives a few typical properties of synthetic fibers, to be compared with similar data on either thermoplastics or vulcanized elastomers (for instance tensile strength in the 0.06–0.09 GPa and 0.02–0.05 GPa ranges, respectively). Polyamides, polyesters, and HS polyethylene fibers are quoted for comparison only, since their relatively low melting points limit their use in most polymers. Aramid fibers are interesting in this respect since such materials do not melt but start to decompose above 420°C. Short synthetic fibers are used as reinforcing materials in general ­purposes (e.g., PP, PE) and technical thermoplastics (e.g., PBT, PA66, PA6/6T). Current loadings are up to 30–40 wt%. Basically the reinforcing effect results from the large difference in tensile properties between fibers and polymers, and with respect to the average tensile strength and modulus of most thermoplastics and (vulcanized) elastomers, short glass, aramid, and carbon fibers are the most interesting synthetic materials for polymer reinforcement (Figure 4.18). Good adhesion between the polymer matrix and the fibers is generally required, and is usually obtained through a chemical process: fibers are treated with chemical agents that promote fiber-to-polymer adhesion. However improvements in mechanical properties are also markedly depending on fibers’ orientation, with respect to the principal strain direction, as we shall see in detail later.

4.4  Short Fibers of Natural Origin Natural fibers are structures of biological origin, which essentially consist in cellulose, hemicellulose and lignin, and smaller quantities of extractible, ­proteins and other inorganic elements. The exact composition depends of course of the very origin (i.e., plant, crop, tree) and the treatment. Depending of the origin, one may distinguish seed fibers (cotton, kapok), leaf fibers (sisal, agave), fruit fibers (coconut), stalk fibers (bamboo, grass, wood flour). From a biological point of view, there are many classes of natural fibers but not all are of interest as fillers for polymers. So-called hard fibers account for approximately 90% of the world production, but only a small proportion is used to manufacture composites with polymers. With respect to the scope of the book, we pay attention only to short natural fibers, i.e., either chopped long fibers or flour obtained through the mechanical reduction of bulk natural products, e.g., wood to their fibrous component. A number of short natural fibers used to prepare polymer composites are byproducts of various industries (e.g., textile, wood, etc.). Properly dispersed in a

2.54–2.60

2.48–2.49

2.62–2.63

1.75–2.10

1.44–1.45

7.65–7.80

1.10–1.16

1.37–1.38

0.96–0.97

S-glass

C-glass

Carbon

Aramids

Steel wire

Polyamides

Polyesters

High strength PE

Specific Gravity (g/cm3)

E-glass

Fiber Type

2.6

0.6–1.2

0.7–1.0

2.0–2.8

2.9–3.8

1.9–5.7

3.1–3.8

4.4–4.8

2.0–3.8

Tensile Strength (GPa)

117

14

6–7

197–200

60–136

240–531

80–81

88–91

73–81

Tensile Modulus (GPa)

3.5

14.5–23.0

18.3–26.0

2

2.4–3.6

0.3–2.1

4.6–4.6

5.4–5.8

4.0–4.9

Elongation at Break (%)

0.38

0.35

0.33

0.28–0.30

0.36

0.23–0.28

0.24–0.27

0.22–0.27

0.17–0.27

Poisson’s Ratio (v)

60

150

120

300–400

300–310

500–600

350–400

350–400

350–400

Maximum Working Temperature (°C)

Typical Properties of (Short) Synthetic Fibers; Compiled from Various Sources (Manufacturers’ Data Sheets, Scientific Publications)

Table 4.12

Typical Fillers for Polymers 73

74

Filled Polymers

30 20 20

Ultra-drawn PE

20 Specific tensile strength, 106 cm

20

Aramids

20

S-glass

18 16

C-glass

14 12

Carbon

E-glass

10 Polyamides

8

Polyesters

6 4

Steel wires

2 0

0

2

4

6 8 10 12 14 16 Specific modulus, 108 cm

Graphite (high modulus)

18

20

22

Figure 4.18 Considering synthetic fibers in terms of specific tensile strength and modulus; specific properties are obtained by dividing the tensile property by the product (density times the gravitation constant g = 9.807 m/s2).

(molten) polymer, such fibers can be considered as fillers for polymers and the resulting mixtures as polymer composites such that, during processing operations, both the fibers and the polymer matrix will flow together. It follows that polymer based products such medium-density fiberboard (MDF, or Craftwood), whilst made with a resin binder, are excluded from our subject. The cellulose/hemicelluloses/lignin ratio exhibits large variation in natural fibers, depending on the vegetal source, but all fibers can be viewed as multilayer composites of a rigid cellulose structure embedded in a lignin matrix, with covalent bonding between both.49 Cellulose is largely crystalline and organized in microfibrils with typical diameters in the 10–30 nm range, with a supramolecular helicoidal structure that gives it an elasticity modulus of around 136 GPa, i.e., twice the value of glass fiber. Hydrogen bonding along and between cellulosic chains plays a role in the mechanical

Typical Fillers for Polymers

75

properties but gives rise also the large variability of physico-chemical properties of natural fibers. Misnamed hemicellulose is in fact a group of polysaccharides (five to six carbon ring sugars) with a large degree of branching and therefore an amorphous nature. Very hydrophilic, it forms the supportive matrix in which cellulose microfibrils are embedded. Lignin is a complex hydrocarbon with both aliphatic and aromatic components of very high molecular weight. Totally amorphous and hydrophobic, lignin provides rigidity to plants and hence natural fibers. Up to five hydroxyl and methoxyl groups, plus a few carbonyls have been identified in lignin, making it soluble in hot alkali, easily oxidized and very reactive with phenols The initial cellulose/lignin ratio can be changed through suitable (chemical) treatments and both cellulose and lignin are organic polymers, the latter being more hydrophobic (because of its aromatic nature) than the former. Generally, tensile strength and modulus of natural fibers increase with increasing cellulose content. Thermal degradation is a limiting factor in using natural fibers as fillers for polymers, as most fibers start loosing stiffness at around 160°C; lignin starts degrading at around 200°C (with oxidative conditions accelerating the process) but destruction of cellulose crystalline order occurs only above 300°C.50 Thermal conductivity of natural fibers is weak, of the order of 0.12–0.22 W/m.K, again a limiting factor for polymer-like processes. And all fibers, not only exhibit dimensional variations when exposed to water but tend also to retain moisture, with associated variations in thermal and mechanical properties. All those factors severely limit the use of natural fibers as fillers for polymers and, as a rule of thumb, all polymers whose melting range is above 180°C are ill suited for preparing natural fibers filled composites. Table 4.13 gives some typical characteristics of a few selected natural fibers of interest for applications in polymer composites49,51–54 Most natural fibers need special preparation processes before they can be used in polymer composites. First, after a sufficient drying of the raw material, there is a mechanical or thermo-mechanical defibrillation step (or grinding in the case of wood flour), followed by chopping to the (average) desired length. Then sorting and screening are made, in order to remove useless solid contaminants, for instance minerals and dried proteinic residues, and to achieve a somewhat controlled fibers size distribution. Depending on the fiber, such operations can be more or less complicated and, of course, involve some degradation, namely in what the lignin part is concerned. Natural fibers often suffer from a large degree of nonuniformity in their characteristics: chemical composition, crystallinity, surface properties, diameter, cross-sectional shape, length, strength, and stiffness.51 Another important issue in using natural fibers in polymer composite is moisture absorption, which generally degrades mechanical properties like tensile strength. Except wood flour in certain parts of the world, were sufficient market developments have been achieved (for instance in USA and Canada for so-called WPCs), constant and guaranteed material specifications for

1.5–5 100–125 10–60 4–60 5–40 >60 2.5–4.5 2.5–3.0 1.0–1.3 1.0–1.3 3–9.105  *

Length [Typical] (mm) 5–25 100–400 12–25 12–30 20–50 25–200 14–33 – – – 10

Diameter (µm)

* Typical length of the thread of raw silk from one cocoon.

Jute Sisal Cotton Flax Hemp Abaca (Manila hemp) Kenaf (close to jute) Pine (flour) Birch (flour) Beech (flour) Silk (Bombyx Mori)

Fiber

Natural Fibers as Potential Fillers for Polymers

Table 4.13

60–70 65 85–90 71 68 52–63 72 – – –

Cellulose (%wt) 14–20 12 5.7 19–20 15 20–25 20 – – – –

Hemicellulose (%wt) 20–55 9–22 6–12 28–70 30–70 12–35 53–60 11.3–11.5 14.5–14.7 13.6–13.8 0.65–0.75

Tensile Modulus (GPa)

1.5–2.0 2–14 6–8 1.3–3.3 1.7–2.7 2.1–2.4 1.7–2.1 n.a. n.a. n.a. 18–20

Elongation at Break (%)

76 Filled Polymers

Typical Fillers for Polymers

77

natural fibers are still a serious obstacle to sizeable industrial applications, despite encouraging progresses in line with concerns about sustainable development. There are many parameters which may affect the performance of a given fiber in a given polymer, and with respect to extremely large variety of possible fiber-reinforced polymer systems, an exhaustive coverage is nearly impossible, despite the already considerable research literature on the subject. Only a few basic trends can be identified. In addition to their own properties and whatever is the polymer (system) considered, all natural fibers must be properly dispersed in the matrix and adequate fiber-polymer interfacial effects must be achieved. Indeed undispersed fiber bundles are obviously failure initiation sites and a basic principle for fiber reinforcement is that effective load transfers occur at the fiber-matrix interface, as clearly demonstrated through micromechanical considerations (as will be detailed in Chapter 7). It is quite logical to expect that both fiber dispersion and fiber-matrix interfacial effects can be improved through suitable chemical means. Simple considerations about the basic chemical aspects of natural fibers help understanding where and how improvements can be expected when either chemically modifying natural fibers or using socalled bonding systems. As mentioned above, from a chemical point of view, cellulose, hemicellulose, and lignin are, in varied proportions, the main components of ­natural fibers, with sometimes minor quantities of heteropolysaccharides (e.g., pectins), waxes (natural polyalcohols), and other chemicals. The three main components have hydroxyl groups with varied degree of accessibility or chemical reactivity that conditions their basic chemical properties and hence the possibility for chemical modification or interaction. Cellulose has strong microcrystalline regions, enhanced by internal hydrogen bonding; is resistant to strong alkali, relatively resistant to oxidizing agents but easily hydrolyzed by acids. Hemicellulose is noncrystalline by nature, is very hydrophilic, soluble in alkali and easily hydrolyzed by acids. Lignin is totally amorphous and hydrophobic, cannot be hydrolyzed by acids, but is soluble in hot alkali, easily oxidized and very reactive with a number of organic chemicals, for instance phenols. It is quite clear that the chemical virtues of the main components of natural fibers dictate the possible chemical modifications and/or the choice of bonding systems. As recently reviewed by Jacob John and Anandjiwala,49 there has been a considerable research during the past decades with the objective to understand and somewhat optimize the fiber–polymer interaction through chemical means. Only a few common techniques will be screened hereafter. There are essentially two approaches that can be used to chemically promote or enhance interactions between natural fibers and polymer matrix, either through a modification of the fiber surface by a suitable pretreatment or through the use of a suitable bonding system during the compounding operations. A priori, the latter approach would be more economical than

78

Filled Polymers

Table 4.14 Chemical Approaches to Promote Natural Fiber–Polymer Matrix Interactions Fiber Surface Modifications (Pretreatment) Organic solvent extraction Steam treatment Alkali treatment (mercerization) under various conditions (temperature, pressure, steam,…) Esterification with various anhydrides (acetic, butyric, methacrylic, etc)

Acetylation, propionylation, benzoylation, stearation, cyanoethylation,… Grafting with isocyanate derivatives    with acrylates or polyacrylic acids    with acrylonitrile

Silanation (various organosilanes used); most reactions occurring above 70°C Peroxide treatment (benzoyl peroxyde) Latex coating (Helium) plasma, γ-irradiation

Bonding Systems (When Compounding) Maleic anhydride grafted polymers    MA-g-Polypropylene    MA-g-Polyethylene    MA-g-Polystyrene    MA-g-Poly(ε-caprolactone) Resins systems : Phenol-formaldehyde Resorcinol-formaldehyde Resorcinol + hexamethylenetetramine SiO2 + resorcinol + hexamethylenetetramine Commercial adhesive systems based on acrylics, epoxydes, urethanes or cyanocrylates technologies Bi-functional silanes: bis(triethoxysilylpropyl)tetrasulfane (TESPT) g-methacryloxy-propyl trimethoxy silane, etc. Thiols e.g., 3-(trimethoxysiliyl)-1-propanethiol Titanate based systems Isocyanates, diisocyanates

Enzymatic treatment, fungal modifications

the former, and achievable during compounding operations. Table 4.14 describes several possibilities that have been frequently studied with certain types of fibers, while only a few ones have apparently lead to sizeable industrial usages. Various results have been reported using these approaches, certain giving significant improvement in mechanical properties, others giving very little benefits, if any. It is fairly clear from Table 4.14 that, when chemical reactions are expected in using these various approaches, the main target are the hydroxyl groups of (amorphous) cellulose, hemicellulose or lignin in order to render the fiber surface more hydrophobic. SEM microphotographs of fractured specimen surfaces are often used to demonstrate fiber–polymer bonding but, in the author’s opinion, such “evidence” could also at best suit enhanced wetting of the fibers by the matrix, without necessarily the occurrence of chemical (i.e., covalent) bonding. The actual role of the fiber–­matrix interface in improving certain mechanical properties remains however hardly understood.

Typical Fillers for Polymers

79

References



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80

Filled Polymers

19. A.I. Medalia. Effective degree of immobilization of rubber occluded within carbon black aggregates. Rubb. Chem. Technol., 45, 1171–1194, 1972. 20. P. Bertrand, L;T; Weng. Carbon black surface characterization by ToF-SIMS and XPS. Rubb. Chem. Technol., 72, 384–397, 1999. 21. A.M. Gessler, W.M. Hess, A.I. Medalia. Reinforcement of elastomers with carbon black. Part IV. Interaction between carbon black and polymer. Plast. Rubb. Proc., 3, 141–156, 1978. 22. E.M. Dannenberg. Primary structure and surface properties of carbon black— Part 1: a study of four ISAF type carbon blacks of varying structure. Rubber Age, 98, 82–90, 1966. 23. E.M. Dannenberg. Primary structure and surface properties of carbon black— Part 2 : properties of commercial carbon blacks of varying structure and particle size. Rubber Age, 99, 81–85, 1966. 24. A.K. Manna, P.P. De, D.K. Tripathy, S.K. De, M.K. Chatterjee. Effect of ­surface oxidation of carbon black on its bonding with epoxidized natural rubber in the presence of silane coupling agent. Rubb. Chem. Technol., 72, 398–409, 1999. 25. M.J. Wang, S. Wolff, J.B. Donnet. Filler-elastomer interactions. Part I: silica surface energies and interactions with model compounds. Rubb. Chem. Technol., 64, 559–576, 1991. 26. C. Saint-Flour, E. Papirer. Gas-solid chromatography : a quick method of estimating surface free energy variation induced by the treatment of short glass fibers. J. Colloid Interface Sci., 91, 69–75, 1983. 27. J.R. Conder, C.L. Young. Physico-chemical Measurements by Chromatography. J. Wiley & Sons, New York, NY, 1979. 28. E. Papirer, H. Balard, A. Vidal. Inverse gas chromatography: a valuable method for the surface characterization of fillers for polymers (glass fibres and silicas). Eur. Polym. J., 24 (8), 783–790, 1988. 29. M.J. Wang, S. Wolff, J.B. Donnet. Filler-elastomer interactions. Part III: carbon black surface energies and interactions with elastomer analogs. Rubb. Chem. Technol., 64, 714–736, 1991. 30. H. Darmstadt, C. Roy, S. Kalliaguine, H. Cormier. Surface energy of commercial and pyrolytic carbon blacks by inverse gas chromatography. Rubb. Chem Technol., 70, 759–768, 1997. 31. H. Darmstadt, N.Z. Cao, D.M. Pantea, C. Roy, L. Sümmchen, U. Roland, J.B. Donnet, T.K. Wang, C.H. Peng, P.J. Donnelly. Surface activity and chemistry of thermal carbon blacks. Rubb. Chem. Technol., 73, 293–309, 2000. 32. A. Schröder, M. Klüppel, R.H. Schuster. Oberflächenaktivität von Furnacerußen. I. Bestimmung der Oberflächenrauheit mittels statischer Gasadsorption, Monolagenbereich. Kautsch. Gummi. Kunstst., 52, 814–822, 1999. 33. A. Schröder. Charakterisierung verschiedener Rußtypen durch systematische statische Gasadsorption; Energetische Heterogenität und Fraktalität der Partikeloberfläche. PhD Thesis, University of Hannover, Germany, 2000. 34. A. Schröder, M. Klüppel, R.H. Schuster. Characterisierung der Oberflächenaktivität. II. Bestimmung der Oberflächenrauheit von Furnacerußen mittels statischer Gasadsorption, Multischichtenbereich. Kautsch. Gummi. Kunstst., 53, 257–265, 2000. 35. A. Schröder, M. Klüppel, R.H. Schuster, J. Heidberg. Energetic surface heterogeneity of carbon black. Kautsch. Gummi. Kunstst., 54, 260–266, 2001.

Typical Fillers for Polymers

81

36. T. Zerda, W. Xu, H. Yang, M. Gerspacher. The effects of heating and cooling rate on the structure of carbon black particles. Rubb. Chem. Technol., 71, 26–37, 1998. 37. M. Gerspacher, C.P.O’Farrell. Tire compound materials interactions. Kautsch. Gummi. Kunstst., 54, 153–158, 2001. 38. A.L. Barabasi, M. Araujo, H.E. Stanley. Three-dimensional Toom model: connection to the anisotropic Kardar-Parisi-Zhang equation. Phys. Rev. Lett., 68, 3729–3732, 1992. 39. G. Heinrich, M. Klüppel. A hypothetical mechanism of carbon black formation based on molecular ballistic deposition. Kautsch. Gummi. Kunstst., 44, 419–423, 1991. 40. M. Pontier-Johnson. Noir de carbone au four : mécanismes de formation des particules. PhD Thesis, University of Haute Alsace, Mulhouse, France, 1998. 41. M. Pontier-Johnson, J.B. Donnet, T.K. Wang, C.C. Wang, R.W. Locke, B.E. Brinson, T. Marriott. A dynamic continuum of nanostructured carbon in the combustion furnace. Carbon, 40 (2), 189–194, 2002. 42. A. Vidal, S.Z. Hao, J.B. Donnet. Modification of carbon black surfaces—effects on elastomer reinforcement. Kautsch. Gummi. Kunstst., 54, 159–165, 2001. 43. P. Liu. Polymer modified clay minerals: A review. Appl. Clay Sci., 38, 64–76, 2007. 44. M.R. Mushack, A.W.Backmann. Neuburg silica: a natural functional filler. Intern. Polym. Sci. Technol., 23 (9), 5–10, 1996. 45. A.I. Medalia. Filler aggregates and their effects on dynamic properties of rubber vulcanizates. (See Figure 2, p. 63). In International CNRS Colloquium, Le Bischenberg-Obernai, France, Sept. 24–26, 1973. CNRS, Paris, France, 1975. ISBN 2-222- 01749-1. 46. D. Göritz. Carbon Black und Silica : Gemeinsamkeiten und Unterschiede. 8th Fall Rubber Colloquium, DIK, Hannover, Germany, Nov. 26–28, 2008. Communication nr 55. 47. M.P. Wagner. Reinforcing silicas and silicates. Rubb. Chem. Technol., 49, 703–774, 1976. 48. R.J. Young, R.J. Day, M. Zakikhani. The structure and deformation behaviour of Poly(p-phenylene benzobisoxazole) fibres. J. Mater. Sci., 25 (1A), 127–136, 1990. 49. M. Jacob John, R.D. Anandjiwala. Recent dvelopments in chemical modification and characterization of natural fiber-reinforced composites. Polym. Compos., 29 (2), 187–207, 2008. 50. D-Y Kim, Y. Nishiyama, M. Wada, S. Kuga , T. Okano. Thermal decomposition of cellulose crystallites in wood. Holzforschung, 55 (5), 521–524, 2001. 51. A.K Bledzki, J.Gassan. Composites reinforced with cellulose based fibres. Prog. Polym. Sci., 24, 221–274, 1999. 52. R.M. Rowell, R.A Young, J.K. Rowell. Paper and Composites from Agro-based Resources, R.M. Rowell, Ed. Lewis Publishers, Boca Raton, FL, 1997. 53. P. Jodin. wood: Engineering material. Arbolor, Nancy, France, 1994. ISBN-10: 2907086073. 54. E. Spaˉrnin¸ š. Mechanical properties of flax fibers and their composites. Licentiate Thesis, Luleå Univ. Technol., Sweden, 2006. 55. M.J. Wang. Effect of filler-elastomer interaction on tire tread performance. Part III. Kautsch. Gummi, Kunstst., 61 (4), 159–165, 2008. 56. M.J. Wang, M.D. Morris, Y. Kutsovsky. Effect of fumed silica surface area on silicone rubber reinforcement Kautsch. Gummi, Kunstst., 61 (3), 107–117, 2008.

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Filled Polymers

Appendix 4

A4.1  Carbon Black Data A4.1.1  Source of Data for Table 4.5











1. http://www.degussa.com: Data on Corax carbon blacks. 2. http://www.columbianchemicals.com: Data on rubber blacks. 3. A.I. Medalia, F.A. Heckman. Morphology of aggregates. II. Size and shape factors of carbon black aggregates from electron microscopy. Carbon, 7, 567–582, 1969. 4. A.I. Medalia. 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. J. Colloid Interf. Sci., 32, 115–131, 1970. 5. P.A. Marsh, A. Voet, T.J. Mullens, L.D. Price. Quantitiative micrography of carbon black microstructure. Carbon, 9, 797–805, 1971. 6. K.A. Burgess, C.E. Scott, W.M. Hess. Vulcanizate performance as a function of carbon black morphology. Rubb. Chem. Technol., 44, 230–248, 1971. 7. J.B. Donnet, A. Voet. Carbon Black. Marcel Dekker, New York, NY, 1976. 8. G.R. Cotten. Influence of carbon black on processability of rubber stocks II. Extrusion shrinkage. Intern. Rubb. Conf. RUBBERCON ‘77, paper 31, 1–13, 1977. 9. G.R. Cotten, J.L. Thiele. Influence of carbon black on processability of rubber stocks III. Extensional viscosity. Rubb. Chem. Technol., 51, 749–763, 1978. 10. A.M. Gessler, W.M. Hess, A.I. Medalia. Reinforcement of elastomers with carbon black. I. The nature of carbon black. Plast. Rubb.: Processing, 3 (1), 1–13, 1978.

Note: All appendices were made with the calculation software MathCad® 8.0 (MathSoft Inc., now part of PTC, Parametric Technology Corporation). Reproduced as such in the software, active worksheets would be obtained and would give the same results providing the layout is strictly respected since the software operates the mathematical formulas from left to right, from top to bottom.

Typical Fillers for Polymers

11. A.I. Medalia. In Carbon Black—Polymer Composites. E.K. Sichel Ed. Marcel Dekker, New York, NY, 1981. 12. G.R. Cotten. Mixing of carbon black with rubber. III. Analysis of the mixing torque curve. Kautsch., Gummi, Kunstst., 38 (8), 705–709, 1985. 13. ASTMD-1765-86a. Carbon blacks used in rubber products. 14. A.C. Patel, K.W. Lee. Characterizing carbon black aggregate via dynamic and performance properties. Elastomerics, 122 (3), 14–18, 1990. 15. A.C. Patel, D.C. Jackson. Carbon black characterization—Part 1: Effects of dynamic parameters on the behaviour of carbon black in rubber. J.M. Huber Corp. (data presented on a poster at IKT’91, June 25, 1991, Essen, Germany). 16. M-J. Wang, S. Wolff, J.B. Donnet. Filler-elastomer interactions. Part III. Carbon black surface energies and interactions with elastomer analog. Rubb. Chem. Technol., 64, 714–736, 1991. 17. F. Ehrburger-Dolle, S. Misono. Characterization of the morphology of rubber grade carbon blacks by thermoporometry. Carbon, 30 (1), 31–40, 1992. 18. C.R. Herd, G.C. McDonald, W.M. Hess. Morphology of carbon black aggregates: Fractal versus Euclidean geometry. Rubb. Chem. Technol., 65, 107–129, 1992. 19. E. Custodero. Caractérisation de la surface de noirs de carbone; nouveau modèle de surface et implication pour le renforcement. PhD Thesis, University of Haute-Alsace, Mulhouse, France, 1992. 20. M-J. Wang, S. Wolff, E-H. Tan. Filler-Elastomer interactions. Part VII. Study on bound rubber. Rubb. Chem. Technol., 66, 163–177, 1993. 21. M-J. Wang, S. Wolff, E-H. Tan. Filler-Elastomer interactions. Part VIII. The rôle of the distance between filler aggregates in the dyanmic properties of filled vulcanizates. Rubb. Chem. Technol., 66, 178–195, 1993. 22. T.C. Gruber, C.R. Herd. Anisometry measurements in carbon black aggregate populations. Rubb. Chem. Technol., 70, 727–746, 1997. 23. A. Schröder. Charakterisierung verschiedener Rußtypen durch systematische statische Gasadsorption; Energetische Heterogenität undFraktalität der Partikeloberfläche. PhD Thesis, Universität Hannover, Germany, 2000. 24. A. Weigert. Rastertunnelspektroskopie an Füllstoffrußen. PhD Thesis, Universität Regensburg, Germany, 2005.

83

84

Filled Polymers

A4.1.2 Relationships between Carbon Black Characterization Data

Compressed DBPA, dm3/kg

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

0.0

0.5

1.0 DBP absorption, dm3/kg

1.5

2.0

300

N2 adsorption, m2/g

250 200 150 100 50 0

0

50

100

150 I2 adsorption, mg/g

200

250

300

Comp. DBP absorption, dm3/kg

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

0

20

40

60 80 100 CTAB adsorption, m2/g

120

140

160

85

Typical Fillers for Polymers

A4.2  Medalia’s Floc Simulation for Carbon Black Aggregate Medalia’s floc simulation data (Table II) Area of a particle

5  10   15   25  50   74  25   74  50   50  15   15 5 Data: =   10  25  5  10   25  5  10   25  50  5  10   25  50

381  572  803   1265  2145   2970  1386   2893 3 2453   2497  605   1089  337   572  1265   331 1 532  1329   357  635   1236  2301   291  496   1289  2214 

Using the published value of 0.87 for the scaling exponent leads to an overestimation of the projected area; with ε = 0.847 the ­estimation is correct

Particle projected area is expressed in “square units”; the area Ap of a particle is 78.5 square units Ap: = 78.54 One has : Ap =

π.d 2 4

= > d: =

4 . Ap π

d = 10

Medalia has thus considered flocs of particles with diameter equal to 10 units

Ap :=

π ⋅ d2 4

Ap = 78.54

A: = sort (Data<1>)

Np: = sort (Data<0>)

Medalia published scaling exponent

Correct exponent

(from nonlinear fitting of Medalia’s data)

εM: = 0.87

εC: = 0.847

ε AM : = Ap ⋅ N p M

AC : = Ap ⋅ N p εC

4000 Floc projected area, square unit

Number of particles

 [Source: A.I. Medalia, J. Colloid Interf. Sci., 24, 393–404, 1967.]

3000 2000 1000 0 0

20 40 60 Number of particle (of d = 10 unit) Medalia's data Published exponent Correct exponent

80

86

Filled Polymers

A4.3  Medalia’s Aggregate Morphology Approach Projected area A of one aggregate [Source: A.I. Medalia, J. Colloid Interf. Sci., 24, 393–404, 1967]: d: diameter of primary particle Np: number of primary particle ε: empirical exponent from floc simulation

d2 A = π ⋅ ⋅ N pε 4



 

 4⋅ A  = > NP =   π ⋅ d 2 

1 ε

d

 em: in Medalia’s R paper, ε = 0.87, but using his data from floc simulation, in which d is set equal to 10 (nm), one finds in fact ε = 0.8471

A

Equivalent sphere model for a single aggregate [Source: A.I. Medalia, J. Colloid Interf. Sci., 32, 115–131, 1970] Ves =

π ⋅ D3 π  4 ⋅ A  = ⋅ 6 6  π 

3 2

 VES: volume of equivalent sphere D: diameter of equivalent sphere A: projected area of aggregate

d

D

3 4 Ves = ⋅ A2 3⋅ π

A

Volume VS of solid (carbon) within an aggregate:

Vs = Np⋅Vp

Vp =

π ⋅ d3 6

VP: volume of one primary particle Ap: projected area of one primary particle 1

=>

π ⋅ d3  4 ⋅ A  ε π ⋅ d3 Vs = N p ⋅ ⋅ =  π ⋅ d 2  6 6



Ap =

π ⋅ d2 4

87

Typical Fillers for Polymers

Medalia’s relationship between void ratio [ratio Ves/Vs] and DPB absorption: ec = CF ⋅

Ves 1 + vf ⋅ .g – 1 = DBPA × ρ ⋅ 0.0115  CF = 0.765: correction factor Vs C accounting for difference

between the projected area of the equivalent sphere and the projected area of the aggregate (around 8.5% reduction in D)

 vf = 0.46: void fraction for randomly packed spheres C = 1.4: correction for partial fusion of primary particles V C in aggregate => es = (1 + DBPA × ρ ⋅ 0.0115) ⋅ Vs CF ⋅ g ⋅ (1 + vf) g = 0.94: anisometry correction factor for non-perfect aligment of aggregate’s main axis with projection plan ρ = filler specific gravity   [carbon black: ρ = 1.86 g/cm3] C 1.4 = = 1.333 DPBA = di-butylphtalate CF ⋅ g ⋅ (1 + vf) 0.765 ⋅ 0.94 ⋅ (1 + 0.46) absorption (cm3/100 g filler) 0.0115: correction for DBPA end point (<0.85% of actual void volume) One has thus: 3  4  ⋅ A 2      3⋅ π 

Volume of equivalent sphere : Ves = 

3  4  ⋅ A2  Ves =   3⋅ π  1    4 ⋅ A  ε π ⋅ d3  ⋅ Vs    π ⋅ d 2  6   

=> 1 ε

3 Volume Vs of solid (carbon): Vs =  4 ⋅ A  ⋅ π ⋅ d 2

 π⋅d 

6

 3

d2 The projected area is: A = π ⋅ ⋅ N p 4

=>

Ves = Vs

1 [2 ⋅ π

 3  2 

⋅ ]

( π ⋅ d 2 ⋅ N pε ) 2  4⋅  ε  ε1  3  ( N p ) ⋅ d  



88

Filled Polymers

 −1  Ves 1 ε 2 ε  21  = ⋅ N p ⋅ ( d ⋅ N p ) ⋅ ( N pε ) ε  Vs d



=>

ε

  Ves  ε + − 1 = N p 2  VS

ε

  ε + −1 Ves C  2  = N = ( 1 + DBPA × ρ ⋅ 0.0115 ) ⋅ Therefore: p VS CF ⋅ g ⋅ ( 1 + vf )

=>

ε    ε + 2 − 1

N p

= (1 + DBPA × ρ ⋅ 0.0115) .1.333 2

N p = ((1+DBPA × ρ ⋅ 0.0115) .1.333) 3

1

with ε = 0.87 N p = (( 1 + DBPA × ρ ⋅ 0.0115) .1.333) 0.305 1

with ε = 0.847 N p = (( 1 + DBPA × ρ ⋅ 0.0115) ⋅ 1.333) 0.27

  

N p = (( 1 + DBPA × ρ ⋅ 0.0115) .1.333)

  

N p = (( 1 + DBPA × ρ ⋅ 0.0115) .1.333)

⋅ ε−2

3.279

3.697

ρ = 1.86 DBPA: = 30,32..160 [i.e., from N990:38cm 3 /100g to N356:157 cm 3 / 100g] 1

N p⋅Med (DBPA): = (1.333 ⋅ (1 + DBPA ⋅ ρ ⋅ 0.0115)) 0.305

  

1

N p (DBPA): = (1.333 ⋅ (1 + DBPA ⋅ ρ ⋅ 0.0115)) 0.27 1

N p⋅Med2 (DBPA): = (1.25 ⋅ (1 + 0.0115 ⋅ ρ ⋅ DBPA)) 0.305

<= explicit from A.I. Medalia, L.W. Richards, J. Colloid Interf. Sci., 40, 233–252, 1972; g correction was not applied (i.e. g = 1)

800 700

N330 : DBPA = 102cm3/100g

600 500

Np.Med(DBPA)

400

Np(DBPA)

300

Np.Med2(DBPA)

200 100 0 0

90 DBPA

180

Np.Med(102) = 114.124 Np(102) = 210.899 Np.Med2(102) = 92.435

89

Typical Fillers for Polymers

A4.4  Carbon Black: Number of Particles/Aggregate (Medalia vs. fractal approaches) i = 0…17 N2 absorpt. m 2 /g

DPB absorpt. cm 3 /100g

137.97  N1110      126.00  N121   116.75  N220      123.26  N234   106.50  N229      82.57  N326      79.45  N330   92.10  N339      71.43  N351  Data: =  88.05  N358      84.30  N550 0     41.12  N630   34.00  N650      35.75  N660   34.85  N762      28.47  N774      29.64  N990   9.09





114 133 115 124 123 76 102 120 121 157 155 117 78 124 94 66 74 38

Aggreg. Elem.Part. diam. diam, nm nm

Data:

68.3 17.8   76.7 18.8  77.3 20.7   80.1 19.7  92.5 24.3   86.8 27.0   105.0 30.1  103.0 25.8   129.0 31.4  136.0 30.22   136.0 30.2   234.0 56.8  220.0 58.3   271.0 60.9  252.0 67.1   255.0 102.0   228.0 87.2  483.0 291.0 

ρ: = 1.86 : carbon black specific gravity, g/cm3 ε: = 0.847 : scaling exponent

F: = 2.44 : (mean) mass fractal dimension α: = 11 : front factor

DBPi : = (Data 1 ) i : DBP absorption, cm3/100g

Di : = (Data 2 )i

: mean aggregate diameter, nm

di : = (Data 3 )i

: mean particle dia meter, nm

 2    1.15   3⋅ε − 2     N1pi : = round  1.333.  1 + DBPi ⋅ ρ ⋅ , 0 : Number of particlees/aggregate    100    according to Medalia 

  D F  N2 pi : = round α ⋅  i  ,0  : Number of particles/aggregate   di   from fractal descripttion

x : 0, 100..600

800

 0  x: =    500 

600 400

N1p x

200 0

0

200

N2 p, x

400

600

5 Polymers and Carbon Black

5.1  Elastomers and Carbon Black (CB) 5.1.1 Generalities Whatever the filler, there are at least six families of parameters to consider when studying how the properties of a given polymer are modified: • The nature of the polymer, its chemistry, its macromolecular and structural characteristics • The chemical nature of the filler and its surface chemistry • The average size and size distribution of particles • The geometry and structure of particles • The polymer/filler concentration ratio (or the filler volume fraction) • The effects of the other ingredients in the compounds No overall, coherent understanding of all those effects exists yet and only several, partial aspects are documented, in various extent, for certain classes of fillers. Due to its major role in rubber technology, CB is surely the material that has so far captured most of the attention and therefore, deserves a specific interest, whilst certain conclusions drawn in studying polymer-CB composites are not necessarily applicable to other systems. In most of their applications, elastomers are used in compounds where CB plays the key role in reinforcing mechanical properties. Not all properties are improved when adding CB particles to an elastomer and a balance of properties has to be achieved with respect to the application. In rubber compounds, CB amounts to 20–30% volume fraction. In rubber technology, a compounding formulation is generally described in phr (part per one hundred rubber) of component. Let us consider a typical general-purpose formulation for styrene-butadiene rubber (SBR) (Table 5.1).

91

92

Filled Polymers

Table 5.1 Typical Carbon Black Filled SBR 1500 Compound Ingredient

phr

Specific Gravity (g/cm3)

Volume Fraction

SBR 1500 N330 carbon black Zinc oxide* Stearic acid Processing oil Antiheat Antioxidant Total

100 50 5 3 5 2 1 166

0.90 1.80 5.57 0.92 0.98 1.08 1.17 1.10

0.736 0.184 0.006 0.036 0.020 0.012 0.006 1.000

* Zinc oxide is nowadays often used as an 80% weight predispersion in EVA; specific gravity of such a predispersion is 2.90 g/cm3, so 5 phr ZnO correspond to 6.25 phr predispersion.

The specific gravity of the compound is calculated as follows:

ρcpd

∑w = w ∑ρ

i

i

i



(5.1)

i

i

where wi and ρi are, respectively the quantity (phr) and the specific gravity of the ith ingredient. The volume fraction of any ingredient i is calculated using:

Φi =

wi × ρcpd wcpd × ρi

(5.2)

where wcpd is the overall quantity (phr) of all ingredients in the formulation (for instance 166 phr for the formulation described in Table 5.1). It is fairly obvious that macroscopic properties of any complex polymer system depend on the degree of dispersion of ingredients, in other words of their spatial distribution within the volume of the material. This stresses the importance of the preparation process, i.e., mixing operations that require adequate equipments and procedures. With CB, the reinforcing element is the aggregate. Therefore, optimum mixing is achieved when all the filler is well dispersed at this level; ­ideally no more agglomerates are present and breaking down aggregates into elementary particles, if arising, has been limited. Basic considerations on optimum dispersion allow estimating the most likely distance between aggregates. Let us indeed consider the most compact spatial disposition of spheres of equal diameter; it corresponds to the face centered cubic lattice (Figure 5.1). One may consider that an aggregate occupies a spherical volume whose diameter is half the so-called Stokes

93

Polymers and Carbon Black

δ

ε

da

Figure 5.1 Estimating inter-aggregates distances through a face-centered cubic lattice model.

diameter* da. Two typical distances between spheres (or aggregates) can then be calculated with respect to filler volume fraction Φ, i.e.: the longest distance:    2  ε= − 1  da  3 12 Φ    π



the shortest distance:



    2 δ= − 1  da  3 3Φ 2    π

Table 5.2 gives results for a series of CB grades, with respect to volume fractions 0.10 and 0.20, i.e., below and above the so-called ­percolation level of around 0.13. As we will see later, significant changes occur in a number of macroscopic properties, when filler content is above this level. In fact, most practical rubber compounds have filler content significantly above the percolation level (for instance Φ = 0.184 for the 50 phr N330 SBR compound described in Table 5.1). * The Stokes diameter is the volume of gyration of an anisometrical particle sedimenting in a fluid, as measured using laser techniques.

94

Filled Polymers

Table 5.2 Estimating Inter-Aggregates Distances with a Face-Centered Cubic Lattice Model Carbon Black Grade 

Volume Fraction = 0.10

Volume Fraction = 0.20

Equivalent Stokes Diameter (nm)

Shortest Distance (nm)

Longest Distance (nm)

Average Distance (nm)

Shortest Distance (nm)

Longest Distance (nm)

Average Distance (nm)

85 102 97 100 106 131 104 126 138 240 251 266 275 261 436

40.3 48.4 46.0 47.5 50.3 62.2 49.4 59.8 65.5 113.9 119.1 126.2 130.5 123.9 206.9

74.6 89.6 85.2 87.8 93.1 115.0 91.3 110.7 121.2 210.8 220.4 233.6 241.5 229.2 382.9

57.5 69.0 65.6 67.6 71.7 88.6 70.3 85.2 93.3 162.3 169.8 179.9 186.0 176.5 294.9

23.2 27.9 26.5 27.4 29.0 35.8 28.4 34.5 37.7 65.6 68.7 72.8 75.2 71.4 119.2

50.5 60.6 57.6 59.4 63.0 77.8 61.8 74.8 82.0 142.5 149.1 158.0 163.3 155.0 258.9

36.9 44.2 42.1 43.4 46.0 56.8 45.1 54.6 59.9 104.1 108.9 115.4 119.3 113.2 189.1

N110 N220 N234 N299 N326 N330 N339 N351 N356 N550 N660 N683 N762 N774 N990

Table 5.3 Interaggregates Distances in Compounds Developed for Optimal Reinforcement Carbon Black Grade N110 N220 N234 N326 N330 N339 N351 N375

SBR 1500

SBR1712

Optimum Volume Fraction

Interaggregates Distance (nm)

Optimum Volume Fraction

Interaggregates Distance (nm)

0.201 0.206 – 0.250 – 0.216 0.230 –

21.4 23.2 – 24.0

0.207 0.211 0.204 0.253 0.230 0.215 0.228 0.216

20.2 22.0 19.7 23.4 24.3 20.3 22.3 21.9

20.6 21.8 –

If one considers reinforcing blacks, for instance series N110–N356, with a volume fraction of 0.20 (around 55 phr filler), typical shortest distance δ ­varies from 23 to 38 nm, and ε from 50 to 82 nm. As shown in Table 5.3, such data are comparable with measurements made by Patel and Byers1 on ­several compounds whose CB level was selected for optimal ­reinforcement, i.e., the best balance of properties with respect to tire applications.

95

Polymers and Carbon Black

This simple exercise demonstrates that in most well dispersed CB filled rubber compounds, average inter-aggregates distances are close to the average quadratic diameter of the polymer random coil, of the order of 50 nm for diene elastomers with molecular weight of around 400,000 g/mole. In other terms, in most rubber compounds, certain chains of elastomer are in contact with at least two aggregates, whilst others are not. 5.1.2 Effects of Carbon Black on Rheological Properties Generally the (shear) viscosity of a filled polymer increases with filler content, as easily understood through simple hydrodynamic considerations. However the flow behavior of filled polymers remains relatively simple only at very low shear rate, with low structure materials of large particle size, for instance glass micro-spheres, or thermal blacks with particle size larger than 500 nm. Typical effects of carbon blacks on the shear viscosity function are illustrated in Figure 5.2, drawn using data published by Montes et al.2 Three types Natural rubber/carbon black compounds at 100°C Effect of filler level

Effect of filler structure 1012

1012 20 phr N326

108

10 Phr N326

30 phr N326

106 Gum 104

108

N990(20 phr)

N110(20 phr)

106 Gum 104 102

102 100

N326(20 phr)

1010 Shear viscosity, kPa.s

Shear viscosity, kPa.s

1010

102

103 104 105 Shear stress, Pa

Low rate range M “Sandwich” rheometer

106

100 102

Medium rate range

Mooney viscometer

105 103 104 Shear stress, Pa

106

High rate range Capillary rheometer

Figure 5.2 Effects of carbon black on the shear viscosity function. (Data from S. Montes, J.L. White, N. Nakajima, J. Non-Newtonian Fluid Mech., 28, 183–212, 1988.)

96

Filled Polymers

of viscometers were necessary to generate the data, all applying simple shear flow. The low rate range was investigated with a (prototype) “sandwich” rheometer, the medium range with a variable speed Mooney viscometer and the high rate range with a capillary rheometer. One notes incidentally that the viscosity is plotted vs. the shear stress. The effect of filler level is shown in the left graph. At 100°C, gum natural rubber (NR) exhibits the expected behavior of a pseudo-Newtonian plateau at low shear stress (and hence low shear rate), then a shear thinning behavior. As filler content increases, the Newtonian plateau progressively disappears and above 20 phr, the viscosity variation with decreasing shear stress is such that a yield stress behavior is suggested. The effect of the structure is shown in the right graph. As can be seen, as the filler structure is more and more complex (from N990 to N326, then N110), the Newtonian plateau disappears, and again a yield stress behavior is suggested for the reinforcing carbon blacks. There is thus, a striking parallelism between the effect of filler content at constant structure, and the effect of filler structure at constant level. We note that the η = F(σ) representation of the viscosity function is important in observing the above effects, particularly the occurrence of a yield stress. Using the (more traditional) plot of η vs. the shear rate ( γ ), only the progressive disappearance of the Newtonian plateau would have been detected, with maybe a slight concave curvature appearing at very large filler level, as illustrated in Figure 5.3. Much has been published about the yield stress behavior of CB filled compounds but it is worth underlining here the experimental difficulties in assessing the shear flow behavior of highly viscous materials in the very low shear rate (or stress) range. No commercial equipment exists for viscosity measurement below γ = 10−2 s-1, and as shown in Figure 5.2, Montes et al. used a prototype plan-plan rheometer, developed at the Polymer Engineering Institute, University of Akron, OH and operated through a creep technique. Drag flow was obtained by moving a plan through the action of a dead weight; a cathetometer was used to measure displacement over various time intervals. It is obvious that cathetometer accuracy reading and operator patience were key elements of the experiment. Any yield stress value that can be derived from such experiments is an extrapolated value; so strictly speaking, one does not measure it, as quoted in the controversial literature on the subject.* Recently Barrès and Leblanc3 developed a prototype ­sliding cylinder rheometer that uses an optical transducer with the capability to detect displacements in the micron range, and a computer to record * Note that there are many (complex) fluids that do not exhibit a Newtonian plateau at low shear rate (stress) and whose shear viscosity function feeds the controversy on the existence of a yield stress. As noted by Barnes, when the flow is so slow than ages are necessary to detect it, at least one could consider that the yield stress is an engineering reality (H. Barnes. The yield stress—a review or “παντα ρει” —everything flows? J. Non-Newtonian Fluid Mech., 81, 133–178, 1999.)

100

101

102

103

104

101 10–2

102

10–1

10–1

100

100

101 102 –1 Shear rate, s

Shear viscosity • function η = f(γ)

•

Flow curve σ = f(γ)

101 102 Shear rate, s–1

Figure 5.3 Shear viscosity plots vs. carbon black level.

10–1 10–2

Shear viscosity, kPa.s

Shear stress, kPa

103

103

103

5×103

5×103

Shear viscosity, kPa.s 10–1 2×101

100

101

102

103

104

102 Shear stress, kPa

Shear viscosity function η = f(σ)

: Φblack = 0.20

: Φblack = 0.10

: Gum SBR

SBR 1500 – N220 carbon black T = 100°C

103

Polymers and Carbon Black 97

98

Filled Polymers

High cis-1,4 polybutadiene/N330 (50 phr) - T = 100°C 106

Shear viscosity, kPa.s

105 104 103 2

10

•

v M

101

Shear viscosity, kPa.s

106

105

η = f(σ)

104 103 102 101 100

10–1

10–2 104

100

105 Shear stress, Pa

10–1 10–2

10–5 10–4 10–3 10–2 10–1 100 101 Shear rate, s–1

102

103

104

106

SCR creep mode SCR constant rate Mooney viscometer Capillary rheometer

Figure 5.4 Shear viscosity function of a carbon black filled BR compound.

data over very long periods of time (up to 24 h, in certain cases). Shear viscosity data obtained with this instrument compare well with (extrapolated) ones generated with both the Mooney viscometer and a capillary rheometer. As shown in Figure 5.4 for a CB filled BR compound, they did not detect any Newtonian plateau down to 10−5 s−1, but the η = F(σ) plot would hardly be considered as suggesting the existence of a yield stress, at the least in the shear rate range investigated. As a matter of fact, there are a number of complex systems that, providing the appropriate equipment is used (for instance the so-called “controlled stress” rheometers), are found to exhibit a drastic fall of the viscosity, by several orders of magnitude, over a narrow range of shear stress. When approaching this critical region from the high stress range, it seems thus that the viscosity goes to infinity at a certain minimum stress. Thanks to their relatively lower viscosity, food and cosmetic products allow this type of behavior to be easily documented, when performing experiments with controlled stress rheometers. There are commercial versions of such instruments, essentially rotating systems (parallel disks or cone-and-plate), which have the capability to measure extremely small rotation rates (in the 10−8 rad/s, i.e., one revolution in 20 years!). Experiments performed in such conditions are called creep testing using controlled torques as low as 10−7 Nm with a resolution of 10−9 Nm. Figure 5.5 shows an example of the shear viscosity function (vs. shear stress) measured on a typical cosmetic product (body cream), using such a controlled stress rheometer. As can be seen, measurements made at stress higher than 50 Pa and extrapolated toward lower shear stress would allow detecting an (apparent) yield stress. In fact, a drastic fall of viscosity (around two millions time) occurs at

99

Polymers and Carbon Black

Controlled stress rheometer 107

σc = 44.6 Pa η1 = 3.16×106 Pa.s

106

Drastic fall of viscosity for a slight variation of stress

σ1 = 13.8 Pa

104 × 2.000.000

Shear viscosity, Pa.s

105

103 102 101 100

10–1

100

σ2 = 79.5 Pa

Body cream (room temperature) ×2 101

Shear stress, Pa

η2 = 0.39 Pa.s 102

103

Figure 5.5 Typical shear viscosity function of a (complex) cosmetic material.

a critical stress of 44.6 Pa, splitting the flow behavior in two nearly constant viscosity regions, one in the 106 Pa.s range, the other in the 10−1 Pa.s range. Such a behavior might also exist with filled high molecular weight ­polymer systems, for instance rubber compounds, but it has so far never been observed (to the author’s knowledge). Consequently, even if the yield stress exists only because the shear rate/shear stress range of observation does not allow the above behavior to be observed, simple yield-stress-containing equations for viscosity properties are useful, at least over a limited range of stress/rate. A number of mathematical models have been proposed for yield stress fluids, not all perfectly coherent however. If a shear thinning material is tested with one of several rheometers and the results plotted in terms of shear stress σ vs. shear rate γ (the so-called flow curve) using linear scales, one is nearly bound to the conclusion that there is a yield stress σc and that the best manner to model the observed behavior consists in considering the following equality: σ = σ c + f ( γ ) . As we will see below, nearly all proposed models follow this approach, but with respect to the definition of the shear viscosity function, i.e., η = σ/γ = f ( γ ) = F(σ ), we consider that a criterion of coherence, if not validity, of such models is that they allow equations to be derived, either as η = f ( γ ) or as η = F(σ). Herschel and Bulkley4 combined the power law model with a yield stress σc:

σ = σ c + K γ n

(5.3)

100

Filled Polymers

so that the following equations are obtained for the shear viscosity function: σc + K γ ( n−1) γ



η = f ( γ ) : η( γ ) =



 K  η = F(σ): η(σ ) =   σ − σ c 

(5.3a)

1/n

σ

(5.3b)

As illustrated in Figure 5.6, the Herschel–Bulkley equation corresponds to viscosity functions without Newtonian region, which exhibit a significant upward curvature as the shear rate/shear stress is decreasing. The η = F(σ) shows indeed a viscosity that goes to infinity as the stress decreases toward a critical value. It is fairly obvious that, when fitting shear viscosity data with an equation that explicitly states that there is a yield stress (Equation 5.3, for instance), a value for σc will be obtained, which has very limited meaning, if any, when the investigated shear stress range is excessively far from where the fit parameter is obtained. In the high rate/stress region, the shear Herschel–Bulkley equation

•

15

Flow curve σ = f (γ) K = 0.9 n = 0.4

σ = σc+ Kγn

Shear stress

10 •

η(γ) = 5

200 Shear rate

400

Viscosity function η = f (σ) 105

σc = 0.50 Shear viscosity

Shear viscosity

Viscosity function η= f (γ)

101 100 10–1

σc = 0.25

103

100

101 Shear rate

102

103

Figure 5.6 Herschel–Bulkley model for yield stress fluids.

σc = 0.50

101

10–1

10–2 10–3 –1 10

+ K γ• (n–1)

K n η(σ) = ( σ – σc ) σ

•

102

•

γ

1

σc = 0.5 0 0

σc

10–3 –1 10

σc = 0.25 100 101 Shear stress

102

101

Polymers and Carbon Black

thinning behavior met by the Herschel–Bulkley equation corresponds to the power law model, not always suiting the downward curvature of the viscosity function observed in practice with complex polymer systems. The so-called Bingham fluid obeys to the following equation:

σ = σ c + η ⋅ γ

(5.4)

In other terms, above a critical shear stress, it flows as a Newtonian fluid of (constant) viscosity η. It follows that a fluid obeying the Herschel–Bulkley model is sometimes called a generalized Bingham fluid, since with n = 1 and K = η in Equation 5.3, one obviously obtains Equation 5.4. The three fit parameters of the Herschel–Bulkley equation can be reduced to two, when considering that n = 0.5. This was in fact the approach used by Casson5 in proposing the following model:

σ = σ c + K γ   or  σ = σ 0c .5 + 2 ⋅ σ 0c .5 ⋅ K ⋅ γ 0.5 + K 2 ⋅ γ

(5.5)

Explicit formulas for both shear viscosity functions are easily derived, as follows:

σ σc η = f ( γ ) : η( γ ) = c + 2 K + K2 γ γ



 K  η = F(σ): η(σ ) =  σ 2 + σ × σ c + 2 σ σ − σ c    σ c − σ 

(5.5a) 2

(5.5b)

As illustrated in Figure 5.7, the Casson model corresponds indeed to a η( γ ) function without Newtonian plateau and an upward curvature in the low shear rate region significantly depending on the magnitude of the yield stress; but the shear thinning behavior does not correspond to observations on filled polymer systems. Similar comments are made for the η(σ) function. White et al.6 submitted a model (the White–Wang model) that was used with some success in a number of works made at the Institute of Polymer Engineering, Akron, OH. With respect to the flow curve, this model writes as follows:

σ = σc +

A γ 1 + B γ 1− n

(5.6)

where A and B are fit parameters whose ratio A/B corresponds to the parameter K of a power law σ = K γ n used to model the high shear rate range, and n is the flow index. An explicit equation is obtained for the viscosity function η( γ ), i.e.:

η = f ( γ ) : η( γ ) =

σc A + γ 1 + B γ 1− n

(5.6a)

102

Filled Polymers

•

Flow curve σ = f (γ)

40

Casson equation

K = 0.2

•

√σ = √σc + K√γ

Shear stress

σ = √σc + 2√σc K√γ +K 2γ • η (γ) = σ + 2√σ K + K 2

20

•

c

γ

•

γ

•

•

c

η(σ)= (σ2+ σ. σc + 2 σ√σ– σc) σc= 5 0

0

100 Shear rate

Viscosity function η= f (σ)

•

102 Shear viscosity

σc= 5 100 10–1 10–2

σc= 1

100

101

102 Shear rate

c

200

Viscosity function η= f (γ)

101 Shear viscosity

2

(σ K– σ )

103

101

σc= 5

100 10–1 10–2

σc= 1

100

101

102

103

Shear stress

Figure 5.7 Casson model for yield stress fluids.

but there is no single solution for η(σ) Indeed, by substituting γ = σ/η in Equation 5.6, one obtains:

η = F(σ): η(σ ) σ n−1 + η(σ )n B =

A σn (σ − σ c )

(5.6b)

Figure 5.8 illustrates the capabilities of Equations 5.6 and 5.6a. Despite its limitations, the White–Wang model suits well certain observations made on CB filled rubber compounds, particularly the behavior of the shear viscosity function η( γ ) in the low shear rate region, as reported in a number of publications. Even if the White–Wang model remains open to discussion, the yield stress data that can be derived from experimental flow curves by fitting Equation 5.6 are likely to be similar to values that would be obtained with other models, for instance the Herschel–Bulkley equation. It is consequently interesting to pay attention to yield stress data that were obtained at the Polymer Engineering Institute, using this model.

103

Polymers and Carbon Black

Flow curve σ = f (γ• )

15 10

Shear stress

White–Wang equation

A=2 B = 0.8 n = 0.3

σ = σc+

η(γ• ) =

5 σc = 0.5 0

0

Shear viscosity

102 101 100

50 Shear rate Viscosity function η= f (γ• )

+

A 1 + B γ• (1–n)

Viscosity function η = f (σ)

σc= 0.50

No single solution for η = f (σ)

σc= 0.25

10–1

γ•

1 + B γ• (1–n)

100

η(σ) . σ n–1 + η(σ)n.B = A . σ n (σ – σc )

10–1 10–2 10–2

σc

Aγ•

100 101 Shear rate

102

Figure 5.8 White–Wang model for carbon black filled compounds.

Table 5.4 is a compilation of yield stress data for various rubber compounds. As previously described (see Figure 5.2) three rheometrical techniques were used to obtain the shear viscosity data, and the yield stress values were derived by fitting results with Equation 5.6a. Rubber nature and CB type are indicated, as well as the volume fraction ΦBlack and the crushed di-butyl phthalate (DBP) absorption numbers (cDBPA) of CB. As can be seen, both the filler level and the reinforcing character of CB do affect σc. One might consider that the yield stress behavior of CB filled compounds does reflect the response to (shear) stress of some kind of tri-dimensional structure, either occurring through contacts between aggregates or, most probably, resulting from rubber–filler interactions and the associated complex morphology. We will elaborate further this comment in subsequent sections, but would such considerations be valid, one should expect the yield stress to be commensurate with a complex parameter in which the volume fraction and an assessment of the aggregate structure are involved. A very simple manner to probe this argument consists in plotting yield stress data vs. the corresponding product ΦBlack × cDPBA. As shown in Figure 5.9, a loose relationship is indeed observed.

0.2 0.2 0.2 0.2 0.2 0.2

N220 N550 N762 N326 N330 N330

SBR Emulsion Goodyear Plioflex 1500 SBR Sol (23.5 Styrène) Firestone Duradene 706 EPDM Chloroprene rubber CR

0.2 0.1 0.2 0.3 0.2 0.3 0.2 0.2 0.2

N326 N326 N326 N326 N330 N330 N220 N550 N762

Polyisoprene Cis-1,4 (Natsyn 2200) SBR1500

0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.2

Volumic Fraction Φ

N326 N220 N550 N762 N110 N326 N326 N326

Carbon Black Type

Natural rubber (RSS1) Natural rubber (SMR L) Natural rubber (SMR 5) Polyisoprene Cis-1,4 (Shell IR305)

Elastomer

Yield Stress Data on Various Filled Rubber Compounds

Table 5.4

0.99 0.84 0.59 0.67 0.87 0.87

0.67 0.67 0.67 0.67 0.87 0.87 0.99 0.84 0.59

0.67 0.99 0.84 0.59 1.01 0.67 0.67 0.67

cDPBA (dm3/kg)

0.198 0.168 0.118 0.134 0.174 0.174

0.134 0.067 0.134 0.201 0.174 0.261 0.198 0.168 0.118

0.134 0.198 0.168 0.118 0.202 0.134 0.201 0.134

Φx cDPA

130 82 62 100 58 56

42 6 13 100 63 138 133 83 63

60 119 79 62 140 44 170 38

Yield Stress (kPa)

118, 2

119

116 6, 117

116 118 2 116

Source

104 Filled Polymers

105

Polymers and Carbon Black

Apparent yield stress, kPa

180 160 140 120 100 80 60 40 20 0

0

0.1 0.2 Volume fraction × cDBPA

0.3

Figure 5.9 Relationship between yield stress (from flow curves at 100°C), carbon black structure, and level in filled rubber compounds.

As we have already underlined, adding CB to an elastomer gives a compound that has a strong nonlinear viscoleastic character, appearing amongst other effects, by the quasi-suppression of the linear region in the shear viscosity function (which no longer exhibits a Newtonian plateau at low shear rate or stress). It follows that other typically nonlinear effects are exacerbated by the presence of CB particles and, as can be expected, the higher the reinforcing character of the filler, the stronger those effects. Several typical examples allow this aspect to be illustrated. Let us ­consider the so-called stress overshoot effect, i.e., the peak stress observed in stress growth experiments when the suddenly applied shear rate is very high. This effect is typically non linear and has been well documented through ­laboratory experiments on a number of simple polymer systems.7 With filled rubber materials, experimental data on stress overshoot are scarce, likely due to ­difficulties in handling such very stiff materials in conventional ­rheometers (i.e., parallel disks or cone-and-plate). Experiments by Montes et al.2 are therefore, worth considering. These authors used a variable speed Mooney viscometer to perform stress growth experiments on NR and SBR 1500 compounds with various types and levels of CB. Figure 5.10 describes their experimental approach and gives typical results on NR compounds, expressed as ratio of the “peak” viscosity over the stabilized viscosity. The stress overshoot is amplified by the presence of CB, and the more reinforcing the filler, the stronger the effect. Such experiments, whilst obtained in laboratory conditions, provide results that correspond very well (at least qualitatively) with observations on factory floor. Rubber engineers are indeed well aware of the difficulties associated with the starting-up of extrusion lines when processing highly reinforced compounds. Machine temperature profile and extruder screw speed must be carefully monitored right from the

106

Filled Polymers

Torque

Variable speed Mooney viscometer (Montes, White, Nakajima–1988) Peak

Maximum viscosity ηmax Stabilized viscosity η∞

Cst shear rate 1.6 1.5

ηmax

1.4

Time Natural rubber (SMR 5) Carbon black : 20% Temperature : 100°C

η∞ 1.3

N326

1.2

N990

1.1 1.0 0.02

N110

0.1

Shear rate, s–1

1

2

Figure 5.10 Stress overshoot experiments on carbon black filled compounds.

beginning of the operation in order to avoid excessive head pressure, and quite long periods (while wasting material) are sometimes necessary before the extrusion line is stabilized. Other easily observed effects of CB on the elastic character of rubber compounds are made when considering the so-called “extrudate swell” (or postextrusion swelling). All other parameters constant, extrudate swell decreases when the CB level increases, 8 as illustrated in Figure 5.11. From a technological point of view, this means that in rubber technology, postextrusion swelling is at least one order of magnitude below what is commonly observed with pure polymers (for instance polyolefins). Consequently extrusion die design and postextrusion equipment are notably different. At constant CB level, the more complex the filler structure the lower the extrudate swell9 as illustrated by Figure 5.12. An immediate practical consequence of such effects is that the dimension stability of highly reinforced tire tread bands is relatively easy to achieve on the factory floor, by controlling the longitudinal shrinkage associated with the swelling (normal to the extrusion axis). Such effects, obviously beneficial and exploited in rubber processing technology, remain however difficult to fully explain. Indeed, whilst extrudate swell decreases with increasing CB volume fraction, the die entrance pressure

107

Polymers and Carbon Black

Natural rubber (SMR10)/N220 carbon black Capillary rheometer at 100°C

60

10 phr

(Relaxed) extrudate swell, %

50

25 phr

40

40 phr

30 20 60 phr

10 0

0

0.1

0.2 0.3 Shear stress (MPa)

0.4

Figure 5.11 Effect of carbon black level on postextrusion swelling. 100

SBR 1500 Carbon black (50 phr)

Extrudate swell, %

90 80 70 60 50 40 30 0.2

0.4

0.6 0.8 1 DBP absorption (cm3/g)

1.2

1.4

Increasing complexity of filler particle structure Figure 5.12 Effect of carbon black structure on postextrusion swelling.

loss significantly increases with higher filler content. Such a behavior is totally different from what is observed on compounds made with CB and saturated polymers. For instance, Robertson et al.10 reported experiments on the nonlinear rheology of hyperbranched polyisobutylene, notably capillary rheometry at 205°C on linear and branched polyisobutylene compounded with 50 phr N339 and 5 phr processing oil. Figure 5.13 is drawn using extrudate

108

Filled Polymers

Extrudate swell

1.50

Capillary flow at 205°C; 1.5 mm dia. dies PIB cpds (50 phr N339; 5 phr oil)

1.25

1.00 0.75

0.50

Linear polyisobutylene Branched polyisobutylene 0

2

4 Entrance pressure, MPa

6

8

Figure 5.13 Postextrusion swelling vs. entrance pressure drop, as observed on compounds of carbon black with saturated polymer melts.

swell data (measured after nine days resting at room temperature) and the ­corresponding entrance pressure drop data Pent (assessed through the Bagley method). Despite a large scatter on Pent, it appears that there is direct proportionality between postextrusion swelling and entrance pressure drop. One notes incidentally that the structure of the polymer has no significant effect on the relationship observed. Robertson et al. give no information regarding likely interactions between CB and polyisobutylene and one might presume that their observations reflect only hydrodynamic effects. Either linear or branched, polyisobutylenes are polymers with a high chain flexibility, like any elastomer, but unlike diene rubbers, there is no chain unsaturation that could, on one hand significantly increases the mean diameter of the random coil (and hence increases the probability for one macromolecule to be in contact with at least two CB aggregates), and on the other hand favors interaction of topological nature between polymer segments and the surface of filler particles, as we shall discuss below. 5.1.3  Concept of Bound Rubber (BdR) When an elastomer and a reinforcing filler are mixed, strong interactions occur in such a manner that, at room temperature, a good solvent of the ­polymer can extract only a free rubber portion, leaving a highly swollen rubber–filler gel. BdR is, by definition, the rubber content of that gel, and the most obvious proof that a heterogeneous, complex structure has been formed during the mixing operations. Known for more than 80 years in the case of CB filled compounds, it is considered one of the major factors in reinforcement and often a global measure of surface activity of the filler, because of the striking parallelism between the amount of BdR and the reinforcing capabilities of the black. Rubber–filler ­interactions readily occur in the early stages of the mixing process and there are consequently direct relationships

109

Polymers and Carbon Black

Table 5.5 Effect of Carbon Black Size on Bound Rubber; Literature Data Carbon Black (50 phr) N110 N220 N330 N339 N347 N351 N375 N440 N550 N539 N568 N650 N660 N765 N774

Bound Rubber (%) SBR (Source) 42.2 ± 0.8 (12,111,112) 36.0 ± 1.2 (12,49,111–114) 32.1 ± 1.4 (12,111,112) 36.1 ± 2.9 (12,111,112) 34.7 ± 1.0 (111,112) 32.6 ± 3.2 (12,49,111–114) 36.8 ± 3.4 (12,49) 22.9 ± 0.4 (12,111) – 24.4 ± 1.1 (12,111) 25.2 ± 0.1 (12,111) 19.9c – 21.5 ± 0.1 (12,111) –

BR (Source) 30.0 (113) 27.0 ± 2.1 (113,114) 27.1 ± 4.6 (113,114) – 27.0 (113) – 29.1 (113) – 20.1 (113) – – 12.9 (114) – 11.0 (114)

between BdR and rheological properties of rubber compounds, as outlined in a recent review paper.11 For compounds made with a given elastomer and at constant filler level (with all other compounding ingredients constant), BdR increases with the reinforcing character of CB. For instance, BdR increases with decreasing elementary particle size, and increases as the structure is more and more complex, as clearly seen when compiling literature data (Table 5.5 and Figure 5.14), despite some differences between authors. In Table 5.5, the particle size effect is readily seen when comparing how BdR evolves in BR compounds with 50 phr of CB of the N110, N220, N330, N550, and N660 grades. Within the same family, for instance the N300 series, increasing aggregate complexity is considered through N330, N339 and N347 in SBR compounds. Figure 5.14 compares data reported by different authors12,13 on SBR 1500 compounds filled with 50 phr CB. As can be seen, the effect of specific area of CB (which reflects the complexity of aggregates) on BdR is quite consistent: the more reinforcing the CB, the higher the BdR. Providing the filler content is above or near a critical level of 12–13%, BdR measurement yields a swollen, coherent sample at the end of the extraction process. This suggests that there are rubber–filler ­interactions, strong enough to resist the solvatation process, which consequently lead to a tri-dimensional morphology in uncured filled rubber compounds. Microphotography evidences of such a 3D structure were published in the early 1970s14,15 and

110

Filled Polymers

45

SBR 1500 compounds 50 phr carbon black

40

N339 N220 N285 N219 N121 N330 N234 N327 N568 N110 N375 N242 N339 N220 N326t N539 N440 N330 N765 N539 N326n N683 N550 N765 N660 N762 Dannenberg (1986)

Bound rubber, %

35 30 25 20 15 10 5 0

N110

N375

N351

N347

Wolff, Wang, Tan (1993) 0

20

40

60 80 Specific area, m/g2

100

120

140

Figure 5.14 Effect of carbon black structure on bound rubber; literature data.

several authors used advanced nuclear magnetic resonance (NMR) techniques to obtain detailed information about how the molecular flexibility is modified by the presence of filler particles. Three distinct regions in the filled rubber, with different degrees of molecular mobility exhibited by the constituent rubber chains are generally reported, irrespective of the chemical nature of the rubber.16–18 In a well-dispersed, uncured compound, isolated aggregates induce restrictions to mobility of rubber chains, in such a manner that three rubber regions can be distinguished. Indeed NMR relaxation experiments reflect the heterogeneity of filled rubber materials, with a fast decay in the 20 µs range that is assigned to the immobilized part of the BdR shell. At longer times (above 100 µs), there is mobile fraction that contributes to the relaxation and in between (in the 50 µs range), there is an intermediate mobility component. NMR relaxation curves are decomposed into components, using the appropriate number of distribution functions. For instance, Yatsuyanagi et al.19 performed pulse NMR measurements at 100 MHz on a series of SBR 1502 compounds with various levels of HAF (~N330 and ~N339) and SAF (~N212) and analyzed the results with respect to BdR content (72 h extraction in toluene at room temperature). By considering solid echo sequence, the proton spin–spin relaxation time T2 was obtained and reduced in several components by fitting the signal with a set of Weibull functions. Above −30°C, filled compounds exhibit three ­components in T2, a long T2L, an intermediate T2M and a short one T2S that are assigned to three different types of BdR. The unfilled compound shows a simple signal that corresponds to T2L and, as illustrated in Figure 5.15, there are clear relationships between T2M and T2S and BdR content. In agreement with other publications, the T2 components can therefore be considered

111

Polymers and Carbon Black

500 400

Bound rubber components, % gum

40 T2L

T2 components, µs

300 200 100 80 60 40 20 0

T2M T2S 0

10 20 30 40 Bound rubber, % gum

50

phr : 30 60 90 N330 N339 N121

30

Extractable rubber

20

10

0

Loosely BdR Tightly BdR 0

10 20 30 40 Bound rubber, % gum

50

Figure 5.15 Bound rubber and NMR results.

Loosely bound rubber 10–36% elastomer; thickness : 3.0–6.6 nm

Unbound (extractable) rubber 60–90% elastomer

Connecting filament

Tightly bound rubber 2–4% elastomer; thickness : 0.4–1.3 nm

Filler particle Equiv. spherical diameter : 100 nm

Figure 5.16 Tridimensional representation of the morphology of carbon black filled rubber compounds.

as a clear indication of three types of rubber in a filled ­compound: the ­extractable (i.e., unbound) fraction and two types of BdR, with different molecular mobility. There are thus various experimental results and common sense observations that support the morphology of an uncured CB filled compound as depicted in Figure 5.16.20

112

Filled Polymers

Very close to filler particles, there is a thin layer of tightly BdR, which is likely to behave in a flow field exactly as the aggregate. Then there is a region of loosely BdR, i.e., chains attached to the particles—through the tightly BdR region—but able to undergo very large deformation during flow. This region eventually forms connective filaments between rubber–filler aggregates. The third portion is the unbound rubber, so-called because it can readily be extracted from the uncured compound by a good solvent of the elastomer. Obviously this rubber region does not interact with the filler particles. Extractable rubber accounts for 70–90% of the gum elastomer of the compound, depending on the formulation. The tightly and loosely BdR fractions are up to 30% of the gum. The connecting filaments, which are readily seen on published microphotographs,14,15 ensure the coherence of the swollen rubber-black gel in a good solvent. 5.1.4 Bound Rubber at the Origin of Singular Flow Properties of Rubber Compounds It is important to underline here that the schematic description of rubber–CB systems given in Figure 5.16 has to be considered as an “instant” view of materials, which remain essentially (pseudo) fluids above their glass temperature, i.e., practically in their whole processing and application temperature windows. We mean that rubber–CB interactions are likely to be dynamic, through continuous adsorption–desorption processes, at equilibrium at any given temperature. BdR content is therefore nothing else than the assessment of an equilibrium state at the temperature of the test, and therefore, is expected to decrease as extraction temperature increases (as readily observed indeed). If it were not the case, such complex materials would not have the capability to flow. Materials which such a complex structure cannot of course exhibit a simple rheological behavior, and relatively easy arguments may be produced to explain—so far qualitatively—how this complex structure affects most of the flow singularities of rubber compound. As illustrated in Figure 5.17, ­typical nonlinear effects observed with CB filled compounds appear as logical consequences of such a soft three-dimensional network of complex rubber–aggregate entities with connective filaments. The suppression of the Newtonian plateau in the shear viscosity function when the filler content is above 12–13% can be seen as the extension in the low rate range of the nonlinear viscoelastic character, otherwise observed on pure, homogeneous polymers in the high rate range. However such a ­nonlinearity is “internal” (i.e., morphology induced, or “intrinsic”) to filled compounds, a part of their basic character, in contrast to the “external” (i.e., strain induced) nonlinear viscoelasticity of pure polymers that appear when stress conditions overcome a certain limit. With respect to Figure 5.16, it is easy to understand that as the applied stress or rate of strain reduce, the system responds more and more as an elastic network and, therefore, the

Enhanced flow anisotropy effects (converging flow)

Enhanced wall slip

Reduced chaoticity (smoother melt fracture)

Soft three-dimensional network of bound rubber–filler particle units with connective elements

Observed according to conditions and rate of deformation

Stress dissipation effects (lower extrudate swell)

Essentially non linear flow properties

Figure 5.17 Carbon black filled rubber compound morphology and nonlinear flow properties.

Suppression of Newtonian plateau (yield stress behavior)

Bound rubber

Carbon black aggregate

Unbound rubber

Morphology of carbon black filled rubber compounds

Polymers and Carbon Black 113

114

Filled Polymers

apparent viscosity tends to excessive values. Compounds with high levels of reinforcing fillers exhibit practically no flow before stress values as high as 105 Pa are reached, as a consequence of the relative amount of BdR, i.e., elastomer chains with restricted mobility, that is increasing with both the CB content and/or the aggregate structure. The decrease in extrudate swell magnitude with higher CB content, as well as the lower severity of turbulent flow defects (the so-called extrudate melt fracture) can be assigned to a dampening effect due to the filler, through stress dissipation effects involving the complex rubber–­aggregate units and their connective filaments. In their displacement (in the main flow direction), units influence each other and either limit the storage of elastic energy by increasing the viscous dissipation term, or favor the elastic release through local microrelaxation processes. The slippage at the wall of processing equipment is a common sense ­observation on factory floor, likely affected by minor compounding ingredients such as oil, plasticizers, processing aids, and other low molecular weight chemicals. Local drag flow mechanisms of rubber–aggregate flow units, not only throughout the bulk of the material, but also close to the wall, obviously enhance this effect. Flow anisotropy effects are important and easily observed in a number of rubber processing operations and the so-called “grain effect” in calendering is a well-known example. In injection molding, flow anisotropy effects have been shown to persist after vulcanization, as illustrated in Figure 5.18. Such effects arise from interactions between rubber–filler units in such a manner that self-organization of the soft network occur in the main flow direction, particularly when converging or diverging conditions prevail. In a sense the rubber–filler soft network can be viewed as an “elasticity dissipation structure” since such an organization process (which by the way has an antientropy character) consumes a part of the strain energy, otherwise stored in an elastic (i.e., recoverable) manner. 5.1.5  Factors Affecting Bound Rubber Qualitatively, literature data on BdR are quite coherent but, quantitatively, a large scatter is observed, as illustrated by Table 5.6 with data on SBR 1500 compounds from different authors. For a given CB grade, data reported by different authors are within a 5–6% bracket, but the expected effect of filler grade is clearly seen. Simple in its principle, BdR measurement is in fact affected by a number of factors, as follows:

1. The exact nature and formulation of the compound (elastomer type, its molecular weight and MWD, unsaturation level, chemical nature and stability, filler type, its characteristics, level, and dispersion state; effects of other mixing ingredients)

115

Polymers and Carbon Black

0 0 45 90 Angle between flow and stretched directions, ° 100 Chlorobutyl rubber compound 80 60 Shore A 60 Vulcanization at 160°C 40 20 0 0 45 90 Angle between flow and stretched directions, °

2 mm

Test sample cut 0° vs. flow direction

EPDM compound 60 Shore A Vulcanization at 160°C

100 80 60 40 20 0

300% modulus, kPa

180 mm

300% modulus, kPa

Test sample cut 90° vs. flow direction

Tensile test on cured samples 100 80 60 Natural rubber compound 40 60 Shore A Vulcanization at 160°C 20

300% modulus, kPa

Flow anisotropy in injection moulding

0 45 90 Angle between flow and stretched directions, °

Figure 5.18 Flow induced anisotropy effects in filled rubber compounds. (Drawn using data from K. Nakashima, H. Fukuta, M. Mineki, J. Appl. Polym. Sci., 17, 769–778, 1973.)

Table 5.6 Bound Rubber of 50 phr Carbon Black SBR 1500 Compounds; Literature Data Bound Rubber (% Initial Gum) Carbon Black N220 N330 N339 N539 N765



Literature Data (Source) 34.8 (115) 29.7 (115) 34.2 (115) – –

34.8 (112) 30.5 (112) 37.2 (112) – –

37.4 (12) 32.2 (12) 38.3 (12) 26.3 (12) 21.4 (12)

26.3 (13) 22.9 (13) 27.0 (13) 17.6 (13) 13.7 (13)

Average

Standard Deviation

32.8 28.5 34.2 21.9 17.6

5.8 4.9 6.0 6.1 5.4

2. The compound preparation procedure (mixing parameters, energy, time, temperature, storage conditions, duration, temperature, pressure) 3. The test method (protocol, extraction temperature, extraction duration)

A critical comparison of a number of methods described in literature has been published elsewhere13 and we will only draw here the attention on a

116

Filled Polymers

Experimental method

+ Solvent at time ti

Glass device for extraction kinetics Stopper

Extraction during a period ti+1 – ti

Solvent Steel wire cage Collecting at time ti+1

Sample PTFE valve

% Extract vs. initial gum content

100

Extract at time ti+1

% BdR

80 60 40 20 0 0

Weighing residue Extracted mass at time ti+1

[%Extr]t = (100 – [%BdR]) ×[1 – exp(–bt)] 50 100 150 Extraction duration, h

Vacuum evaporation of solvent

200

Bound rubber at time ti+1

Figure 5.19 Extraction kinetic method for bound rubber assessment.

technique, based on the extraction kinetics, that offers the advantage to give access to an “absolute” value of BdR (at the measurement temperature).22 As illustrated in Figure 5.19, a special glass device is used to perform the swelling of test samples in a fixed quantity of solvent for well-defined periods. The sample, cut in small pieces, is weighed in a steel wire basket and disposed in the glass vessel with a polytetrafluoroethylene valve at the bottom. A know quantity of solvent (100 ml) is poured in the vessel and left in contact with the sample; after various periods of time, the solvent (which contains some extracted species) is collected through the valve and another portion of pure solvent is introduced in the vessel for a further extraction period. The procedure is repeated until complete extraction is achieved. Aliquot quantities of collected extracts are evaporated under vacuum at 50°C and the extracted quantity assessed by weighing the dry residue. Test data are thus the extracted rubber (g) for various extraction periods (h). Except for binary formulations (i.e., rubber  +  black), a correction is needed for nonrubber soluble ingredients, generally the oil, the stearic acid, and certain chemicals, by considering that at any time ti, those ingredients are extracted in quantities proportional to the labile rubber. After this correction is made, results are expressed in percent extracted rubber vs. the initial gum rubber content in the compound, and used to draw the extraction kinetics curve (see Figure 5.19). A model for the extraction kinetics is then used to fit such data by non linear regression, according to:

117

Polymers and Carbon Black

[%extracted]t = (100 − [BdR]) × {1 − exp(−β t)}



(5.7)

where [%extracted]t is the percent extracted rubber at extraction time t, [BdR] the BdR content for an infinite extraction time, β an extraction kinetic parameter and t the time. Figure 5.20 shows typical results obtained with this extration kinetic method. CB filled compounds (N330, 50 phr) prepared with either high cis1,4 polybutadiene, ethylene-propylene rubber, or nitrile rubber, and stored at room temparature (under cover) for 28 days, were extracted with toluene at room temperature, according to the method described above. Extraction data were fitted with Equation 5.7 and yielded the results given in Table 5.7. As shown in Figure 5.20, quite long extraction periods are needed before an extraction equilibrium is reached and how fast is this equilibrium reached is clearly depending on the chemical nature of the rubber. Whatever is the rubber however, Equation 5.7 has the capability to perfectly fit experimental data and to yield significant and unambiguous BdR data.

Cumultated extracted rubber, %

100 BdR (%) 20.23 23.31 31.52

80 60

100 80 60 40 20 0

40 20 0

50 phr N330 cpds

0

0

10

20

EPDM NBR BR

30

100 200 Extraction time, h

300

Figure 5.20 Extraction kinetic data on N330 carbon black filled compounds; extractions in toluene at room temperature; formulation: rubber, 100; carbon black, 50; ZnO, 5; stearic acid, 3; oil, 5; antidegradants, 2; elastomers: EPDM, 57.5% ethylene, 4.7% ENB; NBR, 34.5% ACN; BR, 98% cis-1,4.

Table 5.7 Bound Rubber Content from Toluene Extraction Kinetic Method Rubber

Bound Rubber (% Initial Gum)

Extraction Kinetic Parameter β

r2

EPDM NBR BR

20.23 23.32 31.52

0.159 0.470 0.176

0.999 0.988 0.994

118

Filled Polymers

As can be seen, the nature of the rubber significantly affects the BdR c­ ontent, and also the extraction kinetics as expressed through the parameter β. Such results do comply with contemporary theoretical views on the origin of BdR in CB filled systems, as a topological constraint effect, as will be described hereafter. The extraction kinetic method does not only provide the absolute BdR content (at the extraction temperature considered) with a de facto compensation for experimental scatter, but also allows additional information to be obtained by analyzing extracts, for instance by Gel Permeation Chromatography (GPC). The residue can also be recovered, dried under vacuum, and the BdR content crosschecked by thermogravimetry analysis (TGA). The % BdR assessed through the extraction kinetic method is quite reproducible (± 1%) when the test material is well mixed. The above method is applicable only if the swollen rubber–filler gel remains coherent during the extraction process, which is generally the case providing the filler level is high enough, i.e., higher than or close to the so-called percolation level, ~12–13 vol.%, and the dispersion quality satisfactory, i.e., no more agglomerates. When the swollen sample disintegrates, the extracted solution is no longer clear and in extreme cases, it becomes very difficult, if not impossible, to recover the rubber–filler complex. In agreement with Figure 5.16, BdR can be viewed as a gel of (carbon) ­particles with the bonding agent consisting of the longer polymer molecules. Such a consideration is the root of many theories, intending to explain the very origin of the phenomenon, with the goal to derive mathematical models that would allow the BdR content of a given rubber formulation to be calculated from a set of fundamental material parameters. Most of those theories are consistent and lead to useful conclusions (see a detailed discussion elsewhere13) but, in the author’s opinion, ignore important aspects, such the chemical nature of the elastomer, the (known) effects of mixing as well as storage effects.23–25 Figure 5.21 shows Brabender EP2 mixer Chamber volume = 90 cc Rotor speed = 50 RPM Temperature = 80°C

Bound rubber, %

40 30 20

OESBR 1712 - 50 phr N339

OEBR 441 - 50 phr N339

10 0

0

5

10 Mixing time, min

15

20

Figure 5.21 Kinetic aspects in bound rubber formation. (    - Data from G.R. Cotten, Rubb. Chem. Technol, 57, 118 (1984); G.R. Cotten, Plast. Rubb. Proc. Appl., 8, 173 (1987).)

119

Polymers and Carbon Black

Bound rubber (% initial gum)

40

Compounds with 40 phr N330 and other usual ingredients [Leblanc, Hardy (1991)] Natural rubber (SMR 5CV)

35 30 25

Polybutadiene (NeoCis BR40)

20 15

EDPM (Dutral TER054/E)

10

EPR (Dutral CO054)

5 0

0

20

40 60 80 100 120 140 Storage maruration* period (days) * Storage at room temperature under dark plastic cover

160

Figure 5.22 Effects of elastomer and storage on bound rubber. (Data from J.L. Leblanc, P. Hardy, Kautsch. Gummi, Kunstst., 44, 1119–1124, 1991; curves are drawn using a model described hereafter.)

for instance data from Cotten, which suggest that BdR does not form instantaneously during mixing, but according to a ­kinetics that will be discussed later. Moreover, as illustrated in Figure 5.22 both the chemical nature of the elastomer and the storage time (at room temperature) affect the BdR. Let us consider that rubber–CB interactions have a pure physical nature. Then such interactions are likely to be topological constraints exerted on chain segments by the appropriate (geometrical) elements on the surface of filler particles. This simplistic view was considered by Leblanc26 who assumed that, from a strong interaction to occur, the surface topology of a given CB particle must locally encounter the conformation of a chain segment equal to at least three structural units typical of the elastomer considered (Figure 5.23). This would be possible provided the polymer segment of structural units and the filler topological site have the corresponding reciprocal geometry, in the appropriate orientation, and at the right time. During mixing, the probability of such favorable events is obviously quite high. Once this topological interaction has taken place, it is quite obvious that, in order to release it, the free portions of the chain must exert on the constrained units not only sufficient stresses but also in the appropriate direction; a process that would require quite high energy level to be statistically significant for BdR to vanish. The key point in this reasoning is that a logical estimate can now be offered for the size of an “active” site on the filler particle, with respect to the dimensions of the monomer unit; for instance it can be considered as two to three times the half lateral surface of a monomer unit, easily calculated with C–C and C–H bonds length. The chemical nature of the elastomer plays thus, its role through the length of the segment needed for a strong interaction to occur, with chain unsaturation involved through

120

Filled Polymers

Rigid chain motif blocked by the appropriate site on filter surface

Dangling segment Surface of carbon black particle

C–C bond half-free rotation

Dangling segment

Steps-like structure at edges of broken graphic plies Graphitic layers

Figure 5.23 Rubber–carbon black interaction viewed as a topological constraints effect.

the associated local segment rigidity. An appropriate explanation for the low BdR values obtained with ethylene-propylene rubber compounds is in the meantime obtained, i.e., longer segments, ~C24, must conform their morphology with the particle surface, an event which obviously has a lower probability. By somewhat extending an approach by Cohen–Addad for Silica-PDMS27 the variation of BdR vs. storage time (at constant temperature), can be considered as follows:

 M0 c Sp  BdR(t) =  M n (0) + [ M n (∞) − M n (0)]( 1 − exp(−βt))  A0 N Av 

(5.8)

where M0 is the weight of one skeletal bond, c the filler concentration (g/g of gum polymer), Sp the specific surface area of CB (m²/g), A0 the ­average area of one interactive site (nm²), NAv the Avogadro number, M n (0) and M n ( ∞ ) the number average molecular weight (g/mol) of chains involved in BdR, respectively immediately at dump and after an infinite storage time, t the time and β a kinetic parameter describing the storage ­maturation. Data in Figure 5.22 were fitted with this equation. The view that BdR is at best the adsorption–desorption balance of rubber segments on CB sites at a given time (and temperature) is clearly supported by the above model. As long as an equilibrium is not reached however, the adsorption–desorption mechanism evolves while the compound is at rest, hence the observed variations upon storage. How fast this equilibrium BdR is reached depends on the chemical nature of the rubber, on the compound formulation, on the mixing and the dump compound storage conditions.

121

Polymers and Carbon Black

In the topological constraints effects behind rubber–filler interactions, there is a competition between short and long chain segments. During ­mixing, owing to the strong flow fields prevailing in the mixer, one considers that all polymer fractions have equal probability to have segments arriving in close contact with the appropriate sites on filler particles, in order to develop a topological interaction. After dump, when the material is at rest, segments from short chains are progressively displaced from active sites by segments from long chains, leading to an increase in the average molecular weight of BdR. This is exactly the mechanism (mathematically) expressed by the last right member of Equation 5.8. 5.1.6  Viscosity and Carbon Black Level How the addition of small solid particles to a liquid affects its viscosity (and other properties) is a long lasting problem, with many theoretical attempts. With respect to the above sections, that underlined specific aspects due to CB nature and characteristics, it is clear that no global understanding can yet be expected, that would be valid whatever the fluid and the filler. Let us consider that the fluid is a polymer (melt); then the complexity of the problem can nevertheless be stated in the form of a general functional, as follows:

Pcpd = Ppolym × F ( Φ, d , S, X , T ,…)

(5.9)

where Pcpd is the property (for instance the shear viscosity) of the mixture polymer  +  filler, Ppolym the same property of the pure polymer, F() an appropriate mathematical function, Φ, d , S, X , T ,... the variables of this function, respectively the filler volume fraction Φ, a parameter d related to particle size, another one S describing the structure of the filler, then the rate of deformation X and the temperature T, etc. All theoretical approaches to describe the effect of filler fraction on compound viscosity can then be considered as attempts to partially describe the functional F().  and T. In other terms, one As a first approach, one may fix parameters S, X, considers how the viscosity of a Newtonian fluid, in isothermal conditions, is affected by the volume fraction of equal diameter spherical particles. With ηpolym and ηcpd the viscosities respectively of the pure polymer and of the filled compound, the following quantities are defined: • The specific viscosity: ηsp =

ηcpd − ηpolym ηpolym

• The relative viscosity (of the compound): ηrel = It follows that ηsp = ηrel −1.

ηcpd ηpolym

122

Filled Polymers

Historically, one of the earliest approaches was made by Einstein who considered a Newtonian medium [of viscosity ηpolym ≠ f ( X ) ] in which rigid spherical particles of equal diameter are suspended. Providing the number of particles is small enough for no interaction to occur between spheres, such a suspension flows macroscopically in an apparently simple manner, when submitted to a shear stress. However, when carefully observed on a finer scale, it becomes obvious that, in the neighborhood of a particle, the flow is not homogeneous because the liquid must flow around it. Thus, the local shear rate of the liquid itself varies from point to point, and the average value is larger than the overall rate of shear of the whole suspension. Consequently, the global viscosity of the suspension is greater than the viscosity of the suspending liquid. Einstein considered a suspension so dilute that no hydrodynamic interaction occurs between different particles, and treated the liquid as a continuum in which ordinary laws of hydrodynamics apply. He then established the well-known equation:



ηcpd = ηpolym (1 + 2.5 Φ)    or   ηsp =

ηcpd − 1 = 2.5 Φ ηpolym

(5.10)

where Φ is the volume fraction of suspended (spherical) particles. With respect to hypotheses in Einstein’s analysis, the applicability of Equation 5.10 is limited (very dilute suspension of rigid spherical particles) and ­several authors,29–36 by using similar arguments, have considered other particle geometries (ellipsoids, rods, disks) and/or slightly larger volume fraction, as summarized in Table 5.8. Owing to their mathematical simplicity, quadratic (and cubic) models (see Figure 5.24) for rigid spheres suspension are attractive for low reinforcing carbon blacks (e.g., N990) for which, for instance, the well known Guth, Gold, and Simha equation fit well data up to a volume fraction equal to 0.20. The cubic model by Vand was found to give good fit for glass spheres suspensions (0.013 cm diameter) up to Φ = 0.37. With reinforcing carbon blacks, the best spatial envelope for an aggregate is the ellipsoid and, owing to its mathematical simplicity, models proposed by Guth and Gold for spheres and rigid revolution ellipsoids drew the attention of several authors. For instance, White and Crowder37 considered the following equation:

2 ) ηcpd = ηpolym (1 + 2.5 × f × Φ eff + 14.1 × f 2 × Φ eff

(5.11)

where f is an anisometry factor and Φeff an effective filler volume fraction that takes into account the BdR, i.e., Φeff  = Φblack  +  ΦBdR, where ΦBdR is the ­volume fraction of BdR.

L  : anisometry factor D

L = large axis D = small axis

f=

Rigid rods

Rigid revolution ellipsoids

Rigid spheres with equal diameter

Type of Filler

   f2 f2 14   ηsp = + +  Φ   ( Simha )  15  ln(2 f ) − 3  5  ln(2 f ) − 1  15      2  2 

    f2 ηsp =   Φ    (Eisenschitz) 45  15 ln(2 f ) −   2 

f2   ηsp =  2.5 +  Φ    (Huggins, Kuhn) 16  

f   ηsp =  + 2  Φ    (Jeffery) 2 ln( 2 f)− 3  

ηsp = 0.67 f Φ + 1.62 f 2 Φ 2    (Guth, Gold)

ηsp = 2.5Φ   (Einstein)

Low Volume Fraction

Effect of Fillers on (Newtonian) Viscosity—Theoretical Approaches

Table 5.8

(Guth, Gold, Simha)

K = constant, close to 1

(Continued)

Kf 2   2 f   Φ    (Guth, Gold) ηsp =  + 2 Φ +  − ln( ) ( ln( 2 2 f 3 2 2 f ) − 3)2    

ηsp = 2.5 Φ + 7.17 Φ 2 + 16.2 Φ 3 (Vand)

ηsp = 2.5Φ + 12.6Φ 2 (Simha)

ηsp = 2.5 Φ + 14.1 Φ 2

Medium Volume Fraction

Polymers and Carbon Black 123

f=

L  : anisometry factor D

Rigid disks

Type of Filler

Table 5.8  (Continued)

 16 f  ηsp =   Φ    (Simha)  15 a tan( f ) 

4f   ηsp =   Φ    (Guth, Jeffery)  3 a tan( f ) 

2

 4f  ηsp =   Φ    (Jeffery)  3 a tan( f ) 

Low Volume Fraction

Medium Volume Fraction

124 Filled Polymers

125

Polymers and Carbon Black

Specific (Newtonian) viscosity

6

(Guth, Gold, Simha) ηsp = 2.5 Φ + 14.1 Φ2 (Simha) ηsp = 2.5 Φ + 12.6 Φ2 ηsp = 2.5 Φ + 7.17 Φ2+ 163 Φ3 (Vand)

4

2

0

ηsp =

0

ηcpd ηpolym

ηsp = 2.5 Φ

–1

0.1

(Einstein) 0.2 0.3 Volume fraction Φ

0.4

0.5

Figure 5.24 Effect of filler content on (Newtonian) viscosity; models for rigid spherical particles.

5.1.7 Effect of Carbon Black on Mechanical Properties Once they have been shaped into a suitable object, rubber compounds are vulcanized in such a manner that the full development of their mechanical properties is achieved, without creep phenomena that are normally exhibited by all polymer materials when on their rubbery plateau. In other terms, vulcanization extends toward infinity the rubbery plateau of the relaxation modulus function G(t). Furthermore, reinforcing fillers somewhat increase the magnitude of the modulus at a given time. The combination of vulcanization and reinforcing effects induces quite complex changes in material functions of polymers, as easily demonstrated through purposely simple calculations. Let us consider, for instance, the relaxation modulus function G(T) of a pure SBR, as reported by Nielsen38 (Figure 5.25). The effect of CB loading (at constant temperature) can be approached by rewriting the Guth, Gold and Simha equation as follows:

Gcpd = Gpolym (1 + 2.5 × Φ + 14.1 × Φ 2 )

(5.12)

where Gpolym and Gcpd are the moduli, respectively of the pure polymer and of the compound with a filler volume fraction Φ. The filler incorporation effect can be simulated by calculating point by point Gcpd with several values of Φ, for instance from 0 to 0.4. The Gcpd (T) functions obtained for all the Φ considered are drawn vs. an arbitrary scale of the process. To simulate changes imparted by vulcanization, we may consider that the modulus varies from its actual value in the so-called flow (or terminal) zone, i.e., Gterm, to a value corresponding to the rubbery plateau, i.e., Gvulc, using a simple sigmoid function, for instance:

126

Filled Polymers

Shear modulus (M

Pa)

1e+4 1e+3 1e+2 1e+1 1e+0 1e–1 1e–2 1e–3 1e–4 1e–5 1e–6 1e–7 1e–8 –60–40 –20 0 20 Tem 40 per 60 80 atu re ( 100 °C) 120

Carbon black filled SBR 1500 compound

Scale = 100 Shear modulus function for vulcanized cpd 100 ) 80 ale sc 60 y r ra Modulus variation 40 bit r a with vulcanization 20 e( tag s s 0 es oc r Modulus increase P due to filler addition Scale = 0 Shear modulus function for gum rubber

Figure 5.25 Expected variation of modulus function during filler loading and vulcanization.



G(t) = Gterm +

Gvulc − Gterm  t 1+    τ

α



(5.13)

where G(t) is the value of the modulus for a given vulcanization time t, τ the necessary (curing) time to obtain G(τ) = (Gterm + Gvulc)/2, and α a ­parameter related to the vulcanization rate. The G(T) curves so obtained have been drawn vs. the arbitrary scale of the process. As can be seen in Figure 5.25, filler loading and vulcanization drastically modify the modulus function, with the largest changes occurring during vulcanization. As previously underlined (Chapter 3), CB reinforcement must be appreciated with respect to a balance of properties, some of them ­modified in antagonistic manner when filler loading is changed. This commands to consider first how specific mechanical properties are modified through CB addition. Tensile modulus, and elongation are, by far, the easiest mechanical properties to measure. If globally, ultimate tensile properties (i.e., stress strength and elongation at break) are improved through reinforcing CB addition, the magnitude of the effect depends on both the chemical nature of the elastomer and the temperature. Figure 5.26 compares the effect of

127

Polymers and Carbon Black

30

30

Stress at break, MPa

Stress at break, MPa

NR gum NR + 30 phr N330 20

10

Styrene-Butadiene rubber SBR gum SBR + 30 phr N330

20

10

Natural rubber 0

0

40

80 120 Temperature, °C

160

0 0

40

80 120 Temperature, °C

160

Figure 5.26 Effect of temperature and carbon black on tensile stress at break.

30 phr N330 CB in NR and SBR vulcanizates.39 The tensile strength (TS) of the unfilled NR compound varies with temperature, in such a manner that a drastic decrease is observed at around 80°C. This behavior is explained by a strain crystallization effect of NR below this temperature. It follows that the effect of a reinforcing CB depends on temperature. The addition of 30 phr N330 increases the TS by 20% when the temperature is lower than 80°C, and by 200% above 80°C. SBR does not exhibit strain crystallization effects and therefore only a slight decrease of the TS with temperature is observed. A room temperature, adding 30 phr N330 increases the TS by a factor of 10, and only doubles it when the temperature is above 120°C. Tensile stress softening (TSS), the so-called Mullins effect,40 is an important phenomenon with CB filled rubber compounds, which describes the fact that all elastomers, either vulcanized or not, either filled or not, require a higher stress at equal strain during the first extension than during further ones. As schematically described in Figure 5.27, a vulcanized rubber sample is first stretched until an elongation ε1 is reached, then the stress is released. From ε1 to ε = 0, the stress–strain curve is below the trace recorded during the first extension, and there is a residual strain. The first extension produces much larger changes in the stress–strain curve than subsequent extensions, in such a manner that after three to four cycles, traces are practically superimposed. A common explanation for the Mullins effect is that, between the first and the second extension, the structure of the material has changed and it is generally observed that the higher the CB loading, the larger this effect. TSS is not solely due to the presence of the filler, whilst the latter enhances the effect. When a vulcanized rubber sample, which has been stretched twice, is submitted to a thermal treatment (for instance several hours at 80°C), it recovers

128

Filled Polymers

20 Extension : 1

2

NR gum

Stress, MPa

15 10 5 0

0

1

2

3

20 Extension : 1 2 NR/N330 (60 phr)

Stress, MPa

5

10

7

2nd extension 0

1

2 Strain

8

1st extension

5 0

6

“Mullins” effect 2nd and further extensions

Stress

15

4 Strain

3

Residual strain 4

ε1

Elongation

Figure 5.27 Tensile stress softening (Mullins effect).

essentially the first strain behavior, at least with certain types of vulcanization systems (mono-sulfide systems or peroxide curing). This means that no rupture process is involved in the TSS effect, which is therefore attributed to a quasi-irreversible rearrangement of the molecular network, because local deformations are not affine. By extending the works of Guth, Gold, and Simha on viscosity–filler fraction effects, one can express the effect of CB loading on modulus with a similar expression, but using an appropriate anisometry factor to somewhat take into account the nonspherical shape of CB aggregates, i.e.:

Ecpd = Epolym ( 1 + 2.5 × f × Φ + 14.1 × f 2 × Φ 2 )

(5.14)

This approach implicitly considers that CB effect on modulus is essentially hydrodynamic. The anisometry factor (or form factor) reflects the ratio large axis/small axis of a revolution ellipsoid, considered as the best envelope for an aggregate. When using f = 6, it is found that Equation 5.14 fit well experimental data with N330 CB up to Φ = 0.3. Note that alternative equations have been proposed, for instance by Guth:

Ecpd = Epolym (1 + 0.67 × f × Φ + 1.62 × f 2 × Φ 2 )

(5.15)

129

Polymers and Carbon Black

If however, one considers f as an adjustable parameter and one uses equation Equation 5.14 (or Equation 5.15) to fit (using nonlinear regression algorithm) modulus data derived from first and second stress strain curves of a Mullins type experiment, then one notes that the parameter f has changed between the two stretching. This suggests considering that there is a kind of transient rubber–CB structure which is affected by the strain. Medalia proposed another approach, through the equation:

2 ) Ecpd = Epolym (1 + 2.5 × Φ eff + 14.1 × Φ eff

(5.16)

where Φeff is the effective filler volume fraction, considered either with respect to BdR content,41 i.e.,

 1 + 0.02139 × DPBA  Φ eff = Φ black   1.46 

(5.16a)

Φeff = Φblack + ΦBdR

(5.16b)

or using:

where DBPA is the dibutylphthalate adsorption number of the CB considered. Medalia proposed Equation 5.16a on the basis of his concept of equivalent spherical envelope for CB aggregates. The constants in this equation have been derived from Transmission electron microscopy (TEM) observations on a selection of ­carbon blacks, combined with several reasonable hypotheses. At best the Φeff so calculated is an estimation and does not take into consideration the mere fact that BdR level is strongly depending on the rubber nature. It is therefore always preferable to measure BdR and to use Equation 5.16b. A similar but slightly different equation, i.e., Φeff = Φblack[(1 + 0.0181 × cDPBA)/1.59] was proposed when “crushed DBPA” is used to characterize the CB.43 In order to explain the effect of CB on mechanical properties, Mullins introduced an interesting concept: strain amplification (Figure 5.28). At microscopic level, a filled rubber compound consists of “hard regions”, i.e., where CB concentration is locally high, and of “soft regions”, i.e., with locally low black level. Such a view, in fact readily seen by microscopy with a suitable magnification, corresponds very well with the fact that a part of the elastomer in unvulcanized mixes is extractable with a good solvent, and therefore, is not interacting with the filler. It follows that, when a stress is applied to a vulcanized compound, soft regions essentially support the overall strain, before hard ones are affected (in which case the TSS effect is observed). The presence of CB particles brings therefore, an amplification of the average strain supported by the elastomer.

130

Filled Polymers

Unfilled vulcanizate ×2 Stretching

10

Filled vulcanizate Soft phase (elastomer)

Strain amplification factor

×2 Stretching

2.000

2.429

2.000 Hard phase (carbon black) "

1.000

"

10

7.285 1.000

Effective strain

λ´ =X λ

Measured strain Non reinforcing carbon black (spherical) X=1 + 2.5 Φ + 14.1 Φ2

4.857 Only the soft phases support the strain

1.000 2.429

The strain supported by the elastomer phase is larger in the filled than in the unfilled material

Reinforcing carbon black (anisometrical) X=1 + 0.67 f Φ + 1.62 f 2Φ 2

(empirical) an isometry parameter

Figure 5.28 Concept of strain amplification in filled elastomers.

By considering an ideal network, the elasticity theory provides a relationship for stress dependence upon strain, found applicable at (very) low strain, i.e.:

σ=

 E0  1 1 λ − 2  = NkT  λ − 2  3  λ  λ  

(5.17)

where E0 is the Young’s modulus, N the fraction of active elastic chains in the network, k and T the Bolztmann constant and the temperature respectively and λ the strain (λ = L/L0; L0 = initial length of test sample; L = stretched length). For larger strain (i.e., up to λ ≈ 3.0), the semiempirical equation by Mooney and Rivlin is applicable, i.e.:

1 C    σ = 2  C1 + 2   λ − 2   λ  λ 

(5.18)

where the constant C1 and C2 (to be determined experimentally) are ­respectively a function of the fraction of active elastic chains and of the type and concentration of CB, as demonstrated by experimental data (Table 5.9).44 As can be seen, C1 does not depend on the type and level of CB, contrary to C2. In fact, the C2 term in the Mooney–Rivlin equation somewhat expresses the deviation with respect to the ideal elastic network (as readily seen when comparing Equations 5.17 and 5.18). When using this equation

131

Polymers and Carbon Black

Table 5.9 Effect of Carbon Black Type and Level on C1 and C2 Constants in Mooney–Rivlin Equation Effect of Volumic Fraction ϕ

Filled Carbon Black Compounds SBR 1500 Carbon black ZnO Stearic acid Sulphur CBS

100.00 variable 5.00 1.00 1.75 1.00

Black – N330

Effect of Black Type at ϕ = 0.20

N550

Black

C1 (MPa)

C2 (MPa)

N220 N330 N550 N660 N770

0.20 0.20 0.20 0.21 0.26

0.96 0.86 0.74 0.67 0.57

λmax 1.5 1.6 1.7 1.8 1.8

N762

ϕ 0 0.05 0.10 0.15 0.20

C1 (MPa) 0.13 0.15 0.16 0.17 0.20

C2 (MPa) 0.21 0.28 0.45 0.62 0.86

λmax 3.0 2.5 2.0 1.8 1.6

0.05

0.15

0.24

2.5

0.10

0.15

0.43

1.8

0.15 0.20 0.05 0.10 0.15 0.20

0.16 0.20 0.15 0.16 0.16 0.18

0.63 0.74 0.23 0.60 0.52 0.61

1.6 1.7 3.0 2.0 1.8 1.6

in the case of a system filled with CB, it is implicitly accepted that the mean effect of the filler consists in increasing the effective strain of the elastomer phase. And because CB aggregates are rigid, the local strain of the rubber matrix is larger that the overall measured deformation. Within identical validity limits, Mullins and Tobin45 have shown that the stress–strain behavior of black-loaded rubber vulcanizates corresponds to the stress–strain response of pure gum vulcanizates multiplied by a suitable strain amplification factor X, which expresses the fact that the average strain supported by the rubber phase, is increased by the presence of filler. In other terms, the effective strain of the elastomer matrix λ′ is given by: λ′ = λ × X, where λ is the overall measured deformation of the filled material. The strain amplification factor is obviously depending on the filler loading and likely on the filler characteristics, firstly the structure. For a nonreinforcing CB (i.e., N990), which consists essentially of spherical ­particles with mean diameter of about 400 nm, Mullins and Tobin showed that an appropriate equation for the strain amplification factor is:

X = 1 + 2.5 × Φ + 14.1 × Φ 2

(5.19)

However, for a finer reinforcing black, a more suitable equation is:

X = 1 + 0.67 × f × Φ + 1.62 × f 2 × Φ 2

(5.20)

132

Filled Polymers

where f is an empirically determined value for the CB grade considered. For instance f ≈ 0.65 was found suitable for N330, which consists of aggregates of up to 300 nm (longest dimension). Both equations are essentially empirical, with an obvious implicit reference to the works of Guth, Gold, and Simha. Recently, advanced Atomic Force Microscopy (AFM) techniques have provided quite a ­convincing demonstration that in unfilled NR vulcanizates, tensile deformation is not homogeneous;46 in other words, the deformation is far to be affine. It is fairly obvious that if nonaffine deformation is observed on unfilled systems, ­nonhomogeneous deformation is surely the rule with filled rubber vulcanizates. Of course the “strain amplification” concept of Mullins and Tobin somewhat recognizes the macroscale occurrence of such inhomogeneities and, moreover, the BdR concept is also an explicit demonstration that, close to CB particles, there are rubber segments that are immobilized, so that they are in a pseudo-glassy state. Except maybe at excessive strain, the tightly BdR is not expected to support the strain, so that only the loosely BdR and the extractible rubber ­support it. But of course the stress is mainly supported by the filler particles and the tightly BdR. Local nonhomogeneous deformation is thus the rule in rubber materials, so that all theoretical considerations that either implicitly or by hypothesis, consider affine deformation are bound to fail or, at least, to have a very narrow applicability range. The TSS softening effect, when observed with an unfilled rubber vulcanizate, is thus attributed to the nonaffine displacement of the junction points of the network and to the incomplete recovery of their original positions after the strain is back to zero. It follows that the TSS effect is larger in CB filled systems (see Figure 5.27 for instance) because of the strain amplification effect. It must be noted here that other explanations have been proposed for the reinforcing effect of CB. Blanchard47 considered that CB strongly modifies the nature of the elastomer network with, amongst other features, strong and weak links between rubber segments and appropriate sites at the surface of particles. Recent considerations are close to this concept, as we shall see later. Up to this point, the strain amplification factor can be viewed as a mere empirical approach to assign the modulus increase in CB filled compound to filler level. Equation 5.19 above essentially resulted from considerations on the hydrodynamic effects induced by the presence of solid particles ideally dispersed in a matrix with a considerably lower modulus. The empirical factor f in Equation 5.20 adds nothing in this respect and it is well known that both equations do not suit at all either highly loaded compounds, whatever is the grade of CB, or moderately loaded materials with high structure blacks. Over the last decades, several authors have developed theoretical considerations to model the likely effect of a socalled filler network structure and the associated energy dissipation process when filled compounds are submitted to increasing strain.

133

Polymers and Carbon Black

Earlier works by Medalia and coworkers demonstrated that certain key mechanical properties of carbon filled compounds were related to the product of the filler volume fraction × the overall rubber-filler interfacial area per unit volume of compound.48,49 More recently, Wang et al.43 observed a good correlation between dynamic properties, namely tan δ, and an ­interaggregate distance, which was calculated with respect to a random distribution of equivalent spheres and the concept of occluded rubber. They concluded therefore that filler particle attractive forces and interaggregate distance were controlling factors for the so-called filler network and the resulting enhancement of compound’s modulus. A CB filled rubber compound can therefore be considered as a kind of soft three-dimensional network in which aggregates would act as “anchoring knots” for several elastomer chains. One part of the rubber would consequently plays a particular role in connecting CB aggregates, in other words, in making “junctions” between them. Such considerations are obviously well in agreement with the concept of “BdR,” as discussed above (Section 5.1.3). By paying attention to this “junction rubber,” Ouyang et al.50,51 developed an interesting approach, generally referred as the network junction (NJ) theory, in which the interaggregate distance is the central argument. There are unfortunately a number of misprints and unit inconsistencies in the published equations, which prompted the author to reconsider the theoretical development in parallel with parametrical verifications (see details in Appendix 5.1). Let us consider a given mass MCB of a carbon black at the end point of the so-called “crushed” dibutyl phtalate adsorption test (ASTM D-3493; see Section 4.1.3 above). At this stage, all the aggregates of the CB sample are likely exhibiting the most compact arrangement in a DBP matrix and, with respect to the overall volume of the DBP + CB mixture, the filler volume fraction is maximum and can be assessed from:



Φ maxCDBP =

MCB ρCB

MCB + cDBP ⋅ MCB ρCB



(5.21)

where ρCB is the specific gravity of the CB and cDBP its crushed DBP adsorption number (cm3 DBP/100 g of filler). As we have seen (Section 4.1.4, Equation 4.7), the solid volume of an aggregate is given by: Vs = Np(πd3/6), where Np is the number of elementary particles of diameter d. The number of aggregates NaCB in MCB grams of CB is thus:

NaCB =

MCB π d3 Np ρCB 6

(5.22)

134

Filled Polymers

and therefore, the number of junctions (or contact) points between ­aggregates is NaCB(ς/2) if ς is the average number of contact points between ­neighboring aggregates. If one makes the (strong) hypothesis that in a rubber compounds, dispersed aggregates keep the same compact arrangement as in DBP, except that the distance between neighboring aggregates is expanded by a distance hgap, and if a cubic arrangement of aggregates is considered for the sake of simplicity (Figure 5.29), it comes that the CB volume fraction is given by:

Φ CB =



MCB ρ 1/3 1/3   MCB ζ    + CDBP ⋅ M + h ⋅ Na ⋅ CB  gap    ρ  CB 2    

Compact cubic arrangement of carbon black aggregates (cDBP absorption test)

Compact cubic arrangement aggregates in rubber matrix

3



(5.23)

d

D

hgap

Aggregates of Np elementary particles 150

100 Particles 200 Particles 300 Particles

100

0 0 5 10 Number of contact points

N330, Ф = 0.20

Junction gap width, mm

Junction gap width, mm

150

100

50

0

2 Junctions 5 Junctions 10 Junctions

0 100 200 300 Number of primary particles/aggregate

Figure 5.29 Network junction model; the two graphs were calculated with Equation 5.25 and the following data: CTAB = 41.32 m2/g, ρCB = 1.86 g/cm3, Φmax = 0.38, ΦCB = 0.20.

135

Polymers and Carbon Black

In the rubber compound, the maximum CB volume fraction is thus, lower than in the mixture with DBP because rubber chains move apart neighboring aggregates by the distance hgap. An expression can then be derived for the junction distance, i.e.:



hgap

 3ς  =  π N p 

− ( 1/3 )

 1 d3 −  Φ CB

3

1  Φ max 

(5.24)

where ς is the average number of contact points between neighboring aggregates, Np the number of elementary particles of diameter d in an aggregate, ΦCB and Φmax, respectively the filler volume fraction and the maximum packing fraction of the CB grade in the rubber compound. How the junction gap width varies with both the number of contact points and the number of primary particles per aggregate is illustrated by the graphs drawn at the bottom of Figure 5.29. If one considers that an aggregate consists of a simple assembly of touching elementary particles, one has the following relationship between the specific surface area (in m 2/g) and the elementary particles diameter d: Ssp = 6/dρ. Equation 5.24 can consequently be rewritten as:



 3ς  hgap =   π N p 

− ( 1/3 )

6  1 3 − ρ Ssp  Φ CB

3

1  Φ max 

(5.25)

So, according to this approach, in a given compound, the junction gap hgap is inversely proportional to the specific surface area of the CB, as measured for instance by the adsorption of CTAB (ASTM D3765; see Section 4.1.3 above). In addition, higher structure (i.e., higher CDBP adsorption) blacks will give shorter junction gap because the Φmax would be smaller. To readily use Equation 5.25, one needs characterization data on CB, namely the specific area, the (average) number of particle per aggregate and the number of junctions per aggregate. Characterization data as compiled from various sources were previously given in Table 5.5 and Np can be estimated with Equation 4.10 (Section 4.1.4). Providing a fair hypothesis is made about ς, for instance six junctions per aggregate with an explicit reference to a simple cubic arrangement of aggregates, then the data in Table 5.10 can be calculated (see details in Appendix 5.1). Note that the maximum volume fraction as calculated from CDBP adsorption (Equation 5.21) was used. In agreement with the theoretical considerations above, the more reinforcing the CB the smaller the junction gap width. It is worth noting that a diene rubber with a molecular weight in the 200–400,000 g/mole range has a

136

Filled Polymers

Table 5.10 Junction Gap Width as Calculated with Equation 5.25 for 20% Volume Fraction Carbon Black and Six Junctions per Aggregate

Carbon Black N110 N220 N330 N550 N660 N774 N990

Elementary Particle Diameter (nm)

Number of Particles per Aggregate

Φmax

hgap (nm)

18.32 20.95 30.50 50.58 59.28 87.20 294.90

278 285 209 298 170 97 26

0.354 0.354 0.382 0.390 0.424 0.460 0.592

28 32 44 100 101 114 286

random coil diameter in the 20–40 nm range. It means that for the reinforcing CB grades, one macromolecule is sufficient to establish a junction between two neighboring aggregates. For less or no reinforcing grades, the junction will always involve entangled rubber chains with, as a consequence, lower reinforcing effects. A relationship for the strain amplification factor can be derived from the NJ theory by considering the analysis made by Gent et al.52,53 of the behavior of a rubber volume bonded between two rigid spheres. Indeed when a compressive (or a tensile) force F provokes a small displacement ∆x of one sphere with respect to the other, there is a compression (or tensile) ­stiffness which consists of two parts: (1) the rubber layer compressed (or stretched) between the two spheres, (2) the restraints at the bonded surfaces of the spheres. Gent and Park developed the following theoretical equation, i.e.:



π  F h  D + h  2 D2 + D h − 2 h 2  + = ⋅  1 +  ⋅ ln    h  2 h (D + h) E0 ⋅ D ⋅ ∆x 8  D 

(5.26)

where E0 is the modulus of the rubber, D the diameter of the spheres and h the initial gap. Experimental data, as well as finite-element method results confirm the validity of the model but its mathematical form makes it not easy to handle. However, Equation 5.26 essentially predicts an inverse dependence of the stiffness upon the ratio h/D for relatively large separation, i.e., in a layer thickness range down to one-tenth of the sphere radius. When using logarithmic scales, such a dependence appears to correspond to a straight line with a negative slope. It is therefore, attractive to approximate Equation 5.26 with a simple power law (see Appendix 5.1 for details). Using data ­published

137

Polymers and Carbon Black

by Gent et al., it can indeed be shown that a much simpler equation can be considered, i.e.: π  h F = α ⋅ ⋅  8  D E0 ⋅ D ⋅ ∆x



−β



(5.27)

with α ≈ 1.22 and β ≈ 0.96. If one considers a volume V of filled compound, the overall number of CB aggregates is N/V = ΦCB × ρCB × NaCB/MCCB, where MCB is the mass of CB in the compound that corresponds to the volume fraction ΦCB. With respect to Equation 5.22, it follows that in a cross sectional area of the sample (i.e., 2  3 V  ), the number of aggregates is:  N  V 



2/3

 6 Φ CB  =  π N 

2/3

p

⋅

1 d2

(5.28)

When the sample is stretched (of compressed) along one axis by a strain ∆X, there will be (N/V)2/3 junction gaps contributing to the stress, in addition to the response of the unfilled rubber fraction. With respect to Equation 5.27, the junction gaps contribute to a force over the cross section V 2/3 and, because obviously ∆X = ∆x ⋅ (N/V)1/3, the NJ contributes to the modulus by a quantity: FJ  N = ∆X  V 



1/3

π  hgap  ⋅ E0 ⋅ Des ⋅ α ⋅  8  Des 

−β



(5.29)

where Des is the equivalent sphere of an aggregate, with reference to Medalia’s floc simulation (see Section 4.1.4). When expressing Des in terms of number of elementary particles and substituting Equations 5.24 and 5.28 into Equation 5.29, it comes:

FJ π  6 Φ CB  = E0 ⋅ α ⋅ ⋅  ∆X 8  π N p 

( 1+β )/3

⋅N

ε (β + 1)/2 2 p

 ζ ⋅   2

β/3

  Φ CB  1/3  ⋅ 1 −      Φ max  

−β

(5.30)

If one considers that, over the cross section, there is also a contribution from the rubber matrix alone, then the associated stiffness is estimated as: FR /3 = E0 ⋅ ( 1 − Φ 2CB ) ∆X



(5.31)

The strain amplification factor Xf is eventually given by: π  6 Φ CB  Xf = α ⋅ ⋅  8  π N p 

( 1+β )/3

⋅N

ε (β + 1)/2 p

 ζ ⋅   2

β/3

  Φ CB  1/3  ⋅ 1 −      Φ max  

−β

/3 + ( 1 + Φ 2CB B ) (5.32)

138

Filled Polymers

Equation 5.32 is quite different from the equation published by Ouyang for at least two reasons: first we have used the Medalia’s aggregate equivalent sphere diameter in exploiting the two spheres model of Gent et al., second misprints may be suspected in Ouyang’s publications because several of his equations are suffering from unit inconsistencies. It is worth comparing the effectiveness of Equation 5.32 both in meeting experimental data and with respect to the well known Guth and Gold approach, as considered by Mullins and Tobin. In the author’s opinion, such a comparison is all the more valid if data from the literature are used. Unfortunately not many full sets of data are available with all the necessary information to perform such an exercise. The graphs in Figure 5.30 were drawn using G′ data from Payne and Whitakker39 on butyl rubber compounds with HAF CB (that we assimilated to N330, in order to calculate Np and Φmax with the average DBP and CDBP data from Table 5.5), E′ data from Caruthers et al.48 on SBR1500 compounds with N347 (from which we extrapolated the unfilled compound value, not given by the authors) and G′ data from Wang.54 In using Equation 5.32, the parameters α and β were respectively taken as 1.223 and 0.964 and two junctions per aggregate were considered (i.e., ζ = 2) ; details of the calculation are given in Appendix 5.1. As can be seen, when compared with the Guth and Gold equation, the NJ model predicts generally a sharper strain amplification factor as CB volume fraction increases, but fits better the ­experimental data in the case of SSBR compounds data. In applying Equation 5.32 however, parameters α, β, and ζ were fixed and the only CB dependent parameters were Np and Φmax, readily calculated from available DBP and cDBP data, respectively. As shown in Figure 5.30b, when using the maximum theoretical volume fraction for a cubic arrangement of spheres of equal ­d iameter (i.e., Φmax = 0.524), and only playing with the value of Np considerably improves the manner the model is meeting experimental data. The model is not perfect but, in the author’s opinion, the above comparison nevertheless demonstrates the validity of the NJ theory, particularly with respect to all the assumptions and simplifications made. Indeed, in a given CB sample, aggregates are varying in shape and size and do not conform really to equal diameter spheres. The effective maximum packing fraction for a given black is thus, likely different. Similar comments apply to the calculated number of particles per aggregate but in a lower extent because when considering either 100 or 300 particles/aggregate the terms involving Np in Equation 5.32 vary from 0.50 to 0.42. Finally in the calculations above, considering only two junctions per aggregate was an arbitrary choice; using six instead of two would give around 50% more weight to the associated term.

139

Polymers and Carbon Black

G´ modulus, MPa

(a) 10

Butyl rubber/N330 cpds G´0 = 0.2 MPa Guth & Gold equation Network junction model Np = 209 Φmax = 0.390

5

0

0

0.1 0.2 Carbon black volume fraction

0.3

E´ modulus, MPa

(b) 40

SBR1500/N347 cpds E0´ = 4.625 MPa Guth & Gold equation Network junction model Np = 333 Φmax = 0.357

20

0

Network junction model Np = 200 Φmax = 0.524 0

0.1 0.2 Carbon black volume fraction

0.3

G´ modulus, MPa

(c) 20

SSBR/N234 cpds G´0 = 0.769 MPa Guth & Gold equation Network junction model Np = 348 Φmax = 0.347

10

0

0

0.1 0.2 Carbon black volume fraction

0.3

Figure 5.30 Comparing the strain amplification factor as calculated from the network junction theory with experimental data and the Guth and Gold equation: Butyl rubber compounds data are from A.R. Payne, R.E. Whittaker, Rubb. Chem. Technol., 44, 440–478, 1971; SBR1500 compound data are from J.M. Caruthers, R.E. Cohen, A.I. Medalia, Rubb. Chem. Technol., 49, 1076–1094, 1976; SSBR compounds data are from M.J. Wang, Rubb. Chem. Technol., 71, 520–589, 1998; For the three carbon blacks considered, average DBP and cDBP data from Table 4.5 were used in calculating the number of elementary particles per aggregate and the maximum filler fraction.

140

Filled Polymers

5.1.8 Effect of Carbon Black on Dynamic Properties The significant effects of CB on the dynamic properties of elastomers are ­particularly important with respect to tire technology. Indeed carbon blacks (as well as other type of fillers) modify dynamic modulus functions in regions which are particularly relevant for the tire behavior and therefore the driving comfort and safety of an automotive vehicle. Let us consider the typical dynamic functions G′(ω) and G′′(ω) of a rubber material (Figure 5.31). The rolling resistance of a tire is depending of the relative magnitudes of the elastic and viscous moduli in the 100–1000 Hz frequency range; the lower the tan δ, the lower the viscous dissipation and hence the rolling resistance. But the wet skid resistance of a tire tread is depending of the relative G′ and G″ in the 100–10,000 KHz region in such a manner that the higher the tan δ, the safer the tire. In other terms, improving both the rolling and the wet skid resistance requires to play with the dynamic moduli in antagonistic manner with respect to the frequency range concerned. Many attempts were made in the 1970–1980s by modifying the (structure) of the elastomer, through advanced synthesis techniques, without real success however as it proved not really possible to induce the desired antagonistic changes in the tan δ vs. frequency function by only changing the macromolecular architecture. The solution came from a (radical) change in the nature of the filler, i.e., from CB to (silinated) silica, as we will see later. How carbon blacks do affect the dynamic properties in both the rubbery plateau and the transition zone is therefore a problem with tremendous technological implications, particularly when “large” amplitude strain are concerned (in practice when γ  >  0.01 [i.e., 1%]).

Log G´ (Pa)

Log G´´(Pa)

10

Terminal zone

Rubbery plateau

Transition zone

Glassy zone

8

6

0 GN

G΄ G΄΄

4 –5 0 Region related with rolling resistance of a tire

5

10

Log ω (Hz)

Region related with wet skid resistance of a tire tread

Figure 5.31 Dynamic properties of rubber materials with respect to tire technology requirements.

141

Polymers and Carbon Black

As may be expected, dynamic properties are affected in a complex ­manner by CB type and level, frequency and temperature. No overall understanding is presently available, but a certain coherence emerges when reviewing the most significant effects of carbon blacks on dynamic properties, as reported in literature. 5.1.8.1 Variation of Dynamic Moduli with Strain Amplitude (at Constant Frequency and Temperature) The general observation, reported by many authors, is that the addition of CB to an elastomer induces typical non-linearity in the viscoelastic behavior. Figure 5.32, drawn using data on SSBR compounds published by Wang,54 shows typical variations of both G′ and G′′ with strain amplitude (at constant frequency and temperature). Qualitatively the same observation is made on many filled systems, whatever the chemical nature of the rubber or its macromolecular characteristics, and irrespective of the grade of CB used. At very low strain, the elastic modulus G′ increases with increasing CB fraction and there is little or no sensitivity on strain amplitude, a characteristic of the linear viscoelastic behavior. As strain increases a drop in G′ is observed and the higher the filler content the larger this drop. This effect is typical of many filled systems and is referred as “dynamic stress softening (DSS)” (or “Payne” effect, with regard to the important contribution of this author). It is also observed that the viscous modulus G′′ passes through a maximum

N234 content Φ phr SSBR Duradene 715 100 N234 carbon black Variable 0.000 0 Zinc oxide 3 Stearic acid 2 0.124 30 0.159 40 Antioxidant 1 Sulphur 1.75 0.191 50 CBS 1.25 0.220 60 MBT 0.2 0.248 70

9.0 6.0 3.0 0.0 10–1

1.6

100 101 102 Double strain amplitude, %

Dynamic spectrometer II (Rheometrics Inc.) Temperature : 70°C Frequency : 10 Hz

Viscous modulus G´´, MPa

Elastic modulus G´, MPa

12.0

1.2 0.8 0.4 0.0 10–1

100 101 102 Double strain amplitude, %

Figure 5.32 Strain sweep experiments on SSBR compounds with various levels of carbon black.

142

Filled Polymers

value as the strain amplitude increases and in line with the G′ behavior, the larger the CB content, the higher the G′′ peak. 5.1.8.2 Variation of tan δ with Strain Amplitude and Temperature (at Constant Frequency) At constant filler level, the magnitude of DSS is depending on the CB structure as shown in Figure 5.33. Either at 0 or at 70°C, an unfilled (gum) solution SBR vulcanizate shows no sensitivity to strain amplitude up to 40% strain; the only effect of a temperature increase is to lower both G′ and G′′ and hence tan δ. With CB, the dynamic behavior is more complicated and strongly depends on the reinforcing character of the filler; at low temperature (i.e., 0°C), the more reinforcing the CB, the lower tan δ at low strain. As strain amplitude increases the black structure effect is reduced, until the large (nonlinear) drop is observed. At higher temperature (i.e., 70°C), the reverse effect is observed: the more reinforcing the CB the higher the tan δ peak occurring at around 10% double strain amplitude. 5.1.8.3 Variation of Dynamic Moduli with Temperature (at Constant Frequency and Strain Amplitude) Generally the elastic modulus of filled rubber vulcanizates decreases with increasing temperature, but the magnitude of the variation is strongly depending on the strain amplitude, as illustrated in Figure 5.34, in the case of

0.70

N234 N347 N660 Gum

10 Hz; 0°C

0.50

0.24

0.40

0.00

10 Hz; 70°C

0.20

0.30

100 50 3 2 1 1.75 1.25 0.2

0.16

0

100 1 10 Double strain amplitude,%

Tangent δ

Tangent δ

0.60

SSBR Duradene 715 Carbon black Zinc oxide Stearic acid Antioxidant Sulphur CBS MBT

0.12 0.08

Dynamic spectrometer II (Rheometrics inc.)

0.04 0.00

0

1 10 100 Double strain amplitude,%

Figure 5.33 Effect of carbon black type and temperature on tan δ of SSBR compounds.

143

Polymers and Carbon Black

Elastic modulus G´, MPa

15

20°C

Natural rubber N330 (32% vol.)

40°C 10

5

60°C 90°C

0 0.001

0.01 0.1 Double strain amplitude

1.0

Figure 5.34 Effect of temperature on the elastic modulus.

N234 content SSBR Duradene 715 100 N234 carbon black Variable Φ phr Zinc oxide 3 0.000 0 Stearic acid 2 0.045 10 Antioxidant 1 0.124 30 Sulphur 1.75 CBS 1.25 0.190 50 0.2 MBT 0.248 70

103 102 101 100

10–1 –60 –40 –20 0 20 40 60 80 100 120 Temperature, °C

Dynamic spectrometer II (Rheometrics Inc.) Double strain amplitude : 5% Frequency : 10 Hz

2.0 1.0 0.5 0.3

Tangent δ

Elastic modulus G´, MPa

104

0.1

0.05 0.03 –60 –40 –20 0 20 40 60 80 100 120 Temperature, °C

Figure 5.35 Effect of carbon black level and temperature on dynamic properties.

a NR compound with 50 phr N330 CB. As can be seen, at (very) low strain, the elastic modulus is roughly divided by two when the temperature increases from 20 to 90°C. For larger strain amplitudes, quite common in fact in many practical applications, the temperature effect becomes negligible. Changing the CB content further complicates the temperature effect. As shown in Figure 5.35 in the case of solution SBR compounds with various levels of CB, G′ slightly increases with CB level at low temperature. Above a certain temperature, a severe drop is observed but increasing CB level somewhat dampens the temperature effect. For instance from

144

Filled Polymers

−40 to + 40°C, the vulcanized gum compound exhibits a nearly three fold drop; the 70 phr compound shows a maximum two fold decreases in the same temperature range. As expected, the G′ vs. T transition corresponds to a peak in the tan δ vs. T curve (right graph in Figure 5.35). Below the transition temperature, changing the CB content has marginal effect, if any; in the higher temperature region, increasing filler content significantly increases tan δ. 5.1.8.4  Effect of Carbon Black Type on G′ and tan δ The stronger the reinforcing character of a given CB, the larger the elastic modulus increase. Because the complexity of the aggregate is the most influential filler characteristic with respect to reinforcement, direct relationships exist between dynamic moduli and parameters related to the structure of CB. This well known aspect is illustrated in Figure 5.36, where the plateau elastic modulus GN0 of 40 phr filled butyl rubber compounds is plotted vs. the N2 specific area. Low structure blacks, i.e., N990 to N550 marginally affect the plateau modulus but the reinforcing effect is clearly related to specific area with higher structure fillers (as clearly seen with ­carbon blacks from the N200 series). In fact, the variation in dynamic properties imparted by a given CB is depending on temperature, as clearly seen when considering tan δ. Figure 5.37 shows the effect of CB type and temperature on tan δ of solution SBR compounds. Above ordinary temperature (i.e., 20°C), the more complex the structure of the CB the higher the tan δ, but the maximum tan δ (occurring at around −20°C, irrespective of filler type or level) tends to decrease as the

Plateau elastic modulus G´0N, MPa

10

Butyl rubber, 40 phr furnace carbon black

8 N210 N242

6 4

N219

N293 N330 N660 N770 2 N880 N550 N990 0 0 20 40 60 80 100 120 Specific area N2 , m2/g

140

160

Figure 5.36 Effect of carbon black specific area on elastic modulus. (Adapted from A.R. Payne, R.E. Whittaker. Rubb. Chem. Technol., 44, 440–478, 1971.)

–60

0.03

0.05

0.1

0.3

0.5

1.0

–40

–20

0

20 40 60 Temperature, °C

Gum

N660

N347

80

100

120

SSBR Duradene 715 100 Carbon black 50 Zinc oxide 3 Stearic acid 2 Antioxidant 1 Sulphur 1.75 CBS 1.25 MBT 0.2 Wet grip Rolling resistance

–80 –60 –40 –20 0 20 40 60 80 100 120 Reduced temperature at 1 Hz, °C

Ice grip

Tangent δ vs. tire performance

Dynamic spectrometer II (Rheometrics Inc.) 5% DSA; 10 Hz

Figure 5.37 Effect of carbon black type and temperature on damping properties of SSBR compounds; relationship with tire performance.

Tangent δ

N234

Tangent δ

2.0

Polymers and Carbon Black 145

146

Filled Polymers

structure increases. Between −20 and  + 20°C, the higher the structure, the lower the slope d tanδ/dt, and below −20°C the carbon type (and level) has no significant effect. The combined effects of CB and temperature on the dampening properties of a filled rubber compounds are thus, complex and, as illustrated by the insert in Figure 5.37, it is therefore the choice of the filler that allows tire engineers to adjust dynamic properties with respect to antagonistic requirements. It is easy to understand that ice and wet grip resistance of a tire are favored by a larger viscous response in dynamic conditions, in other words by an increased tan δ. The requirements are opposite in what the rolling resistance is concerned since a dominant elastic response (i.e., lower tan δ) favors an efficient use of the energy provided by the engine of the automotive vehicle. In terms of tire performance, the reinforcing effects due to a given grade of CB are in fact very complex and the relationships between material functions, such as dynamic moduli, and properties of technological significance are not straightforward. As we said before, many of the relevant properties for tire application evolve in antagonistic manner when the grade (and/or the level) of CB is changed. A study by Byers and Patel55 illustrates the difficulties met by the compounder when trying to improve tire tread performances through the selection of an appropriate grade of CB. Figure 5.38 was drawn using selected data from these authors. As can be seen, the abrasion resistance improves (lower loss under abrasion) as the reinforcing character of the CB increases, and the structure of the aggregate is the key factor, as demonstrated by the ­variation observed when considering the N300 family. The heat build-up under repeated flexion and the rebound evolve in (different) directions with black grade in a manner, which essentially reflects the increasing tan δ with the reinforcing character of the filler, both at room temperature (rebound) and at higher temperature (under repeated flexion). One notes that whatever the technological property considered, there is a kind of plateau in property gain as the finer CB grades are considered, i.e., not much gain is observed when moving from N220 to N110, but the latter black is more difficult to dispersed in a rubber matrix than the former. 5.1.8.5  Effect of Carbon Black Dispersion on Dynamic Properties The poorer the dispersion the larger the DSS (Payne effect). Indeed as shown in Figure 5.39, the elastic modulus drop with increasing strain amplitude tends to decrease as mixing duration increases. But DSS still exists for optimally dispersed compound, as may be expected with respect to curves in the Figure 5.38 which tend to be closer each other as mixing time increases. Such observations are somewhat contradictory with the simple “filler network” interpretation offered by Payne for the strain dependent modulus. Indeed, as the dispersion of CB particles improves, the spatial influence of any expected filler network should widen and the

8

10

12

14

16

18

a

N220

N234

benzothiazol disulfide

100 50 5 3 0.6 2.5

ηcpd Reinforcement factor : RF = η gum

80

(Mooney) viscosity of compound (Mooney) viscosity of gum rubber

N110

NR/black (50 phr)

20 40 60 Reinforcement factor RF

SMR L Carbon black Zinc oxide Stearic acid BTSa Sulphur

0

N347

N339 N375

N351

N330

N326 N347

55

60

65

70

80

0

NR/black (50 phr)

N234

N110

N110

20 40 60 Reinforcement factor RF

NR/black (50phr)

N220

N234

Elastic rebound

20 40 60 Reinforcement factor RF

N351 N347 N326 N339 N330 N375

0

N326

N375

N339

N220

Heat build-up under repeated flexion

N351 100 N330

120

Figure 5.38 Effect of carbon black grade on technological properties relevant in tire tread performances.

Abrasion loss, g/h

Abrasion resistance Running temperature, °C Rebound, %

20

80

80

Polymers and Carbon Black 147

148

Filled Polymers

6

SBR/N220 (70 phr) 1.5 min

Elastic modulus G´, MPa

5

2.0 min 2.5 min 3.0 min

4

16.0 min

3

2

1

0 0.001

0.1 0.01 Double strain amplitude

1.0

Figure 5.39 Effect of mixing duration on the magnitude of dynamic stress softening.

reverse of Figure 5.39 should be observed. But one has nowadays more information than at Payne’s time and if there is indeed a network effect, it concerns both the filler particles and a part of the polymer matrix, strongly bound through intense particle-polymer interactions, as discussed hereafter. It is well known in the rubber industry, particularly in tires manufacturing, that a sizeable reduction in tan δ is obtained through improved microdispersion of fillers, in other terms by loosening-up the filler network that would result from undislocated agglomerates and/or strongly connected aggregates. For instance, benefits of extensive mixing have been documented as well as the positive effects of so-called chemical promoters or coupling agents, whose role is expectedly to enhance/promote interactions between CB particles and rubber chains, therefore, to achieve a better dispersion and hence to reduce hysteresis. Most of those chemicals however either have an ill documented toxicity potential or are not really in line with present environmental concerns. 5.1.9  Origin of Rubber Reinforcement by Carbon Black It is quite difficult to propose an overall coherent explanation for the reinforcing effect of CB and only certain aspects are so far understood. First, the true origin of rubber–filler interactions is not yet clear and second, as we

149

Polymers and Carbon Black

Elastic modulus

have previously underlined, the concept of reinforcement is complex since it concerns a balance of properties, most of them being antagonistic. For instance, in tire technology, would one want to increase the abrasion resistance of a tread band by changing the grade of CB, it is likely that the wet skid resistance will be affected. We will describe hereafter some of the most interesting (or most accepted) theoretical views on the subject. Payne has proposed a classical splitting of all the effects involved in rubber reinforcement which can be further refined with respect to contemporary views (Figure 5.40). In addition to the intrinsic properties of the elastomer network, the CB particles bring first mere hydrodynamic effects, which are further enhanced by strong rubber–filler interactions, and interaggregate interactions which are weaker and depending on strain level. A number of authors58 have followed the Payne suggestion that the elastic modulus decrease with increasing strain could be related to the progressive breakdown of an interaggregate network, assumed to exist in a well dispersed CB filled compound. Accordingly the higher dynamic modulus at low amplitude would result in a stiff CB network that would be destroyed as strain amplitude is rising. Such a model is however unable to explain all aspects of DSS, namely the modulus recovery that is observed when a previously strained material is left at rest for a sufficient period at the appropriate temperature.

Weaker interaggregate interactions = f(carbon black grade and level )

Strong carbon black–rubber interactions = f(carbon black grade and rubber type )

Hydrodynamic effects = f(carbon black type and level) Rubber network properties = f(rubber type and vulcanization system) Strain Figure 5.40 Origin of rubber reinforcement by carbon black particles.

150

Filled Polymers

As we have seen, hydrodynamic effects can somewhat be understood with respect to the works of Einstein, Guth, Gold, Simha, and others, but most of the technologically significant effects are due to rubber–CB interactions. BdR is the most significant evidence for rubber–carbon interactions, which is readily considered through the effective filler fraction, i.e.: Φ eff = Φ filler + Φ BdR



This approach was explicitly proposed by Medalia who adapted the Guth, Gold and Simha equation to modulus (see Equation 5.12), but the effects of CB on mechanical properties are more or less complicated depending on the grade and level of CB and, moreover, one cannot expect much from considerations solely based on hydrodynamic effects. At molecular level, it is fairly obvious that the tightly BdR layer somewhat influences the loosely BdR fraction. There is thus, a reduced segmental mobility in the neighborhood of a CB particle or, in other terms, a “stiffness gradient” as one moves away from the surface of CB aggregates, as explicitly considered by Heinrich and Vilgis59 (see Figure 5.41). Such a view is very similar to the concept of “mesophase” introduced by Theocaris for polymer–glass fiber systems60 and other particle-filled composites.61 This concept was developed to explain why the glass transition Concept of mesophase (Theocaris (1987))

Concept of transition layer (Heinrich & Vilgis (1995))

z

Mesophase

Fiber Interface x y

Carbon black

Bound rubber

Extractable rubber

Transition layer

Matrix phase Chemisorbed molecules (irreversible) Physisorbed molecules (reversible)

Constrained matrix

Figure 5.41 Reduced segmental mobility imparted by the proximity of a rigid body with which strong interactions occur.

Polymers and Carbon Black

151

of composite materials may be reduced in some cases, whereas it may be increased in others. A mesophase is the region, between the matrix ­material and the filler particles, both considered as homogeneous and isotropic, whose thermomechanical properties and fraction are determined from the overall thermomechanical behavior of the composite. Involving BdR in theories on reinforcement is surely attractive, with respect to available experimental data, but because rubber–CB interactions have essentially a physical nature, they are reversible and, at a given temperature, the actual level of BdR (and hence of rubber–filler interaction) is at best an equilibrium level between competitive ­adsorption–desorption processes. As discussed below several quantitative models for the DSS effects have been developed. 5.1.10  Dynamic Stress Softening Effect 5.1.10.1  Physical Considerations One the most typical nonlinear viscoelastic properties of filled rubber compounds is DSS, i.e., the decrease of dynamic moduli (either the complex G* or the elastic G′ modulus) upon increasing strain amplitude at constant frequency and temperature, a phenomenon commonly referred as the “Payne effect,” with respect to the extensive studies of A.R. Payne,62–64 whilst Fletcher and Gent were the first to report it.65 The occurrence of a maximum in the viscous modulus G′′ is generally associated with the decrease in G′ but it must be underlined that, if G′ and G′′ are calculated from G* and tan δ( = G″/G′) measurements, the equality G* = G′ + iG″ is strictly valid in the linear viscoelastic domain only. However, using a ­mechanical spectrometer with a highly precise displacement transducer offering a ­resolution of 0.05 μm (IMass Co. Dynastat Mark II), Roland66 reported that, even at amplitudes for which the modulus is a marked function of strain (γ0 range ≈ 0.001 − 0.01), an undistorted sinusoidal output is obtained in response to a sinusoidal input. Despite this stress/strain proportionality (which for unfilled, pure polymers is typical of the linear viscoelastic domain) the elastic (G′ or E′) and viscous (G′′ and E′′) moduli exhibit a strong dependence on strain magnitude. This amplitude-dependence under dynamic loads has been observed with a large variety of elastomers and fillers, and appears thus, a typical non linear character of filler-loaded vulcanizates. The magnitude of the effect depends on the temperature, the strain history and other factors. Almost all the DSS effect is achieved in a first strain sweep and only minor effects take place in subsequent steps. However, as clearly shown by Wang,67 upon aging at room temperature the softening effect is recovered, at least partially in most cases. Various origins (i.e., physical interpretations) have been assigned to the DSS effect, all calling to local mechanisms involving interactions either between filler particles, or between filler particles and the polymer matrix,

152

Filled Polymers

or a combination of both. Not all interpretations lead however to quantitative workable models. As might be expected, interpretations offered over the time (so far half a century) somewhat reflect not only the on-going developments in technical resources and in the accuracy of investigation means, but also the atthe-time available information regarding the nature of the filler, its structure and surface quality (i.e., chemistry, physics, and topology), as well as the level of understanding of the filler–filler and rubber–filler interactions. However, because very precise measurements have revealed that unfilled materials exhibit also DSS effects providing the strain magnitude is sufficiently large, other mechanisms should be considered as well. A most convincing one, in the case of unfilled vulcanized elastomers is the nonaffinity of the deformation, which essentially means that cross-links move with respect to a coordinate system embedded in the material, with an obvious entropic character.68 DSS is observed with nearly all filled systems and, accordingly, a number of mechanisms have been proposed in the literature that we have (somewhat arbitrarily) sorted out in two main categories: • Destruction (and reforming) of a so-called “filler network” • Destruction (and reforming) of a complex “filler–polymer network” From an engineering point of view, the most useful contributions from the literature are the ones that lead to quantitative workable models, i.e., mathematical relationships that can be used first to fit correctly experimental data, second to (tentatively) interpret fit parameters in view of the physical or physico-chemical processes considered. In discussing the physical interpretations of the DSS effect, quantitative workable models are therefore receiving hereafter privileged considerations. 5.1.10.2 Modeling Dynamic Stress Softening as a “Filler Network” Effect The destruction (and reforming) of a so-called “filler network,” assumed to exist in the material, was initially considered by Payne himself.69 In this model, adjacent CB aggregates are considered as sufficiently close for their respective surfaces to interact through interfacial forces of the London–Van der Waals type.* Such forces would have a significant contribution to modulus at very low strain but would vanish when the separation between interacting surfaces increases. Their contribution to modulus is thus expected to decrease upon increasing strain in dynamic tests. Filler network breakage is the background of the well known phenomenological model developed by Kraus70 (see below), in which, under dynamic deformation, physical bonds between * Van der Waals forces are short distances interaction forces of the order of 4 kJ/mole, to be compared with hydrophobic interactions (around 8 kJ/mole) and hydrogen bonds (20–40 kJ/ mole, depending of chemical groups involved).

153

Polymers and Carbon Black

(log) G´

G´0 G´

G´´ G´∞

tan δ G´´

Tan δ

(Log) strain amplitude Figure 5.42 Dynamic strain softening effect (schematic).

filler particles are continuously broken and reformed. The filler network is an obvious reference to the percolation theory* and is therefore acceptable when the filler volume fraction is above the percolation level, in the 12–13% range for most systems. Because strain dependent dynamic moduli are also observed when the filler content is below this level, any mechanism based on the dislocation of a (pure) filler network is therefore not fully satisfactory. Thanks to progresses in particles observation techniques and to new concepts such “fractal objects” introduced by B. Mendelbrot,71 the filler network approach was considerably enlightened, namely by taking into consideration the three-dimensional structure of carbon blacks. The fractal nature of the filler surface became a key aspect, leading to models that consider aggregates percolation when the volume fraction of filler is sufficiently high. Huber and Vilgis72 considered interactions between aggregates through dynamical processes of breakage and reformation of the resulting network embedded in the polymer matrix. When dynamic strain amplitude increases, the percolated network breaks down in smaller and smaller entities. A number of experimental data have shown that a great variety of filled rubber vulcanizates exhibit the dynamic strain softening effect schematically illustrated in Figure 5.42, at given frequency and temperature.† The ­limiting * It seem that percolation theory has his root in the works of Flory and Stockmayer who used it to study polymerization processes that may lead to gelation; see D. Staufer, A. Aharony. Introduction to Percolation Theory. 2nd Ed., Taylor & Francis, London, UK, 1994. ISBN 0 7484 0253 5. † It is worth noting that dynamic strain softening is observed on filled vulcanizates whatever is the strain mode, i.e., shearing, tensile of uniaxial compression. The shear modulus G* = G′ + iG″ is explicitly considered here, but the same considerations would in principle apply as well to the tensile modulus E* = E′ + iE″ or the compliance J* = J′−iJ.″

154

Filled Polymers

low strain value of the elastic modulus is G0′ and all experimental data suggest that there is a high-amplitude plateau G∞′ . All quantitative workable models so far proposed start from this schematic DSS representation. Apart a few early unsuccessful attempts to assign the effect to the rupture of a network of (spherical) contacting particles, the first quantitative model was suggested by G. Kraus,73 with respect to a thixotropic change in the (filler) microstructure, or in other terms a mechanism of deagglomerationreagglomeration of (CB) aggregates. The basic idea of Kraus was that aggregates must be viewed as “soft spheres” such that, over an infinitesimal range, the interparticle force F increases continuously with the displacement from an equilibrium distance rm. Interaction between “contacting” aggregates can therefore, be viewed as a (nonlinear) phenomenon whose characteristic rate is (∂F/∂r )r =rm. He then referred to an attraction–repulsion mechanism, as suitably modeled with respect to a Lennard-Jones potential, and at the expense of a few approximations, he established that the difference between the two limiting elastic modulus values, i.e., (G0′ − G∞′ ) is proportional to N0, the number of effective interparticle contacts at rest (zero amplitude), the characteristic rate and the equilibrium distance between soft spheres of radius d/2 to which aggregates are assimilated (Figure 5.43), i.e., (see details of the development in Appendix 5.2):

 ∂F  (G0′ − G∞′ ) = C1 N 0   (d + rm )2  ∂r  r = r m

(5.33)

where C1 is a constant of proportionality. To account for the strain-dependence of the dynamic moduli, Kraus considered that, at equilibrium the rates of aggregate deagglomeration and reagglomeration are equal and assumed that power law functions account for the strain dependency, i.e.:

kB N γ 0m = kR ( N 0 − N ) γ 0− m

(5.34)

where kB and kR are respectively, the rate constant for deagglomeration and agglomeration (or “breaking” and “reforming” of the filler network), N0 the number of contacts between aggregates at zero strain, N the number of surviving contacts, γ0 the strain amplitude and m a parameter describing the strain dependency. By defining a critical strain γc as follows:

k  γc =  R   kB 

2m



(5.35)

the following equality is derived from Equation 5.34, i.e.:

N=

N0 γ  1+  0  γc 

2m



(5.36)

155

Polymers and Carbon Black

rm

σ

Potential V(r)

10

rm

0 ε

–10

0

rm

0.5 Interparticle distance r rM

1

Force F(r)

50

d

0

–50 0.3

∂F ∂r r = r m

0.4 Interparticle distance r

0.5

Figure 5.43 Kraus model of deagglomeration–reagglomeration of filler aggregates, assimilated to “soft spheres.”

It is worth making two comments here. First postulating that the same power law function suits adequately the strain-dependency of the deagglomeration and the reagglomeration processes with only a change of sign is a strong hypothesis because it implies a perfect symmetry of both processes. Second the definition of the critical strain postulates also (time) symmetry. Consequently symmetries are being forced in the model equations, as it is indeed the case (see below). In agreement with the attraction–repulsion mechanism described above, the excess modulus of the agglomerate network at any amplitude must be proportional to N, so

G′ ( γ 0 ) − G∞′ = G0′ − G∞′

1 γ  1+  0  γc 

2m

   or   G′( γ 0 ) = G∞′ +

G0′ − G∞′ γ  1+  0  γc 

2m

(5.37)

One sees immediately that γc is the strain amplitude corresponding to (G0′ +G∞′ )/2 .

156

Filled Polymers

Kraus considered that the deagglomaration and the agglomeration ­ rocesses are deemed to dissipate (strain) energy, in addition to the energy p dissipated in straining the vulcanizate. There is thus, an excess loss modulus that may be taken as proportional to the rate of aggregate deagglomeration (or reagglomeration), i.e.: G′′( γ 0 ) − G∞′′= c kB N γ m0



where c is a constant of proportionality. Substituting for N (Equation 5.36) and using Equation 5.33 leads to: G′′ ( γ 0 ) − G∞′′=



C γ 0m (G0′ − G∞′ ) γ  1+  0  γc 

2m



(5.38)

The function G″(γ0) has a maximum at γ0 = γc so that:



G′′ ( γ 0 ) − G∞′′ = (Gm′′ − G∞′′)

γ  2 0  γc 

m

γ  1+  0  γc 

2m

   or   G′′ ( γ 0 ) = G∞′′+

γ  2 (Gm′′ − G∞′′)  0   γc  γ  1+  0  γc 

2m

m

(5.39)

where Gm′′ is the maximum loss modulus at γ0 = γc. The relationship for the loss tangent is then easily obtained, i.e.: 2



 γ  m/2  γ 0  − ( m/2 )  −  G∞′′  0   + 2 Gm′′  γc   γ c   tan δ = m −m  γ0   γ0  G0′   + G∞′    γc   γc 

(5.40)

The above equations require six parameters, G0′ , G∞′ , m, γc, Gm′′, and G∞′′ . Good estimates of G0′ , m, γc, and Gm′′ can generally be directly extracted from experimental data sets but both the elastic and the viscous moduli at infinite strain are at best extrapolated values. Two parameters, m and γc, are common to both G′ ( γ 0 ) and G′′( γ 0 ) . Several authors have applied the Kraus equations to a number of experimental data with fairly good correlation, at least for G′(γ0), but viscous modulus data at very low strain (i.e., < 10−3) are not well accounted for by Equation 5.39. For instance, Ulmer74 tested the Kraus relationships with data on various CB filled compounds, either his own, or from literature. Whatever the testing mode (i.e., shear, compression or torsion), nonlinear fitting of Equation 5.37 on G′ (or E′) data gave correlation coefficient r2 in the 0.99 +  range, at least for CB loadings lower than 60 phr. Fitted values for m

157

Polymers and Carbon Black

G´inf N110 0.659 N330 0.477 N660 0.224

6 4

G´m

γc

m

r2

1.026 0.713 0.300

1.428 1.808 2.174

0.547 0.706 0.491

0.692 0.640 0.709

1.2

2

Viscous modulus, MPa

Elastic modulus, MPa

8

0.8

0 0.01

N110 N330 N660

0.1 G´inf

2.621 2.363 1.825

1 Strain, % G´0

6.209 4.581 2.414

γc 1.376 1.471 1.761

10

100 m

r2

0.614 0.573 0.344

1.000 0.999 0.998

0.4

0 0.01

0.1

1 Strain, %

10

100

Figure 5.44 Kraus model to fit experimental G′ and G′′ data. (Data from M. Gerspacher, C.P. O’Farrell, C. Tricot, L. Nikiel, H.A. Yanga, ACS Rubb. Div; Mtg, Louisville, KY, Oct. 8–11, 1996. Paper 74 on SBR/carbon black (60 phr) compounds.)

were in the 0.5–0.7 range for G′(γ0), in line with m = 0.6 as considered by Kraus when probing his model, but significantly different m values were found for G′′ data. The critical strain γc was found to decrease with increasing CB loading. To further document how the Kraus model meets experimental data, we used a nonlinear algorithm (i.e., Marquart–Levensberg) to fit data obtained by Gerspacher et al.75 on a series of SBR filled compounds (see Appendix 5.2), with Equations 5.37 and 5.39 (Figure 5.44). As can be seen, the model fits reasonably well the experimental G’ data but cannot meet the asymmetric shape of G” vs. strain data. The exponent m is depending on the reinforcing character of the filler, as well as the ­critical strain γc. A similar assessment of the Kraus equations with a silica (40 phr) filled polydimethylsiloxane composite76 yielded the same conclusions regarding the low strain deficiencies of Equation 5.39. For such a silica filled material, fit parameter values were as expected different, i.e., γc ≈ 0.02 and m ≈ 0.45 for G′(γ0) ; γc ≈ 0.01 and m ≈ 0.35 for G″(γ0). As previously commented, the postulate considered by Kraus (i.e., Equation 5.34) imparts symmetries in both the G′(γ0) and the G″(γ0) functions. If indeed, experimental data support an horizontal symmetry for the elastic modulus, with respect to the mid modulus value, no vertical symmetry (with respect to γc) is generally observed for the viscous modulus. The deficiencies of the Kraus model are therefore, embedded in the starting postulate. Various modifications have been proposed to account for the nonsymmetrical behavior of G′′, without changing the physical ideas leading to the model. Using different strain exponents for the deagglomeration and reagglomeration processes (Equation 5.34) was probed by Ulmer who concluded that it

158

Filled Polymers

was not a successful alternative to the Kraus approach. He showed however that adding an (empirical) exponential term, not associated with the filler network but reflecting an additional (exponential) decrease of the viscous modulus upon increasing strain, improved considerably the G″(γ0) description. He then suggested rewriting Equation 5.39 as follows:

G′′ ( γ 0 ) = G∞′′+



γ  2 (Gm′′ − G∞′′)  0   γc  γ  1+  0  γc 

2m

m

 γ  + Gk′′ exp  − 0   γk 

(5.41)

where Gk′′ is a drop in viscous modulus assumed to be proportional to N0 the number of contacts between aggregates at zero strain (i.e., Gk′′ = Ck N 0 ; Ck being a constant) and γk another critical strain. Values for Gk′′ and γk must be obtained through nonlinear curve fitting, which requires obviously excellent quality data in the very low strain range but also suitable initial guess values for the non linear regression algorithm. As pointed by Ulmer, the ratio of the exponential term to the second right term in Equation 5.41 is less than 0.001 in most cases. Therefore, Gk′′ ≈ Gm′′ − G0′′ is a good initial guess value and, whilst somewhat depending on the polymer and the CB loading, the ­corresponding γk guessed value can be taken as considerably smaller than γc. By comparison with the right graph in Figure 5.44, Figure 5.45 ­demonstrates the benefit in using Equation 5.41 to fit G′′ data from Gerspacher et al. (see Appendix 5.3 for details on the nonlinear fitting process). To consider that deagglomeration and agglomeration are two symmetrical processes is surely a strong hypothesis and likely the reason for the deficiencies of the Kraus model. In order to somewhat circumvent it, recent theoretical developments were made with an explicit reference to the fractal description of CB aggregates, and by considering that, above a percolation threshold, highly branched aggregates can flocculate and form a secondary

Viscous modulus, MPa

1.2

0.8

0.4

0 0.01

0.1

1 Strain, %

10

100

G´´∞

G´´m

γc

m

N110

0.120

1.036

1.295

0.519

N330

0.087

0.723

1.351

0.466

N660 (–0.28)

0.300

1.271

0.162

G´´k

γk

r2

N110

0.572

0.199

1.000

N330

0.371

0.156

0.999

N660

0.130

0.042

0.985

Figure 5.45 Kraus model as modified by Ulmer on SBR/carbon black (60 phr) compounds.

159

Polymers and Carbon Black

network. A  kinematics is associated with such an aggregates flocculation ­ rocess, based on the hypothesis that complex tri-dimensional entities like p CB aggregates are not fixed in the space of the material but can perform random motions such that they fluctuate around their mean position in the rubber matrix.77–80 (Note that Klüppel et al. refer to their approach as the “cluster–cluster-­aggregation” (CCA) model; in this book, we use the term “cluster” to describe the large scale structures that silica “string of pearls” develop through hydrogen bonding; CB aggregates entangle through other mechanisms where chemical bonding plays no role; therefore we renamed their model aggregate floculation for consistency.) By considering that, in their working temperature window, elastomers are by definition amorphous polymers on their rubbery plateau and beyond, the hypothesis that aggregates do entangle (or flocculate) through random motions in the rubber matrix is of course reasonable. In the author’s opinion, one of the key aspects of the aggregates entanglement model is the percolation threshold, i.e., the filler volume fraction Φ* below which the stress applied on the filled compound is essentially supported by the polymer matrix around the aggregates that are sufficiently separated from each other to practically not interfere at all. Above Φ*, entangled aggregates form a network which has the capability to support and transmit stress, in such a manner that it is necessary to consider the modulus of this tenuous secondary filler structure (Figure 5.46). In the rubber matrix, flocculated (or entangled) aggregates form a kind of “filler network” whose elastic modulus at small strain amplitude would depend on their fractal connectivity. The low strain elastic modulus of the filler network can be evaluated by considering that entangled aggregates may store energy when strained, through bending–twisting deformation of the inter-aggregates connecting points. An interpretation of the Klüppel et al. proposal, supported by some numerical assessment (see Appendix 5.4) leads to the following practical relationships. The aggregate solid fraction is firstly considered with respect to a fractal description, i.e., (see Figure 5.46, left):

 d Φ A (D) =    D

3− F



(5.42)

where D and d are the diameters of the aggregate and the elementary particle respectively, and F the fractal dimension of the aggregate. Then, with respect to proposals for flocculated materials, the elastic modulus of an aggregate GA is approximated as follows:

( 3 + FB )/( 3 − F )

GA = Gp ⋅ Φ A



(5.43)

where FB is the fractal dimension of the aggregate backbone and Gp an averaged elastic modulus of the aggregate with respect to all its possible

a

Figure 5.46 Aggregates flocculation process according to Klüppel et al.

d

D

160 Filled Polymers

161

Polymers and Carbon Black

angular deformation. In other terms, Equation 5.43 describes the small strain ­modulus of the flocculated aggregates as the product of a local modulus for the ­elastic deformation of aggregate–aggregate connecting points and a geometrical factor, based on the fractal description of the aggregate. In fact with respect to the BdR fraction, it is necessary to consider that there is a layer of immobilized polymer at the (available) surface of the aggregate. For CB aggregates, the typical fractal dimensions are respectively, F ≈ 1.8 and FB ≈ 1.2, so that GA varies with the power ~3.5 of the aggregate solid fraction. The CB concentration dependence of the small-strain dynamic ­elastic modulus G′ above 12–13% filler, as reported by several authors for various rubber compounds (i.e., log G′ ∝ 3.5 · log Φblack) is therefore considered as a strong experimental support for this fractal approach. The mechanically effective solid fraction of the aggregate, i.e., ΦefA(D), must include the tightly BdR. A bound layer of thickness a on the exposed surface of the aggregates can be considered but, because by nature an aggregate is a random object, a precise estimation is impossible. The following approximation gives however reasonable results when running basic numerical assessment, i.e., (see Appendix 5.4):



2π 2  d  3 π  N p (D) ⋅  ( d + 2 a ) − a  3  + a − a   2  6 3    ⋅ Φ efA (D) = π 3 D 6

(5.44)

where Np(D) is the number of elementary particles in an aggregate of diameter D. The term [(π/6)(d + 2a)3] is obviously the surface of an elementary sphere of diameter d + 2a and the term

2π 2  d   a  3  + a − a    3  2 

takes into account the inaccessible surface at the contact point between neighboring particles. Using typical values for N330 CB, i.e., Np(D)  = 200 particles, d = 30 nm, D = 200 nm, and a = 2 nm, the first term yields 20,580 nm3 and the second 410 nm3. This means obviously that the fractal description of CB suits well “open” branched aggregates. Because the tightly BdR layer is relatively thin ( i.e., a << d), a good approximation of Equation 5.44 is:

Φ efA (D) =

N p (D) ⋅ [(d + 2 a)3 − 6 d a 2 ] ⋅ D3

(5.45)

With respect to Equation 5.37, the aggregates flocculation model allows thus, to consider that the low strain modulus of a CB filler compounds

162

Filled Polymers

(whose filler level is above the concentration threshold Φ*) can be estimated from:

 N p (D) ⋅ ( d + 2 a )3 − 6 d a 2     ⋅ Φ G(Φ) = G(0) + Gp ⋅    D3  

( 3 + FB )/( 3 − F )



(5.46)

where G(0) is the modulus of the elastomer and Gp the averaged elastic modulus of the aggregate. Most raw rubbers have low strain modulus of the order of a few MPa, but to offer a reasonable estimate of Gp is obviously quite difficult. Klüppel et al.78 suggested to consider 10 GPa, with respect to the modulus of para-crystalline carbon, but in such a case, Equation 5.46 yields generally excessive values. If for the sake of evaluating the pertinence of the aggregate flocculation model, one considers that between the modulus of a raw rubber and the elastic modulus of the aggregate there is a factor of 1000, then the graph given in Figure 5.47 can be drawn, in comparison with the well known Guth and Gold equation. As can be seen, below the percolation level (around 0.13), the aggregate flocculation model is not relevant, as well as the lower predicted modulus, in comparison with the Guth and Gold approach. For higher filler volume ­fraction, Equation 5.46 predicts a larger increase of the compound’s modulus as filler volume fraction increases, in better agreement with usual experimental data on compounds with reinforcing carbon blacks than the Guth and Gold model. Figure 5.47 supports the comment that, whilst perfectible, the kinetic flocculation of aggregates with a fractal description of the latter might be the right theoretical concept for (CB) filled polymers.

Compound modulus, MPa

10 Klüppel & et al. Guth & Gold, 8% BdR Guth & Gold, 30% BdR 5

0

0

0.05

0.10 0.15 Filler volume fraction Φ

0.20

0.25

Figure 5.47 Comparing the aggregate flocculation model (Equation 5.46) with the Guth and Gold equation; data used : Np(D) = 200 particles, d = 30 nm, D = 200 nm and a = 2 nm, G(0) = 1 MPa, Gp = 1 GPa; the Guth and Gold equation is used by considering either 8% or 30% bound rubber.

163

Polymers and Carbon Black

Flocculated aggregates might be the main source of enhanced compound modulus with higher CB volume fraction but it is quite obvious that as strain increases, such a tenuous secondary structure will be destroyed and will not participate anymore in transmitting the stress through the material. Connecting points in the aggregate flocculation model can be assimilated to the interparticle contacts between soft spheres in the Kraus model and a rate equilibrium between dislocated and flocculated aggregates be considered as well. Heinrich and Klüppel79 used this idea to derive an equation for the DSS, as follows: −τ



  γ 2m G′( γ 0 ) = G∞′ + (G0′ − G∞′ ) ⋅ 1 +  0     γ c  

(5.47)

where τ ≈ 3.6 is the elasticity exponent of percolation, i.e., (3 + FB)/(3 − F), and the only difference with Equation 5.37 derived by Kraus. Unfortunately, no equation is given for the corresponding variation of G′′ with strain ­amplitude. A numerical comparison of Equations 5.37 and 5.47 shows that no much is gained in modeling the DDS with respect to the aggregate flocculation approach (see Appendix 5.4). For equal initial G0′ and final G∞′ moduli, superimposed curves are obtained with substantial differences in models’ parameters. Equation 5.47 still exhibits and horizontal symmetry (as expected since the same rate equilibrium is considered), the upper and lower limits of the term [1 + (γ0/γc)2m]−τ are still 1 and 0 and, at the critical strain γc, this term obviously reduces to 2−τ (≈0.0825 with τ = 3.6), instead of 2 with the Kraus model. With the latter, the critical strain corresponds to the mid modulus value, with Equation 5.47, the critical strain corresponds to a modulus equal to G0′ ⋅ 2 − τ + G∞′ ⋅ ( 1 − 2 − τ ) . In the models described some far, very little consideration was brought to the nature of the matrix in which CB particles are dispersed, except the Ulmer’s modification of the Kraus model, which introduces in the G″(γ) model an additional term, explicitly not associated with the filler network, but reflecting the viscous modulus variation of the polymer upon increasing strain. As we have seen, the Ulmer proposal leads to an excellent description of the asymmetric variation of the viscous modulus with strain amplitude. In the author’s knowledge, van de Walle et al.81 were the first to objectively assign a role to the polymer matrix by considering that the complex moduli of filler rubber compounds result from the superimposition of a linear viscoelastic behavior of the polymer and a nonlinear effect of the aggregate when their contact situation changes upon increasing strain. Firstly they focused on the likely microscopic interaction between two aggregates, by considering how Van der Walls forces might evolve as the particles are displaced from their equilibrium positions (i.e., at zero strain). Such a starting point is not much differing from the Kraus views when he assimilated CB aggregates to interacting “soft spheres,” but van de Wall et al. give explicitly

164

Filled Polymers

a (viscoelastic) role to the polymer matrix. Indeed they consider that upon strain both aggregates of a pair are moved from their respective equilibrium positions, in such manner that a strain is exerted between them, owing to polymer–filler interaction. This strain can be modeled with respect to a spring-and-dashpot system. With the assumption that the relative motion of both aggregates is sufficiently slow for the process to be quasi-static (i.e., near a stationary state), the interaction aggregate-polymer-aggregate can be modeled with a set a very simple equations. As in the Kraus approach, it is considered that, under cyclic deformation, both aggregates are continuously set apart and brought close, but with an increase in the energy loss owing to friction between polymer chains during the separation-and-reformation of the pair of aggregates. Because a spring-and-dashpot model is considered, the polymer role is explicitly viscoelastic nonlinear. A force F exists between both aggregates of the pair, which varies with the strain γ0 and such considerations lead in fact to an idealized model whose microscopic complex modulus has the two standard elastic and viscous components, i.e., g* = g′ + ig″. Therefore, an energy loss E1 is associated with each hysteretic cycle, such that: E1 ∝ g 0 γ b ∆γ



(5.48)

where g0 = dF/dγ0, γb the strain below which there is no energy loss because the hysteresis does not take place and ∆γ the amplitude of the hysteretic cycle. Using the analogy E1 ∝ G′′ γ 20 , one obtains: 0 g ′′( γ 0 , γ b ) = E1 γ 20



if

γ0 ≤ γb

if

γ0 > γb



(5.49)

By considering a linear variation of F vs. γ0 in the interval [0, γ0], an estimate is obtained for the effective g′, which leads to: 3

γ  g ′( γ 0 , γ b ) ∝ g 0  b   γ0 



(5.50)

It follows that the microscopic viscoelastic function g*(γ0) is given by: g0

g* =

3

∆γ  γ b  γ  g0  b  + i g0  γ0  γ b  γ 0 

if

γ0 ≤ γb

if

γ0 > γb

2



(5.51)

The behavior of the overall system is considered as the sum of the individual contributions of an infinite number of pair of aggregates, each having of course a different γb. One must therefore, introduce a weighing function

165

Polymers and Carbon Black

N(γb)d(γ0) which gives the number of pair of aggregates that are split when the strain increases from γb to γb + dγ0. The complex modulus of the macroscopic system is eventually obtained as: ∞

∫

∞

G ( γ 0 ) = G + W ( γ 0 ) dγ 0 + *

* ∞

0

∫ 0

3

 γb   γ  W ( γ 0 ) dγ 0 + i h 0

∞

∫ 0

2

 γb   γ  W ( γ 0 ) dγ 0 (5.52) 0

where G∞* is the complex modulus of the rubber matrix, W(γ0) = g0(γ0)N(γ0) an overall weighing function and h the average ∆γ/γb. On the right side of Equation 5.52, the second and the third terms account for the real (elastic) part of the complex modulus, and the last term for the imaginary (viscous) part. According to the model, only W(γ0) is material dependent and there is a relation between G′ and G′′ since they are both derived from the convolution of the same function W(γ0). The practical use of this so-called van de Walle, Tricot and Gerspacher (VTG) model is not straightforward and was somewhat described in further publications by the authors.75,82 The difficulty is to properly evaluate the weighing function W(γ0) from experimental data. It is in fact more convenient to calculate it from G″(γ) data, using the differential form of the viscous term in Equation 5.52, i.e.:

W (γ 0 ) =

1  dG′′( γ 0 ) 2  G′′( γ 0 )  + h  dγ 0 γ0 

(5.53)

and the calculation procedure is somewhat explained in the original publications.* The capabilities of the above model in meeting experimental G′(γ) data have been documented by the authors, but unfortunately such a demonstration is not offered for G″(γ). With respect to the mathematical form of the last right term in Equation 5.52, one would expect a vertical symmetry for G″(γ), in contradiction with most experimental observation. Apart the merit of introducing the viscoelastic nature of the polymer matrix in the modeling of the DSS effects through the “filler network” approach, the VTG model is not offering real advantages over the Kraus approach. In order to circumvent some deficiencies of the Kraus model, namely the fact that the whole set of model parameters has to be reconsidered if the frequency is changed, Lion et al.83,84 proposed a interesting phenomenological theory which leads also to a six-parameter model for the DSS effect. Both the frequency and the amplitude are taken into account with this model and by interpreting the observed history and recovery effects on the elastic and viscous moduli as manifestations of thixotropy, a so-called * Note that in the referred publications, it seems that there are a number of misprints, likely due to the easy confusion between γb, the critical strain for aggregate–aggregate interaction and γ0, the strain amplitude. Equations 5.52 and 5.53 have been rewritten by the author in agreement with his understanding of the modeling approach.

166

Filled Polymers

“intrinsic time” z(t) is introduced, instead of the physical time t. This ­i ntrinsic time is expected to account for the process-dependent relaxation times of any thixotropic material. In this respect, the Lion et al. approach considers also that DSS is reflecting a filler network effect, whilst it is not stated in such terms by these authors. The theory is formulated in the time domain and therefore can relate the stress to any strain history. Both the elastic and viscous moduli depend on a “reduced frequency” that takes into account the strain amplitude and the angular frequency. It follows that, when the amplitude changes, the moduli curves are not only shifted but also deformed. In one of the latest versions of this theory, 84 a fractional linear model is used that consists of a spring of modulus E1 in parallel with another spring of modulus E2 in series with a “fractional” dashpot of viscosity ηβ = (E2 ⋅ λ)β, with λ a characteristic time and β the fractional exponent. The ­resulting constitutive equations are then formulated with respect to the intrinsic time z(t) of the material, whose current state of microstructure is expressed through an internal variable q(t) that determines the temporal behavior of the intrinsic time z(t). Through quite a rigorous treatment that complies with both the mathematical foundations of fractional calculus and the second law of thermodynamics, the following practical relationships are obtained for the strain dependence of the elastic and viscous moduli:



 ω λ  2 β  ω λ  β  π  E2  + cos  β      2  a(γ , ω )  a ( γ , ω )   G ′( γ , ω ) = E1 + 2β β  ωλ   ωλ   π + 2 cos  β  1+     2  a(γ , ω)  a(γ , ω)

(5.54a)

β



G′′( γ , ω ) =

 ωλ   π E2  sin  β   2  a( γ , ω )   ωλ  1+   a( γ , ω ) 

2β

β

 ωλ   π cos  β  + 2  2  a( γ , ω ) 



(5.54b)

where a(γ,ω) is the intrinsic time depending on both the strain and the ­frequency, as follows:

a( γ , ω ) = 1 +

2 b γ (ω τ)α π

(5.54c)

In the above set of equations, E1 and E2 are elastic (spring) moduli, ω the frequency, λ = η/E2 a characteristic time (with η the dashpot’s fractional viscosity), β the viscosity fractional exponent, b a proportionality constant, τ and α respectively, a characteristic time (≈1 s) and another fractional exponent. It is quite clear from Equation 5.54c that the intrinsic time varies linearly

167

Polymers and Carbon Black

E1 + E2 E1

30 20

E2 2E1 + E2 2 γc

10

0 10–4

ηβ

10–3

E1

10–2 10–1 100 Strain amplitude

101

8 Viscous modulus G´´, MPa

Elastic modulus G´, MPa

40

6

G´´max

4 2

0 10–4

γc 10–3

10–2 10–1 100 Strain amplitude

101

Figure 5.48 Dynamic strain softening as modeled by Lion et al.84; Equations 5.54a and 5.54b were used in drawing the curves with the following set of parameters: E1 = 6 MPa, E2 = 31 MPa, λ = 419.36 s, β = 0.495, ω = 10 Hz, b = 51000, τ = 1 s, α = 0.44.

with the strain amplitude and nonlinearly with the frequency (providing the exponent α is not equal to 1). Figure 5.48 illustrates some basic features of the model, by showing that (E1 + E2) and E1 are respectively, the upper and lower limits for the elastic modulus and that the maximum viscous modulus occurs at a critical strain given by:

γ c (ω ) =

π (ω λ − 1) 2 b (ω τ)α

(5.55)

At the critical strain, the elastic modulus is a mid-modulus value (equal to (2E1 + E2)/2 and thus not depending on frequency). The model predicts that as the frequency increases, both the G′ and G′′ curves are shifted along the strain scale toward higher strain values. From a theoretical point of view, the Lion et al. model has the merit to approach the DSS effect by applying constitutive laws formulated on the basis of fractional calculus, in other terms by formulating the behavior of materials with respect to fractional time derivatives of stress and strain; an approach that in principle requires only a small number of material constants to express the material properties in the time or the ­frequency domain. However, deriving model parameters from experimental data is not straightforward and, for instance Lion et al. had to use a ­stochastic Monte Carlo method to estimate the model parameters for a comparison with experimental data on 60 phr CB filled rubber compound.84 Moreover, mathematical handling of the above equations (see Appendix 5.5) shows that, like the Kraus model, this one exhibits also horizontal symmetry for the G′ curve and vertical symmetry for the G′′ curve, and is therefore not expected to perfectly meet experimental data, at least in its present state of development.

168

Filled Polymers

Modeling DSS as a “filler network” effect is, in the author’s opinion, still far from completion and except the empirical modification introduced by Ulmer, no one of the proposed theories gives satisfactory results in meeting both the experimental G′(γ) and G″(γ) functions. So far collected data on quite a large number of (CB) filled systems demonstrate the universality of the effect and to be fully satisfactory any theory should meet experimental data in all aspects. The major role played by filler particles interacting with each other to control the low strain modulus is surely well demonstrated by the excellent fitting of G′(γ) data by the models described above, with obviously some approaches being (slightly) better than others. The recurrent failure of all the models in meeting the asymmetric shape of experimental G″(γ) curves suggests that any reasoning simply based on an equilibrium between deagglomeration and reagglomeration of (CB) aggregates, whilst correct in what energy storage processes (i.e., elastic) are concerned, is necessarily limited when dissipative phenomena are concerned. In addition to the filler network effect, there is obviously a contribution, which is not simply that of the pure polymer, but has a dissipative nature and is larger at high strain than at low strain. The successful Ulmer’s proposal in well fitting G″(γ) deserves consideration and calls for further theoretical developments. 5.1.10.3 Modeling Dynamic Stress Softening as a “Filler–Polymer Network” Effect The likely role in the DSS effect of a complex soft network, involving filler particles and a part of the polymer matrix, finds its root in the measurement of BdR, otherwise identified as an important aspect of the properties of uncured filled compounds. A number of authors have developed arguments favoring the filler–polymer network approach and have reported convincing experimental evidences, but only a few went to workable quantitative models. Disentanglement of the bulk polymer from the rubber fraction bounded to the filler surface was discussed by Funt85 as a contribution to the effective crosslink density of the vulcanizate. Between the bulk rubber and the tightly BdR shell, there is a transition zone (i.e., the loosely BdR) whose level of entanglement is expected to decrease with increasing strain amplitude. At high strain, when many of the entanglements in the loosely BdR region have been disengaged, the modulus is essentially depending on the crosslink density and less (or not) on the type of filler used. The chain softening of a glassy polymer shell surrounding filler particles as recently discussed by Montes et al.86 exploits a similar idea. It must be noted that such approaches consist in fact in considering locally a Mullins effect. With a reference to the well known Langmuir’s theory87 for the equilibrium adsorption–desorption of gas molecules on solid surfaces, Maier and Göritz88 developed a quantitative model by considering the kinematical aspects of an adsorption–desorption mechanism of rubber chain segments

169

Polymers and Carbon Black

1

Local area on carbon black particle Site occupied by a stable “link” Isolated site, available for an unstable “link”

2

Free site

3

4

Rubber chain Segments

Figure 5.49 Interactions between rubber segments and carbon black surface according to Maier and Göritz.

on appropriate sites on the filler surface. Stable (strong) and unstable (weak) “bonds” between polymer segments and (hypothetical) interactive sites on the filler particles are expected to occur depending on the strain amplitude. As illustrated in Figure 5.49, these authors describe the surface of a CB aggregate in a rubber compound as locally consisting of either free sites, or sites occupied by stable “links” or isolated sites occupied by unstable “links.” Rubber–filler interactions sites are like “knots” whose effect superimposes to chemical reticulation. With reference to the theory for rubbery ­elasticity, in a filled network, the dynamic modulus is proportional to the overall networking density N and the temperature T, according to (see details in Appendix 5.6):

G′ = N kB T

(5.56a)

where kB is the Boltzmann constant and with three contributions for N:

N = N chem + N stable + N unstable

(5.56b)

Nchem is the fraction of the overall networking density due to vulcanization knots, Nstable and Nunstable the fraction due to stable and unstable rubber–­ particle “links.” The adsorption–desorption equilibrium is considered through the following equality:

Φ free × vads = Φ occ × vdes

(5.57)

where Φfree and Φocc are the fractions of free and occupied sites, respectively on CB particles, vads and vdes, the rates of asdorption and desorption.

170

Filled Polymers

The adsorption rate is assumed to be constant, while vdes is proportional to strain γ, i.e., vdes = Kγ, where K is a constant. It follows that: 1 1 Φ occ = = (5.58) Kγ 1 + cγ 1+ vads The fraction of unstable links, which depends on the strain, is related to the fraction of free sites times the number of isolated sites per volume unit, i.e.: Nunstable(γ) = Φocc(γ)  ×  Nisolated, and the following relationship for the elastic modulus is easily derived: G′( γ ) = Gstable + Gunstable × ′ ′



1 1 + cγ

(5.59a)

= ( N chim + N stable ) kB T and Gunstable = N isolated kB T. A similar reawhere Gstable ′ ′ soning leads to the relationship for the viscous modulus, i.e.: G′′ ( γ ) = Gstable × ′′ + Gunstable ′′



cγ (1 + cγ )2

(5.59b)

In order to assess how the Maier and Göritz model meets experimental data, we used a nonlinear algorithm (i.e., Marquart–Levensberg) to fit data obtained by Gerspacher et al.75 on a series of SBR filled compounds (see Appendix 5.6), with Equations 5.59a and b (Figure 5.50). 8

SBR/60 phr black cpds

4

1.2

2

0.661 0.492 0.253

G´´unstable 1.507 0.903 0.191

c

r2

1.243 1.027 0.795

0.970 0.925 0.920

Viscous modulus, MPa

N110 N330 N660

Elastic modulus, MPa

6

G´´stable

0.8

0 0.01

0.1

1 Strain, %

G´stable G´unstable N110 N330 N660

2.328 2.236 1.917

4.072 2.421 0.445

10

100

c

r2

0.679 0.642 0.708

0.997 0.999 0.999

0.4

0 0.01

0.1

1 Strain, %

10

100

Figure 5.50 Maier and Göritz model to fit experimental G′ and G′′ data. (Data from M. Gerspacher, C.P. O’Farrell, C. Tricot, L. Nikiel, H.A. Yanga, ACS Rubb. Div; Mtg, Louisville, KY, Oct. 8–11, 1996. Paper 74 on SBR/carbon black (60 phr) compounds.)

Polymers and Carbon Black

171

As shown in Figure 5.50, Equation 5.59a meets reasonably well the dynamic strain softening of G′; however Equation 5.59b cannot meet the asymmetric shape of experimental G′′ vs. strain curves. As can be seen, the dynamic strain softening of the elastic modulus G′ drop is indeed well fit by the model, which supports the view that the elastic modulus decreases with higher strain because there are less and less unstable links. The maximum in G′′ vs. γ is well taken into account by the model, as could have been expected with respect to the quadratic term in Equation 5.59b but the asymmetric shape of the experimental curves is not at all met by the model. In line with the reasoning by Maier and Göritz, the quantities related to the unstable part of the rubber–filler network, i.e., G′unstable and G′′unstable significantly decrease with the decreasing reinforcing character of the CB. However, for a given compound, the fit parameter c is clearly different for the G′ and the G′′ curves, in contrast with the model’s expectation. When compared to filler network breakdown models for DSS, the proposal by Maier and Göritz attributes a key role to (transient) interactions between chain segments of the rubber network and the filler surface, by splitting the adsorbed chains between stable and unstable “links.” No indication is given to the exact nature of the “links,” except that some (if not all) are not permanent; a view that is well compatible with a number of information about BdR. As for the Kraus model, the postulate and assumptions considered by Maier and Göritz inevitably lead to symmetries in the mathematical form of both the G′(γ0) and the G″(γ0) functions. Again the horizontal symmetry for the elastic modulus is generally well met by experimental data but no vertical symmetry is generally observed for the viscous modulus. The most critical assumption in this respect is the equilibrium between adsorption and desorption of chain segments on filler particle sites. Whilst likely restricted to the case of silica filled systems, a further refinement was brought to this model by Ladouce–Stelandre et al.89 who considered that the mechanism was thermomechanically activated, with a dependence on the amount of “free sites” available at the (silica) filler surface. But, of course in the case of silica filled systems, the interaction sites are well identified, i.e., the silanol groups with their specific chemistry, as we shall see hereafter. The response of a complex soft network involving filler particles and the BdR was also considered by Wang.90 A soft network results from contacts between particles through elastomer layers which, in a glassy state near the filler surface, exhibit a gradually decreasing modulus with the ­distance from the surface. In addition, occluded rubber likely entrapped in filler–rubber clusters behaves mechanically as a filler. It is interesting to remark that the concept of a pseudo-glassy shell of polymer surrounding filler particles and whose rigidity is gradually decreasing as one moves away from the particle surface, is very similar to the basic ideas for the so-called shear lag model, developed for short fiber reinforced systems, as we will see in Chapter 7.

172

Filled Polymers

It must be noted that, apart the initial filler–filler network considerations of Payne and Kraus, the subsequent models essentially recognized the ­fact—today widely accepted—that, except in very highly filled systems, direct filler–filler contacts are very unlikely. Explicitly stated by Aranguren et al.,91 in the case of silica-silicone systems however, it must be noted that successful BdR measurements imply that filler particles surface is completely wetted by the polymer. Therefore, contacts between filler aggregates can occur only through the polymer. The differences in the various models arise either from the description of the filled system or from the manner the local thermodynamics is treated.

5.2  Thermoplastics and Carbon Black 5.2.1 Generalities Contrary to rubber materials, CB is not used in thermoplastics for reinforcement purposes. There are essentially three reasons for CB addition to a thermoplastic: • Tinting or pigmentation (in gray or black) • Reducing UV sensitivity • Modifying the electrical properties (conductivity) The two first applications involve only a few percentage weight of CB, largely below any percolation level above which significant changes would occur in rheological and/or mechanical properties. The latter application implies obviously that a “threshold limit,” evidently related to the percolation level, is reached through CB loading, in order to have, in the thermoplastic matrix, a network of particles to act as an internal electrical conductor. It is indeed widely accepted that, in an isolating (polymer) matrix electrical charges follow the paths of a conducting filler network. Upon increasing the concentration of conductive particles (i.e., CB) in a nonconducting matrix an isolator–conductor transition is thus, expected to occur. In modern versions of the percolation theory two models are considered, both leading to (expectedly) universal scaling relationships for the conductivity and permittivity with respect to a fractal network of conductive particles.92 The so-called lattice gas model describes the conductivity as an anomalous diffusion of charge carriers on the conducting paths The so-called equivalent circuit model consider the fractal particles system as a random mixture of tiny resistors and capacitors assumed to form in the local interphase regions between conductive particles and the surrounding isolating (polymer) matrix. Note that, in order to significantly modify the conductivity of an insulating material it is not required

173

Polymers and Carbon Black

Table 5.11 Typical Coloring Carbon Blacks Carbon Black Type HCFa HCF HCF MCFb MCF MCF MCF LCFc LCF

1 a

Mean Particle Diameter (nm) 8.9 1.3 16.4 16.0 20.0 24.2 29.7 73.5 85.7

N2 1 Specific Area (m2/g) 430 575 525 190 125 95 65 33 25

CTAB2 (m2/g)

DBPA2 (cm3/100 g)

Tinting4 Strength

300 330 270 175 131 102 66 33 29

95 95 98 65 56 55 50 63 75

133 133 135 130 133 121 97 64 48

BET, ASTM D 3037; 2 ASTM D 3765; 3 ASTM D 2414; 4 ASTM D3265. High color furnace; b Medium color furnace; c Low color furnace.

to have actual physical contact between dispersed conductive particles; only close proximity is required because electrons can cross short layers of insulating materials through so-called tunnel effects.93 Measurements on CB filled polymer systems by various authors indicate that gap widths of the order of 2–5 nm between neighboring particles are sufficient for quantum tunneling of electrons, and that the required minimum gap width is independent of the filler volume fraction.94–96 Color differences in materials depend on the interaction of light with electrons. In the graphitic layers of CB particles, electrons are free to vibrate at practically any frequency and therefore absorb all wavelengths of visible light ranging from infrared to ultraviolet. Table 5.11 gives the characterization data for a few typical CB grades used in coloring applications. As can be seen, there are correlations between the size of elementary particles, the structure of the aggregate and the coloring capacity of CB, but the more complex the aggregates the more difficult the ­dispersion process. The UV absorption capabilities of CB are used to protect polyolefins against light exposure; a few percentages offer an adequate protection without much modification of rheological and mechanical properties. 5.2.2 Effect of Carbon Black on Rheological Properties of Thermoplastics Essentially the same (qualitative) effects as with rubber materials are observed when adding CB particles to thermoplastics. As illustrated by Figure 5.51, the disappearance of the Newtonian plateau and a (apparent) yield stress limit is observed when adding increasing quantities of CB to thermoplastic melts.97,98

Filled Polymers

LDPE / N220 carbon black 0% 3 10 1% 5% 10% 102 20 %

10–1

Polystyrene / N110 carbon black 0% 5% 10% 20% 25%

104

103

101 100

105

Shear viscosity, kPa.s

Shear viscosity, kPa.s

174

102 Temp. = 170 °C

Temp.= 150 °C 100

101

101 102 103 Shear stress, kPa

10–2

10–1

101 100 –1 Shear rate, s

102

Figure 5.51 Effect of carbon black loading in the shear viscosity function of LDPE and PS melts. (Drawn using data by C.Y. Ma, J.L. White, F.C. Weissert, K. Min, J. Non-Newtonian Fl. Mech., 17, 275–287, 1985 and V.M. Lobe, J.L. White, Polym. Eng. Sci., 19, 617–624, 1979.)

Extensional viscosity, kPa.s

105

Polystyrene / N110 carbon black 25%

104

20% 0%

103

Temperature = 170 °C Strain rate : 0.063 s–1 102 0.1

0.2

0.5

1

2 5 Time, s

10

20

50

100

Figure 5.52 Effect of carbon black loading in the shear viscosity function of LDPE and PS melts. (Drawn using data by M. Lobe, J.L. White, Polym. Eng. Sci., 19, 617–624.)

Extensional flow properties are important in a number of processing techniques for thermoplastics, such as extrusion blowing and blow molding and accordingly the effect of CB content of the extensional viscosity has been somewhat studied by several authors. As shown in Figure 5.52, it is generally observed that ηE increases with higher CB level, but typical high strain effects, such as strain hardening, seem to be suppressed or at least moved outside the experimental window.

175

Polymers and Carbon Black

In fact the effects of CB type and level on the flow properties of t­ hermoplastics, whilst less documented, appear very similar (at least qualitatively) to what is observed with rubber materials. Accordingly the loading dependence of viscosity can be treated with similar equations, but the complications due to strong polymer–CB interactions seem to be either absent or at least negligible (to the author’s knowledge). 5.2.3 Effect of Carbon Black on Electrical Conductivity of Thermoplastics With conductivities in the 10−5 − 10−19 (Ω.cm)−1 range, polymers are poor electrical conductor, and, as might be expected, a sufficient quantity of CB particles, whose conductivity is in the 102−105 (Ω.cm)−1 range, is likely to impart some electrical conductivity properties to composites. It is fairly obvious that the principal cause of differences in the conductivity of CB–­thermoplastic composites is the structure formed by CB particles, not the intrinsic conductivity of the carbon itself. It follows that the structure of the aggregate and its overall size are important factors and therefore some CB grades are more suitable than others. Because a network of connected or at least sufficiently close particles within the thermoplastic matrix is the ­necessary condition for the composite to exhibit electrical conductivity, the aggregates must be as “open” as possible with the largest number of elementary particles per aggregate a favoring factor. Table 5.12 compares several grades of so-called “conductive” carbon blacks, using characteristics, which may be ­considered as important with respect to electrical properties of composites. Specific area is surely an important factor in obtaining a CB network above a certain level (the so-called percolation level), but acetylene black, with its large number of particles per aggregate, has a very favorable structure for imparting a good conductivity to composites, as discussed by Medalia.99 Special, open-branches aggregates grade are produced by CB manufacturers for preparing conductive filled polymer materials. Table 5.12 Properties of “Conductive” Carbon Blacks

Grade N472 N880 Acetylene Vulcan® XC-72 (Cabot) HiBlack® 420B (Evonik)

N2 Specific Area (m2/g)

DPBA (cm3/100 g)

Elementary Particle Size (nm)

Number of Particles/ Aggregate

Anisometry

238 10 51

193 31 262

22.1 192.0 42.2

481 4.9 1070

2.00 1.21 2.00

220

170

30

n.a.

n.a.

88

n.a.

24

n.a.

n.a.

176

Filled Polymers

When the suitable quantity of CB particles is dispersed in a single polymer, the resulting black network is sensitive to processing because of converging flows which may, in the best cases induce strong anisotropy effects (and likely a good electrical conductivity along the processing flow) or in the worst cases result in a disruption of the conductive network. This explains the recent interest for conductive materials prepared by ­loading polymer alloys with CB. Indeed most incompatible polymer blends are made with finely dispersed components and therefore, have a co-­continuous heterogeneous structure. Because CB has not the same “affinity” for all polymers, a careful selection of blend components with the appropriate blending procedure allow to somewhat control the localization of particles in the interphase region,100–103 as illustrated in Figure 5.53. Preparing conductive polymer blends with selected localization of CB has attracted considerable research interested with a near continuous stream of publications whose detailed review is outside of the scope of the present book. The thermodynamic and kinetic factors that govern the localization of CB particles at interfaces in polymer blends are rather complex since systems far from equilibrium state are obtained, as recently considered by several authors.104,105 Different results are obtained when using short carbon fibers For instance, Zhanga et al.106 prepared high-density polyethylene (HDPE)/isotactic 1012

1 : Carbon black in PS phase 2 : Carbon black in PE phase

Resistivity (Ohm.cm)

1010

3 : Carbon black in PE phase of co-continuous PS/PE blend

108

4

3

4 : Carbon black at interface of co-continuous PS/PE blend

1

2

106

Lowest percolation level

104

Carbon black particles localize at interfaces PS

102 100

PE 0

2

4

6

8

10

12

14

16

Carbon black volume fraction, % Percolation level for system 1 Figure 5.53 Controlling the electrical conductivity through carbon black localization in thermoplastic blends. (Drawn using data from F. Gubbels, E. Vanlathem, R. Jerome, R. Deltour, Ph. Teyssié. 2nd International Conference on Carbon Black, Mulhouse, France, Sept. 27–30, 1993, 397.)

Polymers and Carbon Black

177

polypropylene (iPP) blends filled with vapor-grown carbon fibers (VGCF). They observed that the VGCF percolation threshold was only 1.25 phr, a much lower level than with the individual polymers. SEM micrographs revealed that the effect is essentially due to the selective location of the carbon fibers in the HDPE phase. It follows that a double percolation is the basic requirement for the conductivity of such composites, i.e., first a percolation of carbon fibers in the HDPE phase, then the continuity of this phase in the polymer blends. Preparing a blends with incompatible polymers and selectively locating CB in the interphase region is obviously quite difficult and how the morphology of such complex materials evolves during the processing/shaping operations is generally not well documented. Therefore, it is worth underlining that, much simpler systems are exhibiting quite interesting electrical conductive properties that are also assigned to a (natural) selective location of CB. For instance CB filled poly(ethylene-co-butyl acrylate) is used in semi-conducting power cable shielding but filler loadings must be above the critical value of the percolation theory, i.e., Φc > 0.14–0.17 (up to 40% by weight) to ensure the adequate electrical properties and an optimal dispersion is required. Poly(ethylene-co-alkyl acrylate)s are semicrystalline materials whose degree of crystallinity depends on the level of acrylate.107 At room temperature; such materials naturally develop an heterogeneous structure such that crystallites form only in ethylene rich regions. Butyl acrylate (BA) rich regions are essentially amorphous. In composites made by mixing CB particles with poly(ethylene-co-butyl acrylate) (EBA), semiconducting properties depend not only on the network microstructure of connected CB particles but also on specific interactions between the filler and the amorphous BA-rich regions of the matrix. When EBA/CB composites are hold above their melting temperature for a sufficient time, the filler microstructure can somewhat evolve, with consequently modified electrical properties after cooling.108 Such systems exhibit also a set of interesting nonlinear viscoelastic properties, not only in the solid (i.e., at room temperature), but also in the molten state.109 Through a series of advanced rheological measurements, Leblanc and Jäger110 came to the conclusion that CB particles essentially concentrate in amorphous regions, leading thus to a highly complex morphology with at least three phases: PE rich crystallites (no black in it), BA rich amorphous and CB (Figure 5.54). In terms of modulus, the ranking is obviously CB > crystallites > amorphous. It follows that, in the solid state, EBA CB composites exhibit viscoelastic properties that are essentially dominated by the CB rich BA phase, whilst in the molten state, the PE rich phase contributes significantly to the initial response in such a manner that a certain strain limit has to be reached before the filler reinforcing effect in the BA phase is observed. Because of specific interactions between the acrylate rich regions of the material and carbon black particles, a complex structure naturally occurs, similar to what can be obtained by carefully controlling the location of CB in incompatible polymer blends.

1

10

10

Strain amplitude, %

100

118.7

(Фblack = 0.275) (Фblack = 0.248) (Фblack = 0.226) (Фblack = 0.203) Virgin EBA

Molten state

Molten PE rich regions (deformable)

Strain

Carbon black/butyl acrylate rich amorphous regions

Strain

Behavior in solid and molten states PE cristallites (underformable) Solid state

Figure 5.54 Strain sweep tests (rubber process analyzer, updated for fourier transform rheometry) on molten virgin and carbon black filled poly(ethylene-co-butyl acrylate) composites and likely morphology in both the solid and molten states.

Complex modulus G*, MPa

100

RPA-FT; 200°C; 1 Hz; 2 tests

178 Filled Polymers

Polymers and Carbon Black

179

References

1. A.C. Patel, J.T. Byers. The influence of tread grade carbon blacks at optimum loadings on rubber compound properties. Rubb. India, 34 (4), 9–13, 1982. 2. S. Montes, J.L. White, N. Nakajima. Rheological behaviour of rubber carbon black compounds in various shear histories. J. Non-Newtonian Fluid Mech., 28, 183–212, 1988. 3. C. Barrès, J.L. Leblanc. Recent developments in shear rheometry of uncured rubber compounds. part 1: design, construction and validation of a sliding ­cylinder rheometer. Polymer Testing, 19, 177–191, 2000. 4. W.H. Herschel, R. Bulkley. Konsistenzmessungen von Gummi-Benzollösungen. Kolloid Z., 39, 291–300, 1926; Measurement of consistency as applied to rubberbenzene solution. Proc. Am. Soc. Testing Mat., 26, 621–633, 1926. 5. N. Casson. In Rheology of Disperse Systems, C.C. Mill, Ed. Pergamon, London, UK, 84, 1959. 6. J.L. White, Y. Wang, A.I. Ysayev, N. Nakajima, F.C. Weissert, K. Min.Modeling of shear viscosity behavior and extrusion through dies for rubber compounds. Rubb. Chem. Technol., 60, 337–360, 1987. 7. See for instance C.W. Macosko. Rheology, Principles, Measurements and Applications. VCH Publishers, New York, NY, 416, 1994. 8. A.K. Bagchi, K.K. Sirkar. Extrusion die swell of carbon black filled natural rubber. J. Appl. Polym. Sci., 23, 1653–1670, 1979. 9. G. Kraus. In Science and Technology of Rubber, F.R. Eirich Ed. Acad. Press, London, UK, 1978. 10. C.G. Robertson, C.M. Roland, J.E. Puskas. Nonlinear rheology of hyperbranched polyisobutylene. J. Rheol., 46, 307–320, 2002. 11. J.L. Leblanc. Rubber-filler interactions and rheological properties in filled compounds. Prog. Polym. Sci., 27, 627–687, 2002. 12. E.M. Dannenberg. Bound rubber and carbon black reinforcement. Rubb. Chem. Technol., 59, 512–524, 1986. 13. S. Wolff, M-J. Wang, E-H. Tan. Filler-elastomer interaction. Part VII. Study on bound rubber. Rubb. Chem. Technol., 66, 163–177, 1993. 14. J. Kruse. Rubber microscopy. Rubb. Chem. Technol., 46, 653–785, 1973. See Figures 29 and 30 in the paper. 15. L.L. Ban, W.M. Hess, L.A. Papazian. New studies of carbon rubber gel. Rubb. Chem. Technol., 47, 858–894, 1974. See Figure 3 in the paper. 16. S. Kaufman, W.P. Slichter, D.D. Davies. Nuclear magnetic resonance study of rubber–carbon black interaction. J. Polym. Sci., A2 (9), 829–839, 1971. 17. J. O’Brien, E. Cashell, G.E. Wardell, V.J. Mc Brierty. An NMR investigation of the interaction between carbon black and cis-polybutadiene. Macromolecules, 9, 653–659, 1976. 18. J.C. Kenny, V.J. McBrierty, Z. Rigbi, D.C. Douglass. Carbon black filled natural rubber. 1. Structural investigation. Macromolecules, 24, 436–443, 1991. 19. F. Yatsuyanagi, H. Kaidou, M. Ito. Relationship between viscoelastic properties and characteristics of filler-gel in filled rubber system. Rubb. Chem. Technol., 72, 657–672, 1999. 20. J.L. Leblanc. Insight into elastomer-filler interactions and their role in the ­processing behaviour of rubber compounds. Plast. Rubb. Proc. Technol., 10 (2), 110–129, 1994.

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Appendix 5

A5.1  Network Junction Theory [M.J. Wang, S. Wolff, E.H. Tan. Rubb. Chem. Technol., 66, 178–195, 1993. G.B. Ouyang, N. Tokita, M.J. Wang. ACS, Rubb. Div. Mtg., Cleveland, OH, Oct. 17–20, 1995. Paper 108.] A5.1.1  Developing the Model Aggregates consist of Np elementary particles of diameter d, and are assimilated to equivalent spheres of diameter D. At the end point of the crushed DBP absorption measurement, the volume of the absorbed DBP is the sum of the void volume within the aggregates and the volume between equivalent spheres. The volume fraction of equivalent spheres in the system differs according to the type of packing. The loosest packing for a touching sphere system is the cubic arrangement, for which the theoretical volume fraction of spheres is π  = 0.524. Compact cubic arrangement of CB aggregates (cDBP 6 absorption test) d D hgap

Data: ρ: = 1.86.

cDBP : = 0.87 ⋅

gm : filler density cm 3 (CB)

In a rubber matrix, CB aggregates keep the same arrangement as in DBP but junction points are widened by a distance hgap.

Np = number of elementary ­particles in an aggregate d = elementary particle diameter D = equivalent sphere diameter

103 ⋅ cm 3 : crushedDBP absorption (here N330)   kg

MCB : = 150 . gm: mass of CB sample

cDBP = 0.87 

cm 3 gm

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Maximum filler volume fraction at the end of cDBP absorption test:

Φ max : =

MCB ρ MCB + cDBP ⋅ MCB ρ

[1]

Φ max = 0.382

Note: such a ­maximum volume fraction is quite conform to the cubic arrangement of spheres.

The overall volume of the cDBP + CB mixture is thus [MCB(ρ−1 + cDBP)]3 If one considers that, in a rubber matrix, CB aggregates keep the simple cubic arrangement, the distance between neighboring aggregates is expanded by a distance hgap, such that for a fraction Φ of equivalent spheres of diameter D, one has: −1 3 π  hgap =  ⋅ Φ 3 − 1 ⋅ D  6 



The overall volume of a rubber + CB compound is thus: 3

1 1   3 3 1 ζ        M ⋅ + cDBP  + ⋅ ⋅ h Na gap      CB  ρ  CB 2     



[2]

where NaCB is the number of aggregates in MCB grams of CB:

NaCB ⋅

ζ = 2

MCB ζ ⋅ π ⋅ d3 2 Np ⋅ ⋅ρ 6

[3]

ζ = average number of contact points between neighboring aggregates (minimum is 2 obviously) By substituting the denominator of the right member of [1] by [2] one obtains thus, an expression for the volume fraction of CB in a filled compound, i.e.: φ CB =

MCB ρ 1 1     M ⋅  1 + cDBP  3 + h ⋅  Na ⋅ ζ  3  CB  gap  CB        ρ 2   

3

=

MCB ρ 1 1     M ⋅  1 + cDBP  3 + h ⋅  3 ⋅ ζ ⋅ MCB  3  CB gap       ρ  Nρ ⋅ π ⋅ d 3 ⋅ ρ    

3

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φ CB =

1 1 1    3 3.ζ  1  3  ρ ⋅   + cDBP  + hgap ⋅   ρ   N p ⋅ π ⋅ d 3 ⋅ ρ    

Using [1] it comes: φ CB =

3

1 1 1   3 3   ζ . 1 3      φ max  + hgap ⋅  N p ⋅ π ⋅ d 3    

 = > Aggregate junction gap width: hgap

 3.ζ  = d ⋅  N ⋅ π  p

−1 3

3

1 1    1 3  1 3  ⋅  −  φ   [4]  φ CB  max   

The elementary particle diameter is not an easily accessible parameter and even tedious TEM analysis gives at best average values. However if one considers that an aggregate consists of a simple assembly of touching elementary particles, one has the following relationship between the specific surface area (in m2/g) and the elementary particles diameter d: Ssp =



Therefore:

hgap

6 = ρCB ⋅ Ssp

 3 ⋅ζ  ⋅  N ⋅ π  p

−1 3

6 d ⋅ ρCB 1 1    1 3  1 3  ⋅  –    φ CB   φ max    

[5]

The NJ model leads to several interesting conclusions, i.e. for a given CB grade (i.e. specific surface area and number of primary particles): 1. The junction gap with decreases with increasing number of contact points 2. The junction gap increases with the number of particles per aggregate

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A5.1.2  Typical Calculations with the Network Junction Model i := 0 ⋅⋅6

DBPA dm 3 /kg

 1.14   N110      N 220  1.15     N330   1.02    cm 3    N550  DBPA : = 1.17  . gm  N660   0.94      N774    0.74       0.38   N990 

cDBP dm 3 /kg  0.98     0.98   0.87   cm 3  cDBP : =  0.84  .  0.73  gm    0.63     0.37 

CTAB m 2 /g

d n−m

125.45    111.56   80.54   m2  CTAB : =  41.32  ⋅  37.52  gm    30.37     9.70 

 18.32     20.95   30.50    d : =  50.58  ⋅ 10−9 ⋅ m  59.28     87.20     294.90 

Estimating the number of particles/aggregate: ε: = 0.847 2   N pi : = round [1.333 ⋅ (1 + DBPA i ⋅ 102 ⋅ ρ ⋅ 0.0115)] 3⋅ε − 2 , 0   

 278   285     209    N p =  298   170     97   26   

 N110   N220     N330     N550   N660     N774   N990   

 0.354   0.354     0.382    =  0.39   0.424     0.46   0.592   

 N110   N220     N330     N550   N660     N774   N9990   

 9.011.1016   5.877.1016     2.597.1016    =  3.994.1015   4.349.1015     2.395.1015   2.31.1014   

 N110   N220     N330     N550   N660     N774   N990   

Rem: DBPA (dm3/kg) × 100 = DBPA (cm3/100 g)

Maximum volume fraction in cDBP absorption test:



Φ maxi : =

MCB ρ MCB + cDBPi ⋅ MCB ρ



Φ max

Number of aggregates in MCB grams of CB:



NaCBi : =

MCB π ⋅ (di )3 N pi ⋅ ⋅ρ 6

‘ NaCB

189

Polymers and Carbon Black

Junction gap width:   ΦCB: = 0.20   : filler volume fraction A hypothesis is necessary about the number of junctions per aggregate, let’s say ζ: = 6

hgapi

 3 ⋅ζ  :=   π ⋅ N pi 

−1 3

⋅

3 3 1 6 1   ⋅ − ρ ⋅ CTAB i  Φ CB Φ maxi 

hgap

 27.827   31.552     44.061    =  99.537  10−9 ⋅ m  100.878     113.176     285.892 

 N110   N2220     N330     N550   N660     N774     N990 

Let us consider a given CB, i.e., N330    i: = 3    CTAB: = CTAB3     Φmax: = Φmax3     ΦCB:  =  0.20    ζ: = 2..10   Np: = 10..300  

N330, 20% vol. fraction

Junction gap width, mm

150

Junction gap width, mm

150

 3 ⋅ζ  hg (ζ , N p ): =   π.N p 

−1 3

⋅

3

 6 1 ⋅ − ρ ⋅ CTAB  Φ CB

3

1  Φ max 

N330, 20% vol. fraction

100

100

500

5 Number of contact points 100 particles 200 particles 300 particles

10

50

0 0 100 200 300 Number of primary particles/aggregate 2 junctions 5 junctions 10 junctions

190

Filled Polymers

A5.1.3  Strain Amplification Factor from the Network Junction Theory A5.1.3.1 Modeling the Elastic Behavior of a Rubber Layer between Two Rigid Spheres When a compressive force F provokes a small displacement Δx of one sphere toward the other, there is a compression stiffness which consists of two parts: - The rubber layer compressed between the two spheres - The restraints at the bonded surfaces of the spheres ∆x

One has: F = F1 + F2 then, with: A = 1 +

D

F

F x

h D+ h = D D h

Gent and Park derive the following equations:

F1  A    π  –1 = E0 ⋅ D ⋅   ⋅  A ⋅ ln   A –1    2  ∆x

[Gent and Park, Equation 5]

F2 1 1 1    π  + + 3.A ⋅ ln  1 −   = E0 ⋅ D ⋅   ⋅  3 +   8  ∆x 2 ⋅ A A –1 A                           

[Gent and Park, Equation 8]

1 1 1  F   A    π   π  − 1 +   ⋅  3 + + + 3 ⋅ A ⋅ ln  1–   =   ⋅  A ⋅ ln    A  A − 1   8   2 ⋅ A A –1 E0 ⋅ D ⋅ ∆x  2  



which simplifies to:

π  F  A  5 ⋅ A –2 ⋅ A 2 –1  + = ⋅  A ⋅ ln   A –1  2 ⋅ A ⋅ ( A –1)  E0 ⋅ D ⋅ ∆x 8 

Gent and Hwang write a different equation for the same problem: π  1 1  F  A  + –1 + = E0 ⋅ D ⋅ ⋅  A ⋅ ln  [Gent and Hwang,   A –1  8  2 ⋅ A A –1  (E0 ⋅ ∆x ⋅ D) Equation 3]



191

Polymers and Carbon Black

which, after rearrangement, leads to the same equation: π  F  A  5 ⋅ A –2 ⋅ A 2 –1  + = ⋅  A ⋅ ln   A –1  E0 ⋅ D ⋅ ∆x 8  2 ⋅ A ⋅ ( A –1)  h D+h A = 1+ = D D

=>



[6]

F π  h   (D + h)  2 ⋅ D2 + D ⋅ h – 2 ⋅ h2  + = ⋅  1 +  .ln   2 ⋅ (D + h) ⋅ h E0 ⋅ D ⋅ ∆x 8  D   h  

A5.1.3.2  Experimental Results vs. Calculated Data Exp [1]

Calc [1]

F/E0 D∆x

Calc [2]

F/E0 D∆x

h/D

F/E0 D∆x

h/D

h/D

FEM [2] F/E0 D∆x h/D

0.064 7.1 0.01 40.90 0.01 40.77   0.069 3.63 0.02 21.01 0.02 20.916  0.075 6.15 0.05 8.904 0.05 8.704   0.14 3.52 0.1 4.749 0.1 4.519  0.219 2.38 0.2 2.579 0.2 2.399   0.235 2.24 0.5 1.171 0.5 1.122   0.323 1.71 1 0.643 1 0.648  0.518 1.135 2 0.347 2 0.358   0.674 0.905 2 0.347 2 0.358 

 0.064 7.2   0.069 5.1  0.075 6.02  3.6  0.14 Data-:  0.2219 2.54   0.235 1.99   0.323 1.6  0.518 0.929   0.674 0.957

 

Sources of data [1] Measurement with two spheres of D=41.7 mm spaced with cured silicone rubber and calculations with model; A.N. Gent and B. Park, Rubb. Chem. Technol., 59, 77, 1986. [2] Calculated with model and using a finite-element method; A.N. Gent and Y.C. Hwang, Rubb. Chem. Technol., 61, 630, 1988.

Extracting data for nonlinear regression 100

x1: = stack(Data<0> , Data<2> )

x: = stack( x1,x 2) y 1: = stack(Data<1> , Data<3> ) y 2: = stack(Data

<5>

, Data

<7>

Stiffness

x 2: = stack(Data<4> , Data<6> )

10

1

)

y: = stack( y 1,y 2) With log scales, data exhibit an inverse dependence so that one can consider a much simpler approximate model :  

0.1 0.01

0.1

h/D

π  h F = α ⋅ ⋅  8  D E0 ⋅ D ⋅ ∆x

1

10

–β

   

[7]

π ( − C)1     C0 ⋅ 8 ⋅ x   1.2  π - C1  C : =    Function to fit with partial derivatives:  F( x ,C): =  ⋅x    0.8  8   π  −C0 ⋅ ⋅ x − C1 ⋅ ln( x)   8

Initial guessed parameters for nonlinear fitting

192

Filled Polymers

Res: = GenFit (x, y, C, F)   : calling nonlinear fitting algorithm Z: = Re s 0

π Res1 , x   < = fitting equation 8

 1.223  R2: = corr(Z,y)   < = correlation coefficient r2   Res =      R2 = 0.998  0.964 

Stiffness

100 10

A simple power law with a negative exponent and the appropriate prefactor fits well the data in the h/D range of interest.

1

0.1 0.01

0.1

h/D

1

10

A5.1.3.3 Comparing the Theoretical Model with the Approximate Fitted Equation h: = 0.001, 0.002..10

D: = 100 Z1( h,D): =

β: = Res1

 π  h  –β  π  h  (D + h)  2 ⋅ D2 + D ⋅ h – 2 ⋅ h2    ⋅  1 +  ⋅ ln  + Z h D = α ⋅ 2 , : ( )  ⋅    8  2 ⋅ (D + h) ⋅ h D  h   8  D  

1.105 Difference (theory-fitting)

1000

1.104 Stiffness

α: = Res0

1.103 100 10 1 1.103 0.01

0.1 1 10 h D = 100 (Theory) D = 100 (Fitting) D = 200 (Theory) D = 200 (Fiting)

500

0 1.10–3

0.01

0.1 h D = 100

1

10

D = 200

Except at very low separation gap, the empirical fitting equation gives similar results to the theoretical model in the layer thickness extending down to one-tenth of the sphere diameter.

193

Polymers and Carbon Black

A5.1.3.4  Strain Amplification Factor Let us consider a volume V of filled compound: the overall number N of CB aggregates in this sample is given by: N NaCB = Φ CB ⋅ ρ ⋅ V MCB



2

2 With respect to Equation [3], in a cross  N  3  6 ⋅ Φ CB  3 1 section of the volume V, the number of    =  ⋅      [8] V  π ⋅ N p  d 2 aggregates is:

When the sample is strained in one direction x by an amount ΔX, the junctions across a section perpendicular to the stretching direction contribute to the overall stress, in addition to the contribution of the rubber matrix. With respect to Equation [6], it follows that, over the cross section, the junctions contribute to a force given by: 2

π  h gap   N3 FJ =   ⋅ ( E0 ⋅ Des ⋅ ∆x ) ⋅ α ⋅ ⋅  V 8  Des  Obviously ΔX = Δx (N/V)1/3 and the network junctions contribute to the  modulus by a quantity:

−β

where Des is the diameter of the equivalent sphere of an aggregate 1

FJ  N  3 π  hgap  ⋅ (E0 ⋅ Des ) ⋅ α ⋅ ⋅  = ∆X  V  8  Des 

−β

From Medalia’s flocε simulation (see Chapter 4, section 4.1.4, Equation 4.6), one has: Des = d. N p 2 with Np the number of particles of diameter d in an aggregate, and ε = 0.847 Substitutions are made with respect to Equations [4] and [8]:

 

−1 1 1      d.  3 ⋅ ζ  3 ⋅  1  3 –  1  3   1       N p .π   Φ CB   Φ max    ε  6.Φ CB  3 1  FJ  π     3 ⋅ ⋅ E0 ⋅  d ⋅ N p   ⋅ α ⋅ ⋅  =  ε   8 ∆X  π ⋅ N p  d    d ⋅ Np 2       1

  =>



 ε  6 ⋅ Φ CB  3 1   π  −2ε  2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ α Np E d N 0 p  π ⋅ N  d     8   p 

=>

FJ π  6 ⋅ Φ CB  = E0 ⋅ α ⋅ ⋅  ∆X 8  π ⋅ N p 

1+β 3

 3 ⋅ζ  ⋅  N ⋅ π 

⋅ Np

p

ε⋅(β + 1) 2

−1/3

–β

1 1    1  3  1  3   ⋅  –  Φ     Φ CB  max  

1 β    ζ  3   Φ CB  3  ⋅   ⋅ 1–    2    Φ max    

–β

–β

194

Filled Polymers

Over the cross section, there also a contribution from the matrix rubber alone, so that the resulting stiffness can be estimated as:

2  FR  = E0 ⋅  1–Φ CB 3    ∆X

The overall stiffness is thus,

F F F = J + R ∆X ∆X ∆X

  =>

ECpd

π  6 ⋅ Φ CB  = E0 ⋅ α ⋅ ⋅  8  π ⋅ N p 

1+β 3

⋅ Np

ε ⋅(β + 1) 2

1 β    ζ  3   Φ CB  3  ⋅   ⋅ 1–    2    Φ max    

–β 2

+ E0 ⋅ (1 – Φ CB 3 )

So that the strain amplification factor Xf due to CB presence is:  

ECpd E0

π  6 ⋅ Φ CB  = Xf = α ⋅ ⋅  8  π ⋅ N p 

1+β 3

⋅ Np

ε⋅(β +1) 2

–β

1 β   2    ζ  3   Φ CB  3  ⋅   ⋅ 1–  +  1 – Φ CB 3    2    Φ max      

[9]

A5.1.4 Comparing the Network Junction Strain Amplification Factor with Experimental Data G′ [A ]

E ′ [B ]

E ′ [B ]

Φ CB

Φ CB

Φ CB

N330

N347

N327

 0   0.048  0.092  Data2: =  0.132  0.168   0.202   0.232

0.2 0.4 0.7 1.2 2 4.5 9.8

0 0.086 0.159 0.175 0.22 0.274 0.248

5.625 6.875 10.781 9.531 17.1875 29.375 22.4

0 0.086 0.159 0.22 0.239 0.274 0.191

5   6.25  6.875   8.906  10.625   14.0625   7.5 

Source of data: [A] A.R. Payne, R.E. Whitakker. Rubb. Chem. Technol., 44, 440–478, 1971. Figure 3; data on butyl rubber cpd, at 0.05% strain. [B] J.M. Caruthers, R.E. Cohen, A.I. Medalia. Rubb. Chem. Technol., 49, 1076–1094, 1976. Figure 6; data on SBR 1500 cpds, with N347 and N327 carbon blacks, at 60°C, 0.25 Hz, 5% strain; note that the last points in columns 4 and 5 are cubic interpolation data, in order to complete the matrix.

Strain amplication factor according to Guth and Gold: Φ: = 0,0.01..0.25

XfGG (Φ) : = (1 + 2.5 ⋅ Φ + 14.1 ⋅ Φ 2 )

Strain amplication factor according to NJ theory: Data used : i: = 0 .. 2

195

Polymers and Carbon Black

gm ρ = 1.86 ϒ 3 cm

ζ: = 2

DBP cDBP N p Φ max

1.02 0.87 209 0.390   α = 1.223   β = 0.964   TT: = 1.22 0.97 333 3 0.357   0.96 0.68 179 0.442 

 N330     N347   N3227 

Np and Φmax were calculated as in above using the corresponding DBP and cDBP data

  Np: = TT<2>   Φmax: = TT<3> 1+β β  ε .(β +1)  ζ 3  π  6⋅Φ  3 2 ⋅ XfNJ (Φ ,N p ,Φ max ): = α ⋅ ⋅  N ⋅ p  2  8  π ⋅ N p   



1    Φ 3 ⋅ 1–     Φ max    

–β

 2  + (1 – Φ 3 )  

E10: = 0.2 G’ data on butyl rubber/N330 cpds

5

0 0

0.1 0.2 Carbon black volume fraction Guth & Gold

0.3

5

0 0

0.1 0.2 0.3 Carbon black volume fraction Network junction Data

Data

E20 : = 5.625 E’ data on SBR 1500/N347 cpds 40

E’ data on SBR 1500/N347 cpds

Modulus, MPa

Modulus, MPa

40

20

20

0 0

G’ data on butyl rubber/N330 cpds

10 Modulus, MPa

Modulus, MPa

10

0.1 0.2 Carbon black volume fraction Guth & Gold

Data

0.3

0

0

0.1 0.2 Carbon black volume fraction Network junction

Data

0.3

196

Filled Polymers

E20 : = 5.625 E’ data on SBR 1500/N347 cpds 40

E’ data on SBR 1500/N347 cpds

Modulus, MPa

Modulus, MPa

40

20

20

0 0

0.1 0.2 Carbon black volume fraction Guth & Gold

0.3

0

0

0.1 0.2 Carbon black volume fraction

Data

Network junction

Data

A5.2 Kraus Deagglomeration–Reagglomeration Model for Dynamic Strain Softening [G. Kraus. J. Appl. Polym. Sci.: Appl. Symp., 39, 75–92, 1984.] A5.2.1  Soft Spheres Interactions Aggregates are viewed as “soft spheres” in such a manner that, over a range of distance rm – rM, the interparticle force increases with the displacement from the equilibrium distance rm

rm

Dynamic modulus varies with strain because of a deagglomeration–reagglomeration process of soft spheres.

d

 = >  difference between the two limiting elastic modulus values:

C1 is a constant   Ge0 : limiting low strain modulus         Geinf : limiting high strain modulus kB =

∂F  = rate constant of the deagglomera∂r r = rm tion process Note: G′ = Ge; G″ = Gv

rm

Force F(r)

   (Ge0 –Geinf ) = C1 ⋅ N 0 ⋅ kB ⋅ (d + rm )2

0

rM

∂F ∂r r = r m

Interparticle distance r

0.3

197

Polymers and Carbon Black

A5.2.2  Modeling G′ vs. γ0 A power law model is assumed to suit the strain dependency of the deagglomeration and agglomeration processes deagglomeration:   γ0m   reagglomeration :γ0−m Rem: this means that both processes are perfectly symmetrical, as illustrated below:

γ0: =  0,0.01..5   m: = 0.5  

f ( γ 0 ): = γ 0m    g( γ 0 ): = γ −0 m

10

< = deagglomeration 1

0.1

< = reagglomeration 0

1

2

3

4

g(γ0)

1

γ0

f(γ0)

5

At equilibrium the rates of aggregate deagglomeration and reagglomeration are equal, i.e.: kB ⋅ N ⋅ γ m0 = kR ⋅ ( N 0 − N ) ⋅ γ 0− m

=> N =

kR ⋅ γ −0 m ⋅ N 0     or: kB ⋅ γ m0 + kR ⋅ γ −0 m

kB: rate constant of deagglomeration kR: rate constant of reagglomeration N0: number of contacts at zero strain N: number of surviving contacts   N =

γ −0 m ⋅ N 0 kB m k R − m ⋅γ0 + ⋅γ0 kR kR

One defines a critical strain that corresponds 2⋅m k to the deagglomeration–reagglomeration equilibrium:   γ c =  R   kB  It follows:  N =

N0 γ  1+  0   γc 

2⋅m

With respect to the attraction-repulsion mechanism between soft spheres (aggregates), there is an excess modulus due to the agglomerate network at any amplitude that must be proportional to N, i.e.: (Ge(γ0)−Geinf) ⋅ N0 = (Ge0−Geinf) ⋅ N Note: here G* = Gx; G’ = Ge; G″ = Gv

198

=>

Filled Polymers

Ge( γ 0 ) – Geinf = Ge0 – Geinf

1 γ  1+  0   γc 

or:   Ge( γ 0 ) = Geinf +

2⋅m

(Ge0 − Geinf )   γ 0  2⋅m  1 +      γ c  

Numerical illustration:

γ0 : = 0.001,0.002,..10   : strain range for calculation



Geinf: = 2   Ge0: = 22   m: = 0.55   γc: = 0.03   : model parameters

Ge( γ 0 ): = Geinf +

Ge0 − Geinf γ  1+  0   γc 

100 γc

2⋅m

Ge0 + Geinf = 12 2 Ge( γ c ) = 12

10

1 1.10–3

0.01

0.1 γ0

Ge(γ0)

1

10

Ge0+Geinf 2

There is indeed horizontal symmetry with Ge0 + Geinf respect to the mid modulus at 2 the critical strain γ . c

A5.2.3 Modeling G″ vs. γ0 There is an excess loss modulus that may be taken as proportional to the rate of deagglomeration (or reagglomeration), i.e.: Gv( γ ) − Gv inf = c ⋅ kB ⋅ γ 0m ⋅ N

[1]

Gvinf: limiting high strain viscous modulus Rem: Kraus gives no indication regarding the values of Gvinf except that Gvinf << Gm.

One has (at deagglomeration–agglomeration equilibrium):

N=

N0 γ  1+  0   γc 

2⋅m

   [2]

199

Polymers and Carbon Black

From the soft spheres approach, one has: Ge0−Geinf = C1⋅N0⋅kB⋅(d + rm)2    where C1 is a constant



It follows:    N 0 ⋅ kB =



Ge0 − Geinf c ⋅ (d + rm )2

Equations [2] and [3] are substituted in Equation [1], then:   Gv( γ ) − Gv inf =

All terms in



C1

[ c ⋅ (d + rm )2 ]

⋅ γ m0 ⋅

Ge0 − Geinf γ  1+  0   γc 

2⋅m

C1 are constant, then: ⋅ ( c d [ + rm )2 ]

Gv( γ ) = Gv inf +

C ⋅ γ m ⋅ (Ge0 − Geinf ) γ  1+  0   γc 

The function Gv( γ ) = Gv inf + for γ0 = γc

Gv m = Gv inf +

[3]

C ⋅ γ mc ⋅ (Ge0 − Geinf ) γ  1+  c   γc 

2⋅m

    with C = 

C ⋅ γ m ⋅ (Ge0 − Geinf ) γ  1+  0   γc 

2⋅m

C1 [ c ⋅ (d + rm )2 ]

has a maximum value Gvm

which gives for the constant: C = 2 ⋅

2 ⋅m

(Gv m − Gv inf )

[ γ mc ⋅ (Ge0 − Geinf )]

It follows: Gv ( γ ) = Gv inf + 2 ⋅

=>

( Gv m − Gv inf ) ( Ge0 − Geinf ) ⋅γm ⋅ 2⋅m [ γ c m ⋅ ( Ge0 − Geinf )]   γ0 1 + γ  c  

( )

γ  2 ⋅ 0   γc 

 

Gv ( γ ) − Gv inf = 2 ⋅

Gv( γ 0 ) − Gv inf = ( Gv m − Gv inf )   γ 0  2⋅m  1 +      γ c  

or:

0 c

m



( γγ )

Gv( γ 0 ) = Gv inf

m

( Gv m − Gv inf )

( )

 γ0 1 + γ  c

2⋅m

  

  γ0 m 2 ⋅      γ c   + (Gv m − Gv inf ) ⋅    γ 0  2⋅m  1 +      γ c  

200

Filled Polymers

Numerical illustration:

γ0 : = 0.0001, 0.002,..10   : strain range for calculation

Gvinf: = 0.001   Gvm: = 1.6   m: = 0.55   γc: = 0.03   : model parameters   γ0 m 2 ⋅      γ c   Gv(γ 0 ) : = Gv inf + (Gv m − Gv inf ) ⋅    γ 0  2⋅m  1 +      γ c   Gv(γc) = 1.6

10 γc 1

There is indeed vertical symmetry with respect to the critical strain γc.

0.1 1.10–4 1.10–3

0.01

γ0

0.1

Gv(γ0)



1

10

Gvm

lim Gv ( γ 0 ) → 1.0000000000000000000.10−3 = 1 ⋅ 10−3

γ 0 →0

lim Gv( γ 0 ) → 1.0000000000000000000.10−3 = 1 ⋅ 10−3

γ 0 →∞

Rem: the low strain limiting value of the viscous modulus Gv is obviously the value assigned to Gvinf; in agreement with Kraus paper, one has considered Gvinf << Gvm A5.2.4  Modeling tan δ vs. γ0 One has thus:

Ge( γ 0 ) = Geinf +

(Ge0 − Geinf )   γ 0  2⋅m  1 +      γ c  

  

and  

Gv ( γ 0 ) = Gv inf

( )  ( ) 

 γ0 2 ⋅ γ c  + ( Gv m − Gv inf ) ⋅  γ0 1 + γ c 

m

2 ⋅m

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Polymers and Carbon Black

By definition: tan δ = Let: X =

Gv( γ 0 ) Ge( γ 0 )

γ0 γc

m 2⋅m  [ Gv inf ⋅ ( 1 + X ) + ( Gv m − Gv inf ) ⋅ 2 ⋅ X ]    [ Gv inf ⋅ ( 1 + X 2⋅m ) + ( Gv m − Gv inf ) ⋅ 2 ⋅ X m ] ( 1 + X 2⋅m ) tan δ = = [ Geinf ⋅ ( 1 + X 2⋅m ) + ( Ge0 − Geinf )] [ Geinf ⋅ ( 1 + X 2⋅m ) + ( Ge0 − Geinf )] m ⋅ 2 (1 + X )

tan δ =

( Gvinf + Gvinf ⋅ X 2⋅m + 2 ⋅ X m Gvm − 2 ⋅ X m ⋅ Gvinf ) = X m ⋅ ( 2 ⋅ Gvm + Gvinf ⋅ X m − 2 ⋅ Gvinf + Gvinf ⋅ X − m ) X m ⋅ ( Geinf ⋅ X m + Ge0 ⋅ X − m ) ( Geinf ⋅ X 2⋅m + Ge0 )

tan δ =

2 ⋅ Gv m + Gv inf ⋅ X m − 2 ⋅ Gv inf + Gv inf ⋅ X − m 2 ⋅ Gv m + Gv inf ⋅ ( X m − 2 + X − m ) = ( Geinf ⋅ X m + Ge0 ⋅ X − m ) Geinf ⋅ X m + Ge0 ⋅ X − m

m

−m

m −m 2 ⋅ Gv m + Gv inf ⋅ (X 2 –X 2 )2 m − 2 + X − m ) = ( X 2 − X 2 )2 or  tan δ =   since  ( X Geinf ⋅ X m + Ge0 ⋅ X − m

2



=>

m −m    γ0  2  γ0  2   Gv inf ⋅   −   + 2 ⋅ Gv m  γ c   γc     tan δ( γ 0 ) = −m m γ   γ  Geinf ⋅  0  + Ge0 ⋅  0   γc   γc 

γ0: = 0.001, 0.002..100

Gvinf: = 0.001    m: = 0.55    γc: = 0.03    Gvm: = 1.6 2



m −m    γ0  2  γ0  2   + 2 ⋅ Gv m Gv inf ⋅   −    γ c   γc     tanδ( γ 0 ) : =    tanδ(γc) = 0.133 −m m  γ0   γ0  Geinf ⋅   + Ge0 ⋅    γc   γc 

202

Filled Polymers

1

Gv m = 0.133 Ge( γ c )

γc

0.1

The tan δ curve exhibits also a (vertical) symmetry but not with respect to γc.

0.01 1.10–3 0.01

0.1 γ0

tanδ (γ0)

1

10

100

Gvm Ge (γc)

A5.2.5  Complex Modulus G* vs. γ0 One has:  Gx( γ 0 ) = Ge( γ 0 )2 + Gv( γ 0 )2   Rem: absolute value of the c­ omplex modulus

 

 [ Geinf ⋅ (1 + X 2⋅m ) + (Ge0 − Geinf )]   [ Gv inf ⋅ (1 + X 2⋅m ) + (Gv m − Gv inf ) ⋅ 2 ⋅ X m ]  ⇒ Gx( γ 0 ) =    + (1 + X 2⋅m ) (1 + X 2⋅m )     2

2

with X =  

Gx ( γ 0 ) =

γ0 γc

1 ⋅ (Geinf ⋅ X 2⋅m + Ge0 )2 + (2 ⋅ Gv m ⋅ X m + Gv inf + Gv inf ⋅ X 2⋅m − 2 ⋅ Gv inf ⋅ X m )2 (1 + X 2⋅m )

this term is suitably rearranged as: 2 ⋅ Gv m ⋅ X m + Gv inf ⋅ X m ⋅ (X − m + X m − 2)



m

2 ⋅ Gv m ⋅ X m + Gv inf ⋅ X m ⋅ (X 2 − X



)

−m m   X m ⋅  2 ⋅ Gv m + Gv inf ⋅ (X 2 − X 2 )2   



 

−m 2 2

=> Gx( γ 0 ) =

m −m 1    ⋅ (Geinf ⋅ X 2⋅m + Ge0 )2 +  X m ⋅  2 ⋅ Gv m + Gv inf ⋅ (X 2 − X 2 )2   ⋅ m 2 1+ X   

2

i.e.: m −m 2  2   2⋅m      γ 0  m   γ0   γ 0  2  γ 0  2    ⋅  Geinf ⋅   + Ge0  +   ⋅  2 ⋅ Gv m + Gv inf ⋅   −   Gx(γ 0 ) = 2⋅m  γ c   γ c     γc    γ 0      γ c       1 +        γ c  

1

2

203

Polymers and Carbon Black

Alternatively one can consider:   Gx = Ge2 + Gv 2

=> Gx = Ge2 + tanδ 2 ⋅ Ge2

=> Gx = Ge ⋅ 1 + tanδ 2

Substitution gives: −m 2 m     2 2 γ γ         0 0    Gv inf ⋅  γ  −  γ   + 2 ⋅ Gv m    c c   (Ge0 − Geinf )     Gx( γ 0 ) =  Geinf + ⋅ 1+   2⋅m  –m m   γ0    γ0   γ0      Geinf ⋅   + Ge0 ⋅   1 +              γ γ γ c c c         

2

Numeric illustration γ0: =  0.001, 0.002..10 Ge0 = 22   Geinf = 2   m = 0.55   γc = 0.03   Gvm = 1.6   Gvinf = 1.10−3 m 2   2⋅m     γ 0  m   γ0   γ 0  2 ⋅ ⋅ + + ⋅ ⋅ + Gx( γ 0 ) : = Ge Ge 2 Gv G v ⋅   0 m inf  inf     γ c   γ c    γ 0  2⋅m     γ c    1 +        γ c  

1

 

lim

γ 0 →∞

γ  – 0   γc 

−m 2

   

2

    

2

Gv( γ 0 ) → 1.0000000000000000000.10−3 < = one recovers obviously the value assigned to Gvinf

Gv0: = Gvinf   Gvinf = 1.10−3 Gx 0 : = Ge0 2 + Gv 0 2

100 γc

Gx0 = 22   < = obviously = Ge0 Gx inf : = Geinf 2 + Gv inf 2

10

Gxinf = 2   < = obviously = Geinf 1 1.10–3

0.01 Gx(γ0)

0.1 γ0

Ge(γ0) Ge0

1

10 Gx(γc)

Geinf

204

Filled Polymers



Mid modulus:

Gx 0 + Gx inf = 12 2

The mid complex modulus G* does not coincide with G* at γc  Gx(γc) = 12.106 A5.2.6 A Few Mathematical Aspects of the Kraus Model One has thus:

Ge( γ 0 ) − Geinf = Ge0 − Geinf



1 γ  1+  0   γc 

2⋅m

   and  

Gv( γ 0 ) − Gv inf = (Gv m − Gv inf )

γ  2 ⋅ 0   γc  γ  1+  0   γc 

m

2⋅m

Let:

Z1( γ 0 ) : =



γ  2 ⋅ 0   γc 

1     Z 2( γ 0 ) : =   γ 0  2⋅m    γ 0  2⋅m  1 +    1 +      γ c     γ c  

lim Z1( γ 0 ) → 1.    



γ 0 →0

lim Z1( γ 0 ) → 0    



m

γ 0 →∞

lim Z2( γ 0 ) → 0

γ 0 →0

lim Z2( γ 0 ) → 0

γ 0 →∞

1 γc 0.5

0 1.10–3

0.01

0.1 γ0

Z1(γ0)

1 Z2(γ0)

10

The actual shapes of G*(γ0), G′(γ0) and G″(γ0) are essentially dictated by the mathematical functions used to express the dependence on γ0 and by the upper and lower limits.

205

Polymers and Carbon Black

Numerical illustration γ0: = 0.0001, 0.0002..10 Ge0: = 15.106⋅Pa   Geinf: = 3⋅106 ⋅ Pa  Data: G. Heinrich, M. Kluepel. Adv. Polym. Sci., 160, 1–44 , 2002; Figure 1; NR + N110 cpd, vulcanized.

Gvm: = 1.7 ⋅ 10 ⋅Pa 

Gvinf: =  0.1 ⋅ 10 ⋅Pa

m: =  0.55    

γc: = 0.013

6

6

  Gx = Ge ⋅ 1 + tanδ 2    Gx = A ⋅ 1 + B   The 1 + tan δ 2 term can be viewed as the viscous character, weighed with respect to elasticity. Let:



 (Ge0 − Geinf )     A( γ 0 ) : =  + Geinf    γ 0  ( 2⋅m)   1 +      γ c     

=>

−m 2 m      γ0  2  γ0  2      Gv inf ⋅  γ  −  γ   + 2 ⋅ Gv m  c c     B( γ 0 ) : =   –m m  γ0   γ0    Ge Ge ⋅ + ⋅ inf  0     γ c   γ c     

2

Gx( γ 0 ) := A( γ 0 ) ⋅ 1 + B( γ 0 )

2.107 γc

A(γ0) is the elastic modulus and the variation of both G′ ( = Ge) and G* ( = Gx) with strain are similar.

1.107

1.10–4 1.10–3 0.01 0.1 γ0 A(γ0)

Gx(γ0)

1

10 Gx(γc)

1.04

γc

1.02

1

1 + B( γ c ) = 1.018

1.10–4 1.10–3 0.01 0.1 γ0 1+B(γ0)

1

10

1+B(γc)

206

Filled Polymers

The Kraus model implies thus that, at the critical strain, the viscous character is maximum 2% of the elastic one and less than 0.1% at very high strain. Such limits somewhat depend on the value of the critical strain γc. The variations of G′ and G″, as modeled by Kraus model, essentially reflect the mathematical forms of the γ0/γc functions.



Z1( γ 0 ) : =

1

  

  γ 0  2⋅m  1 +      γ c  

Z 2( γ 0 ) :=

γ  2 ⋅ 0   γc 

m

  

  γ 0  2⋅m  1 +      γ c  



Zx( γ 0 ): = Z1(γ 0 )2 + Z 2(γ 0 )2

ZX(γc) = 1.118 1 γc

γc

0.5

1

0 1.10–4 1.10–3 0.01 0.1 γ0 Z1(γ0)

1 Z2(γ0)

0.8 1.10–4

10

   

1.10–3 0.01 0.1 γ0 Zx(γ0)

1

10

Zx(γ0)

A5.2.7  Fitting Model to Experimental Data DSS data on SBR/60 phr black cpds [M. Gerspacher, C.P. O’Farrell, C. Tricot, L. Nikiel, H.A. Yang. ACS Rubb. Div; Mtg, Louisiana, KY, 1996. Paper 74.]

207

Polymers and Carbon Black

Strain sweep experiments N660 N330 N110 Strain G ′ G ′′ G′ G ′′ G′ G ′′ % MPa MPa MPa MPa MPa MPa  0.091   0.19  0.28   0.38  0.48   0.55   0.66  0.74   0.84  0.93   1.07  Data: =  1.22  1.48   1.7   1.9  2.8   3.7  4.6   5.5   6.5  7.6   8.5   9.5

2.34 2.31 2.3 30 2.26 2.24 2.23 2.22 2.21 2.20 2.18 2.16 2.15 2.14 2.12 2.11 2.06 2.05 2.04 2.02 2.00 1.99 1.97 1.95

0.265 0.275 0.280 0.290 0.295 0.296 0.297 0.297 0.297 0.298 0.2298 0.299 0.299 0.298 0.298 0.294 0.290 0.294 0.285 0.280 0.275 0.271 0.269

4.45 4.41 4.30 4.24 4.10 4.05 3.95 3.89 3.82 3.75 3.61 3.55 3.46 3.40 3..34 3.08 2.95 2.85 2.78 2.71 2.65 2..62 2.57

0.630 0.630 0.650 0.660 0.670 0.685 0.692 0.7 700 0.710 0.716 0.720 0.720 0.719 0.718 0.717 0.690 0.660 0.631 0.608 0.581 0.560 0.542 0.530

6.02 5.95 5.80 5.61 5.50 5.32 5.18 5.05 4.95 4.83 4.65 4.48 4.31 4.20 4.08 3.71 3.49 3.30 3.18 3.08 3.01 2.95 2.90

0.915   0.938  0.942   0.960  0.9 980   0.993   1.000  1.010   1.026  1.030  1.032   1.033  1.031   1.030   1.020  0.968   0.914  0.870   0.828   0.782  0.752   0.722  0.700 

Kraus model Elastic modulus Ge(γ 0 ) = Geinf +

(Ge0 − Geinf )   γ 0  2 ⋅m  1 +      γ c  

Geinf = G′inf: elastic modulus for an infinite strain Ge0 = G′0: initial low strain elastic modulus γc: critical strain Viscous modulus   γ0 m 2 ⋅     γ c   Gv(γ 0 ) = Gv inf + (Gv m − Gv inf ) ⋅    γ 0  2 ⋅m  1 +      γ c  

Gvinf = G″inf: viscous modulus at infinitely low or infinitely high strain Gvm = G″m: maximum value of viscous modulus (occuring at the critical strain) m = exponent related to the rates of deagglomeration and reagglomeration of filler particles γ0 = strain amplitude

A5.2.7.1  Modeling G′ vs. Strain Extracting data   Strain (%)   N660 G′ (MPa)   N330G′ (MPa)   N110G′ (MPa)

j: = 0..2    γ: = Data<0>    Ge0: = Data<1>    Ge1: = Data<3>    Ge2: = Data<5>

n: = length(γ)   n = 23   :number of data

Guess parameters for nonlinear fitting algorithm (GenFit function)

208

 (Ge j )n−1     (Ge j )0  Cj : = γ   round n   2    0.6 

Filled Polymers

< = extracting guess parameters from experimental data, i.e., - The lowest measured modulus for Ginf ′ - The highest measured modulus for G0′ - The mid range strain for γc - The common value considered by Kraus

C1 − C0   C0 + 2⋅C3    γ    1+      C2    1   1−     γ  2⋅C3    1+        C2     1       γ  2⋅C3  Model equation and   1+       C2   partial derivatives:    F(γ ,C): =    2⋅C3 C3 C1 − C0  γ    ⋅  ⋅ 2⋅   2⋅C3 2  C  C2 2   γ     1 +        C 2             2⋅C3     − γ γ C C     1 0 − 2 ⋅  ⋅ ln ⋅          γ  2⋅C3  2  C2    C2     1 +            C2  

N660

N330

N110

 1.95   2.34   C0 =   1.48     0.6 

 2.57   4.45   C1 =   1.48     0.6 

 2.9  6.02   C2 =   1.48     0.6 

Resj: = GenFit(γ,Gej,Cj,F)

:initial guessed parameters as extracted from the experimental data sets

: calling the nonlinear regression algorithm

209

Polymers and Carbon Black

Fitting equation

Correlation coefficient r2

     (Res j )1 − (Res j )0   Ga j : = (Res j )0 + 2⋅( Res j )3   γ    1 +     ( Res ) j 2    

         (Res j )1 − (Res j )0  Gar2 j : = corr  (Res j )0 + , Ge j  2⋅( Res j )3     γ   1+      ( Res ) 2 j      

Result: fit parameters    N660



N330

N110

:Ginf ′  2.621  1.825   2.363   6.209   2.414   4.581     : G0′     Res1 =      Res 2 =  Res 0 =   1.376   1.761   1.471  : γc       0 . 614 :m 0 . 344 0 . 573      

   Gar20 = 0.998

Gar21 = 0.999

Gar22 = 1

: r2

Elastic modulus, MPa

8 6 4 2

0 0.01

0.1

1 Strain, %

Data N660 Fit N660 Data N330 Fit N110

10

100

Fit N330 Data N110

A5.2.7.2  Modeling G″ vs. Strain Extracting data  Strain(%)  N660G″ (MPa)  N330G″ (MPa)  N110G″ (MPa) J: = 0..2      γ: = Data<0>  Gv0: = Data<2>    Gv1: = Data<4>    Gv2: = Data<6> Guess parameters for nonlinear fitting algorithm (GenFit function) Finding max G″ and corresponding γ

210

Filled Polymers

gc j : = A ← max ( Gv j ) for i ∈1.. n − 1    γi  B←  if (Gv j )i = A Gv ( ) j i   A←B A

(Gv j )n −1    (gc j )1   Cj : =  (gc j )0     0.5 

< = extracting guess parameters from experimental data, i.e.: - The measured modulus at the lowest strain for Ginf ′′ - The maximum ­measured modulus for Gm ′′ - The strain for the maximum G” - The common value considered by Kraus

Model equation and partial derivatives: C    γ  3 2 ⋅     C2    C0 + ( C1 − C0 ) ⋅ 2⋅C3    γ  1+      C2    C3   γ     2 ⋅     C2  1−   2⋅C3  γ    1+      C2    C3    γ  2 ⋅     C2      Q(γ ,C): =  2⋅C3    γ  1+       C 2     2⋅C3 C3    γ   γ       C   C   C3    2 2 ⋅ 2⋅ − 1   2 ⋅ ( C0 − C1 ) ⋅ ⋅ C2   γ  2⋅C3     γ  2⋅C3     1 +     1 +            C C 2 2          2⋅C3    γ   γ      ln   C  C    C2   γ  3   2 ⋅ 1− 2⋅  2 ⋅ ( C1 − C0 ) ⋅   ⋅  C2   γ  2⋅C3     γ  2⋅C3       1 +    1 +        C2      C2     

211

Polymers and Carbon Black

N660

N330

N110

 0.269   0.299   C0 =   1.48     0.5 

 0.53   0.72   C1 =   1.22     0.5 

 0.7  : initial guessed parameters 1.033  as extracted from the experi C2 =   1.22  mental data sets    0.5 

Resj: = GenFit(γ,Gvj,Cj,Q)

: calling the nonlinear regression algorithm

Fitting equation



 ( Res j )3    γ    2⋅  (Res j )2       Gb j := (Res j )0 + (Res j )1 − (Res j )0  ⋅ 2⋅( Res j )3   γ    1+     ( Res ) j 2    

Correlation coefficient r2



 ( Res j )3     γ     2⋅    ( R es )  j 2   Gbr2 j := corr  (Res j )0 + (Res j )1 − (Res j )0  ⋅ , Gv j  2⋅( Res j )3     γ   1+       (Res j )2     

    N660



N330

N110

:Ginf ′′  0.224   0.477   0.659   0.3   0.713   1.026  ′′     Res1 =      Res 2 =      : Gm Res 0 =   2.174   1.808   1.428  : γc       :m  0.491  0.706   0.547 

    Gbr20 = 0.709

Gbr21 = 0.64

Gbr22 = 0.692

: r2

Viscous modulus, MPa

212

Filled Polymers

The Kraus model fits reasonnably well the experimental G′ data but cannot meet the asymmetric shape of G” vs. strain data.

1

0.5

0 0.01

0.1

1 Strain, %

10

Note: the results of the nonlinear algorithm are very sensitive to the initial guess values, particularly for γc; using the strain position of Gmax ′′ somewhat reduces this problem.

100

A5.3 Ulmer Modification of the Kraus Model for Dynamic Strain Softening: Fitting the Model Elastic modulus

Ge( γ 0 ) = Geinf +

( Ge0 − Geinf )   γ 0  2⋅m  1 +      γ c  

Viscous modulus (Ulmer’s modification)



Gv( γ 0 ) = Gv inf +

γ  2 ⋅ ( Gv m − Gv inf ) ⋅  0   γc  γ  1+  0   γc 

2⋅m

m

 −γ  + Gv k ⋅ exp  0   γk 

Dynamic strain softening data on SBR/60 phr black cpds [M. Gerspacher, C.P. O’Farrell, C. Tricot, L. Nikiel, H.A. Yang. ACS Rubb. Div; Mtg, Louisiana, KY, 1996. Paper 74]

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Polymers and Carbon Black

Strain sweep experiments N660

N 330

N110

Strain %

G′ MPa

G ′′ MPa

G′ MPa

G ′′ MPa

G′ MPa

G ′′ MPa

0.091   0.19  0.28   0.38  0.48   0.55   0.66  0.74   0.84  0.93   1.07  Data :=  1.22  1.48   1.7  1.9   2.8   3.7  4.6   5.5  6.5   7.6   8.5  9.5

2.34 2.31 2.330 2.26 2.24 2.23 2.22 2.21 2.20 2.18 2.16 2.15 2.14 2.12 2.11 2.06 2.05 2.04 2.02 2.00 1.99 1.97 1.95

0.265 0.275 0.280 0.290 0.295 0.296 0.297 0.297 0.297 0.298 0.2298 0.299 0.299 0.298 0.298 0.294 0.290 0.294 0.285 0.280 0.275 0.271 0.269

4.45 4.41 4.30 4.24 4.10 4.05 3.95 3.89 3.82 3.75 3.61 3.55 3.46 3.40 3..34 3.08 2.95 2.85 2.78 2.71 2.65 2..62 2.57

0.630 0.630 0.650 0.660 0.670 0.685 0.692 0.7700 0.710 0.716 0.720 0.720 0.719 0.718 0.717 0.690 0.660 0.631 0.608 0.581 0.560 0.542 0.530

6.02 5.95 5.80 5.61 5.50 5.32 5.18 5.05 4.95 4.83 4.65 4.48 4.31 4.20 4.08 3.71 3.49 3.30 3.18 3.08 3.01 2.95 2.90

0.915  0.938  0.942   0.960  0.9980   0.993   1.000  1.010   1.026  1.030   1.032   1.033  1.031   1.030  1.020   0.968   0.914  0.870   0.828  0.782   0.752   0.722  0.700 

Geinf = G’inf: elastic modulus for an infinite strain Ge0 = G’0: initial low strain elastic modulus Gvinf = G″inf: viscous modulus at infinitely low or high strain Gvm = G ″m: maximum value of viscous modulus (occuring at the critical strain) Gvk = G ″k drop in viscous modulus assumed to be proportional to the number of contacts between aggregates at zero strain m = exponent related to the rates of deagglomeration and reagglomeration of filler particles γ0 = strain amplitude γc = critical strain for G ″m γk = critical strain for G ″k

A5.3.1  Modeling G′ vs. Strain (same as Kraus) Extracting data  Strain (%)  N660 G′ (MPa)  N330 G′ (MPa)  N110 G′ (MPa) j : = 0..2      γ: = Data<0>  Ge0: = Data<1>   Ge1: = Data<3>   Ge2: = Data<5>

n: = length(γ)   n = 23   : number of data

Guess parameters for nonlinear fitting algorithm (GenFit function)

214

Filled Polymers

 ( Ge j )n −1     ( Ge j )0   Cj : =   γ round  n    2       0.6 

< = extracting guess parameters from experimental data, i.e., - The lowest measured modulus for G ′inf - The highest measured modulus for G’0 - The mid range strain for γc - The common value considered by Kraus

C1 − C0   C0 + 2⋅C3    γ    1+  2  C      1   1−     γ  2⋅C3    1 +        C2     1   2⋅C3     γ     1 +    F(γ ,C) : =     C2     2⋅C3 C3 C1 − C0  γ    2⋅ ⋅  ⋅   2⋅C3 2  C  C 2 2   γ     1 +        C2             2⋅C   γ  3  γ  C1 − C0  − 2 ⋅ ⋅    ⋅ ln        γ  2⋅C3  2  C2    C2     1 +          C2    

Model equation and partial derivatives:

N660  1.95   2.34   C0 =   1.48     0.6 

N 330  2.57   4.45   C1 =   1.48     0.6 

Resj: = GenFit(γ, Gej, Cj, F)

N110  2.9  6.02   C2 =   1.48     0.6 

:initial guessed parameters as extracted from the ­experimental data sets

: calling the nonlinear regression algorithm

215

Polymers and Carbon Black

Fitting equation

Correlation coefficient r2

               (Re s j )1 − (Re s j )0  (Re s j )1 − (Re s j )0   , Ge j  Ga j : = ( Re s j )0 + Gar2 j : = corr  ( Re s j )0 + 2⋅(Re s j )3  2⋅(Re s j )3     γ   γ     1+  1+               (Re s j )2   (Re s j )2  

Results: fit parameters



N660

N 330

N110

 1.825   2.414   Res 0 =   1.761     0.344 

 2.363   4.581  Res1 =   1.471     0.573 

 2.621  6.209   Res 2 =   1.376     0.614 

: Gi′nf : G0 : γc :m

Gar 2 0 = 0.998

Gar 2 1 = 0.999

Gar 2 2 = 1

:r 2

Elastic modulus, MPa

8 6 4 2 0 0.01

0.1

1 Strain, %

Fit N660 Data N660 Fit N110 Data N330

10

100

Fit N330 Data N110

A5.3.2  Modeling G′′ vs. Strain Extracting data   Strain (%)   N660 G″ (MPa)   N330 G″ (MPa)   N110 G″ (MPa)

j: = 0..2      γ : = Data<0>  Gv0 : = Data<2>   Gv1 : = Data<4>   Gv2: = Data<6> Guess parameters for nonlinear fitting algorithm (GenFit function)

216

Filled Polymers

Finding max G′′ and corresponding γ gc j := A ← max ( Gv j ) for i ∈1..n − 1  γi  B←  if ( Gv j )i = A ( Gv j )i  A←B A

C    γ  3 2 ⋅ (C1 − C0 ) ⋅      C2   −γ   C + + C ⋅ exp 4  C   2⋅C3  0 (Gv j )n −1  5  γ    1+      C2    ( gc ) j 1     C3  (gc )   γ    j 0  C      2        C j :=  0.55  1− 2⋅   2⋅C3   γ       + 1         C  (Gv j )0  2      (gc )  C3   0 j γ          C2   100    2⋅     γ  2⋅C3    1 +         Model equation and   C2     3⋅C C Q( γ , C) :=  partial derivatives:  γ  3  γ  3      −  C    2 ⋅ (C − C ) ⋅ C3 ⋅  C2  2 0 1 Extracting guess parameters   2⋅C3 2 C2    γ    from experimental data, i.e.: 1+         C 2       3⋅C C  - The measured modulus at the  γ  3  γ  3   −     highest strain for G″inf  C2    2 ⋅ (C − C ) ⋅ ln  γ  ⋅  C2  0 1  C    2⋅C 2 - The maximum measured 2   γ  3   ″ + 1   modulus for G m       C2     - The strain for the maximum    −γ  G″   exp      C5  - The common value consid  ered by Kraus γ  −γ    exp ⋅ ⋅ C 4 2    C5  - The difference G ″m –G″0 (meaC ( ) 5  

sured modulus at the lowest strain) for G″k - γc/100 for γk

217

Polymers and Carbon Black

N660  0.269   0.299     1.48  C0 =    0.55   0.265     0.015 

N 330

N110

 0.53   0.72     1.22  C1 =    0.55   0.63     0.012 

 0.7   1.033     1.22  C2 =    0.55   0.915     0.012 

Resj: = GenFit (γ, Gvj, Cj, Q)

: initial guessed parameters as extracted from the experimental data sets

: calling the nonlinear regression algorithm

Fitting equation:  (Re s j )3    γ    2⋅   (Re s j )2    −γ   + (Re s j )4 ⋅ exp  Gb j : =  (Re s j )0 + (Re s j )1 − (Re s j )0  .  2⋅(Re s j )3   γ    (Re s j )5     + 1          (Re s j )2 

 

r2:

 

    (Re s j )3      γ   2⋅    −γ     ( Re s j )2         Gbr2 j : = corr   Re s j + ( Re s j )1 − ( Re s j )0  . + ( Re s j )4 ⋅ exp , Gv j  2⋅( Re s j ) 0     3 Re s     j 5    γ    1+      Re s j       2  

(

)

(

N660



(

)

N 330

N110

 − 0.284   0.3     1.271  Res 0 =    0.162   0.13     0.042 

 0.087   0.723     1.351  Res1 =    0.466   0.371     0.156 

 0.12   1.036     1.295  Res 2 =    0.519   0.572     0.199 

Gbr 2 0 = 0.985

Gbr 2 1 = 0.999

Gbr 2 2 = 1

)

: Ginf ′′ : Gm ′′ : γc :m : Gk′′ : γk : r2

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Filled Polymers

Viscous modulus, MPa

1

0.5

0 0.01

0.1

1 Strain, %

10

100

The Ulmer’s additional term to the Kraus equation for G ″ vs. strain gives a considerably improved fit of experimental data.

A5.4 Aggregates Flocculation/Entanglement Model (Cluster–Cluster Aggregation Model, Klüppel et al.) Aggregate solid fraction  d φ A (D) =    D

3− F

 F = fractal dimension of aggregate ( = 1.8)

a

D

Elastic modulus of aggregate:  3+ FB   

GA = Gp ⋅ φA  3− F  FB = fractal dimension of aggregate backbone ( = 1.2)

d

Gp = Averaged elastic modulus of the aggregate with respect to all its possible angular deformations (bending, twisting) Gp is in fact controlled by the BdR which consists of a layer of immobilized rubber, of thickness a F: = 1.8 FB: = 1.2 :typical fractal dimensions for CB  = >

3 + FB = 3.5 3− F

219

Polymers and Carbon Black

A5.4.1  Mechanically Effective Solid Fraction of Aggregate Np (D): = 200

: number of primary particle of size d in the aggregate of size D d: = 30⋅10 –9⋅m : primary particle diameter (e.g. = 30 nm in N330) D: = 200⋅10 –9 ⋅m : aggregate size a: = 2 ⋅ 10 –9⋅m : tightly BdR layer thickness (typical 2 nm)



 d Aggregate solid fraction:    φ A (D) : =    D

Solid volume of Np particles of diameter d:

3− F

φ A (D) = 0.103

3  d3  N p (D) ⋅  π ⋅  = 2.827 ⋅ 106  ( 10−9 ⋅ m )  6

Diameter of equivalent sphere (according to Medalia) Des: =  d⋅Np(D)0.424 Volume of a sphere with diameter Des:

   

 d3     N p (D) ⋅  π ⋅ 6   = 0.237  Des 3   π ⋅ 6 

π⋅

Des3 = 1.195 ⋅ 107  (10−9 ⋅ m)3 6

 = > the aggregate solid fraction as ­calculated from fractal condiseration is likely underestimated

The effective fraction of the aggregate must include the tightly BdR fraction: 2⋅π 2   d π   N p (D) ⋅  ⋅ (d + 2 ⋅ a)3 − ⋅ a ⋅  3 ⋅  + a − a    2 6 3    φe A (D) : = π 3 ⋅D 6

 d    N p (D) ⋅ (d + 2 ⋅ a)3 − 4 ⋅ a 2 ⋅  3 ⋅  + a − a   2     φe A (D) : = D3

 

or:  

< =  Equation 71 given by Heinrich and Klüppel [Adv. Polym. Sci., 160, 1–44, 2002] φe A (D) : =

 = > effective fraction of the aggregate



N p (D) ⋅ (d 3 + 6 ⋅ d 2 ⋅ a + 6 ⋅ d ⋅ a 2 ) D3

: φe A(D) = 0.963

2⋅π 2   d   ⋅ a ⋅  3 ⋅  + a − a  is expected to take into account the vol3  2   ume resulting from the intersections of the rubber layer of thickness a, at the The term

220

Filled Polymers

contact points between two neighboring particles; however this corrective term is in fact relatively small, as suggested by calculation results with the typical values given above, i.e.:



π ⋅ (d + 2 ⋅ a)3 = 2.058 ⋅ 10 4  (10−9 ⋅ m)3 6

20, 580 nm 3

2⋅π 2   d  3  ⋅ a ⋅  3 ⋅  + a − a  = 410.501ϒ( 10−9 ⋅ m )   3 2  

410 nm 3

According to Heinrich and Klüppel an approximation for the effective aggregate fraction is possible if a << d, as follows:

φefA (D) : =

N p (D) ⋅ [(d + 2 ⋅ a)3 − 6 ⋅ d ⋅ a 2 ] D3

=>

φefA (D) = 0.965

Rem: one notes that (d + 2⋅a)3–6⋅d⋅a2 = d3 + 6⋅d2 ⋅a + 6⋅d⋅a2 + 8⋅a3; so this approximation means that the term 8.a3 is neglected φefA (D) − φA (D) = 8.398 φ AD

BdR increases the mechanically effective fraction of the aggregate by some 8% (with respect to numerical values used)

φ = φA(D)   : space filling condition Gp: = 1 ⋅ 1 09⋅Pa

: elastic modulus of aggregate (para-crystalline carbon: 1 GPa)

GR: = 1 ⋅ 106 ⋅ Pa

: elastic modulus of a gum rubber (typical)

A5.4.2  Modulus as Function of Filler Volume Fraction  3 + FB     N p (D) ⋅ [(d + 2 ⋅ a)3 − 6 ⋅ d ⋅ a 2 ]   3 − F    G(φ) : = GR + Gp ⋅  ⋅ φ   D3    

BdR : = 1, 1.01...1.4

φ: = 0, 0.01..0.25 Klüppel and Heinrich model based on fractal considerations.

GGG(φ, BdR): = GR⋅[1 + 2.5⋅(BdR⋅φ) + 14.1⋅(BdR⋅φ)2] : Guth and Gold equation

221

Polymers and Carbon Black

Compound modulus, MPa

10

When compared with the Guth and Gold equation (which is based on mere hydrodynamic considerations) the Klüppel and Heinrich model predicts a stronger dependence on filler volume fraction.

5

0 0

0.05 K&H

0.1 0.15 Volume fraction G&G, 8% BdR

0.2

0.25

G&G, 30% BdR

A5.4.3  Strain Dependence of Storage Modulus [G. Heinrich, M. Klüppel. Adv. Polym. Sci., 160, 1–44, 2002. Equation 76] −τ

  γ  2⋅m  Ge( γ ) = (Ge0 − Ge f ) ⋅ 1 +    + Ge f   γ c  

τ = elasticity exponent of percolat τ =

3 + FB 3− F

γ c = critical strain γ: = 0.0001, 0.0002..2   Ge0: = 8.4 ⋅ 106 ⋅ Pa   Gef: = 1.1 ⋅ 106 ⋅ Pa  

Ge0 + Ge f = 4.75 ⋅ 106 Pa 2

Comparison with Kraus model Heinrich and Klüppel γcHK: = 0.20    mHK: = 0.22    τ: = 3.5 −τ

  γ  2⋅mHK  GeHK ( γ ) : = (Ge0 − Ge f ) ⋅ 1 +   + Ge f     γ cHK  γ cK : = 0.007

   

Kraus :

m K : = 0.27

1  GeK (γ ): = ( Ge0 − Ge f ) ⋅  2⋅m K 1+  γ    γ cK 

  + Ge f  

222

Filled Polymers

Elastic modulus, MPa

10

γcK

γcHK

The upper and lower limits of the term

5

  γ  2⋅mHK  1 +      γ c  

0 1.10–4 1.10–3

0.01 0.1 1 Strain Heinrich & Klüppel H&K at critical strain Kraus Kraus at critical strain

−τ

are 1 and 0

respectively, and, at the critical strain, this term equals 0.088 ( = 2-τ)  = > the critical strain γcHK is the strain for which the elastic modulus is equal to Ge0⋅2-τ + Gef(1 – 2–τ).

10

 

1

  γ cHK  2⋅mHK  1 +      γ cHK  

−τ

2

= 0.088 1 −τ

= 0.0088

γ

γcHK

2.mHK

–τ 0.5

2–τ

GeHK(γcHK) = 1.745⋅106 Pa

0 1.10–4 1.10–3 0.01 0.1 γ

Ge0⋅2–τ + Gef(1 – 2–τ) = 1.745⋅106 Pa

1

10

A5.5  Lion et al. Model for Dynamic Strain Softening [A. Lion, C. Kardelky, P. Haupt. Rubb. Chem. Technol., 76, 533–547, 2003] A5.5.1  Fractional Linear Solid Model (Rem: analog to the Zener model where the dashpot viscosity has a ­fractional exponent) ηβ

Model parameters: E1: = 6 ⋅ 106 ⋅ Pa

: First spring modulus

E2: = 31 ⋅ 106 ⋅ Pa

: Second spring modulus

E1

η: = 1.3 ⋅ 1010 ⋅ Pa⋅sec : fractional dashpot viscosity β: = 0.495

E2

: fractional exponent  

223

Polymers and Carbon Black

λ:=

η E2

λ = 419.355 sec

: model characteristic timee

ω : = 1 · Hz.. 100 · Hz  : frequency range  γ : = 0.0001, 0.0002…10  : strain range Strain and frequency dependency of the intrinsic time scale: b: = 51000   : constant   τ: = 1 · sec   : characteristic time α: = 0.44    : fractional exponent a( γ , ω ) : = 1 +



Intrinsic time scale vs. strain at fixed frequency

Intrinsic time scale vs. frequency at fixed strain 3.104 Intrinsic time scale

Intrinsic time scale

3.106

2.104

2.106 1.106 0

2 ⋅ b ⋅ γ ⋅ (ω ⋅ τ)α π

1.104

0 1 Hz

5 Strain 10 Hz

10 100 Hz

0

0

50 Frequency

100

Strain : 0.001 Strain : 0.01 Strain : 0.1

The fractional exponent a gives a nonlinear variation of the intrinsic time scale with frequency (at constant strain). A5.5.2  Modeling the Dynamic Strain Softening Effect Elastic modulus:



2⋅β β  λ  λ    π  + ⋅ E2 ⋅  ω ⋅ ω ⋅ cos  β ⋅       2    a( γ , ω )   a( γ , ω )  Ge( γ , ω ) : = E1 + 2⋅β β λ  λ     π + 2 ⋅ω ⋅ ⋅ cos  β ⋅  1+ ω ⋅  2  a( γ , ω )   a( γ , ω ) 

224

Filled Polymers

γ c (ω ) : =

Elastic modulus, Pa

4.107

γc(1)

3.107

π ⋅ (ω ⋅ λ − 1) 2 ⋅ b ⋅ (ω ⋅ τ)α

γ c (1 ⋅ Hz) = 0.013

γc(100)

Ge( γ c (1 ⋅ Hz), 1 ⋅ Hz) = 2.15 ⋅ 107 Pa

2.107

γ c (100 ⋅ Hz) = 0.17

1.107

Ge( γ c (100 ⋅ Hz), 100 ⋅ Hz) = 2.15 ⋅ 107 Pa

0 1.10–4 1.10–3 0.01 0.1 1 10 Strain 100 Hz 1 Hz 10 Hz G' at critical strain

2 ⋅ E1 + E2 = 2.15 ⋅ 107 Pa 2

Viscous modulus: β

Gv( γ , ω ) : =



Viscous modulus, Pa

8.106 6.106

γc(1)

λ    π E2 ⋅  ω ⋅ ⋅ sin  β ⋅   2  a( γ , ω )  λ   1+ ω ⋅  a( γ , ω ) 

2⋅β

β

λ    π + 2 ⋅ω ⋅ ⋅ cos  β ⋅   2  a( γ , ω ) 

Gv( γ c (1 ⋅ Hz), 1 ⋅ Hz) = 6.349 ⋅ 106 Pa

γc(100)

Gv( γ c (100 ⋅ Hz), 100 ⋅ Hz) = 6.349 ⋅ 106 Pa

4.106 2.106 0 1.10–4 1.10–3 0.01 0.1 Strain 1 Hz 10 Hz

1 100 Hz

10

Frequency effects are well captured by the model.

225

Polymers and Carbon Black

Elastic modulus at 10 Hz Elastic modulus at 10 Hz

4.107

γc(10)

3.107 2.107 1.107 0 1.10–4 1.10–3 Ge(γ,10) E1

0.01

γ

0.1

1

10

Ge (γc(10.Hz), 10.Hz) E1+E2

According to the model, the upper and lower limits for the elastic modulus are E1 + E2 and E1 respectively, and the mid modulus value corresponds to the strain for the maximum value of the viscous modulus; the model exhibits horizontal symmetry with respect to the mid modulus value. Viscous modulus at 10 Hz

Viscous modulus

8.106 6.106

Viscous modulus at 10 Hz

γc(10)

4.106 2.106 0 1.10–4 1.10–3 0.01 0.1 1 10 Strain G" at 10 Hz max G" at the critical strain

The model gives a vertical symmetry of the G″(γ) curve with respect to the critical strain for the maximum G″ value.

226

Filled Polymers

A5.5.3 A Few Mathematical Aspects of the Model  

 π cos  β ⋅  = 0.713  2

 π sin  β ⋅  = 0.702 : thesee two terms are close because of the particcular  2 value of the fractional exponent β

For β = 0.5 they would be equal ( = 0.707)



λ   Terms  ω ⋅  a( γ , ω ) 



1.103

2⋅β

β

and

λ    ω ⋅ a( γ , ω )  vs. strain

Frequency = 10 Hz

100

γc(10)

10 1 0.1 0.01 1.10–3 1.10–4 1.10–3

0.01 0.1 Strain 2 beta

1

10

Beta

λ   Denominator 1 +  ω ⋅  a( γ , ω ) 



2⋅β

At low strain, the 2 β term is the dominant one; at high strain, the β term is the dominant one; at the critical strain, both terms are equal. β

λ    π + 2 ⋅ω ⋅ ⋅ cos  β ⋅  vs. strain   2  a( γ , ω ) 

1.104

Denominator

1.103

100 10 1 1.10–4 1.10–3 0.01 0.1 1 10 Strain at 10 Hz at 100 Hz

At high strain, the denominator becomes asymptotic to 1.

227

Polymers and Carbon Black

A5.6  Maier and Göritz Model for Dynamic Strain Softening [P.G. Maier, D. Göritz. Molecular interpretation of the Payne effect. Kautsch. Gummi, Kunstst., 49, 18–21, 1996] A5.6.1  Developing the Model 1

Rubber–filler interaction sites are considered as “knots” whose effect superimposes to chemical networking. The dynamic modulus is proportional to the overall networking density N and the temperature T (theory of rubber elasticity)

Local area on carbon black particle Site occupied by a stable "link" Isolated site, available for an unstable "link"

2

Free site 3

4

Ge = N⋅kB⋅T

Rubber chain segments

kB = Boltzmann constant



Three contributions are considered for N: N = Nchem + Nstable + Nunstable Nchem

: vulcanization knots

Nstable

: stable rubber-filler knots

Nunstable : unstable rubber-filler knots [ = f(strain)] An equilibrium is assumed between adsorption and desorption of rubber segments on filler sites, i.e.:

Φfree⋅vads = Φocc⋅vdes   [1]   Φfree: (volume) fraction of free sites

Φfree + Φocc = 1           Φocc: (volume) fraction of occupied sites vads vdes

: adsorption rate, assumed to be constant : desorption rate, assumed to be proportional to strain amplitude, i.e., vdes = K⋅γ [2] It follows: Φ occ =



Φ free v ⋅ vads = (1 − Φ occ ) ⋅ ads K⋅γ K⋅γ

=> Φ occ ( γ ) =

1 1 = K ⋅γ 1+ c⋅γ 1+ vads

[3]

228

Filled Polymers

The number of isolated sites Nisolated, available for unstable links, times the fraction of free sites is in fact the number of unstable rubber-filler knots, which depends on the strain amplitude, i.e.: N unstable ( γ ) = N isolated ⋅ Φ occ ( γ ) = N isolated ⋅



1 1+ c⋅γ

There are thus, two contribution in the dynamic modulus, one due to the vulcanization and the stable rubber-filler knots, the other owing to unstable rubber-filler knots, thus, dependending on strain: 1   Gestable =  N chem + N stable + N isolated ⋅ ⋅kB⋅T  1 + c ⋅ γ 

or:

Ge(γ ) = Gest + Geun ⋅

1 1+ c⋅γ

The viscous modulus is considered proportional to the fraction of occupied sites times the probability P that an attached segment is able to slide into a near (free) site, i.e. G″(γ)∼Φocc(g)P. This probability depends on the quantity of free sites at the surface of the particle, i.e., P∼Φfree(γ) There are therefore, two contributions to the viscous modulus, one from the stable links and one from the unstable links, the latter proportional to the product Φfree × Φocc. From Equations [1] and [2], one has K⋅γ =   Φ free ( γ ) = Φ occ ⋅ thus: vads

1 K ⋅γ ⋅ K ⋅ γ vads 1+ vads

Therefore: Gv(γ) = Gvstable + Gvunstable⋅Φocc(γ)⋅Φfree(γ)

Gv(γ ) = Gv stable + Gv unstable ⋅



or:  Gv(γ ) = Gv stable + Gv unstable ⋅

1 K ⋅γ 1+ vads

1 K ⋅γ   ⋅  K ⋅ γ vads  1+  vads  

c⋅γ (1 + c ⋅ γ )2

with

c=

K vads

229

Polymers and Carbon Black

A5.6.2 A Few Mathematical Aspects of the Model γ: = 0.0001, 0.0002, .. 2

: strain range for calculation

Gest: = 0.74 ⋅ 106 ⋅ Pa   c: = 40.15   Gvst: = 0.10 ⋅ 106 ⋅ Pa

Data for butyl/N330 cpd (f N330 = 0.233) used by Maier and Göritz when probing their model

Geun: = 9.52 ⋅ 106 ⋅ Pa   Geun: = 4.55 ⋅ 106 ⋅ Pa Ge(γ ): = Gest + Geun ⋅

Gv(γ ):= Gv st + Gv un ⋅

1 c

5.106

1



c 1.106

5.105

0 1.10–4 1.10–3 0.01 0.1 Strain

10

c is the reverse of the strain for which G′ = 0.5 × G′unstable + G′stable.  1 Ge   = 5.5 ⋅ 106 Pa  c

( 1 + c ⋅ γ )2

1

1.107

0 1.10–4 1.10–3 0.01 0.1 Strain

c⋅γ

1.5.106 Viscous modulus, MPa

Elastic modulus, MPa

1.5.107

1 1+ c⋅γ

1

10

c is the reverse of the strain for which G″ is maximum and equal to 0.25 × G″unstable + G″stable.

 1 Gv   = 1.238 ⋅ 106 Pa  c

Gest + 0.5 ⋅ Geun = 5.5 ⋅ 106 Pa

Gv st + 0.25 ⋅ Gv un = 1.2238 ⋅ 106 Pa

Rem: one notes also that the 1st derivative of

1 −c is 1+ c⋅γ ( 1 + c ⋅ γ )2

Due to its starting hypotheses, the Maier and Göritz model has a mathematical form that leads to symmetries in both the G′ vs. strain and the G″ vs. strain functions; the former exhibits an horizontal symmetry with respect to a mid- modulus value, the latter shows vertical symmetry with respect to a critical strain = 1/c. Both the elastic and the viscous moduli at the critical strain are simple combinations of the “stable” and “unstable” links contributions.

230

Filled Polymers

A5.6.3  Fitting the Model to Experimental Data Dynamic strain softening data on SBR/60 phr black cpds [M. Gerspacher, C.P. O’Farrell, C. Tricot, L. Nikiel, H.A. Yang. ACS Rubb. Div; Mtg. Louisiana, KY, 1996. Paper 74] Strain sweep experiments N660 Strain %

 0.091   0.19  0.28   0.38  0.48   0.55   0.66  0.74   0.84  0.93   1.07  Data: =  1.22  1.48   1.7  1.9   2.8   3.7  4.6   5.5  6.5   7.6   8.5  9.5

G′

G ′′

2.34 2.31 2.330 2.26 2.24 2.23 2.22 2.21 2.20 2.18 2.16 2.15 2.14 2.12 2.11 2.06 2.05 2.04 2.02 2.00 1.99 1.97 1.95

0.265 0.275 0.280 2.290 0.295 0.296 0.297 0.297 0.297 0.298 0.2298 0.299 0.299 0.298 0.298 0.294 0.290 0.294 0.285 0.280 0.275 0.271 0.269

MPa

MPa

Maier and Göritz model Elastic modulus

N330 G′

MPa

4.45 4.41 4.30 4.24 4.10 4.05 3.95 3.89 3.82 3.75 3.61 3.55 3.46 3.40 3..34 3.08 2.95 2.85 2.78 2.71 2.65 2..62 2.57

N110 G ′′

MPa

0.630 0.630 0.650 0.660 0.670 0.685 0.692 0.7700 0.710 0.716 0.720 0.720 0.719 0.718 0.717 0.690 0.660 0.631 0.608 0.581 0.560 0.542 0.530

G′

MPa

6.02 5.95 5.80 5.61 5.50 5.32 5.18 5.05 4.95 4.83 4.65 4.48 4.31 4.20 4.08 3.71 3.49 3.30 3.18 3.08 3.01 2.95 2.90

G ′′

Ge( γ ) = Gest + Geun ⋅

MPa

0.915   0.938  0.942   0.960  0.9980   0.993   1.000  1.010   1.026  1.030   1.032   1.033  1.031   1.030  1.020   0.968   0.914  0.870   0.828  0.782  0.752   0.722  0.700 

1 1+ c⋅γ

Gest = G′stable: elastic modulus due to chemical crosslinks + stable rubber-filler interactions Geun = G′unstable: elastic modulus due to unstable rubber–filler interactions [ = f(strain)] Viscous modulus Gv( γ ) = Gv st + Gv un ⋅

c⋅γ

( 1 + c ⋅ γ )2

Gvst = G″stable: viscous modulus due to chemical crosslinks + stable rubber-filler interactions Gvun = G″unstable: viscous modulus due to unstable rubber–filler interactions [ = f(strain)] c = constant related to the rate of adsorptiondesorption of rubber segments on appropriate sites on filler particles γ = strain amplitude

231

Polymers and Carbon Black

A5.6.3.1 Modeling G′ vs. Strain Extracting data Strain(%)

 

j: = 0..2

γ : = Data

N660G ′(MPa) Ge0 := Data

<0>

<1>

n: = length( γ )

N330G′(MPa) Ge1 := Data n = 23

N110 G′(MPa) Ge2 := Data<5>

<3>

: number of data

Guess parameters for nonlinear fitting algorithm (GenFit function)  ( Ge j )0    C j : =  ( Ge j )n− 1    1   γ   n  round 2  

< = extracting guess parameters from experimental data, i.e., - The highest measured modulus for G′stable - The lowest measured modulus for G′unstable - The reverse of the mid range strain for c Model equation and partial derivatives:

      N660

N330

C1    C0 + 1 + C ⋅ γ  2   1     1 F ( γ , C) : =    1 + C2 ⋅ γ    −C1  ⋅γ  ( 1 + C ⋅ γ )2  2  

N110

 6.02   2.34   4.45        C0 =  1.95      C1 =  2.57      C2 =  2.9   0.676   0.676   0.676 



Resj: = GenFit(γ,Gej,Cj,F) : calling the nonlinear regression algorithm Fitting equation

Correlation coefficient r2

  ( Re s j )1  Ga j : = ( Re s j )0 +  1+ ( Res j )2 ⋅ γ  

  ( Re s j )1 , Ge  Gar2 j : = corr  ( Res j )0 +  j   1 + ( Re s j )2 ⋅ γ    

   N660



N330

N110

:Gstable ′ 1.917   2.236   2.328  ′ Res 0 =  0.445     Res1 =  2.421     Res 2 =  4.072     :Gunstable  0.708   0.642   0.679  :c

   Gar20 = 0.997

Gar21 = 0.999    Gar22 = 0.999

:r 2

232

Filled Polymers

Elastic modulus, MPa

8

6

4

2

0 0.01

0.1

1 Strain, %

10

100

A5.6.3.2  Modeling G″ vs. Strain  

Extractingdata

Strain(%)

N660 G ′′(MPa)

N330 G ′′ ( MPa)

N110G ′′ ( Mpa)

j = 0..2

γ : = Data<0>

Gv 0 := Data<2>

Gv1 := Data<4>

Gv 2 := Data<6>

Guess parameters for nonlinear fitting algorithm (GenFit function) Finding max G″ and corresponding γ gc j : = A ← max ( Gv j ) for i ∈1..n − 1  γi  B←  if ( Gv j )i = A ( Gv j )i  A←B A

Model equation and partial derivatives:

Extracting guess parameters from experimental data, i.e.: - The measured modulus at the highest strain for G″stable - The highest measured modulus for G″unstable - For c, the reverse of the strain corresponding to the max G″ data

   Gv ( j )n−1    C j : ( gc j )1     1     ( gc j )0 

C ⋅C ⋅ γ   c0 + 1 2 2   (1 + C2 ⋅ γ )     1   C2 ⋅ γ Q( γ , C) : =   2   (1 + C2 ⋅ γ )     C1 ⋅ γ γ2 − 2 ⋅ C1 ⋅ C2 ⋅  3 2 (1 + C2 ⋅ γ )   ( 1 + C2 ⋅ γ )

233

Polymers and Carbon Black

       N660

N330

N110

 0.269   0.53   0.7      C0 =  0.299  C1 =  0.72  C2 = 1.033   0.676   0.82   0.82         



Resj: = GenFit(γ,Gvj,Cj,Q)

: calling the nonlinear regression algorithm

Fitting equation

Correlation coefficient r2

  ( Res j )1 ⋅ ( Res j )2 ⋅ γ  Gb j : = ( Res j )0 + 2  1 + ( Res j ) ⋅ γ    2

  ( Res j )1 ⋅ ( Res j )2 ⋅ γ , Gv   Gbr 2 j : = corr  ( Res j )0 + j 2  1 + ( Res j ) ⋅ γ      2

   N660

N330

N110

: G′′ stable  0.254   0.492   0.661  : G′′ unstable Res 0 =  0.188  Res1 =  0.903  Res 2 = 1.507   0.799  1.027   1.243  :c          : r2    Gbr20 = 0.968 Gbr21 = 0.925 Gbr22 = 0.92

Viscous modulus, MPa

1

The Maier and Göritz model fits reasonnably well the experimental G′ data but cannot meet the asymmetric shape of G″ vs. strain data.

0.5

0 0.01

0.1

1 Strain, %

10

100

6 Polymers and White Fillers

6.1  Elastomers and White Fillers 6.1.1 Elastomers and Silica 6.1.1.1  Generalities In contrast with carbon black, silica has particular surface properties that bring a number of problems when using such materials as reinforcing fillers, particularly in hydrocarbon elastomers. Indeed the surface of silica, either fumed or precipitated, is strongly polar and hydrophilic, owing to its polysiloxane structure with numerous silanol groups. This particular surface chemistry of silica has several immediate consequences:



1. Silica surface can adsorb significant quantities of water, as reflected by the well known usage of the material as drying agent (or moisture absorber) 2. Moist silica is very difficult to dry 3. Interparticle interactions are very strong because of hydrogen bonding 4. Silica is the ideal filler for silicone polymers, for instance polydimethylsiloxane 5. When used as filler for diene elastomers, chemical modification of silica surface is required firstly to promote mixing through decreased inter-particles interactions and secondly to establish ­adequate ­rubber–filler interactions.

Most of the specific properties of silica (and silicates) were known more than three decades ago, as reviewed by Wagner in 1976,1 who somewhat foresaw the tremendous developments that followed. Indeed, in the last decades, significant progress was made in using silica in diene elastomers, supported by the peculiar dynamic properties this filler brings to (diene) elastomers. It permitted the development of the so-called “green tire,” because a lower rolling resistance is obtained, when compared with carbon black reinforced tires. 235

236

Filled Polymers

6.1.1.2  Surface Chemistry of Silica The siloxane group Si–O–Si is nonpolar and therefore would provide a hydrophobic character to the silica surface. But, as illustrated in Figure 6.1, there are various types of hydroxyl groups that are strongly hydrophilic, from isolated (or free) silanols to silanediols and silanetriols (rare, if any). All these groups are easily identified by infrared analysis. Vicinal silanols can develop hydrogen bonding and eventually siloxane groups through water elimination. Free silanols are more reactive than vicinal groups and therefore promote inter-particles hydrogen bonding. They are also the prime reactive site for organic molecules, namely organo-silanes. The average surface density is around 3–6 silanol/nm2, depending on the specific area. Thanks to various equilibrium reactions involving hydrogen interactions, silica can adsorb significant quantities of water, in such a manner that all silanol groups on the surface are easily saturated. When silica is heated above 100°C, such a physisorbed water is removed and becomes negligible above 250°C. Between 200 and 500°C, condensation of vicinal silanols occurs to yield siloxane groups, and above 600°C, free silanols start also to condensate. When compared with carbon black, the surface chemistry of silica is the key parameter in reinforcement with this filler. Indeed silica surface is occupied by sizable quantities of siloxane and silanol groups, giving rise to hydrogen interaction with either “free” or “bound” water. Free water is easily removed by drying at 105–250°C, while bound water is only released at 900–1000°C and results in fact from the condensation of vicinal silanol groups. As can be expected the free moisture content of silica strongly affects the rheological and curing properties of silica filled rubber compounds. Another important consequence of the oxygen rich surface chemistry of silica are the strong interparticles interactions through hydrogen bonding that, on one hand give rise to poorer dispersibility than carbon black (in nonpolar polymers), and on the other hand permit the use of suitable chemical promoters, e.g., bis(triethoxysilylpropyl)tetrasulfane, TESPT, to form covalent bonding with (unsaturated) elastomer. Of course, certain specialty elastomers such as H

H

O

O

O

Si

Si

Si

O

O O O O

O

O

OH

Vicinal silanols

Figure 6.1 Surface chemistry of silica.

Si O

OH Si

OH OH Si

OH

Si

O

OH Si

Si

O O O O O O O O O O Free silanol Siloxane Silanediol Silanetriol => hydrogen bonding (rare, if any) between particles

O

237

Polymers and White Fillers

polydimethylsiloxanes naturally interact with silica, thank to their similar chemistry. It is for instance long known that, with silicone polymers, the silica surface chemistry can be varied nearly at will, by controlling the degree of adsorbed water, the hydroxyl population and the degree of organophilicity,2 all aspects largely exploited by the silicone rubber industry.3 6.1.1.3  Comparing Carbon Black and (Untreated) Silica in Diene Elastomers Silicone rubber and, in general polar polymers, are by nature materials of choice for preparing silica filled systems; however limited to niche applications, with respect to the range of properties that such specialty polymers may offer. In order to develop optimum reinforcing performance with more common diene elastomers, silica must be chemically treated as we will see below, because contrary to carbon blacks, silica particles do not develop spontaneous strong interactions with nonpolar polymers. It is nevertheless interesting to see that, even with comparable size and structure, pure silica does not affect the mechanical properties of vulcanized rubber compounds in the same manner as carbon black. This was clearly demonstrated in the excellent review paper by S. Wolff 4 who studied the effects of two comparable series of silica and carbon black in 50 phr filled natural rubber (RSS1) compounds, vulcanized with peroxide (Note that such a vulcanization system was chosen because there is no interference between silica and peroxide curing). Table 6.1 gives typical size and structure data for the two series of filler considered. It is worth underlining that there are no standard methods for characterizing silica. Either an existing method can be used as such because it does not depend on the filler Table 6.1 Comparable Series of Precipitated Silica and Carbon Black

Filler Grade Precipitated silica

Furnace blacks

a

1 2 3 4 5 6 N660 N550 N326 N330 N356 N220 N110

DBP or TEA Absorptiona (ml/100 g)

N2 Specific Area (m2/g)

Uncompressed

30 48 123 167 172 173 36 40 76 78 88 110 139

100 164 192 227 188 204 95 123 70 100 153 114 115

Crushed (24M4)

DBP, di-butylphthalate for carbon black; TEA, triethanolamine for silica.

64 74 90 90 96 93 70 86 64 85 113 94 94

238

Filled Polymers

considered, or the method must be modified to take into account the surface chemistry of silica. For instance the specific area of silica can be assessed through nitrogen adsorption (BET method) but for aggregate structure, dibutylphtalate adsorption is not convenient (because DPB does not “break” interparticles hydrogen bonding). Adsorption of triethanolamine gives correct results, comparable to data obtained on carbon black with DPB. Comparing mechanical properties imparted by either carbon black or silica in a purposely simple natural rubber formulation allows several interesting conclusions to be drawn. Figure 6.2 shows for instance the 100% and 200% modulus, both affected by the size and the structure of the filler. At low strain, i.e., 100%, most precipitated silica and several high structure blacks exhibit similar reinforcing capabilities; at higher strain however, all silica are clearly less reinforcing than carbon black. Wolff attributed this effect to a “silica network” which is destroyed when straining vulcanizates. Low strain amplitude dynamic properties reveal quite an interesting aspect of silica reinforcement. As shown in Figure 6.3, high structure silica (i.e., with crushed TEA adsorption values higher than 80 ml/100 g) give NR compounds with higher complex modulus G* and lower tan δ than carbon blacks of similar structure. This advantages of high structure silica over carbon blacks is also observed when performing technological dynamic tests, for instance rebound resilience test. However, in line with tensile modulus data, silica gives larger compression sets than carbon blacks. Despite the fact that the compounds investigated were (purposely) oversimplified with respect to industrial practices, the key information in the experiments reported by Wolff is that silica filled compounds exhibit definitely better dynamic properties that carbon black filled ones, namely higher rebounds and lower heat build-up. In addition the higher the specific area of fillers, the larger the differences between silica and carbon black loaded materials. Freund and Niedermeier made a similar

100% Modulus, MPa

4.0

Carbon black Silica

3.5 3.0 2.5 2.0 1.5 1.0 60 80 100 120 40 Crushed DBP or TEA absorption, ml/100g

20 200% Modulus, MPa

4.5

15

NR (RSS1) 100 Filler 50 Peroxide (DCP) 2.03

10 5 0 40 60 80 100 120 Crushed DBP or TEA absorption, ml/100g

Figure 6.2 Effect of carbon black and silica structure of tensile properties. (Drawn using data from S. Wolff, Rubb. Chem. Technol., 69, 325–346, 1996.)

239

Polymers and White Fillers

26 20

Carbon black Silica Frequency : 5 Hz Temperature : 23°C

14 8 2 60 80 100 120 40 Crushed DBP or TEA absorption, ml/100g

0.20 0.15

NR (RSS1) Filler Peroxide (DCP)

100 50 2.03

Tan delta

Complex modulus E*, MPa

32

0.10 0.05 0 60 80 100 120 40 Crushed DBP or TEA absorption, ml/100g

Figure 6.3 Effect of carbon black and silica structure of low strain dynamic properties. (Drawn using data from S. Wolff, Rubb. Chem. Technol., 69, 325–346, 1996.)

comparative study on carbon black and silica compounded in nonpolar and polar elastomers.5 From their investigation of the dynamic strain softening effects, they concluded that reinforcement by carbon black and silica essentially proceed from different micro-mechanisms. Polymer adsorption prevails in carbon black filled systems, while filler particle networking is the key aspect in silica filled systems. Performed some 20 years ago, such basic studies (and many others) clearly indicated that highly structured silica (i.e., very large specific area) were surely interesting alternative fillers for carbon black in highly demanding dynamic applications, for instance tire technology. However, when compared to carbon blacks, silica have serious drawbacks, essentially arising from their peculiar surface chemistry. Besides the strong inter-particles interactions which give dispersion difficulties in hydrophobic polymers (notably all diene rubbers, i.e., around 90% of the overall elastomers consumption), the chemically active surface of silica has a strong potential for interacting/interfering with curing systems, particularly when basic accelerators are used, giving lower cure rates and lower crosslink densities (i.e., modulus).4,6,7 6.1.1.4  Silanisation of Silica and Reinforcement of Diene Elastomers In order to fully exploit the promising capabilities of silica in the reinforcement of diene elastomers, it is essential to consider their surface chemistry and to accordingly proceed to a number of changes, at various levels of rubber technology, in terms of formulation, compounding, mixing, and processing. First the vulcanization chemistry must be modified in order to take into account the high chemical reactivity of silica particles surface and the likely modification of the structure of the networking bonds, with respect to the experience gained over the years with carbon blacks. Second particle– particle interactions are very strong with silica, due to hydrogen bonding

240

Filled Polymers

and the hydrophobic character of diene elastomers is surely not a favorable aspect in what silica dispersion is concerned. Third rubber–filler interactions of physical origin, as occurring with carbon blacks, cannot be expected with silica with respect to the shielding effect of silanol groups. It follows that a chemical approach has to be considered in order to create covalent bonding between rubber and silica particles. The benefit in using silanes as coagents in (diene) rubber compounding has been recognized for long1,8 and many organo-silanes were studied. A number of investigated organo-silanes were found either limited in their interest, difficult or inconvenient to use, or too expensive for industrial applications. Eventually, the reduction of silica interparticle interactions, in association with the development of suitable (chemical) bonds with diene elastomers were obtained through the use of so-called “reinforcement promoters,” essentially bifunctional silanes of general formula:

(RO)3–Si–(CH2)n–X

One end of such molecules is expected to specifically react with silanols on silica surface, whilst the other end is expected to eventually interact with the vulcanization system (essentially sulfur based) in order to provide chemical bonding with the rubber network. It worth underlining again the fundamental difference between carbon black and silica reinforcement: no chemistry is needed with the former but is essential with the latter. Many organo-silanes have been synthesized and tested over the years (mainly in the 1970s) and essentially two chemicals were found of interest in the rubber (tire) industry: • Bis(3-triethoxysilylpropyl)tetrasulfane (TESPT); note that TESPT is in fact a mixture of different polysulfane with an average S chains of four9)

(C2H5O)3–Si–(CH2)3–S4–(CH2)3–Si–(OH5C2)3 frequently referred under its commercial name Degussa (Evonik) Si69 • 3-thiocyanatopropyl-triethoxy silane (TCPTS)



(C2H5O)3–Si–(CH2)3–SCN also referred under the commercial name Degussa (Evonik) Si264.

In principle, one may either pretreat silica with such silanes (usually in solution/suspension, with subsequent elimination of solvents) and then use the modified silica in compounding, or consider silane as a compounding

Polymers and White Fillers

241

ingredient and proceed to the silanization during mixing operations. Obviously pretreated silica are expensive products since notwithstanding the cost of such fine chemicals as organo-silanes, solvent elimination, product drying and conditioning bring uncompressible costs. In situ silanization became therefore the preferred approach despite the challenging difficulties in controlling a chemical reaction in a highly viscous medium, i.e., during mixing. In other terms, quite a complex set of physico-rheological processes had to be mastered in equipment that at first were essentially developed for preparing carbon black compounds. Chemistry in highly viscous media is not an issue in carbon black reinforcement and therefore, controlling the variation of temperature during mixing is essentially considered in terms of limitation of the warming up associated with the process, mainly due to viscous heat dissipation effects as arising when shearing a viscoelastic material. With in situ silanization of silica (i.e., during mixing), the problem is completely different since elementary considerations allows the following requirements and difficulties to be a priori identified:





1. An even dispersion of all reactive ingredients has first to be achieved in a highly viscous medium, which means than reactive species displacement is an issue. 2. Reactions between silica and silane must be activated (usually by reaching the appropriate temperature) and completed (by maintaining the appropriate temperature conditions for a sufficient time. 3. Premature reactions between the rubber and the silane must be avoided during mixing operation.

Directly studying silanization chemistry during rubber mixing is very challenging and has never been made (or even tempted) to the author’s knowledge. But a number of very elegant studies of silanization in solution or in suspension have been performed9–13 that eventually confirmed earlier proposals for a likely reactional scheme,14 and allowed to understand certain aspects of the in situ process and the interference with vulcanization. Investigations using rubber compounds15 essentially confirmed the conclusions of such basic studies. The silica modification with a bi-functional organosilane (either TESPT of TCPTS for instance) and the subsequent development of rubber–filler bonding during vulcanization is essentially considered as follows:16

1. Silanisation (Figure 6.4): first one ethoxy group reacts quickly with an isolated silanol (around 85% on silica surface) or a silanediol (15%); then there is hydrolysis of the remaining ethoxy groups, which produce a reticulation of silane molecules through siloxane bonding.



2. Vulcanization (Figure 6.5): the tetrasulfane group (with TESPT silanated silica) is broken and forms rubber–filler covalent bonds with the

242

Filled Polymers

Si

OH

Si

OH

+

C2H5O C2H5O C2H5O

Si (CH2)3 S4 (CH2)3 Si

OC2H5 OC2H5 OC2H5

-CHOH

Si

O

Si

O

Si

O

Si Silica reaction with TESPT

O

Si

Si

OC2H5 (CH2)3 Sa (CH2)3 Sb OC2H5

Silanated silica (a+b = 4)

Figure 6.4 Silica modification with bi-functional organosilane.

Si

O

Si

O

Si

O

Si

O

Si

Si

OC2H5 (CH2)3 Sa (CH2)3 Sb

+ Rubber

Si

O

Sulphur S8

Si

O

Accelerator

Si

O

Si

O

OC2H5

Si

Si

OC2H5 (CH2)3 Sa (CH2)3 Sb OC2H5

Figure 6.5 Silica–rubber bonding during vulcanization.

polymer during the rubber networking. Note that bonds between silanated silica and rubber are either mono- or disulphidic. In fact, the silanization itself occurs in two steps: first there is a reaction between the silanol groups on silica surface with the alkoxy group of the silane, likely through hydrolysis of the alkoxy groups followed by a condensation reaction with the silanols, but direct condensation is also possible. Hydrolysis then condensation is supported by the beneficial influence of the moisture content of silica on the rate of silanization. The second step is a condensation reaction between adjacent molecules of the silane on the silica surface, and a hydrolysis step is also likely occurring. The result is a significant decrease of the hydrophilic degree of silica particles and hence an easier dispersing in hydrophobic elastomers (i.e., most diene rubbers used in tire technology). Detailed investigations on the kinetics of this complex set of reactions have demonstrated that the activation energy of the first step is nearly twice of what is needed for the second step (i.e., 47 kJ/mole vs. 28 kJ/mole) but the secondary step is around 10 times slower that the first one.13 Recently reported results, obtained by using a model silane and time resolved IR spectroscopy in a microreactor with infrared transparent windows brought a very elegant confirmation of such a two steps mechanism.17 It was indeed shown that the silane interacts first by hydrogen bonding with isolated silanol groups. This first step is very fast and the so-immobilized species react dissociatively with silanols to give covalent bonding with the silica surface, while alcohol is released. Hydrogen bonded silane is less stable

243

Polymers and White Fillers

than when covalent bonded. It was also found that vicinal silanol groups do not react with silane, likely owing to a lower reactivity or to steric hindrance. Such results explain why only 25% of the total hydroxyl groups on silica surface are involved in the silanization process. Both sequences of reactions are acid as well as alkaline catalyzed and the rate constant for the primary reaction decreases as the silane content increases. It is likely that the lower accessibility of silanol groups on the filler surface as the silane content increases, and the decrease of H2O available for the hydrolysis step are responsible for this effect. It follows that the optimal loading of the silane is around eight parts of TESPT for 100 parts of silica. Details on this complex chemistry can be found in the referenced papers. The reactional scheme described above prompts several remarks: first ethanol is produced during silanization reactions (around 2 moles of ethanol per mole of TESPT), and must be eliminated of captured by the appropriate formulation ingredient, otherwise there will be porosity in the vulcanized product; second with TESPT, highly reactive tetrasulfane groups are formed which may give thermo-activated reactions with the polymer if the temperature is too high. It follows that controlling the temperature during mixing is a crucial aspect of the operation: it must be high enough for the silanization reaction to be activated and low enough for tetrasulfane networking or premature vulcanization to be avoided. As may be expected the nature of the alkoxy group in the organo-silane is playing a role in the silanization process (Figure 6.6); whilst very fast, the methoxy group cannot be used for obvious toxicological reasons and the reaction rate is decreasing with the size of the group, leaving the ethoxy as the best choice. Ethanol formation during the in situ silanization is readily an issue in practical compounding since for each gram of silane used, around 0.5 g of alcohol would be produced if all ethoxy groups were reacting. Not all ethoxy groups are reacting however13 but, on the factory floor, considerable amount of ethanol are produced, which besides potential health and toxicity hazards, can readily decrease the efficiency of the mixing process, through recondensation in the mixer chamber and hence wall slippage of the compound. The nature of the rubber has been found to affect the silica silanization process, as demonstrated by Table 6.2. It was also established that, at constant mixing time, the reaction efficiency increases with the (dump) temperature, but in the mean time higher mixing temperature increases the risk of premature vulcanization. Lengthening the mixing cycle and/or using several (re)mixing steps offers several advantages, most likely because it favors the volatilization of ethanol. Dierkes has

Methoxy group not used for toxicological reasons

CH3O- >

C2H5O-

> C3H70- > C4H9O > ...

Decreasing rate

Figure 6.6 Effect of the alkoxy groups in the silanization efficiency of organo-silanes.

244

Filled Polymers

Table 6.2 Effect of Rubber Type on the Silica Silanisation Reaction Rubber or Rubbers Blend

Mole Ethanol/mole TESPT

S-SBR/BR NR NR/BR E-SBR NBR EPDM

1.50 1.75 1.80 2.25 0.90 2.30

Source: Data from U.Görl and A.Parkhouse Kautch. Gummi Kunstst., 52, 493–500, 1999. Note: Experimental conditions: TESPT content: 6.5% of silica; 5 min. mixing with dump at 160°C.

0.30

Frequency : 5 Hz Temperature : 23°C

Frequency : 5 Hz Temperature : 23°C

Natural rubber cpd

30

Silica

N110 carbon black 0.20 Tan δ

Elastic modulus E´, MPa

40

20

10

Silica + TESPT

0.10

N110 carbon black

Silica

Silica + TESPT 0

–3

–2 –1 Log double strain amplitude

0

0

–3

–2 –1 Log double strain amplitude

0

Figure 6.7 Effect of silanization on the reinforcing properties of NR compounds (Data from S. Wolff, Rubb. Chem. Technol., 69, 325–346, 1996.)

thoroughly investigated the industrial mixing of silica filled compounds and considered several approaches to overcome such practical problems.18 Silanisation has profound effects on the reinforcing character of silica and allows to obtain vulcanized rubber systems which exhibit certain benefits with respect to corresponding carbon black filled compounds. As expected silica interparticle interactions are considerably reduced through silanization, as reflected by the large reduction in the dynamic strain softening effect. Figure 6.7 shows for instance the dynamic properties of NR

Polymers and White Fillers

245

compounds filled with carbon black, silica, or silica + TESPT, through strain sweep experiments performed at 5 Hz frequency. As can be seen, the treatment of silica with TESPT reduces the elastic modulus drop upon increasing strain amplitude; this indeed corresponds to a strong reduction of interparticles interactions but the reinforcing effect is also reduced (lower modulus that the reference carbon black filled compound. However the interest in using treated silica appears on tan δ which is slightly increased through silanization but remains significantly lower than the homologous carbon black filled system. In other words, silanization reduces the dynamic strain softening effect but keeps the lower viscous dissipation of silica under dynamic strain. These characteristics of silane treated silica are at the origin of the development of silica filled tread bands in automotive tires (so called “green tires” because with the lower viscous dissipation imparted by silica reinforcement, the energy produced by the engine is more efficiently used in moving the vehicle). The complex chemistry associated with the in situ silanization of silica, as mastered today by tire manufacturers, is surely worth detailed considerations, particularly with respect to the processing behavior. The mixing procedure and conditions, the temperature control, the elimination of ethanol, etc. are key issues on the factory floor, so-far typical of each mixing plant and essentially monitored through pragmatic engineering practices, about which tire manufacturers remain relatively discreet. Silanols can react with several compounding ingredients such as stearic acid, polyalcohols, and amines and, obviously would compete with silane and reduce the silanization efficiency. The order of addition of compounding ingredients is consequently of prime importance; indeed the mixing procedure must be such that the whole amount of TESPT is consumed during the primary reaction.15 In situ silanization is a very important subject in contemporary rubber technology, actually well mastered through the appropriate (and complicated) engineering practices,7,19 but outside the very scope of this book. What must be kept in mind with respect to the scope of this book, is first that for silanes to be effective “promoters” of mineral fillers, active chemical groups on filler particles are needed in quite large quantities, second that specific conditions must be met and maintained for a sufficient time, for the silanization to be complete, third that the polymer must have a reactivity potential with some chemical functionalities of the silane. Silica is surely the right filler for the effective use of silanes, but using such chemicals with other minerals having no or a poorer surface chemistry (e.g., kaolin, mica, talc, …) is obviously not expected to bring the same benefits on reinforcement. Nevertheless, as coated mineral particles are generally easier to dispersed in diene rubbers than their uncoated equivalent, silane-treated talc, mica, and kaolin grades permit fine dispersion to be obtained, with obviously less clustered particles that are always potential failure initiation sites. Any silica-filled rubber formulation may benefit from in situ silanization during the compounding operations and, in the tire industry, it is now well established that partial or total substitution of carbon black by silanated

246

Filled Polymers

(precipitated) silica in tread formulation gives the best compromise in terms of rolling resistance and wet grip. It is interesting to note that most of the ­silica-filled tread compounds in use today are essentially developments of the original Michelin formulation patented by Rauline,20 in which a mediumor high-vinyl solution styrene butadiene copolymer (S-SBR) is the main elastomer. S-SBR is produced by anionic polymerization whose capabilities to control the molecular weight distribution and the level of branching are well known. All such characteristics are of course of importance as they provide the rubber matrix the required performances with respect to tire tread dynamic behavior. However, as recently pointed out by Heinrich and Vilgis,21 the reason why S-SBR (instead of less expensive emulsion E-SBR) is the polymer of choice for silica-filled tread compounds is not fully understood. When comparing the tan δ vs. dynamic strain amplitude functions (at 30°C) of two S-SBR and E-SBR compounds with equal silica content (80 phr) and the same hardness, these authors note that the S-SBR system has a significantly lower tan δ (around 20% drop). Such a difference can hardly be explained with respect to the microstructure (cis, trans and 1,2-vinyl) and MW distribution of both elastomers, and Heinrich and Vilgis develop an argument based on the confinement of polymer segments in pores that would exist in amorphous precipitated silica, seen as clusters of elementary spherical particles. Polymer segments could be immobilized (or confined) in such pores if, indeed, their size, in the nanometer range, is close to typical dimensions of the polymer chain, for instance the so-called Kuhn length (from the theoretical view of a polymer chain seen as made of N segments of Kuhn length b, so that each segments are freely jointed with each other; the so-called contour length of the polymer is then L = N b). Because E-SBR is somewhat more branched than S-SBR, it would saturate the external surface of silica clusters, without penetrating much into silica pores. On the reverse, S-SBR segments would have the capability to completely fill pore volume and remained confined, thus giving strong polymer–filler interaction, that would of course superimposed to the chemical interaction imparted by the silanization. Such polymer segment confinement would lead to a hindered polymer dynamics within nano-scale ranges, playing a key role in the frequency domains associated with either wet skid or rolling resistance. Further works are needed to fully support such proposals, but recently published data document indeed the very special dynamic–mechanical properties of silica-filled S-SBR compounds, with very small effects assigned to silanization (with TESPT) up to 70 phr silica. 6.1.1.5  Silica and Polydimethylsiloxane Polyorganosiloxanes, the so-called “silicones,” whose general formula is R′–(SiOR2)n–R′, are a family of polymer materials with unique and interesting properties.3 Silicone polymers have the alternating –Si–O– type structure as part of their backbone chain and although silicon is in the same group as carbon in the periodic table, it has quite a different chemistry. Various

Polymers and White Fillers

247

organic groups, e.g., methyl, benzyl, etc., may be bonded to the silicon to yield polymer materials that are by nature water repellent, heat stable, and very resistant to chemical attack. Poly(dimethylsiloxane) or PDMS is by far the most common silicone polymer, whose flexibility is due to the inorganic siloxane backbone, with a very low surface energy imparted by the methyl groups. This results in a glass-transition temperature of less than –120ºC, and consequently quite a large usage temperatures window, from below –40ºC to above 150ºC. Various PDMS materials are obtained through synthesis and hydrolysis of chlorosilanes, then polycondensation, according to the following schema:

Si + 2 ClCH3 → SiCl2(CH3)2



n SiCl2(CH3)2 + 2 n H2O → Si(OH)2(CH3)2 + 2 n HCl



n Si(OH)2(CH3)2 → [SiO(CH3)2]n + n H2O

The value of n fixes the molecular weight, and hence the viscoelastic nature of the material, from low viscosity oils up to high MW polymers. Vulcanizable elastomers are obtained by introducing reactive sites, for instance vinyl groups. Mechanical properties of silicone polymers are improved through the addition of fillers, with silica the most obvious choice. Globally the same reinforcing effects as in other elastomers are observed with silicone/silica compounds, with the level and the structure of the filler playing qualitatively the same roles. However there are a few singular aspects in polysiloxane/ silica systems worth discussing in details because certain well established scientific knowledge can be somewhat extrapolated to other systems. Interactions between organosiloxanes and silica particles, either fumed or precipitated is a long studied subject, either for purposely promoting grafting chemical reactions on the particles23 or as an approach for understanding the interactions between siloxane and silanol groups.24 One would a priori consider that, owing to their close chemistry, polysiloxanes and silica are naturally compatible and that it is relatively easy to disperse the latter in the former. This is globally true but it has been observed for long that, once a silica and a silicone polymer have been mechanically mixed, the very adsorption process of polymer chains on the surface of silica particles is relatively slow, even if it tends to accelerate at higher temperature. For instance, at 70°C, three months are necessary for a PDMS sample to fully saturate the silica surface. The adsorption properties and kinetics in silica/PDMS systems, the structure of the adsorbed layer and other singular aspects were investigated by a number of authors and their findings allow to somewhat understand certain engineering practices that were pragmatically developed by manufacturers of silicone products, e.g., mastics, sealants, and other self­vulcanizing liquid silicones. Early observations revealed that in useful silica/

248

Filled Polymers

silicone dispersions, the polymer on the silica substrate is approximately ten times more concentrated than what would correspond to a monolayer of water.25 Then, by combining chemical microanalysis, Nuclear Magnetic Resonance (NMR) relaxation and swelling measurements, it was shown that directly after mechanical mixing, PDMS chains are strongly adsorbed on only a quarter of the silica surface, in the form of adsorbed islets.26 It was first established that, during the slow saturation of silica surface, which lasts around three months at 70°C, the average number of contact points of a chain with the silica surface is proportional to N b (Nb is the number of skeletal bonds in one chain).27 Eventually using NMR relaxation and diffusion studies, the structure of silica/PDMS systems was rationalized by a three-state model: close to silica particle, there is a (strongly) bound polymer layer, then polymer chains are entangled or restricted by the adsorbed layer and eventually there is the free bulk polymer.28 Recent works with modern sophisticated analytical techniques essentially confirm previous observations.29,30 It is worth noting here the remarkable similarity between such a description of the structure of silica/silicone systems and the established morphology for carbon black filled rubber compounds (refer to Figure 5.16). Thanks to their similar chemistry, silica and silicone polymers develop thus instant interactions but the full capabilities of such systems are obtained only when the structure of the adsorbed polymer is completely developed. The adsorption kinetics is consequently an issue of industrial importance with silica/silicone systems and was consequently thoroughly investigated. Cohen-Addad et al.31 studied the time dependence of the adsorption of siloxane chains on silica aggregates and proposed the following empirical model:

[Qsat − Q(t)] = [Qsat − Q0 ] × exp  −

t  tad 

(6.1)

where Q(t) is the amount of bound polymer at time t (in g/g of silica), Q 0 and Qsat respectively, the initial (i.e., directly after mechanical mixing) and final (i.e., at saturation) amounts of adsorbed polymer (in g/g of silica), t the time and tad a characteristic time. Experiments at room temperature with either hydroxyl or methyl terminated polymers and 29%wt silica gave very high values for tad, i.e., 7.1 × 107 s (around 2.25 years) and 6.2 × 108 s (nearly 20 years), respectively. Whatever their chain end, polydimethylsiloxanes adsorb on silica particles according to simple equivalent laws of adsorption, proportionally to Mn .32 For a given silicone polymer, there is a specific silica concentration for a tri-dimensional silica-polymer morphology to be obtained, with the associated viscoelastic behavior (i.e., gel or no-gel).33 Equation 6.1 is essentially an empirical model, likely selected for its convenience in fitting experimental data, using a linear algorithm for instance the least square method. It is worth noting that considering the variation of

Polymers and White Fillers

249

the adsorbed amount of polymer with respect to the square root of time is an obvious reference to a Fickean process. Providing nonlinear fitting algorithms are available, an equivalent but more explicit model is:

Q ( t ) = Q0 + Qinf 1 − exp ( − λ t )

(6.2)

where Q 0 is the initial adsorbed polymer (i.e., directly after mechanical mixing) and Qinf the additionally adsorbed polymer after an infinite time (i.e., at saturation), both in g/g of silica), t the time and λ is a parameter related with the characteristic time tad for the adsorption process (i.e. λ = t-0.5 ad). The overall bound polymer for an infinite time is then given by (Q 0 + Qinf), which corresponds obviously to Qsat in Equation 6.1. As shown in Figure 6.8, experimental data on silica/PDMS systems34 were fitted with Equation 6.2, using a nonlinear fitting algorithm (see details in Appendix 6.1). As can be seen the model meets well experimental data and the initial and the final quantities of polymer are directly obtained as fit parameters. The initial quantity of adsorbed polymer is somewhat depending on the mechanical mixing conditions (unfortunately not precisely documented in the source of data) but it is quite clear that, at equal silica content, the higher the molar mass of the polymer, the larger the bound polymer content at the end of the mixing step. Increasing the silica loading gives expectedly a higher initial bound polymer but also a lower quantity of adsorbed polymer at saturation. The value of the characteristic time tad (or the parameter λ in Equation 6.2) gives an insight on the time scale of the adsorption process. Even if increasing the temperature somewhat speeds up the process, quite long maturation periods are necessary for the silica surface to be fully saturated. As clearly seen in Figure 6.8, higher Mn polymers tend to mature faster and silicone product manufacturers obviously take advantage of this effect in tailoring their products. But slow maturation processes mean also that some “ageing” either on storage, or during the life of the material can be expected, as a mere result of polymer chain dynamics in the vicinity of silica particles. For instance, DeGroot and Macosko investigated the aging behavior of silica/ PDMS systems, by measuring the rheological properties, the bound rubber, and the state of dispersion as a function of time.35 They observed softening rather than hardening as typically reported for silica filled systems. Polymer adsorption onto the surface plays obviously an important role in determining the overall stability of these systems and the addition of a surface treating agent (e.g., hexamethyldisiloxane and hexamethyldisilazane), either physically adsorbed or covalently bound to the silica surface, can inhibit the adsorption of polymer. A better stability of the silica dispersion is then observed and therefore variation in rheological properties are reduced. With respect to the silica surface chemistry (see Figure 6.1) and the chemical nature of polysiloxane, the interaction sites are clearly identified since they

0

0.5

1

1.5

2

0

r2

Qinf (g/g) λ tad (h)

Q0 (g/g)

Mn, g/mole :

1000

0.257 0.588 0.044 514.8 0.991

43000 0.320 1.321 0.053 350.7 0.987

73000

2000 Time (h) 0.733 1.483 0.061 273.1 0.992

300000

3000

Mn = 43,000 g/mol

Mn = 73,000 g/mol

Mn = 300,000 g/mol

4000

Adsorbed polymer Q (g/g of filler) 0

0.5

1

1.5

0

0.103 1.702 0.041 588.7 0.980 r2

Qinf (g/g) λ tad (h)

0.049

Q0 (g/g)

2000 Time (h)

Silica fraction :

1000

PDMS Mn = 43000 g/mol 2

0.266 1.042 0.022 2020 0.993

0.103

3000

0.255 0.614 0.039 648.2 0.985

0.204

Φ = 0.204

Φ = 0.103

Φ = 0.049

4000

Figure 6.8 Adsorption kinetics of polydimethylsiloxane on silica particles. (Experimental data from L. Dujourdy, PhD Thesis, University of Grenoble, France, 1996; nonlinear fitting of Equation 6.2.)

Adsorbed polymer Q (g/g of filler)

Filler weight fraction = 0.20 2.5

250 Filled Polymers

251

Polymers and White Fillers

essentially concern hydrogen bondings, either at the ends of the polymer chain (if they are hydroxylated) or through any of the –O– of the chain. It follows that the area of an interaction site is perfectly known, by calculating the “area” of one silanol group, around 0.55 nm2 (55.10–20 m2). In comparison with carbon black/elastomer systems, having a clear identification of the fillermatrix interaction sites is a tremendous advantage in developing a theoretical approach of the adsorption process. Indeed, starting from the percolation theory with only two assumptions, Cohen-Addad developed a model for the bound PDMS polymer on silica at saturation.36 The assumptions are (1) that PDMS chains obey Gaussian statistics, (2) that there is hydrogen bonding at each PDMS-silica contact point. The model is written as follows: Qsat =



M0 c Sp ⋅ ⋅ Mn A0 ε 0 N Av

(6.3)

where Qsat is the bound polymer at saturation (g/g of silica), M0 the mass of the monomer unit [75 g/mole for -Si(CH3)2-O-], A0 the area of one interaction site (i.e., 0.55 nm2), c the filler concentration (g of filler/g of polymer), Sp the specific surface of silica, NAv the Avogadro number, Mn the number average molar mass of the polymer and ε0 a factor for the stiffness of the chain ( ≈ 1 in first approximation). Figure 6.9 shows how experimental data34 on various 1 [1 – exp(– 0.065 √Mn × ΦSil)]

Qsat × ΦSil

0.8

0.6

0.4

Ssp (m2/g) 150 50

0.2

0

300

0

100

200 √Mn × ΦSil

300

Figure 6.9 Adsorption of polydimethylsiloxane on silica particles; Cohen–Addad percolation model for maximum adsorbed polymer at saturation (Equation 6.3) vs. experimental data from L. Dujourdy PhD Thesis, University of Grenoble, France, 1996.

252

Filled Polymers

silica/PDMS systems, largely varying in polymer molar masses and/or silica contents, fall on a single curve when calculating the maximum adsorbed polymer at saturation and suitably reducing the scales. Having the interaction site well identified in a filled polymer system, in terms of chemical activity and surface, and a clear picture of the nature of the polymer–filler interaction allow quite convincing theoretical models to be developed. Such a favorable situation is however restricted to a few cases, namely silica/polysiloxane systems. With other systems, either the nature of the polymer–filler interaction is badly known or the size of the interaction site cannot be clearly quantified, or both. In such case however, the successful silica/PDMS case provides some interesting guidelines when assuming that, whatever are the respective chemical natures of the filler and the polymer, at least the physics is the same. As we have seen the author has successfully adapted this model to the case of carbon black/rubber systems, with however the additional difficulty that the surface area of the interaction site A0 cannot be known a priori (see Chapter 5, Section 5.1.5). Owing to the interactions described above, gels are easily obtained when adding silica particles to liquid PDMS, and without further cross-linking, such gels find niche applications such as protective materials for fiber-optic cables and as encapsulants for semiconductor devices. As may be expected, the rheological and mechanical properties of silica-PDMS gels are quite complex, namely with respect to the viscoelastic behavior, but are well documented in the research literature, likely because it is relatively easy to prepared silica filled polysiloxane systems with standard laboratory equipment. Hysteretic effects upon increasing shear rate in both the viscosity and the first normal stress difference, as well as significant overshoot in the stress growth function, were reported by Caruthers and colleagues,37,38 and interpreted in terms of interparticles interactions via entanglements of the polymer adsorbed on the (fumed) silica surface. After an applied shear is step changed, the shear stress relaxes or grows in a complex manner that depends on the shear history. Consequently, PDMS molecular weight and silica volume fraction play an important role in such effects, as well as the surface chemistry when modified in the appropriate manner, but the preparation procedures (mixing technique and time) and the sample age further affects the rheological behavior of such systems.39–42 Precipitated silica develop also strong interactions with PDMS, and bring similar effects, but less than that of fumed silica.43 At high volume concentrations of precipitated silica (0.128 and 0.160), a yield behavior is evident from the storage modulus measurements. It is worth underlining that the key for understanding all those effects is the physical adsorption of PDMS on the silica surface, without chemical bonding, as clearly demonstrated by several authors.44,45 Suitably cross-linked, an unfilled, high molecular weight polydimethylsiloxane exhibits very modest mechanical properties, for instance a tensile strength (TS) in the 0.35 MPa range, largely insufficient for most applications. But the addition of a reinforcing filler, such as a high structure silica, increases

253

Polymers and White Fillers

the same property by a factor of around 40, yielding products with TS in the 13–14 MPa range, and around 600–700% elongation at break. Such reinforcing effects are by large more important than what is currently achieved with reinforcing fillers (e.g., carbon black and silica) in common hydrocarbon elastomers. This unusually high degree of reinforcement observed with silica/ polysiloxane systems has long been attributed to the particular polymer– filler interactions previously described, which persist after exposure to high temperature curing.46 No chemical bonding has been demonstrated between silica and (uncrosslinkable) polysiloxane but, in a cured PDMS-silica system, one cannot exclude a combination of chemical and physical bonds. The former are likely covalent bonding occurring upon vulcanization; the latter are hydrogen bonding and van der Waals forces, indeed favored by a high structure of filler particles. There are of course a variety of finely divided minerals that can be used as fillers for (curable) polysiloxanes, for instance finely grinded quartz, or zinc, titanium and iron oxides, or calcium carbonate, but amorphous silica in the 150–400 m2/g surface area range provides the best reinforcement. In order to have easy-to-process systems, it is however necessary to prevent certain detrimental polymer–filler interaction prior to vulcanization by using suitable plasticizers, for instance low molecular weight polysiloxane oligomers. Figure 6.10  shows typical dynamic properties of vulcanized PDMSsilica systems, as investigated through strain sweep experiments at constant frequency and temperature.47 As can be seen, dynamic strain softening is observed in a qualitatively similar manner to other filled polymers. It follows that models, which successfully fit conventional filled rubbers (e.g., carbon black filled compounds), are expected to well suit such data. This is indeed the case, as shown by the curves in Figure 6.10, drawn by fitting the Kraus–Ulmer equations, i.e.,

G′ ( γ 0 ) = G′inf +

G0′ − G′inf γ  1+  0  γc 

2m



(6.4a)

and



G′′ ( γ 0 ) = Ginf ′′ +

γ  2 (Gm′′ − Ginf ′′ )  0   γc  γ  1+  0  γc 

2m

m

 γ  + Gk′′ exp  − 0   γk 

(6.4b)

The fitting parameters are given in Table 6.3. In agreement with the physical reasoning that supports the model, the critical strain γc decreases with

Elastic modulus G´, MPa

10–3

10–2 10–1 Strain amplitude

100

10+1

0 10–4

2

4

6

8

10

10–3

10–2 10–1 Strain amplitude

100

10+1

Figure 6.10 Dynamic strain softening as observed on PDMS-silica systems; 0.078% vinyl-pendant PDMS (Mn=140,000 g/mol; Mw=390.000 g/mol); 300 m2/g silica; peroxide crosslinked. (Experimental data from L. Dujourdy, PhD Thesis, University of Grenoble, France, 1996.)

0 10–4

2

4

6

8

Filler fraction 0.04 0.08 0.12 0.15 0.18 0.21 Viscous modulus G´´, MPa

PDMS + Silica; 1Hz; 25°C 10

254 Filled Polymers

255

Polymers and White Fillers

Table 6.3

Modeling the Dynamic Strain Softening Effect on PDMS-Silica Systems with the Kraus–Ulmer Equations Φsilica

0.18

0.21

G′ vs. Strain Amplitude (Equation 6.4a) 0.038 0.403 0.327 0.236 0.457 0.698 1.459 3.008 0.3398 0.0740 0.0392 0.0346 0.144 0.671 0.284 0.229 0.9872 0.9969 0.9979 0.9985

0.616 4.423 0.0235 0.308 0.9986

1.010 8.252 0.0195 0.385 0.9990

Modeling G′′ vs. Strain Amplitude (Equation 6.4b) (–0.108) 0.200 (–0.023) 0.451 G″inf 0.106 0.301 1.046 2.523 G″m 0.0382 0.0063 0.0415 0.0302 γc m 0.201 0.100 0.295 0.417 G”k 0.071 0.295 0.624 0.762 0.0022 0.0001 0.0009 0.0010 γk r2 0.8642 0.7777 0.9504 0.9961

1.023 4.224 0.0288 0.476 1.000 0.0011 0.9966

2.780 8.010 0.0300 0.611 1.154 0.0021 0.9990

Modeling G′inf G′0 γc m r2

0.04

0.08

0.12

0.15

Note: Fit parameters; note that negative values for G″inf have no physical meaning, likely reflect experimental inaccuracy and could be replaced by very low positive numbers.

increasing filler fraction and is not much different for G′ and G′′. However the exponent m is typically depending on the silica content and is nearly twice as large for G′′. The additional Ulmer term in Equation 6.4b allows to meet the G′′ vs. γ behavior at low strain, with the main result that the critical viscous modulus G′′k significantly increases with the filler fraction. The above data allows however to demonstrate how really strong are the PDMS-silica interactions. Indeed, using the fit parameters in Table 6.3 and the Kraus-Ulmer equations, one easily calculates low strain (let’s say 0.001) values of G′ and G′′, in order to draw Figure 6.11. Since the dynamic properties of the pure polymer were not given in the source data, G′(Φ = 0) and G′′(Φ = 0) were obtained by second degree extrapolation. The left graph shows that 20% silica increase the elastic and viscous moduli by a factor of respectively, 42 and 95. But because the elastic modulus of the pure polymer is considerably larger than the viscous modulus, filled materials still exhibit a strong viscoelastic character. In the right graph, the normalized complex modulus is plotted vs. the filler fraction. The complex modulus is calculated as G * = G′ 2 + G′′ 2 and normalized with respect to G * of the pure, unfilled material. In other terms, one plots the functional for the silica effect, in order to compare it with the well known Guth and Gold term for filled systems, when only hydrodynamic effects occur, i.e., (1 + 2.5 Φ + 14.1 Φ2). Of course, when significant interactions exist between a

0

1

2

3

4

5

6

7

0

0.05

0.1 0.15 Filler fraction Φ

: G´ : G´´

0.2

Normalized complex modulus G*(Φ)/G*(0) 0

10

20

30

40

50

0

0.05

0.1 0.15 Filler fraction Φ

0.2

Guth & gold

Figure 6.11 PDMS-silica systems; variation of the low strain (γ = 0.001) dynamic properties with filler fraction; the dash curve in the right graph is the Guth and Gold term for mere hydrodynamic effects.

Dynamic moduli G´ and G´´ at 0.001 strain, Pa

PDMS + Silica; 1 Hz; γ = 0.001; 25°C 8

256 Filled Polymers

Polymers and White Fillers

257

polymer matrix and dispersed mineral particles, one does not expect the Guth and Gold model to meet experimental data, but the right graph in Figure 6.11 is an impressive (and simple) demonstration of the exceptional reinforcing effect of high structure silica in polysiloxanes. As we will see later, other (white) fillers are far to give effects of such a magnitude. 6.1.2 Elastomers and Clays (Kaolins) Kaolins have the finest particle-size range of all naturally occurring white fillers. Compared to (silanated) silica, kaolins or clays have relatively mild reinforcing properties but satisfactory however in a number of applications, for which specific effects are obtained owing to their platy structure. Clays in rubber allow very high hardness parts to be fabricated, essentially because of their plate-like particles, in contrast with spherical particles of equivalent dimensions. So-called “hard clay” grades have reinforcing capabilities, i.e., higher modulus, tensile strength and resistance to abrasion. “Soft clays” give lower physical properties and therefore are rather extenders.48 There are essentially four qualities of kaolin, depending on the recovery process and/or the after mining treatment: air-floated, water-washed, delaminated, or calcined. Airfloat hard clay is a general purpose white extender in rubber applications. Other grades are used for specific purposes, e.g., (clear) color enhancement, better mechanical properties and  abrasion resistance for waterwashed and delaminated clays. Calcined and surface-treated clays are used for improved electrical properties in low and medium voltage power cables, with the insulation properties maintained under wet operating conditions owing to the low moisture adsorption of the calcined and coated particles. Rubber extrusion and calendering generally benefit from the processing aid capabilities of calcined clays, so that hoses, profiles, and sheets can be manufactured with a very smooth finish and an excellent dimensional stability. With certain specialty rubbers, e.g., polychloroprene, butyl rubber and chlorinated polyethylene, calcined kaolins give very low levels of mill sticking. Light colored and chemically inert, calcined clays allow to prepare compounds for applications where only high quality materials are permitted, for instance pharmaceutical closure applications. There are many grades of kaolins available worldwide, certain only locally. As illustrated in Table 6.4, it is not easy to compare the various products on the market, essentially because there is no standard description of such materials. Most suppliers give (in their published technical data sheets) a limited number of common information that would allow some comparison. Specific gravity is the same whatever the origin, essentially because the mineral composition is fairly constant, approximately 45–55%wt SiO2, 35–41 Al2O3 and smaller quantities of various oxides, certain obviously affecting the degree of whiteness (e.g., Fe203). In what the mechanical and rheological properties are concerned, the exact mineral composition of kaolin is not a relevant information; it is clearly the particle size and the

VANDERBILT (USA)

SERINA (South Africa)

IMERYS (Worldwide)

Producer

14 n.a.

2.6 2.6 2.62

SpeswhiteTM Ultrafine powder

SupremeTM Powdered

GlomaxTM LL High temperature treated

n.a

2.62 2.62

Dixie Clay® (Hard clay)

Par® Clay (Hard clay)

n.a.

21.25

2.6

Ultrafine Water washed, Screened

18.73

2.6

Filler grade Water washed

9

n.a.

2.62

PoleStarTM 450 Calcined clay

8.5

Specific Surface Area, BET (m2/g)

2.6

Specific Gravity (g/cm3)

PoleStarTM 200R Calcined clay

Brand Name

Selected Commercial Kaolin Grades from Various Suppliers

Table 6.4

40

42

57–63

45–50

53

n.a.

42

60

n.a.

Oil Absorption (g/100 g)

99.3% < 78 0.2 µm (median)

99.3% < 78 0.2 µm (median)

98% < 20 90% < 10 48% < 2 99% < 20 98% < 10 70% < 2

1.5 (median)

0.2 > 10 77% < 1

0.5 > 10 80% < 2

1.5 (median)

44% < 2

Particle size (Distribution) (µm)

258 Filled Polymers

SOKA (France)

2.6

2.6

2.6

Blankalite 78 Powdered

Metasial V800 Calcined clay

2.62

Peerless® 1

Sialite Powdered

2.62

LangfordTM (Soft clay)

6–7

15

9

n.a.

n.a.

48

36

41

30

36

100% < 20 96% < 10 70% < 5 35% < 2

99% < 20 97% < 10 76% < 5 51% < 2

99% < 20 85% < 10 63% < 5 38% < 2

99.6% < 78 1.2 mm (median) 30

99.0% < 78 h 1.3 mm (median)

Polymers and White Fillers 259

260

Filled Polymers

particle size distribution that are the key characteristics,49 unfortunately not documented in a comparable manner by the various suppliers. 6.1.3 Elastomers and Talc Ground talc can be considered as a functional additive for elastomers, because its specific structure and aspect ratio may provide interesting enhancements of certain properties, namely mechanical properties, i.e., modulus, tensile strength and elongation at break, hot tear resistance, useful for easy demoulding parts with complex geometries, and barrier properties. In tire technology, the use of talc in innerliner and white side-wall compounding (the latter application no-longer of fashion today) has been reported.48 Talc (and other platy fillers) is known for long to restrict the swelling of rubber parts exposed to automotive fuels, oils and greases, an effect essentially due the correct orientation of the platy particles, in such a manner that a strong swelling anisometry is obtained, because of obvious barrier effects. Indeed talc particles have large planar surfaces that are hydrophobic and thus preferentially wetted by organic substances such as elastomers. Talc incorporation in elastomers is thus easy and, owing to their aspect ratio, talc platelets easily self-orient along flow lines. This give talc the role of a processing aid through internal lubricating effects between elastomer chains. Easier extrusion is therefore obtained, as well as reduced extrudate swell. Talc’s lamellar structure give barrier effects, substantially reducing air and fluid permeation in stopper, sealing, membrane and hoses; improved thermal and weathering resistance is also obtained through similar barrier effects. Talc platelets are electrically neutral, so that interesting insulation effects are obtained in wires and cables. Oriented platy fillers provide elastomers quite unique failure mechanisms that, properly exploited, give materials with an enhanced resistance to destructive crack growth. For instance Eldred compared chlorosulfonated polyethylene compounds with equal volume fraction of either N550 carbon black or 6 µm platy talc particles.50 Through the appropriate procedure on a two-roll laboratory mill, samples where prepared with a purposely anisotropy. Using results from trouser tear, peel and De Mattia fatigue tests, Eldred clearly demonstrated that talc filled compounds failed in an interlaminar shear mode, parallel to the applied stress, not resulting however from planes of weakness in the lateral direction, but from a more than five-time increase in the cohesive fracture energy with respect to the same measurements on the carbon black filled control compound. Fractured surfaces observed by scanning electron microscopy reveal that this enhanced crack propagation resistance was due to a more efficient fracture energy absorption, as resulting from a diversion of the fracture paths through the elastomer, owing to the presence of the talc particles. Such results are important because they clearly demonstrate the strong links to (may) exist between the particular rheology imparted to composites by platy fillers and the resulting benefits in mechanical properties, providing an adequate control of flow induced anisotropy effects is achieved.

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261

Extruded or calandered products are obvious practical applications for such effects. There are however some negative aspects which limit the reinforcing capabilities of talc in tire applications. Mouri and Akutagawa51 reported that when substituting 20 phr of N234 carbon black in a typical OESBR/BR tread compound by 20 phr talc (with also a 20–5 phr reduction in process oil content), better wet traction and lower rolling resistance are obtained but with a significant reduction in wear resistance. It must be noted however that the grade of talc used by these authors was considerably coarser than N234 (particle size in the 200–8000 nm range for the former, vs. 17–26 nm for the black) and when more balanced formulation changes are made with an ultra-fine talc grade, substantially different results are obtained, as recently reported by Noel and Meli.52 These authors showed by adding only 5 phr of an ultra-fine talc (Mistron® Vapor R; Luzenac, Rio Tinto Group) to either N220 black filled NR/BR or Silica filled SSBR/BR compounds, significant improvement in filler dispersion were obtained, at equal or even 20% reduced mixing time. Not significant modification were seen in ultimate tensile, dynamic and adhesion properties. Only a very slight decrease in wear resistance was noted with respect to the carbon black filled compound, with however a beneficial effect on De Mattia fatigue resistance. Some possible interferences with the silica silanization process were also noted. It is well known that the basal surfaces talc platelets are hydrophobic, while the edge surfaces are hydrophilic.53 The hydrophobic character of the basal surfaces arises from the fact that the atoms exposed on the surface are linked together by siloxane (Si-O-Si) bonds and therefore cannot develop hydrogen bonding with water. The edge surfaces, however, are made of hydroxyl ions, Mg, Si and substituted cations; all of which may undergo hydrolysis. It follows that the edges are hydrophilic, and can form strong hydrogen bonds with water and/or polar substances. Chemical modification of talc with appropriate chemicals is therefore possible, in order to optimize its compatibility with the polymer matrix, through possibly (and ideally) chemical adhesion. As there is a limit (either technical or economical) in reducing the size of any naturally occurring mineral, adding any type of organic chemical is a common method to produce so-called surface-treated minerals. But simple surface coating is substantially different from surface modification based on reactive functionality available on the mineral particles. A truly surface-modified talc has a coupling agent durably attached by covalent bonds and this can only be achieved through the hydroxyl groups available at the edges of the platelets. Depending if the modifying agent is bifunctional or not, chemical bonding with the polymer could eventually occur through the appropriate chemical reaction or at least chain entanglement may be expected. The most commonly used talc modifiers are the organosilanes, whose general chemical structure is RSiX3, where X is a hydrolysable group, such as methoxy, ethoxy, acetoxy, or chloride, and R is a nonhydrolysable organofunctional group. The most often used silanes are alkoxy derivatives, RSi(OR’)3, with amino, epoxy, methacrylate or vinyl functionality.

262

Filled Polymers

As we have seen, silanization has been most studied in the case of silica which are considerably richer in surface hydroxyl groups, so-called silanols, than talc. Because the effectiveness of silane modification depends on the availability of silanol groups on the mineral surface, some of the silicate fillers are more amenable to silane modification than others. Hydroxyl groups are available on the edge of talc platelets, so that silane treatment is feasible and, indeed, silane treated talc grades are commercially available. Silane treatment levels are typically 0.5–1.0% on mineral weight, to be compared with the 8% recommended for the silanization of synthetic silica. Whilst not comparable with what can be achieved with (silanated) silica, very fine microcrystalline talc grades offer quite attractive reinforcing properties for white rubber compounds, particularly when a pretreatment with a suitable organo-silane is used. Not much published data is available regarding the (industrial) silane modification of talc (as well as mica and kaolin) but commercially available silane-modified grades offer proved technical benefits when used as fillers or additives to polymer systems. With respect to the general chemical structure of the most widely used organo-silanes, it is permitted that industrial modification of talc involves the following four steps. First, hydrolysis of the three labile groups must occur, then condensation to oligomers follows. The oligomers then hydrogen bond with OH groups at the edges of talc platelets. It is eventually during drying or a further thermal treatment that a covalent bond linkage is formed, with a concomitant loss of water. Such reactions are likely to occur simultaneously after the initial hydrolysis step. The relatively low quantities of (supposedly) chemically bound silane appear sufficient to drastically modify certain key properties of the filler, namely its dispersibility in diene rubbers and hence the associated rheological and mechanical benefits. It is worth mentioning at last an important use of talc in the rubber ­industry, whilst not as a filler. Relatively coarse grades of talc (average particle size ≈ 40–50 µm) are used as partitioning agent, either in powder form or as suspension in water, to coat freshly compounded, uncured rubber strips and sheets, in order to avoid stickiness during storage, before further processing operations. Surface coated talc is easily incorporated in the compound during subsequent shaping steps, without generally undesirable effects.

6.2  Thermoplastics and White Fillers 6.2.1 Generalities Most of the minerals used as fillers in thermoplastics are “white” with however a great variety in chemical nature and/or physical forms. Generally fillers are added to thermoplastics for economical reasons with possibly some

Polymers and White Fillers

263

benefits in mechanical or physical properties. Notable exceptions (i.e., where cost is not the prime issue) are when improvements in certain specific properties of thermoplastics are sought, for instance fire resistance or flame retardancy through the addition of finely grinded aluminium tri-hydrate (ATH), Al2O3.3H2O, magnesium hydroxide Mg(OH)2 or antimony tri-oxide Sb2O3. If cost saving is the main reason, then two problems appear: 1. losses (or at least undesired changes) in mechanical properties must be minimized; 2. the rheology and hence the processing behavior of filled compounds must be mastered and/or controlled. Besides cost, white fillers globally affect the following properties of thermoplastics: • Stiffness and hardness generally increase with higher filler content • Impact resistance is either decreased or improved depending on the filler nature and its characteritics, namely particle size • Thermal expansion and shrinkage are reduced • Softening temperature (Vicat, HDT) is increased • Flow properties are modified and either easier or worst processing behavior is obtained • Surface properties (gloss, scratch resistance, etc.) are modified • Fire resistance is improved But the effect of a given (type of) filler is frequently specific to a polymer or to a class of polymers and, quite often, there are antagonistic effects. For instance, benefits in mechanical properties are counterbalanced by increasing processing difficulties, or the fire resistance imparted by the additive (e.g., ATH in polyester or epoxy resins) goes along with an excessive increase in viscosity. A priori, the chemical compatibility between filler surface and polymer segments is critically important in both the wetting and dispersion of particles in the matrix, the processing behavior of molten composites and eventually the (mechanical and physical) performances of final objects. But except in a few cases, the actual surface chemistry of many white fillers is illknown, and at best referred in terms of affinity for water, i.e., hydrophilic or hydrophobic material. Hydrophilic fillers would give maximum interaction with polar polymers, hydrophobic fillers would be preferred with nonpolar polymers. It follows that many commercial fillers, especially of mineral origin, are surface coated or chemically treated with hydrophobic wetting agents, in order to modify their surface chemistry, or at least to alter their wetting character. Dispersion in nonpolar polymers is expectedly easier, likely because wetting agents help in deagglomerating clustered filler particles during the mixing process. This aspect is particularly important for continuous mixing operations for which the viscosity of the molten composites must be kept low, at levels compatible with the performance of the

264

Filled Polymers

equipment. Typical wetting agents are fatty acids and derivatives, polymeric esters, and organosilanes, the latter widely used with respect to their reactive potential with suitable functional groups on the filler surface. Except silica, whose surface is known to be silanols rich, the real (i.e., chemical) effect of silane treatment of many white mineral fillers is either unproved or unclear in the author’s opinion. Table 6.5 illustrates the usages of a few selected white fillers in thermoplastic polymers. Table 6.5 Typical Industrial uses of a Few Selected Mineral Fillers General Purpose Fillers

Polymer

Loading Range (%)

Calcium carbonate (grinded or precipitated) Clays, silicates, kaolin

PP, PS, PVC, ABS, POs, TPE, PU, Epoxy, Phenolics, Fluoropolymers POs, PU, PVC, PA, TPE

5–80

Mica

PP, ABS, POs, PC TPE, PA, Fluoropolymers

5–40

Silica

ABS, POs, PS, PVC, PU, TPE, Epoxy PP, POs, PVC, PS, Phenolics

10–30

Talc

10–40

20–50

Loading Range (%) 9–50

Specialty Fillers Aluminum tri-hydrate

Polymer ABS, LDPE, PVC, TPE

Barium sulfate (barites)

PU

10–30

Wollastonite

PC, PS, TPES, PA

20–50

Improvements/Effects/Application Rheological properties Mechanical properties Surface aspect Numerous applications Rheological properties Surface aspect Wire and cables Automotive parts Mechanical properties Barrier effects Dielectric properties Thermal properties Rheological properties (thixotropy) Extender or thickener Reinforcement Extender Reinforcement (stiffness, tensile) Creep resistance Barrier effects Improvements/Effects/Application Extender Flame retardant Smoke suppressant Wire and cable Specific gravity increase Surface properties (friction) Chemical resistance Mechanical properties (tensile) Dimensional stability Barrier effects Electrical properties Thermal properties

Note: PP: polypropylene; PS: polystyrene; PVC: poly(vinyl chloride), POs: polyolefins; TPE: ­thermoplastic elastomers; PU: polyurethane; PA: polyamides; ABS: acrylonitrilebutadiene-styrene terpolymer; LDPE: low density polyethylene.

Polymers and White Fillers

265

With respect to mechanical properties, the effect of any mineral filler is firstly related to particle size and particle size distribution. For instance, the impact resistance of a PP–talc composite strongly depends on those two filler characteristics, and it is easily understood that particles with excessive dimensions are likely to be fracture initiation sites. For mineral fillers which are extracted from lodes, grinding and sorting are thus key preparation steps, with obviously processing costs that dramatically increase as the targeted (average) particle size decreases. The balance between the benefits in mechanical or physical properties and the extra-cost of filler addition is therefore the bottleneck of all applications for filled thermoplastics. Besides the (beneficial) effects on flow and mechanical properties, not all fine white minerals can be considered as filler for thermoplastics because, depending on their hardness, some can bring excessive wear of processing equipment. It follows that the Mohs hardness is a relevant property in selecting (white) mineral fillers, notwithstanding their other effects. The Mohs scale is one of several definitions of hardness in materials science and conveniently characterizes the scratch resistance of a given mineral through its ability to scratch a softer one. It was established in 1812 by the German mineralogist Friedrich Mohs (1773–1839) using ten minerals, all readily available. Diamond, the hardest known naturally occurring substance, is at the top of the scale, and talc at the bottom. The hardness of a given material is measured against the scale by finding the hardest material it can scratch, and/or the softest material that can scratch the given material. For instance a material that is scratched by calcite but not by gypsum has a Mohs hardness between 2 and 3. The Mohs scale is a so-called ordinal or successive scale, thus somewhat arbitrary, and does not measure the actual hardness. The Mohs scale is roughly linear with respect to the logarithm of the absolute hardness, as measured with a “sclerometer,” a special instrument invented in 1896 by T. Turner and used by mineralogists. It consists essentially in measuring with a microscope the width of the scratch made by a diamond when drawn under fixed load across the face of the specimen under test. The wider the scratch the softer the material and the results are expressed with respect to an increasing scale, starting from 1 (talc) up to 1500 (diamond). Table 6.6 gives the Mohs scale with respect to the absolute hardness. Common metals have Mohs hardness between 2 and 5 (Al: 2.75; Cu: 3.0; iron: 4–5) so that with respect to wearing of polymer processing equipment, only soft minerals close to talc and calcite are of interest as fillers. Ground calcium carbonate, mica (Mohs hardness in the 2–2.5 range) and clays (Mohs hardness about 2) give moderate wearing problems providing they do not contain hard impurities (e.g., sand or quartz particles). Synthetic white fillers (e.g., precipitated calcium carbonate) have advantages in this respect. Wollastonite, a form of calcium silicate, is a special case, with a Mohs hardness of 4.8 but with needle-like particles that make it interesting in certain applications. It must be noted however that if the abrasivity of filler particles depends first on their hardness, their size and

266

Filled Polymers

Table 6.6 Mohs Hardness Scale Mineral Talc Gypsum Calcite Fluorite Apatite Orthoclase Quartz Topaz Corundum Diamond

Chemical Composition (Mg3Si4O10(OH)2) (CaSO4·2H2O) (CaCO3) (CaF2) (Ca5(PO4)3(-OH,-Cl,-F) (KAlSi3O8) (SiO2) (Al2SiO4(-OH,-F)2) (Al2O3) C

Mohs Hardness

Absolute Hardness

1 2 3 4 5 6 7 8 9 10

1 2 9 21 48 72 100 200 400 1500

shape are important factors as well. Indeed, particles with sharp edges, e.g., flakes, scales, or rod-shapes particles are more abrasive than smooth and round particles, and molten polymer systems made with large particles are generally more abrasive than with smaller ones. With respect to the Mohs hardness of mineral glass (i.e.: 5.5) short glass fibers filled polymers are likely to give the severest wearing problems, somewhat compensated for however by using suitable surfactants and other additives. Other factors are important, such as the coefficient of friction, surface treatment and energy (of both the filler particles and the metal of the processing equipments), all are somewhat controllable, but the purity of the filler remains a critical one since the most common contaminant in extracted minerals is the highly abrasive sand (i.e., quartz). 6.2.2 Typical White Filler Effects and the Concept of Maximum Volume Fraction It is quite a trivial statement that, owing to their viscoelastic character, the rheological and mechanical properties of polymers differ from those of other materials, and it is quite a common observation that the viscoelastic character of polymers is modified through the addition of foreign materials, e.g., mineral particles. The modification of the polymer viscoelastic behavior results from (at least) three type of parameters: • The filler volume fraction effect • The shape and size of the particles • Interactions between particles and/or between particles and polymer No overall understanding exists for all possible combinations of those effects in any given polymer; moreover effects or benefits obtained with

Polymers and White Fillers

267

a given filler-polymer system cannot generally be extended to another ­system. The preparation mode of the material deeply affects the quality of dispersion of a given mineral in a given polymer and thermo-mechanical degradation occurs rapidly if excessively energetic mixing processes are used. There is however an aspect which is qualitatively common to all ­filler–thermoplastic systems: the linear viscoelastic behavior exhibited by most pure polymers at sufficiently low strain or low rate of deformation ­disappear above a sufficient filler level. For instance, the so-called Newtonian plateau on the shear viscosity function is no longer observed, the dynamic modulus is strongly strain dependent and the terminal region in the elastic modulus function disappears and is replaced by a low frequency plateau. As we have seen, such typical effects are also observed with filled rubber compounds. Let us consider the progressive disappearance, as the filler content increases, of the pseudo-Newtonian region in the shear viscosity function. In extreme cases, an (apparent) yield stress behavior occurs whose origin is assigned to interactions between particles, which of course superimpose to hydrodynamics effects. Particles interactions are nonhydrodynamic by nature and depend on many factors, for instance the electric (ionic) and chemical properties of the particles, the surface area and properties of the particles and the associated effects with the polymer matrix, and the presence of a surfactant (so-called compatibilizer) layer, if any. Yield stress behavior, either apparent or real, is a low deformation rate phenomenon and, at higher deformation rates, the nonhydrodynamic forces on the particles are dominated by the stress field in the polymer matrix, which remains of course affected by interparticles hydrodynamics, depending on the rheological properties of the polymer and the arrangement of particles along flow lines. Different arrangements of particles are expected in different types of flow and it is well known that extensional flow fields induce an organization of particles that is radically different from what is obtained in shear flow fields. Therefore particles arrangements, or in more general terms the flow induced anisotropic distribution of particles, is strongly depending on the actual flow geometry. For instance, in steady shear conditions, flow induced anisotropy in a parallel disk rheometer is differing from the arrangement of particles in a coneplate system. All the more when extensional and shear flows are combined, as in converging flow for instance, quite common in processing equipments and techniques. A common explanation for such phenomena is that the gradient of the deformation rate tensor affects the flow induced anisotropy and hence the stress. It follows obviously that filled (polymer) systems cannot be “simple” fluids since their behavior, by virtue of their true nature, violates the principle of “local action,” which states that the stress in a fluid element is determined by the deformation history of that fluid element and is independent of the history of neighboring elements. This is the main reason for the

268

Filled Polymers

limited applicability of continuum mechanics to filled polymer systems; a limit that can however be somewhat circumvented through micromechanic approaches. Let us consider a filled polymer system in a simple shear flow situation. If indeed nonhydrodynamic interparticles effects dominate the shear flow behavior at low shear rate and vanish at higher shear rates, leaving only hydrodynamic effects, then one could consider that the actual viscosity function of the filled material results from the mere superimposition of two contributions, both expressed with the same convenient mathematical model. Such models must explicitly consider a low shear viscosity plateau and a high shear thinning behavior. The well-known Carreau–Yasuda model meets such requisites and, as illustrated in Figure 6.12, the steady shear flow behavior could be expressed through the following equation:54

a ηcpd ( γ ) = η0 ,1 1 + ( λ 1 γ ) 1   



(n1 − 1)/a1

a + η0 ,2 1 + ( λ 2 γ ) 2   

( n2 − 1)/a2



(6.5)

The first term of the right member of this equation expresses the nonhydrodynamic interparticles effects through parameters η0,1, λ1, a1 and n1; the second term of the right member of this equation expresses the particles

105

Low shear plateau

η0,1

Shear viscosity, Pa.s

Apparent yielding region Intermediate plateau

104

ηc η0,2

103 1 λc

1 λ1 102 10–4

10–3

10–2

10–1 100 Shear rate, s–1

Shear thinning region

1 λ2 101

102

103

Figure 6.12 Modeling the shear viscosity function of filled polymer systems by combining two Carreau–Yasuda equations; the curve was calculated with the following model parameters: η0,1 = 8 × 104 Pa.s; λ1 = 500 s; a1 = 1.9; n1 = 0.4; η0,2 = 3 × 103 Pa.s; λ2 = 0.1 s; a2 = 3; n2 = 0.33.

269

Polymers and White Fillers

hydrodynamic effects through parameters η0,2, λ2, a2 and n2. The physical meaning of parameters η0,1, λ1, η0,2, and λ2 is explicit from Figure 6.12. The flow indices n1 and n2 monitor the shear thinning behavior in the low and high shear regions respectively, and parameters a1 and a2 affect the curvature of the viscosity function in the two transition regions. Such a model allows thus four regions to be distinguished, with respect to three critical shear rates: 1 • A low shear rate region, when γ ≤ λ1 a1   1  1   η0 ,2  n1 − 1  −1 • An apparent yielding region, when λ ≤ γ ≤ λ   η   1 1  0 ,1  

1 • An intermediate plateau, when λ 1

a1   n1 − 1  η   0 ,2 − 1   η0 ,1    

• A high shear thinning region, when

1/a1

1/a1

≤ γ ≤

1 λ2

1  ≤γ λ2

Parameters λ1 and λ2 are characteristic times and, as clearly seen when considering the values of the parameters used to calculate the curve in Figure 6.12, the nonhydrodynamic interparticles effects (i.e., clustering) operate in a time range that is several decades larger that the time range for polymer flow processes. Available experimental data on filled polymers hardly meet such a model in all its aspects, essentially because there is no technique to readily capture the very low shear viscous behavior of very stiff systems. The author has reported very low shear (down to the 10 –4 s–1 range) viscosity measurements on filled rubber materials that do not show a yield stress limit but suggest rather a low shear thinning region with an excessively high viscosity. Similar observations are expected with filled thermoplastics. The very low shear plateau, where nonhydrodynamic interparticles effects dominate the viscous behavior, likely remains out of reach of experimental techniques for the simple reason that in order to establish very low shear rate regimes, one needs excessively long times. However, explicit yield stress σc data can be extrapolated from low shear experiments and, if such data are available, then Equation 6.5 reduces to:



η( γ ) =

σc a ( n − 1)/a + η0 1 + ( λ γ )    γ

(6.6a)

270

Filled Polymers

If there is no intermediate plateau (corresponding to η0,2), one obviously recovers the Herschel–Bulkley equation, i.e.:



η( γ ) =

σc + K γ n − 1 γ

(6.6b)

where the prefactor K has the meaning of the product η(1/λ) × λn–1. Despite its mathematical simplicity, such a model offers large flexibility. For instance, with the other parameters constant, how n1 and a1 values affect the shape of the viscosity function is quite interesting. Indeed, either higher n1 or lower a1 somewhat dampen the intermediate plateau in such a manner that the shape on the function appears very close to what can experimentally be observed with filled polymers (see Appendix 6.2 for a numerical illustration). The flow properties of concentrated suspensions in Newtonian fluids have been studied for long and a few theoretical models have been compiled in Table 5.12. However, not all conclusions of such studies can be extended to filled polymer systems for at least two reasons: 1. only particles hydrodynamic interactions are considered and 2. specific particle–matrix interactions are not taken into consideration. Moreover, polymers are non-Newtonian fluids and their viscoelastic character adds complexity. Certain aspects of such studies are nevertheless interesting, namely particle clustering in shear flow,55,56 and the associated problem of particles packing. For filled systems that consist of even dispersions of non-interacting, rigid spheres of equal diameter in a Newtonian matrix of viscosity η(0), it is indeed quite convenient to consider that there is a maximum volume packing fraction Φm. For a loose cubic packing, it is easy to establish that Φm = π/6 ≈ 0.5236 and for a close hexagonal packing (also called face-centered cubic) of uniform spheres, one has Φ m = π/3 2 ≈ 0.7405. Those two values can be considered at the lower and upper bounds for the packing of uniform spheres. Other arrangements give maximum packing fraction in between these two limits. For instance the body-centered arrangement corresponds to Φ m = π 3/8 ≈ 0.6802 and for a random close packing of uniform spheres, computer simulation yields Φm ≈ 0.64.* It is pretty obvious that with an adequate distribution of spheres’ diameters, the maximum packing distribution with the above ideal arrangements is bound to increase. Vand57 was probably the first to hypothesize that in concentrated suspensions, particles may cluster and that the suspending liquid in the neighborhood of contact points between particles is effectively immobilized and therefore contributes to an “effective” volume fraction * The packing of objects (spheres, ellipsoids, marbles, etc) in a finite volume is a problem of considerable interest, approached by mathematicians and physicists for centuries. It appeared recently that the packing of spheres is apparently the exception rather than the rule, and that as soon as the shape of objects becomes nonspherical, the packing efficiency increases by a surprisingly large amount; see, for instance, D.A. Weitz. Packing in the spheres. Science, 303 (5660) 968–969, 2004.

271

Polymers and White Fillers

of the particles, larger that their true fraction. For clusters made of a small ­number of uniform sphere, simple geometrical arguments allow to estimate the quantity of immobilized liquid. For larger cluster of spheres, it is convenient to consider a “shape factor,” defined as the ratio of the actual volume of the cluster of i particles to the overall volume of the i particles. As the number of particles increases, such a shape factor becomes essentially constant and depends only of the kind of packing. For model arrangements of uniform spheres, it is easy to demonstrate that the shape factor is the reverse of the maximum packing fraction, for instance equal to 1.9098, 1.4702, and 1.3505 for cubic, body-centered, and hexagonal packing, respectively. For such ideal systems as suspensions of spheres of equal diameter, many equations, either theoretical or empirical have been proposed for the relative viscosity as a function of the filler volume fraction. Such a subject is obviously of tremendous importance in many fields. A thorough discussion of suspensions of rigid particles in Newtonian fluids was made by Jeffrey and Acrivos58 and some models available up to 1985 were discussed in detail by Metzner.59 We will consider below only the most referred equations that explicitly consider the maximum packing fraction. One of the oldest proposal was likely made by Eilers60 in order to model the behavior of highly viscous suspensions, i.e.:* 2



 5   1+ Φ  η( Φ ) 2  = ηr ( Φ ) =  η( 0)  1− Φ   Φ m 

(6.7)

Obviously, owing to its simple quadratic form, the Eilers model is not asymptotic to the Einstein’s one and therefore yields excessive values at low volume fraction Φ. An often-quoted model, better in this respect, is the one developed by Mooney for concentrated suspensions of uniform rigid spheres, by considering only first-order interactions between spheres of equal diameter, essentially a crowding effect. For very low spheres fraction, the Einstein’s formula obviously applies and, by considering that a densely packed spheres system would exhibit an infinite viscosity, Mooney established the following equation:61



  5 Φ  η( Φ )  = ηr ( Φ ) = exp  η( 0)  2 1− Φ   Φ m 

(6.8)

* Note that all the equations reproduced from literature are rewritten with respect to the formalism used throughout this book.

272

Filled Polymers

where Φm is the maximum volume packing fraction. Later, Krieger and Dougherty modified the Mooney functional analysis to obtained an equation that fitted well experimental results on polymer latex. Their equation, frequently quoted for suspensions of rigid particles, is:62



η( Φ ) Φ   = ηr ( Φ ) =  1 −  η( 0) Φ  m

− Φm B



(6.9a)

where B is a so-called “intrinsic viscosity” that should in fact match the value 5/2 of the Einstein equation, as explicitly considered by Ball and Richmond,63, i.e.:



η( Φ ) Φ   = ηr ( Φ ) =  1 −  η( 0) Φ  m

5 − Φm 2



(6.9b)

Equations 6.9a or b are generally referred as the Krieger–Dougherty equation. A similar but simpler equation was proposed by Kitano et al.,64 i.e.: η( Φ ) Φ   = ηr ( Φ ) =  1 −  η( 0)  Φm  −2



(6.9c)

With respect to their intensive observations on uniform spheres in simple shear flow, Graham et al.65 paid attention to clustering of particles suspended in Newtonian fluids and derived the following equation:



2    1− Φ   η( Φ )  Φ m − Φ     m = ηr ( Φ ) = 1 − Φ 1 +    1 −  Φ     η( 0) m   Φ m      

−2.5



(6.10)

With respect to the mathematical form of the well-known Guth, Gold, and Simha equation, it is also interesting to mention a model previously developed by Graham:66



 η( Φ ) 5 9 1  = ηr ( Φ ) = 1 + Φ +  η( 0) 2 4  A ( 1 + 0 . 5 Φ ) ( 1 + A )2    Φ   A = 2  1 − 3   Φ  m 

3

(6.11)

Φ  Φ m 

Figure 6.13 shows a comparison of these models, which in fact can be divided in two groups: 1. models predicting a sharp variation of the relative

0

10

20

30

40

0

1

2

3

0

Eilers

0.2

0.2 0.4 Volume fraction

0.1

0.6

Einstein

Guth, Gold, Simha

Kitano

Mooney

Relative viscosity 0

5

10

15

20

0

1

2

3

0

Figure 6.13 Comparing model equations for the relative viscosity of suspensions of uniform spheres.

Relative viscosity

50

0.2 0.4 Volume fraction

0.1 0.2 Graham et al.

0.6

Einstein

Graham

Guth, Gold, Simha

Krieger–Dougherty

Polymers and White Fillers 273

274

Filled Polymers

viscosity upon increasing filler volume fraction, 2. Models predicting a mild variation upon higher Φ. As can be seen, the former group consists of the Eiler and Mooney equations and give predictions which are above the Guth, Gold and Simha model (that does not consider a maximum packing fraction); the later group is the Krieger–Dougherty, the Graham and the Graham et al. equations, which are generally below the Guth, Gold and Simha model, except the former at high volume fractions. Note that in calculating the curves, the maximum hexagonal packing fraction for uniform spheres (i.e., 0.74) was used. Using lower packing fraction somewhat modifies the shape of the curves: the larger Φm, the larger the volume fraction range before relatively viscosity goes to infinity for the Mooney and the Graham et al. models, the lower Φm, the steeper the viscosity variation for the Krieger–Dougherty and the Graham models (see Appendix 6.3). If the filler has some trend to cluster in aggregates, one could somewhat further develop the Krieger–Dougherty type of equations by considering an effective enhanced volume fraction. Such an effective volume fraction could then be related to both the overall size of the cluster D and the size of the single particles d with respect to fractal considerations. Equation 6.8a would then be rewritten as: C  η( Φ ) Φ  D  = ηr ( Φ ) =  1 −    η( 0)  Φ m  d  



− Φm B



(6.12)

where C is the so-called connectivity exponent from the fractal theory that could for instance be taken as ≈ 1.8 (see Chapter 4, Section 4.1.4) It is worth noting that the Krieger–Dougherty equation (Equation 6.8a) can be expended in polynomial form to yield (see Appendix 6.5): η( Φ ) ( B Φ m + 1) Φ 2 + B ( B Φ m + 1)( B Φ m + 2 ) Φ 3 = 1+ B Φ + B 2! Φm η( 0) 3! Φ 2m

+B

( B Φ m + 1)( B Φ m + 2 )( B Φ m + 3) Φ 4 + 



4! Φ 3m

(6.13a)

or, in an abridged form: a−1



η( Φ ) = ηr ( Φ ) = 1 + B Φ + η( 0)

n

∑B a=2

∏(B Φ

m

+ i)

i=1

a! Φ ma− 1

Φa

(6.13b)

Polymers and White Fillers

275

If the series goes to infinity, Equation 6.13a and b match exactly the Krieger–Dougherty equation and if B is taken equal to 2.5, when reducing the polynomial to the first two terms one obtains of course the Mooney equation. This means that, in a very simple manner, expanding the Krieger–Dougherty type of equation yields a polynomial function that appears to somewhat take into account interactions between particles in addition to simple hydrodynamic effects. In agreement with the theoretical reasonning by Einstein and Guld, Gold and Simha in deriving their equations, the three first terms of the polynomial account for simple hydrodynamic effects and further terms account for interparticle interactions, with the packing mode and its maximum packing fraction playing the key roles. It is quite interesting to note that with B = 2.5 and Φm = 0.74 (hexagonal packing of uniform spheres), the third and the fourth terms in Equation 6.13b are respectively, 4.814 and 8.349, i.e., values framing the Φ2 multiplying factor in the relationship developed by Batchelor for a suspension of spherical particles for which Brownian motion is an issue,67 i.e.: ηr = (1 + 2.5 Φ + 6.2 Φ2) When compared with experimental data, the Krieger–Dougherty is generally found to overpredict the filler fraction effect, particularly when one approaches the a priori considered maximum packing fraction. In the author’s opinion, the weakness of the Krieger–Dougherty equation is that, with respect to its mathematical form and the logical limiting variation as Φ → 0, only the maximum packing fraction is an adjustable parameter. This may lead of course to quite unrealistic fitted Φm values. The polynomial equation, Equation 6.13, is quite attractive in this respect because, while keeping the a priori considered values for B (i.e., B = 2.5) and Φm (i.e., with respect for instance to information about the average shape of filler particles), the number of terms of the polynomial allows to easily meet a large variety of experimental observations, as illustrated in Figure 6.14. The number of needed polynomial terms can then be interpreted as an indication of the extent of interparticle interactions. It is clear that the above models are oversimplified with respect to the known complexity of the rheological behavior of suspensions and (obviously) of filled polymers. Their attractiveness is however their mathematical simplicity, whilst to consider that the volume fraction of the dispersed particles is the only variable for the rheological properties is surely incorrect, notwithstanding the temperature, the mode of flow, the stress and the rate of deformation that could however, in a first approximation, be considered as independent variables. In such a case, the shape, the average size and the size distribution of the particles are obviously very influential factors, whose first effect will be to modify the maximum packing fraction. For simple geometrical particle shapes (spheres, rods), Φm can be calculated providing the arrangement is either geometrically defined or considered at random (Table 6.7).

276

Filled Polymers

50

Φm = 0.74

Φm = 0.524

B = 0.25

B = 0.25 Relative viscosity

40 Relative viscosity

Uniform spheres, cubic packing

30

30 20

20

10

10 0

0

0.25 0.5 Filler volume fraction

0

0.75

0

0.2 0.4 Filler volume fraction

Polynomial, 6 terms

Polynomial, 3 terms

Polynomial, 12 terms

Polynomial, 6 terms

Krieger–Dougherty

Krieger–Dougherty

0.6

Guth & Gold Uniform spheres, random packing

30

Prolate ellipsoids, random packing

30

Φm = 0.621

Φm = 0.74 B = 0.25

20

Relative viscosity

Relative viscosity

B = 0.25

10

0

0

0.2 0.4 Filler volume fraction

0.6

20

10

0

0

0.2 0.4 Filler volume fraction

Polynomial, 3 terms

Polynomial, 3 terms

Polynomial, 6 terms

Polynomial, 6 terms

Krieger–Dougherty

Krieger–Dougherty

0.6

Figure 6.14 Polynomial vs. Krieger–Dougherty equations for relative viscosity variations with respect to maximum particle packing fraction.

277

Polymers and White Fillers

Table 6.7 Typical Maximum Volume Packing Fraction for Simple Geometries Particles Shape Spheres all with the same diameter Spheres (bimodal) two diameters D1, D2 Rods All with the same diameter

Ellipsoids a,b,c [a: long axis] Prolate ellipsoids [b >> c]

Fm

Packing/Arrangement Cubic At random Hexagonal At random, D2/D1 = 3.8 At random, D2/D1 = 4.7 At random

0.524 0.601–0.640 0.741 0.68 0.81

f = shape factor

f = 2 = > Φm = 0.676 f = 10 = > Φm = 0.430 f = 100 = > Φm = 0.040

Perfectly aligned

π/(2 3 ) ≈ 0.907

At random At random

0.68–0.71 0.74

1 1.38 + 0.0376 f 1.4

Table 6.7 reveals that aligned rods and randomly dispersed ellipsoids* can pack more densely than spheres and the higher Φm for bimodal spheres is a clear indication that particle size distribution is an important issue. Indeed, it has long been observed that a dispersion of uniform spheres (i.e., same diameter) has a higher viscosity than a suspension of spheres with different diameters, while the volume fraction is kept constant.68,69 This would suggest that a simple volume fraction Φ is insufficient to describe a polydisperse suspension and that a distributive function Φ(d) would be more convenient, with d a representative dimension of the particle. Such a distributive function is likely very difficult to assess and an alternative approach would be to consider that the (experimentally determined) maximum packing fraction is a typical information for a given filler. Indeed, good fitting of ηr (Φ) data for filled polymers are in certain cases obtained if the packing fraction is considered as an adjustable parameter, as we will see later. Equations 6.6 through 6.13 above (and many others in the literature) are a priori valid only if the suspending medium is a Newtonian fluid, i.e., whose viscosity is not affected by the rate of deformation. For nonNewtonian suspending media, one of the simplest approach consists in first accepting the above theoretical views about the role of the particles

* Note that quite complex simulation algorithms are needed to estimate the maximum packing fraction of ellipsoids; for details, see A. Donev, I. Cisse, D. Sachs, E.A. Variano, F.H. Stillinger, R. Connelly, S. Torquato, P.M. Chaikin. Improving the density of jammed disordered packings using ellipsoids. Science, 303 (5660), 990–993, 2004.

278

Filled Polymers

and second in introducing a multiplicative term to express the shear rate dependence, e.g.70

Φ   ηr ( Φ, γ ) = η0  1 −  Φ m 

−2

1 + ( λ cpd γ )2   

n− 1 2



(6.14)

where η0 is the zero-shear viscosity of the polymer matrix and λcpd = λ0 (1–(Φ/ Φm))–2 is a characteristic time of the compound, in fact the characteristic time of the suspending medium multiplied by a function of the filler fraction. In the term expressing the shear rate dependence, one recognizes the Carreau equation for the shear viscosity function of polymer melts. As such Equation 6.14 is relatively limited because it predicts simple vertical shifting of the shear viscosity function of the polymer matrix with increasing filler content. Experimental observation with filled thermoplastics shows that this is not the case and that higher filler content not only shifts vertically the shear viscosity function but also modifies its shape. Equation 6.14 lacks therefore flexibility and obvious changes are first to introduce an explicit yield stress σc and second to refer to the ­Carreau–Yasuda model, i.e.

ηr ( Φ, γ ) =

σc Φ   + η0  1 −  Φ m  γ

−2

1 + ( λ cpd γ )a   

( n − 1)/a



(6.15)

As demonstrated in Figure 6.15 (see also Appendix 6.4), Equation 6.15 has the capability to meet all the observed singularities that fillers impart to the shear viscosity function. It is worth noting that to consider that a filled polymer compound has a characteristic time that depends on the polymer, on the filler fraction and its maximum packing fraction implies that the rheological (and also mechanical) properties of the system are strongly related to a modification of the viscoelastic properties of the polymer matrix itself, namely the spectrum of relaxation times, since by nature filler particles are infinitely rigid when compared to the polymer. Certain experimental results strongly support this point.71 In what filled molten polymers are concerned, it is worth underlining however that the maximum packing fraction remains essentially a theoretical limit whose meaning must be somewhat changed. Indeed if, as it is frequently the case, strong interactions between the polymer and the filler occur, or are purposely initiated, the possibility of a full kinetic aggregation of particles is obviously reduced, as least because of the high viscosity of the matrix. But strong polymer–filler interactions promote also a reduced mobility of polymer segments in the vicinity of particles’ surface. It follows that complex polymer–filler clusters form, in which particles are bounded by polymer segments in a pseudo-glassy state. The actual and effective maximum packing

Shear viscosity, Pa.s

102 10–3

103

104

105

106

102 10–3

103

104

105

10–1 102 Shear rate, s–1

a = 2.0

a = 0.5 a = 1.0

10–1 102 Shear rate, s–1

Φ = 0.5

Φ = 0.1 Φ = 0.3

103

n = 0.3 a = 0.3

σc = 200 Pa

λ0 = 0.01 s

η0 = 3 kPa.s

103

n = 0.3 a=2

σc = 200 Pa

λ0 = 0.01 s

η0 = 3 kPa.s

102 10–3

103

104

105

106

10 2 10–3

103

104

105

106

10–1 102 Shear rate, s–1

σc = 500

σc = 100 σc = 250

10–1 102 Shear rate, s–1

λ0 = 0.60

λ0 = 0.01 λ0 = 0.05

103

n = 0.3 a=2

Φ = 0.3 λ0 = 0.01 s

η0 = 3 kPa.s

103

n = 0.3 a=2

σc = 200 Pa

Φ = 0.2

η0 = 3 kPa.s

Figure 6.15 Capabilities of Equation 6.14 in meeting the typical features of the shear viscosity function of filled polymer systems; fixed parameters: η0 = 3 kPa.s, n = 0.3, Φm = 0.74; variable parameters: Φ, λ0, a, σc.

Shear viscosity, Pa.s

Shear viscosity, Pa.s Shear viscosity, Pa.s

106

Polymers and White Fillers 279

280

Filled Polymers

fraction is consequently lower than the theoretical packing limit of the particles alone. 6.2.3  Thermoplastics and Calcium Carbonates Calcium carbonate, either ground natural (GCC) or precipitated (PCC), is in volume terms, the largest filler material for thermoplastics (over 70% of the worldwide volume consumption), first in PVC but also in other polymers, essentially polypropylene and polyamides. Pure CaCO3 is a relatively soft material (Mohs hardness 3.0) so that wear of filled plastics processing equipment is in principle limited, providing however there is no hard contaminant. Whatever the care and procedure in preparing GCC, it may contain minute quantities of highly abrasive quartz (sand) depending on the deposit; this is not the case for PCC. Depending on the particle size, GCC is cheaper than PCC, or the reverse. Coarse GCC is obviously a cheap material but milling costs rise sharply as finer grades are produced until precipitated calcium carbonate grades become the most economic to produce. A wide range of particle sizes is available to suit many applications, with PCC the finest range, and many grades are treated with hydrophobic agents, generally fatty acids, in order to modify their surface or wetting properties and therefore, to ease their dispersion in molten polymers, which are generally nonpolar. Notable exceptions are polyamides for which uncoated (dry) CaCO3 grades give the best properties. Both dry and wet coating processes are used by calcium carbonate manufacturers. Dolomite is a calcium magnesium carbonate, with widely spread deposits, that properly ground give grades that are used in place of CaCO3 because it is slightly denser (2.85 vs. ~2.76 g/cm3), but it is also slightly harder (Mohs hardness 3.5). Worldwide, ready-to-be-processed CaCO3 filled polymers are available from a number of compounders. Polypropylene compounds, in the 10–40%wt range are available from numerous producers and grades up to 50%wt are also locally offered for special applications. Table 6.8 gives the average properties of typical commercial PP–CaCO3 composites, as compiled from manufacturers’ data sheets (when available). As can be seen, for a given loading in CaCO3, certain properties have averaged values that suffer from such a large standard deviation that twice the tabulated number could be considered as well. This obviously reflects the diversity of polypropylene grades (see also the large standard deviations for the based resins) used in preparing commercial composites, the presence or the absence of coating agents and likely also the various compounding procedures; such information are of course not disclosed by suppliers. As expected the largest deviations are exhibited by fracture related properties, i.e., elongation at break and impact resistance. The melt flow index, the ultimate tensile properties, the flexural yield strength tend to decrease with increasing filler content, and the reverse is observed for the elasticity and flexural moduli.

Unit g/cm3

g/10 min

MPa MPa % % GPa GPa MPa J/cm J

ASTM D792

D1238

D638 D638 D638 D638 D638 D790 D790 D4812 D256

Property Specific gravity

Melt flow Hardness, rockwell R Hardness, shore D Tensile strength, ultimate Tensile strength, yield Elongation at break Elongation at yield Tensile modulus Flexural modulus Flexural yield strength Izod impact, unnotched Gardner impact

0.90 ± 0.02 28 ± 18 63 ± 37 60 ± 18 26.6 ± 7 26.2 ± 8.2 244 ± 35 12 ± 11 1.19 ± 0.35 1.15 ± 0.42 30 ± 11 9.97 ± 8.56 12 ± 11

10

0.99 ± 0.07 13 ± 8 76 ± 22 70 ± 4 22.9 ± 9.8 31.6 ± 18.8 229 ± 105 33 ± 45 1.50 ± 0.52 1.77 ± 0.96 45 ± 31 10.02 ± 4.65 18 ± 20

0.032

Volume fraction Φ.**

0 0.96

%wt

Calculated density*

CaCO3 Content

Commercial PP–CaCO3 Composites; Average Suppliers’ Data

Table 6.8

20

1.04 ± 0.05 28 ± 37 76 ± 24 69 ± 4 23.3 ± 8.4 23.7 ± 9.1 215 ± 117 32 ± 46 1.94 ± 0.73 1.80 ± 0.79 38 ± 16 8.55 ± 4.53 18 ± 20

0.061

1.02

30

1.11 ± 0.07 15 ± 15 78 ± 30 69 ± 4 20.7 ± 7.2 36.2 ± 30 203 ± 97 18 ± 22 2.19 ± 0.81 2.37 ± 1.37 52 ± 40 10.62 ± 5.33 19 ± 20

1.07 0.087 0.090

40

1.26 ± 0.14 14 ± 15 76 ± 29 72 ± 4 20.6 ± 7.1 22.1 ± 9 166 ± 35 29 ± 41 2.59 ± 0.81 2.25 ± 1.51 36 ± 14 7.96 ± 4.45 16 ± 16

1.11 0.112 0.127

45

16 – – – – – – – 2.5 – – 12 ± 0

1.35

1.13 0.124 0.147

50

(Continued)

1.34 ± 0.02 – – – – 18 ± 2.6 143 ± 93 5 ± 0 – 2.30 ± 0.68 – – 16 ± 3

1.15 0.135 0.157

Polymers and White Fillers 281

ISO D648 D648

%wt

J/cm kJ/m2 °C °C °C

2.34 ± 3.83 2.2 ± 15 90 ± 25 52±6 108±50

0 2.17 ± 2.92 10 ± 7.4 92 ± 42 72±37 –

10 2.38 ± 3.41 9 ± 6.3 80 ± 47 56±34 108±40

20 1.94 ± 2.72 7.9 ± 5.5 100 ± 58 71±29 119±30

30 2.36 ± 3.6 2.1 ± 0.1 87 ± 53 61±33 113±39

40

– – – – –

45

0.78 ± 0.41 – 101 ± 7 – –

50

* Theoretical densities were calculated with respect to the given weight percentage of CaCO3 and considering that the filler was coated with fatty acid(s), 10% of its weight; the following densities were used: PP, 0.90; CaCO3, 2.76; fatty acid(s), 0.88 (average value for lauric, linoleic, myristic, oleic, palmitic, and stearic acids). ** The calculated filler volume fraction depends on the exact composition of the composite (not given by suppliers) and on the density of the compound; two values are given when the calculated and given densities are (too) different.

Izod impact, notched Izod Impact, notched HDT, at 0.46 MPa HDT, at 1.80 MPa Vicat softening point

CaCO3 Content

Table 6.8  (Continued)

282 Filled Polymers

Polymers and White Fillers

283

In (isotactic) PP homopolymer, well dispersed fine CaCO3 particles have been reported to give substantial increases in impact strength compared to the unfilled polymer.72 But the toughening effect is strongly depending on the hydrophobicity of the filler and, in this respect, fatty acid treated grades have been found more interesting and the effect was attributed to different interfacial phenomena imparted by the coating layer.73 As shown in Table 6.8, the (average) impact resistance is indeed somewhat improved when loadings below 20% are used, but the large standard deviation suggests that for composites with certain grades of virgin PP, the benefits might be substantially higher. Impact resistance is however difficult to assess without a large experimental scatter and other properties, somewhat related to toughness, show some significant trends with increasing calcium carbonate content, as illustrated in Figure 6.16. The tensile properties, either at yield or at break, steadily decrease with higher filler level, but the flexural and elasticity moduli vary with CaCO3 level slightly above what is predicted by the Guth and Gold equation, i.e., G cpd = G polym(1 + 2.5Φ + 14.1Φ2). This is quite a remarkable observation since it suggests that, in terms of flexural modulus, the effect of calcium carbonate particles is slightly more than just hydrodynamic, with interactions between the dispersed particles and the matrix playing a relatively minor role. The left upper graph shows, in comparison with average commercial data, flexural modulus results from a study with a 5.2 g/10 min MFI PP and various levels of uncoated 4 μm CaCO3 particles (BET: = 18 m 2:g).74 The flexural modulus of the virgin resin is 0.567 GPa and using the Guld and Gold equation, predicted flexural modulus values would be significantly below the observed ones. Generally speaking, the mechanical properties (stiffness, strength, and toughness) of mineral filler-thermoplastic composites depend on three factors: filler particles size, particle/matrix adhesion, and filler loading. Particle/ matrix adhesion is especially important because strength depends on the effective stress transfer between filler and matrix, and toughness/brittleness is controlled by adhesion. The benefit in impact properties of CaCO3 -filled PP have long been investigated and a direct correlation between the impact strength and the matrix-to-filler adhesion has been recognized. Up to a certain filler level, the impact properties increase with increasing filler loading, essentially because there is a weak adhesion of CaCO3 to the PP matrix, as proved by published SEM micrographs of fracture surfaces.75 There are theoretical considerations that may help understanding this effect, for instance the concept of critical interparticle distance76 and the associated preferential orientation of crystal planes in the PP shell surrounding CaCO3 particles.77 If the interparticle distance is lower than a certain critical value, then the composite is expected to exhibit ductile rather than brittle fracture, and when the interparticle distance is short enough, a preferential orientation of crystal planes of the lowest shear resistance occurs between filler particles. Such a crystallographic orientation is believed to lower the local plastic resistance,

284

Filled Polymers

Tensile properties

3.5

70

3.0

60

2.5 2.0 1.5 1.0 0.5 0

Flexural 0

0.2 0.1 Filler volume fraction

At yield

50 40 30 20 10 0

0.3

0

3.5

40

3.0

35 Tensile strength, ultimate, MPa

Elasticity modulus, GPa

Polypropylene + CaCO3

Modulus

Tensile strength, at yield, MPa

Flexural modulus, GPa

4.0

2.5 2.0 1.5 1.0 0.5 0

0.05 0.1 0.15 Filler volume fraction

0.20

At break

30 25 20 15 10 5

Elasticity 0

0.05 0.1 0.15 Filler volume fraction

0.2

0

0

0.05 0.1 0.15 Filler volume fraction

0.20

Figure 6.16 Mechanical properties of commercial PP–CaCO3 composites; ° are averaged suppliers data; the vertical bars indicate the standard deviations; shaded diamonds are data from one single manufacturer; shaded triangles are data from B. Haworth, C.L. Raymond. Proceedings Eurofillers 97, Manchester, UK, Sept. 8–11, 251–254, 1997; the curves in the left part have been calculated with the Guth and Gold equation.

thus allowing larger local plastic deformation to occur and therefore an overall higher toughness of the composite. In compiling data used to fill Table 6.8, no attention could be paid to the quality of the calcium carbonate used in preparing the composites and particularly to the presence of a surface modifier, because this information is

Polymers and White Fillers

285

generally not disclosed by compounders. When using uncoated ­carbonate fillers, impact resistance falls generally quickly with increasing filler level, but when a coated filler is used, a very different behavior is observed: notched impact strength remains higher up to 40%wt loading, i.e., 0.12 volume fraction. Coated CaCO3 (with a fatty acid, for instance) gives impact resistance an order of magnitude greater than that of the unfilled polymer. Easier dispersion and better processing properties of the composites are obviously obtained with coated particles but at the expense of certain mechanical properties. For instance, a PP composite made with 20% (volume) of 8% fatty acid coated CaCO3 exhibit a flexural modulus around three times higher than the virgin polymer. If uncoated CaCO3 particles are used, the composite modulus is more than four times higher. Using 4% of fatty acid gives nearly the same results. But using coated calcium carbonate gives significant increases in impact resistance.74 Coating CaCO3 with a small amount of fatty acid (or a derivative) is generally considered as beneficial, not only in terms of processing behavior, but also in term of balance of mechanical properties, because enhanced impact resistance is generally the main sought benefit. A common explanation for the benefit of fatty acid (or other surface modifiers, e.g., organosilanes) is that a small increase in the crystallization temperature is obtained, whilst maintaining the same cristallinity because coating does not generally change the nucleating power of the filler,78 already excellent if CaCO3 particles are fine enough.79 Most of the benefit is obtained by using maximum 10% weight surfactant (with respect to filler content). It must be noted however that the coating effects may somewhat depend on the structure of the polymer. Indeed, Supaphol and Harnsiri80 investigated recently the effects of CaCO3 particles of varying content, size, and type of surface modification on the rheological and isothermal crystallization characteristics of syndiotactic PP. They observed the usual effects on rheological properties, i.e., steady shear and dynamic viscosity increasing with increasing content, decreasing size, and surface coating of filler particles. The half-time of crystallization was found to decrease with higher CaCO3 content and increases with increasing particles size. When coating the particles with either stearic acid or paraffin, their ability to effectively nucleate the polymer is significantly reduced. When coating calcium carbonate, the industrial approach consists so far in using quite common and inexpensive agents, for instance paraffins, fatty acids, and fatty acid derivatives. More expensive chemicals, e.g., silanes, titanates, seem rarely used in the industry, in contrast with the abundant research literature on the benefits in using sophisticated surface treating agents. Han et al.81 compared the effects of two silane coupling agents (N-octyl triethoxy silane and -aminopropyl triethoxy silane), and of isopropyl triisostearoyl titanate on rheological properties, processability, and mechanical properties of 50% CaCO3-filed PP compounds. All coupling agents give processing benefits (lower melt viscosity and higher melt

286

Filled Polymers

elasticity) but in quite different manners. For instance the two silanes affect the melt elasticity in reverse manners. Mechanical properties of injection molded specimens show the effects on the tensile strength and percent elongation of the filled polypropylenes depend on the specific coupling agent used. Leong et al.82 compared the effects of commercially available neoalkoxy titanate, 3-aminopropyltriethoxysilane and stearic acid as coupling agents in CaCO3 –PP composites. The silane and titanate treatments dramatically increase the elongation at break for both the single-filler and hybrid-filler composites, whereas stearic acid does not. A moderate improvement in the impact strength of the composites is observed, particularly with the titanate product. Wang and Lee compared the benefits of a liquid titanate coupling agent (isopropyl triisostearoyl titanate) and stearic acid in CaCO3 –PP composites.83,84 A small amount of coupling agents is found to give a drastic decrease in the surface energy of CaCO3 particles, and the magnitude of the effect depends on the type of coupling agents. Completely covered particles exhibit low polarity surfaces. Infrared analysis shows that stearic acid reacts extensively with the filler surface to produce chemically bound organic salt compounds. No transesterification reaction between the titanate agent and the filler is observed but IR data suggest adsorption on the particle surface. Surface treated CaCO3 composites generally exhibit higher impact strength than untreated systems and the titanate agent appears to me more effective than stearic acid. On a cost/ performance basis however, the low-cost stearic acid proves to be more effective when dealing with impact properties. In the author’s opinion, the exact mechanisms for the effects of the various coating agents on the mechanical properties remain unclear however whilst some (qualitative) relationships can be considered between the modification of the rheological properties when coated CaCO3 particles are used and the likely origin of the observed effects on mechanical properties. For instance, Price and Ansari85,86 used inverse gas chromatography to characterize the surfaces of CaCO3 particles before and after treatment with sodium polyacrylate or stearic acid, in order to explain the effects of such coating agent on the mechanical properties of filled polypropylene. As expected, they found that the surface treatment reduces the surface polarity but also that modification with stearic acid produces nonpolar, low­energy surfaces. Some mechanical properties of the composites are found to somewhat correlate with the surface energy modifications imparted by the coating agents. Similar conclusions were reached by other authors when considering the effects of titanate coupling agents on the rheological properties of particulate-filled polyolefin melts.87 The main effect of the coupling agent is a considerable reduction of the melt viscosity, at least in the case of CaCO3 –PP systems, with some associated effects on mechanical properties of injection molded specimens, for instance reduced modulus and tensile strength, but increased elongation and impact strength of the filled systems. The lowering of the surface energy of CaCO3 through stearic

Polymers and White Fillers

287

acid coating has been reported by other authors88,89 and the associated benefits in ­processing behavior of PP–CaCO3 composites were assigned to lower ­i nterfacial force between the filler surface and the resin matrix. The benefits in using very fine particles for better impact resistance is well established but, the smaller the particles, the more difficult the dispersion. It is therefore, quite clear that an immediate benefit of coating agents, whatever their chemical nature, is to make CaCO3 particles easier to disperse by decreasing their natural trend to remain clustered. As discussed by Richard et al.90 the dispersion behavior of mineral fillers is strongly depending on both the polar nature of the polymer and the surface properties of filler particles. In a PVC matrix, CaCO3 particles rapidly disperse because there are immediate strong acid-base interactions at polymer filler contacts. In contrast, polypropylene (PP) is essentially a “Van der Waals-force material” and fillers without pronounced acid or base surface characteristics tend to disperse more rapidly and produce mechanically stronger compounds. This would explain the beneficial aspect of coating agents on dispersion mechanisms of CaCO3 in PP, with of course more decisive effects with very fine particle materials, i.e., precipitated calcium carbonate. It is worth underlining here a basic difference between filled elastomers and filled thermoplastics. As we have seen, in the case of the former, the normal loading for a reinforcing filler, e.g., carbon black or high structure silica, is in the 50–60 phr range, that is in terms of volume fraction, largely above the so-called percolation level, i.e., 0.12–0.13. In addition, a number of interesting properties are due to the development of a soft “filler-rubber network” embedded in the vulcanized matrix. Calcium carbonate filled polypropylene grades are presented by manufacturers with respect to the weight percentage of filler. In terms of mechanical (and rheological) properties, only volume fractions of ingredients are relevant and, accordingly, the average data of Table 6.8 were plotted in Figure 6.16 with respect to CaCO3 volume fraction. As can be seen, most of the commercial PP–CaCO3 grades have filler volume fraction below the percolation level. The observation that mechanical properties such as the flexural and the elasticity moduli are slightly above the prediction of the Guth and Gold equation indicates that, in addition to hydrodynamic effects between near spherical particles, which do not interact much with each other (if dispersion is correctly made), there are other phenomena, whose origin is most likely the boundary region between the particles and the matrix. In contrast with carbon black filled rubber systems, the role of the interfacial region in filled thermoplastics has so far received less attention, from both theoretical and experimental points of views. Although the modulus increases upon increasing filler content, the tensile properties (strength at yield and at break in Figure 6.15; the corresponding elongation data are given in Table 6.8) do not follow the same trend. As illustrated above in the case of PP–CaCO3 systems, the tensile properties decrease with increasing filler loading, due to stress concentration effects

288

Filled Polymers

(the so-called strain amplification effect, as seen in Chapter 5, Section 5.1.7), but there are many factors that can affect these properties, for instance size and geometry of the particles, interfacial adhesion, etc. For model systems in which there is a good adhesion between the (spherical) particles and the matrix, some authors91,92 reported that the elongation at yield (or at break) should follow a simple relationship:



 ε cpd = ε pol  1 − 

3

3 2 π

3

 Φ  ≈ ε pol ( 1 − 1.105 

3

)

Φ

(6.16)

where εcpd and εpol refer to the composite and the polymer respectively, and Φ is the volume fraction of the filler. One notes that the prefactor 3 3 2/π refers to a cubic close-pack array of uniform spheres. If there is poor adhesion between the particles and the matrix, one would expect a more gradual decrease in ultimate elongation with higher filler loading than that calculated with Equation 6.16. Figure 6.17 (left) shows that averaged data on commercial PP–CaCO3 systems, despite the very large standard deviation, are not far from the trend predicted by the equation. This would at least mean that, whilst not exactly so, CaCO3 particles in a thermoplastic polymer roughly behave as beads. A rigid filler, finely dispersed in a rigid polymer, is normally expected to decrease the impact strength of the composite, because such a property is largely determined by dewetting and crazing phenomena. Interestingly, the right part of Figure 6.17 shows the reverse: low volume fractions of calcium carbonate somewhat improve the impact resistance, at least when notched samples are used. A common interpretation of such an observation is that nontouching, well dispersed CaCO3 particles would indeed promote craze formation but would also impede crack growth, either through local higher impact energy absorption effects or through crack propagation deviation. At higher CaCO3 loading, this beneficial effect would be lost, as indeed shown by Figure 6.17. Approximately 80% of all the fillers used in PVC is calcium carbonate. Titanium dioxide is second at around 12%, followed by calcined clay at about 5%. The remaining few percent is taken up by other materials, including glass and talc. The performances of CaCO3 in PVC formulations strongly depend on the particle size, the particle size distribution, the loading level and the presence of so-called impact modifiers, for instance acrylic oligomers or rubber-like materials, e.g., nitrile rubber NBR. Above a certain (average) particle size (around 1 µm), PCC is at best a filler, below this size, ultrafine PCC improves the impact resistance. Ultrafine PCC grades are commercially available with particles in the 0.07 µm range. Calcium carbonate, either ground or precipated, has been used as inexpensive filler and extender in flexible and rigid PVC formulations for more than 30 years.93 As filler, both types of CaCO3 are nearly equivalent but

0

100

200

300

400

0.00

0.05 0.10 0.15 Filler volume fraction

0.20

(ISO) Izod impact, notched, KJ/m2 0

2

4

6

8

10

12

14

16

18

20

0

0.05 0.10 0.15 Filler volume fraction

Polypropylene + CaCO3

0.20

Figure 6.17 Ultimate mechanical properties of commercial PP–CaCO3 composites; averaged suppliers data; the vertical bars indicate the standard deviations; shaded diamonds are data from one single manufacturer; the curve in the left part has been calculated with Equation 6.16.

Elongation at break, %

500

Polymers and White Fillers 289

290

Filled Polymers

ultrafine (i.e., submicron particle size) PCC has revealed an interesting role as impact modifier in rigid PVC.94,95 PCC particles are all about the same, uniform in size, in contrast with GCC which has many large particles and many very small particles. The narrow particle size distribution typical of PCC has many advantages, especially where high impact strength is needed. As shown in Figure 6.18, impact strength improves when the median particle size is below 1 µm. At low loading, the benefit of smaller particle is relatively small, but above a critical loading, around 20 phr, the benefit is particularly impressive. Note that CaCO3 contents were recalculated in terms of volume fraction using the formulation and specific gravity data given in the caption of Figure 6.18. It is worth underlining that at 20 phr loading, the filler content is still far from the theoretical percolation level (i.e., around 0.12). As it has been reported that submicron PCC grades were enhancing the gelation of PVC, it is likely that the origin of this beneficial effect on impact resistance is related to improved dispersion not only of the filler particles but also of the other compounding ingredients, namely the processing aid and the lubricants. Indeed a better and more complete gelation would give a composite with fewer defect sites and opportunities for crack propagation. Ductile fracture, rather than brittle failure, is observed when using sufficient levels of ultrafine precipitated calcium carbonate. The improvement in the impact performance is obtained in rigid PVC compounds without the addition of any

Izod impact strength, notched, kJ/m

1.8 1.6

PVC compounds CaCO content phr Φ

1.4 1.2

10

0.046

1.0

15

0.068

0.8

20

0.088 Unfilled

0.6 0.4 0.2 0.0

0

1

2 3 Median particle size, µm

4

Figure 6.18 Effect of CaCO3 particle size on the impact resistance of rigid PVC (data from: Specialty Minerals Inc., Easton, PA); Formulations: PVC: 100 (ρ = 1.39 g/cm3); stabilizer: 1.5 (1.05 g/cm3); acrylic process aid: 1 (1.09 g/cm3); calcium stearate: 0.5 (1.04 g/cm3); fatty acid esters: 1.3 (0.96 g/cm3); CaCO3: variable (2.71 g/cm3).

Polymers and White Fillers

291

organic impact modifier. But in the presence of either acrylic and rubber-type modifiers, this improvement adds from the levels obtained with the modifier alone. However, whereas polymer modified compounds display generally a rapid fall in impact strength a low temperature, coated PCC modified systems still maintain a good impact resistance even at low temperatures. Fine and ultrafine precipitated calcium carbonates are used in a number of rigid polyvinyl chloride applications such as window and door profiles, pipes and fittings, fencing and decking, increasingly replacing more expensive impact modifiers. Besides providing improved impact resistance, small particle PCCs give products with an excellent surface gloss and finish. Coating PCC particles with fatty acid derivatives, e.g., fatty stearates, helps the processing, namely during compounding through easier dispersion, and gives lubrication effects during extrusion. It follows that, besides providing improved impact resistance, small particle PCCs give products with an excellent surface gloss and finish. In PVC cable fabrication, calcium carbonate is used at loadings of around 70 phr. The choice of calcium carbonate depends on the specifications for cable. Higher quality cables benefit from the better mechanical properties (tensile strength and elongation) and electrical properties (volume resistivity) offered by ultrafine coated grades. Cables with lower specification can use uncoated PCC grades. 6.2.4  Thermoplastics and Talc The term “talc” covers a wide range of products, but among its various forms, mostly pure and lamellar grades are used in the plastic industry. Talc is a crystalline form of magnesium silicate, whose uses in polymers is essentially determined by its lamellar structure, essentially magnesium hydroxide sheets between siloxane layers. Layers are bonded by weak Van der Waals forces and therefore talc is a very soft material (Mohs hardness: 1) that, through cleavage, gives high aspect ratio particles. It is the high aspect ratio that is the most important property for its use in polymers. In thermoplastics, the main usage of talc is in polypropylene where the high aspect ratio of particles gives significant increases in stiffness, but aspect ratio is rarely quoted by talc suppliers. Depending on the deposit very pure and very white talcs are available, with a great variety of average particle size and size distribution. Another variety of talc, generally referred as chlorite (a natural mixture of magnesium silicate/hydroxide with iron and aluminium substitution) is also used in polymers. No surface treatment has really been found of interest when using talc as a filler for thermoplastics and consequently nearly all marketed talcs are untreated. Talc is usually lamellar (platy), but the aspect ratio may vary considerably depending on the deposit and the treatment (grinding, sorting) of the filler. There are many deposits and hence numerous producers of talc(s), but around 80% of the world production come from seven countries: China,

292

Filled Polymers

South Korea, USA, India, Finland, France and Brazil. The former three account for more than half of the world production (around 8.5 MioT in 2006). Talc is used in the manufacture of a myriad of products, with papers and ceramics the largest markets worldwide (approximately 30% each). Talc as a filler for thermoplastics (around 15% of the world consumption) is in competition with fine grades of calcium carbonate (ground or precipitated) and find applications in the production of automotive parts, household appliances, cables and engineering plastic parts. Talc has an average growth of around 3% per year, essentially in line with the increasing use of polypropylene, especially for the automobile market where lightweight and recyclability are important issues nowadays. The development of very fine, compacted, submicron grades offers the possibility to enhance the properties of plastic parts. Talc is used to stiffen thermoplastics, mainly polypropylene but also polyethylene and polyamide (nylon). In polypropylene compounds, talc is generally enhancing the following properties: • • • • • • • •

Stiffness (elasticity modulus) Impact resistance (impact strength) Creep resistance Thermal conductivity Heat distortion temperature Nucleation Barrier properties Chemical resistance

Table 6.9 gives the average properties of typical commercial PP–talc composites, as compiled from manufacturers’ data sheets (when available). For a given filler loading, certain properties have averaged values that suffer from a large standard deviation so that twice the tabulated number could be considered as well. This obviously reflects the diversity of polypropylene grades used in preparing commercial composites, the particular grade of talc and possibly some additives not disclosed by the manufacturers. The compounding procedure also plays a role. The largest deviations are exhibited by fracture related properties, i.e., elongation at break and impact resistance. The melt flow index and the elongations at break and at yield tend to decrease with increasing filler content, but the reverse is observed for the elasticity and flexural moduli. The main reason for incorporating talc in polypropylene is to increase the stiffness (modulus of elasticity). The degree of rigidity obtained depends on the filling level, aspect ratio and fineness of the talc, with high aspect ratio grades giving the largest enhancement. Figure 6.19 shows how the flexural and elasticity moduli vary with increasing talc

Property Specific gravity Melt flow Hardness, rockwell R Hardness, shore D Tensile strength, ultimate Tensile strength, yield Elongation at break Elongation at yield Tensile modulus Flexural modulus Flexural yield strength Izod impact, unnotched Charpy impact, notched Gardner impact

Volume fraction Φ.**

Calculated density*

Talc Content

MPa MPa % % GPa GPa MPa J/cm J/cm2 J

D638 D638 D638 D638 D790 D790 D790 D4812

D256

Unit g/cm3 g/10 min

ASTM D792 D1238

%wt

0

12 ± 11

0.90 ± 0.02 28 ± 18 63 ± 37 60 ± 18 26.6 ± 7.0 26.2 ± 8.2 169 ± 83 12 ± 11 1.19 ± 0.35 1.15 ± 0.42 30 ± 11 9.97 ± 8.56 –

Commercial PP–talc Composites; Average Suppliers’ Data

Table 6.9 10

0.98 ± 0.03 17 ± 12 82 ± 20 70 ± 4 25.2 ± 11.1 27.7 ± 10.1 31 ± 21 12 ± 8 1.74 ± 0.82 1.56 ± 0.4 37 ± 16 5.82 ± 3.62 0.45 ± 0.21 2 ± 3

0.964 0.0318 0.0323

20

1.17 ± 0.25 17 ± 12 79 ± 24 71 ± 9 24.6 ± 11.7 27.1 ± 10.4 13 ± 14 10 ± 7 2.26 ± 1.13 2.09 ± 0.83 38.7 ± 17.2 3.91 ± 3.11 1.5 ± 0.07 1 ± 0

1.019 0.0615 0.0707

30

1.19 ± 0.17 12 ± 6 75 ± 28 73 ± 9 24.5 ± 10.6 26 ± 9.5 13 ± 9 9 ± 6 2.49 ± 1 2.18 ± 0.65 44.2 ± 14.7 2.85 ± 2.24 0.22 ± 0.06 3 ± 2

1.071 0.0895 0.0995

40

1.26 ± 0.06 14 ± 15 76 ± 31 74 ± 10 18.9 ± 12.4 35.3 ± 22.5 13 ± 7 3 ± 2 3.26 ± 1.4 3.23 ± 0.5 43.8 ± 22.1 2.37 ± 1.99 0.24 ± 0.14 5 ± 7

1.12 0.1159 0.1304

50

(Continued)

23.5 ± 2.1 30.4 ± 5.8 3 ± 3 3 ± 1 3.75 ± 0.64 4.14 ± 0.86 – – – 2

1.37 ± 0.03 7 ± 6 90 –

1.166 0.1408 0.1655

Polymers and White Fillers 293

D648

Deflection temperature

°C

°C

J/cm kJ/m2 °C 108±50

52±6

2.34 ± 3.83 2.2 ± 15 90 ± 25

0

104±12

72±63

3.65 ± 1.78 6.5 ± 4.2 137 ± 74

10

110±42

86±38

3.54 ± 2.22 2 ± 0 109 ± 30

20

113±72

85±46

2.9 ± 1.72 8 ± 6.3 104 ± 44

30

116±40

93±44

1.14 ± 0.83 6 ± 5.3 126 ± 19

40

158

–

140 ± 2

0.27 ± 0.07 –

50

* Theoretical densities were calculated with respect to the given weight percentage of talc and considering that the filler was uncoated; the following densities were used: PP, 0.90; talc, 2.76. ** The calculated filler volume fraction depends on the exact composition of the composite (not given by suppliers) and on the density of the compound; two values are given when the calculated and given densities are (too) different.

Vicat softening point

ISO D648

%wt

Izod impact, notched Izod impact, notched Deflection temperature

Talc Content

Table 6.9  (Continued)

294 Filled Polymers

295

Polymers and White Fillers

5

5 Elasticity modulus, GPa

6

Flexural modulus, GPa

Polypropylene + talc 6

4 3 2 1 0

4 3 2 1

0

0.10 0.15 0.05 Filler volume fraction

0.20

0

0

0.05 0.10 0.15 Filler volume fraction

0.20

Figure 6.19 Effect of talc volume fraction on the flexural and elasticity moduli of commercial filled polypropylene compounds; clear squares are average moduli data; gray shaded diamonds are data from one selected manufacturer; the dotted curve was calculated with the Guth and Gold equation and Epolym = 1.152 GPa and 1.191 GPa for the flexural and elasticity moduli respectively; the solid curve was calculated with the modified equation and an anisometry factor f = 2.1.

volume fraction. Talc particles are far to correspond to spheres and therefore the variation is larger than predicted with the Guth–Gold equation. A simple manner to account for particles anisometry, while considering that the filler effect is essentially hydrodynamic, consists in introducing an anisometry factor f, i.e.

Ecpd = Epolym ( 1 + 2.5 × f × Φ + 14.1 × f 2 Φ 2 )

(6.17)

As shown in Figure 6.19, this remarkably simple approach suits well average moduli data on talc filled polypropylene compounds. Compared to polymers, talc has a significantly higher thermal conductivity and therefore, heat transfers during processing are more efficient: talc filled PP melts easier and cools faster and high production rates can be obtained. Small quantities of fine talc particles act as a nucleating agent, thus promoting the crystallization of polypropylene.96–99 For instance, exothermic crystallization peaks, as measured by DCS, show that talc particles, whilst having nearly twice as large average particle size are stronger nucleating agents of PP than CaCO3 and kaolin.75 When the crystallinity of the polymer matrix increases, the composite is expected to exhibit a higher modulus,

296

Filled Polymers

an increased tensile strength and a better dimensional stability, as indeed observed. Talc is better than kaolin and CaCO3 in this respect. Crystallization starts at a higher temperature in the presence of talc, compared to unfilled PP. The impact strength is improved but this is primarily due to an increase in the crystallization of the PP and not the mechanical properties of the talc itself. There is also a change in modulus as a result of the change in crystallinity. Mineral fillers do not generally improve impact strength, but fine talc is an exception in PP compounds that explain its use in manufacturing car bumpers. In this case, 5–10% of fine talc give a benefit, which however vanishes at higher loadings, as indeed shown in Figure 6.20, drawn with average Izod data (notched specimen, test at room temperature). Other impact tests (average) data show a similar trend. Talc has a beneficial effect on the deflection temperature of PP parts, but it strongly depends on the aspect ratio of the particles, as easily understood via micromechanics considerations. Mineral fillers impart substantial reduction in creep of plastics and fine platy talc grades give the best performance, with however some synergistic effects when additional different fillers (e.g., calcium carbonate) are used, without or with surfactant additive.100,101 Thanks to its lamellar structure, talc has excellent barrier properties, by reducing for instance transmission rates for water vapor and oxygen, providing however that particles are properly oriented during the processing operations. Lamellar talc particles are mostly orientated in films and help constraining

7

Izod impact, notched, J/cm

6 5 4 3 2 1 0

0

0.1 Filler volume fraction

0.2

Figure 6.20 Effect of talc volume fraction on the impact resistance of commercial filled polypropylene compounds; clear squares are average moduli data; shaded diamonds are data from one selected manufacturer.

297

Polymers and White Fillers

Table 6.10 Typical Industrial Applications of Talc in Thermoplastics

Application

Role

Typical Loading (%)

Typical Grade

Median Particle Size (µm)

Additive in semicrystalline polymers

Nucleating agent

0.5

Very pure

1–2

Plastic films

Antiblocking agent

0.5

Controlled size distribution

2–3

IR radiation retainer

5–10

Very pure

5

White reinforcing filler

30–40

Very white

≈ 10

20

Very white

6–10

20

Very white

10

20

Lamellar

3–6

40

Lamellar, pure

10

15–20

Lamellar

3–5

5–10

Fine, lamellar

2

LDPE/EVA agro-films (greenhouses) Domestic appliances (injection molded parts) Household appliances (PP) Garden furnitures (PP) Industrial PP foils/films Automotives Parts Parts in engine compartment Dashboards Bumpers, exterior parts

Temperature resistance Stiffness, rigidity Stiffness, gas impermeability Temperature resistance Dimensional stability Stiffness, impact strength

the water vapor and oxygen. Talc is water repellent and chemically inert and therefore increases the chemical resistance of PP parts. The above data explain why the automotive and domestic appliances markets are the major ones for talc-filled PP. Talc is the preferred additive for such applications, as it imparts high stiffness, which allows a reduction in wall thickness, at least below a certain loading. Food packaging applications for talc-filled PP systems are due to the higher rigidity and barrier properties obtained. Table 6.10 describes a few typical industrial usages of talc. 6.2.5  Thermoplastics and Mica Mica is considered as the third most important white filler for thermoplastics but has some unique characteristics that sometimes give filled materials exceptional properties not obtained with any other mineral. Mica is a transparent, flexible and elastic phyllosilicate that can be ground to very fine particles with a high-aspect ratio and high mechanical properties, e.g., tensile modulus ≈ 175 MPa, flexural modulus ≈ 220–270 MPa, compressive

298

Filled Polymers

modulus ≈ 190–285 MPa (as measured on sheets of muscovite). Mica is relatively soft, with Mohs hardness of around 2.8–3.2 that helps in minimizing wear in plastic processing equipment. Finely ground mica has a high degree of whiteness (around 83%) and is strongly absorbing UV radiations, thus increasing the service life and decorative value of end products for outdoor applications. In semicrystalline thermoplastics, mica is believed to act as a nucleating agent. Polypropylene is likely the major polymer compounded with mica for a number of applications, mainly in the automobile industry, sometimes in association with other white fillers, calcium carbonate, talc or short glass fibers, to fabricate a large variety of parts, e.g., instrument panels, heater housing, rear-light retainers, ceiling fan blades, etc. Other applications for mica-filled PP compounds are: blow molded containers and consumer goods packaging, extruded fibers, filaments and films, pipes and sheets, wires and cables), injection molded parts, e.g., furniture, house wares, luggage, boxes, etc. Compounded mica filled polypropylenes are available from a number of manufacturers, with however a large offer from India (where are the largest deposits of mica). Grades up to 40% are marketed and Table 6.11 is a compilation of typical average properties, based on manufacturers’ data sheets. As can be seen, both the tensile and flexural properties (strength and modulus) generally increase with higher mica content, with corresponding decreases in elongation (at yield and at break). Mica has a variable and generally negative effect on impact strength. As expected, heat distortion temperature markedly increases with increasing filler content. Flexural and elasticity moduli are worth special attention. Figure 6.21 compares data from Table 6.11 and from a selected compounder with the prediction of the Guth and Gold equation, modified by the addition of an anisometry factor of 3.3. As can be seen, the agreement is excellent which suggests that the prime effect of mica particles is essentially hydrodynamic, with however the flaky shape playing a key role. The incorporation of mica generally reduces thermal expansion and helps eliminating the nonuniform thermal shrinkage in injection molding; both effects are obviously associated with the natural flake-like form of ground mica. Like talc, mica has also excellent barrier properties, all the more if the particles are properly oriented with respect to the trajectory of the permeant. Mica provides similar benefits in a wide range of thermoplastic and thermoset composites including polyolefins, polyamides and styrenics. It is also reported that surface coated mica further increases tensile strength, flexural strength and modulus, and heat deflection temperature. The automobile industry is the main user of mica-filled composites, either with polypropylene or nylon as polymer matrix. Up to 40% mica loadings are used, sometimes in association with calcium carbonate, to produce various injection

MPa MPa % % GPa GPa MPa J/cm J J/cm °C °C

g/cm3 g/10 min

0.90 ± 0.02 28 ± 18 63 ± 37 60 ± 18 26.6 ± 7.0 26.2 ± 8.2 169 ± 83 12 ± 11 1.19 ± 0.35 1.15 ± 0.42 30 ± 11 9.97 ± 8.56 12 ± 11 2.34 ± 3.83 90 ± 25 52 ± 6

0

10

1.6 ± 1.8 0.58 ± 0.37 87 ± 18 54 ± 7

0.97 0.0313 0.0309 0.96 ± 0.01 8 ± 7 65 ± 19 70 ± 1 18.5 ± 7.7 28.4 ± 2 19 ± 3 6 ± 1 1.37 ± 0.38 1.56 ± 0.64 28.8 ± 11 –

20 1.026 0.0606 0.0615 1.04 ± 0.03 11 ± 9 55 ± 22 74 ± 4 24.8 ± 10.7 23.2 ± 10.2 12 ± 5 5 ± 2 2.1 ± 0.46 2.48 ± 0.68 35.6 ± 12.7 3.88 ± 1.27 0.8 ± 0.7 0.77 ± 0.81 116 ± 19 68 ± 13

25

0.39 ± 0.13 124 69

3.62 ± 0.68 3.52 ± 0.1 48 4.81 –

32.6 ± 3.7 34.2 ± 1.1 15 ± 1 4

1.053 0.0747 0.0706 1.08 ± 0 – 97 –

30 1.079 0.0883 0.0925 1.13 ± 0.04 6 ± 5 43 ± 20 71 ± 1 17.2 ± 7.3 31.2 ± 7.1 4 ± 3 2 ± 1 2.86 ± 0.32 3.13 ± 0.94 38.2 ± 5.4 1.67 ± 0.64 0.6 ± 0.4 0.32 ± 0.18 106 ± 13 65 ± 11

40 1.128 0.1143 0.1307 1.29 ± 0.11 16 ± 5 88 ± 23 73 ± 1 26.6 ± 8 38.3 ± 7.1 10 ± 3 4 ± 1 4.69 ± 1.11 5.24 ± 1.67 52.5 ± 9.1 3.47 ± 1.27 1.8 ± 0.7 0.73 ± 0.52 141 ± 10 102 ± 22

* Theoretical densities were calculated with respect to the given weight percentage of (uncoated) mica; the following densities were used: PP, 0.90; mica, 2.82. ** The calculated filler volume fraction depends on the exact composition of the composite (not given by suppliers) and on the density of the compound; two values are given as the calculated and given densities are different.

Density Melt flow Hardness, rockwell R Hardness, shore D Tensile strength, ultimate Tensile strength, yield Elongation at break Elongation at yield Tensile modulus Flexural modulus Flexural yield strength Izod impact, unnotched Gardner impact Izod impact, notched HDT, at 0.46 MPa HDT, at 1.8 MPa

Volume fraction range

Calculated density

Mica, %wt

Commercial PP–mica Composites; Average Suppliers’ Data

Table 6.11

Polymers and White Fillers 299

300

Polypropylene + mica

7

6

6

5

5

Elasticity modulus, GPa

Flexural modulus, GPa

7

Filled Polymers

4 3 2 1 0

4 3 2 1

0.00

0.05 0.10 Filler volume fraction

0.15

0

0.00

0.05 0.10 Filler volume fraction

0.15

Figure 6.21 Effect of mica volume fraction on the flexural and elasticity moduli of commercial filled polypropylene compounds; clear squares are average moduli data; shaded diamonds are data from one selected supplier; the dotted curve was calculated with the Guth and Gold equation and Epolym = 1.152 GPa and 1.191 GPa for the flexural and elasticity moduli, respectively; the solid curve was calculated with the modified Guth and Gold equation and an anisometry factor f = 3.3.

molded PP parts, such as instrument panels, heater housing, ceiling fan blades, tail-light retainers, door panels and seat backs in automobiles. Mica is also used as a filler in fire-retardant polypropylene foams, for instance in side components for underground swimming pools. Mica-filled nylons are used to produce automobile parts such as opening panels, various parts for the engine compartment and also headlight covers. In such applications, mica generally provides higher tensile strength, heat deflection temperature and notched Izod impact than other mineral fillers, such as CaCO3 and wollastonite. 6.2.6  Thermoplastics and Clay(s) Kaolin (clay) is a widely used filler for thermoplastic polymers because of its reinforcing effects on mechanical properties, such as stiffness and strength, with however certain negative effects on impact resistance. Clays are typically used at 20–50% wt in polymer composites, and commercially available grades have aspect ratio between 4 and 12, and average particles in the 0.5–2.3 µm range. Because kaolin has a platelike structure with a high

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301

aspect ratio, particle size and particle sizes distribution are very important parameters in determining the mechanical properties of filled composites. Kaolin improved the stiffness of PP102 and other polymers, and surface treatment of particles can somewhat improve the mechanical properties of the composites,103 likely because either the dispersion is easier during the compounding operations or because a good adhesion is obtained at the filler–matrix interface, or both. Clays are phyllosilicates and therefore, can be delaminated through the use of suitable chemicals and/or mechanical means. By splitting apart the platelet structure, which further increases the available surface area more filler–polymer contacts are obtained at constant loading, and therefore enhanced reinforced properties are obtained. It has been shown that dramatic improvements in mechanical properties can be achieved with very low levels (a few weight percent) of inorganic exfoliated clays. Exfoliated clays have a thickness of around 1 nm and lateral dimensions of ≈ 30 nm to several microns or larger. The large aspect ratios of exfoliated clays are thought to be the main reason for the enhanced mechanical properties of so-called polymer nanocomposites. Considerable research work has been published on polymer nanocomposites, with a few recent reviews104,105 that are worth reading, but sizeable industrial applications remain scarce. The subject is outside the scope of the present book and will not be discussed further here. Combining several (white) fillers sometimes allow to obtain composites with a wider range of properties at acceptable costs. Calcium carbonate provides improved impact properties in polyolefins but relatively modest stiffness. Talc, mica and clay, owing to their aspect ratio, give moderate stiffness but with adverse effects on impact resistance. Short glass fibers provide a high stiffness, somewhat depending on average fiber orientation, but bring a considerable brittleness. Through a judicious and adequate combination of the above filler materials, it is thus in principle possible to achieve a very broad range of properties, but in a highly pragmatic manner. For instance, so-called engineering plastics, e.g., PET, PBT, and ABS, can be filled with up to 30% mica, in association with 20–25% short glass fibers to obtain improved tensile, flexural and compressive strength, stiffness and to impart particular electrical properties. Except for a few very low demanding applications, adding fillers to a thermoplastic polymer does not cheapen formulations, at least on a volume basis because mixing and compounding costs ­generally override savings in raw materials. But in many cases, the overall balance of properties achieved with filled polymers is such that the composite may compete with more expensive, high performance engineering materials. Good examples are filled-reinforced PP to compete with ABS, and short glass ­fiber-reinforced polyamides to compete with more expensive specialty polymers. This likely explains the huge diversity of filled thermoplastics available today, as a demonstration of continuously extended opportunities for filled polymer composites.

302

Filled Polymers

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Appendix 6

A6.1  Adsorption Kinetics of Silica on Silicone Polymers Model of polymer adsorption kinetics Q(t) = Q0 + Qinf ( 1 − exp ( − λ ⋅ t )) Qinf: quantity of adsorbed polymer for an infinite contact time Q 0: initial quantity of adsorbed polymer (i.e., immediately after mixing) λ: kinetic constant for the adsorption ­process (λ = tad–0.5 with tad a characteristic time) A6.1.1 Effect of Polymer Molecular Weight PDMS Mn (g/mole): [Data: L. Dujourdy, PhD Thesis, University J. Fourier, Grenoble, France, 1996] 43,000  0  175   525  1015 Data1 :=  1715   2293  2713   3308 time h

0.264  0.48  0.673   0.704  0.722   0.792  0.785  0.792  Q(t) g/g silica

Silica: 150 m2/g Weight fraction: 0.20

73, 000

 

 0  28   42   245  245   413 Data2: =  413   840   1190  1838   2030  2695   2842

0.405   0.588  0.588   1.092  1.099    1.197  1.25  1.3   1.498  1.567   1.563  1.482   1.514 

300,000  0  70   238  595 Data3 : =  1085  1918  2415   3255

0.775  1.197  1.69   1.901  2.077   2.113  2.116  2.134 

309

Polymers and White Fillers

Model equation and partial derivatives Collecting experimental data

x0 : = Data1<0>

y 0 : = Data1<1>

x1 : = Data2<0>

y1 : = Data2<1>

 C0 + C1 ⋅ ( 1 − exp ( − C2 ⋅ x ))   1   F(x, C): =   1 − exp ( − C2 ⋅ x )    C1 ⋅ x ⋅ exp ( − C2 ⋅ x ) 

x2 : = Data3<0>

y 2 : = Data3<1>

nj : = length ( x j )

j : = 0 .. 2

: datasets

 (yj )  0  : initial guessed parameters  C j : = ( y j ) n − 1  j    0.1  Calling the nonlinear Resj: = GenFit(xj,yj,Cj,F)    fitting algorithm:  

Fa j : = ( Res j )0 + ( Res j )1 ⋅ 1 − exp  − ( Res j )2 ⋅ x j     : fit equation         Characteristic time

R2j: = corr(Faj,yji)

: correlation coefficient

t ad j : =

1

( Res j )  2

 0.257  Res 0 =  0.588   0.044 

2

tad = 514.819

1

0

R2 0 = 0.991

Adsorbed polymer, g/g of filler

Fitting data on PDMS (43,000 g/mol)/silica compound

(Res 0 )0 + (Res 0 )1 = 0.845

 

0 0

1000

2000 Time, h

3000

2

310

Filled Polymers

 0.32  Res1 =  1.321   0.053  t ad1 = 350.675 R2 1 = 0.987

(Res1 )0 + (Res1 )1 = 1.641

Adsorbed polymer, g/g of filler

Fitting data on PDMS (73,000 g/mol)/silica compound 2

1

0

0

1000

2000 Time, h

3000

Adsorbed polymer, g/g of filler

Fitting data on PDMS (300,000 g/mol)/silica compound  0.733  Res 2 =  1.483   0.061 t ad2 = 273.138 R2 2 = 0.992

(Res 2 )0 + (Res 2 )1 = 2.217

 

2

1

0 0

1000

2000 Time, h

3000

A6.1.2 Effect of Silica Weight Fraction Silica weight fraction: [Data: L. Dujourdy, PhD Thesis, University J. Fourier, Grenoble, France, 1996] 0.049  0  38   61  163 Data4: =   469   1388  2000   3327

0.103

0.206   0 0.312   163  0.497   653  0.676   Data5: =  980  1.241  1306   1.5   2286  1.588  3633   1.544 

PDMS, Mn = 43,000 g/mole Silica: 150 m2/g

 0 0.279   184  0.55   510 0.691   1000  0.779  Data6: =  1706  0.882    2245 0.985   2673  1   3265

0.204 0.265  0.456  0.676   0.691 0.721  0.788  0.794   0.794 

311

Polymers and White Fillers

j : = 3..5

: datasets

x3 : = Data4<0>

y 3 : = Data4<1>

x4 : = Data5<0>

y 4 : = Data5<1>

x5 : = Data6<0>

y 5 : = Data6<1>

: collecting experimental data n j : = length( x j )

 ( y j )0  : initial guessed parameters   C j : = ( y j )n j −1       0.1  Calling the nonlinear    Res : = GenFit(x ,y ,C ,F) j j j j fitting algorithm:



Fa j : = ( Res j )0 + ( Res j )1 ⋅ 1 − exp  − ( Res j )2 ⋅ x j  

  : fit equation     

R2j: = corr (Faj,yj)

Characteristic time

: correlation coefficient  t : = ad j

1

( Res j )  2

2

Adsorbed polymer, g/g of filler

Fitting data on 0.049 silica weight fraction compound

 0.103  Res 3 = 1.702   0.041 tad 3 = 588.735 R2 3 = 0.98

(Res 3 )0 + (Res 3 )1 = 1.805

 

2

1

0 0

2000 Time, h

4000

312

Filled Polymers

 0.266  Res 4 = 1.042   0.022  tad4 = 2.02 ⋅ 10 R2 4 = 0.993

3

(Res 4 )0 + (Res 4 )1 = 1.308

Adsorbed polymer, g/g of filler

Fitting data on 0.103 silica weight fraction compound 2

1

0

0

2000 Time, h

4000

2000 Time, h

4000

 0.255  Res 5 =  0.614   0.039  tad5 = 648.22 R2 5 = 0.985

(Res 5 )0 + (Res 5 )1 = 0.869

Adsorbed polymer, g/g of filler

Fitting data on 0.204 silica weight fraction compound 2

1

0 0

A6.2 Modeling the Shear Viscosity Function of Filled Polymer Systems By combining two Carreau–Yasuda equations, an eight parameter model is obtained that would meet all the likely typical features of the shear viscosity behavior of filled polymer systems. a η(γ °) = η 01⋅ 1 + ( λ 1⋅ γ °) 1   

n 1− 1 a1

n 2 -1

a + η 02⋅ 1 + ( λ 2⋅ γ °) 2  a 2  

nonhydrodynamic

interparticless

effectsof filler

hydrodynamic

particles

effectts

 he (very) low shear T plateau is accounted for by parameter η01 and the    high shear thinning region is depending on the flow index n2.

313

Polymers and White Fillers

Nonhydrodynamic effects of filler particles (e.g., filler networking) ­ ominate the low shear behavior in such a manner that an apparent yieldd ing region overrides the pseudo-Newtonian plateau of the polymer matrix ­(corresponding to parameter η02) There is an intermediate plateau when: η 02 = η c = η 01⋅ [1 + (λ 1⋅ γ °cc )a 1 ]

n 1− 1 a1

which corresponds to a critical shear rate γc:  1

 a1    a 1  1  η 02   n 1−1   γ °c = ⋅ −1  λ 1  η 01   

The critical shear rate γc corresponds to a critical characteristic time λc A few mathematical aspects of the model: γ°: = 0.0001⋅sec−1, 0.001⋅sec−1 .. 200⋅sec−1 : shear rate range for calculations Model parameters: η01 : = 8 ⋅ 10 4 ⋅ Pa ⋅ sec

λ 1 : = 500 ⋅ sec

a1 : = 1.9

η02 : = 3 ⋅ 103 ⋅ Pa ⋅ sec

λ 2 : = 0.1 ⋅ sec

a2 := 3

η(γ ° ) : = η 01⋅ 1 + ( λ 1⋅ γ ° 

)

a1

 

n 1− 1 a1

n1 : = 0.4

n2 := 0.33

+ η 02⋅ 1 + ( λ 2⋅ γ ° 

)

a2

γ ° c 1 = 2 ⋅ 10−3°sec −1

η( γ ° c 1 ) = 6.727 ⋅ 10 4 °Pa ⋅ sec

γ ° c 2 = 2 × 10°sec −1

η( γ ° c 2 ) = 3.052 ⋅ 103 °Pa ⋅ sec

 

n 2−1 a2



γ °c l : =

γ °c 2 : =

1 λ1

1 λ2

: Model equation

 1



Critical shear rate corresponding to the intermediate plateau:

γ °c = 0.476 ° sec −1

η c : = η ( γ °c )

 a1    a 1  1  η 02   n 1−1   γ °c : = ⋅ −1  λ 1  η 01   

η c = 6.103 °Pa ⋅ sec   :viscosity at the ­critical shear rate

314

Filled Polymers

η01

: Shear viscosity function with respect to ­parameters η01 and η02.

γ°c2

γ°c1 1.104

η02

1.103 100 1.10–4 1.10–3 0.01 0.1 1 Shear rate, 1/s

 hear viscosity function S with respect to critical shear rate and viscosity:

100 1.103

10

  1.105 Shear viscosity, Pa.s

Shear viscosity, Pa.s

1.105

γ°c

1.104

ηc

1.103 100 –4 1.10 1.10–3 0.01

n1 : = 0.1,0.2 .. 0.5

a η(γ °, n1 ) : = η 01⋅ 1 + ( λ 1⋅ γ °) 1   

n 1− 1 a1

0.1 1 10 Shear rate, 1/s

100 1.103

a + η 02⋅ 1 + ( λ 2⋅ γ °) 2   

n 2−1 a2

Shear viscosity, Pa.s

1.105

    : Effect of parameter n1 1.104 1.103 100 1.10–4 1.10–3

0.01

0.1 1 10 100 Shear rate, 1/s n1 = 0.5 n1 = 0.2

1.103

a a 1: = 0.5, 1.6 .. 3.5 n 1: = 0.3 η(γ º , a 1) : = η 01⋅ 1 + ( λ 1⋅ γ º ) 1   

n 1− 1 a1

+ η 02⋅ [ 1 + (λ 2⋅ γ ° )a 2 ]

n 2 −1 a2

315

Polymers and White Fillers

1.105 Shear viscosity, Pa.s

    : Effect of parameter a1

1.104

1.103

100 1.10–4 1.10–3

0.01 0.1 1 Shear rate, 1/s a1 = 3.2 a1 = 0.8

10

A6.3 Models for the Rheology of Suspensions of Rigid Particles, Involving the Maximum Packing Fraction Φm The general approach consists in considering that the viscosity of the suspension is equal to the viscosity of the suspending medium times a suitable functional whose mathematical form is derived from theoretical considerations. No specifc interactions are considered between the particles and the suspending medium; only nonhydrodynamic crowding effects are considered in such a manner that the viscosity of the systems goes to infinity as the particle volume fraction approaches the maximum packing fraction of the particles. Besides the theoretical reasonning in deriving the model equation, there are two criteria for validity:



1. As the volume fraction decreases, the viscosity must become asymptotic to the straightline of slope 2.5 corresponding to the Einstein’s equation 2. As the volume fraction increases and approaches the maximum packing fraction, the viscosity must go to infinity

All the models considered below meet those two validity criteria. Φ: = 0,0.01 .. 0.5 π Φ m: = 3⋅ 2

: volume fraction range for calculations : maximum packing fraction of particles (here, spheres of equal diameter in the closest hexagonal packing)

316

Filled Polymers

ηE(Φ): = (1 + 2.5⋅Φ)

: Einstein (1906–1911); low volume fraction limit 2

  5 1+ ⋅Φ  2  ηEil( Φ ) :=   1− Φ   Φ m   5 Φ η M(Φ): = exp  ⋅ 2 1− Φ  Φm

Φ   ηKD( Φ ) : =  1 −  Φ m 

Φ   ηK ( Φ ) : =  1 −  Φ m 

−5 ⋅Φ m 2

: Eilers (1941) Guth, Gold and Simha (1948) ηGGS(Φ): = (1 + 2.5⋅Φ + 14.1⋅Φ2)

    

: Mooney (1951)



: Krieger–Dougherty (1959)

−2



: Kitano et al. (1981)

 1   3 3 Φ   1− Φ 1 −   Φ Φ Φ   m m 5 9 ⋅  1 +  ⋅ 1 + 2 ⋅ 3 ηG( Φ ) : = 1 + ⋅ Φ + ⋅  2 ⋅ 3 Φ  2  Φ 2 4    Φm Φm     

ηGSB

(Φ)

2    1− Φ   Φ m− Φ    m : = 1 − Φ ⋅ 1 +   ⋅ 1 −  Φ   m      Φ m   

  2                

: Graham (1981)

−2.5

: Graham et al. (1984)

317

Polymers and White Fillers

Comparing models Models predicting a sharp variation vs. Φ

Models predicting a smooth variation vs. Φ 20

50 40

Relative viscosity

Relative viscosity

15

30

10

20

5

10 0

0

0 0

0.2 0.4 Volume fraction Einstein Eilers Mooney Kitano Guth, Gold & Simha

0.2 0.4 Volume fraction Einstein

  

Krieger & Dougherty Graham, Steele & Bird Graham Guth, Gold & Simha

Effect of maximum packing fraction Φm: = 0.5 .. 0.74 [lower bound: cubic packing = 0.524] [higher bound: hexagonal packing = 0.74]

Mooney (1951):

 5 Φ η M( Φ, Φ m ): = exp  ⋅ 2 1− Φ  Φm

    

Relative viscosity

100

50

0 0

0.2 0.4 Volume fraction

0.6

Max Fract. = 0.524 Max Fract. = 0.6 Max Fract. = 0.7

The larger Φm, the larger the ­ olume fraction range before v ­relative ­viscosity goes to infinity. 

318

Filled Polymers

Krieger–Dougherty (1959):

Φ   η KD( Φ , Φ m ): =  1 −  Φ m 

−5 ⋅Φ m 2

Relative viscosity

40

20

0 0

0.2 0.4 Volume fraction Max Fract. = 0.6 Max Fract. = 0.524 Max Fract. = 0.7

Graham (1981):

 he larger Φm, the smoother T the variation in relative viscosity.



   1   3 3 Φ Φ  5 9  1− 1− ηG ( Φ , Φ m ) : = 1 + ⋅ Φ + ⋅   Φm  Φ Φm 2 4 ⋅  1 +  ⋅ 1 + 2 ⋅ 3 2⋅ 3  2  Φ Φ    Φm Φm   

    2              

Relative viscosity

40

20

0 0

0.2 0.4 Volume fraction

Max Fract. = 0.6 Max Fract. = 0.524 Max Fract. = 0.7



The larger Φm, the smoother the  variation in relative viscosity.

319

Polymers and White Fillers

Graham et al.:

2    1− Φ   Φ − Φ   m η GSB(Φ, Φ m): = 1 − Φ ⋅ 1 +  ⋅ 1−  m     Φ m      Φ m  

−2.5

Relative viscosity

40

20

0 0

0.2 0.4 Volume fraction Max Fract. = 0.6 Max Fract. = 0.524 Max Fract. = 0.7



 he larger Φm, the larger T the volume fraction range before relative viscosity goes to infinity.

A6.4 A  ssessing the Capabilities of the Model for the Shear Viscosity Function of Filled Polymers γ: = 0.001 sec.−1, 0.01 sec−1...1000 sec−1 : shear rate range for calculations Φ: = 0, 0.01 .. 0.5 : volume fraction range for calculations Φm: = 0.64 : maximum packing fraction for a random arrangement of spheres (of equal diameter) 1 = 100 °sec −1 η0: = 3 ⋅ 103 ⋅ Pa ⋅ sec : zero-shear viscosity of polymer matrix λ 0 λ0: = 0.01 ⋅ sec : characteristic time of polymer matrix n: = 0.3 : flow index (in high shear region) a: = 2 : Yasuda exponent (curvature in transition region for the polymer matrix) σc: = 200 ⋅ Pa : yield stress of filled compound a −2 −2      σ Φ      Φ   η( Φ ,γ  ,λ 0 , a,σ c ) : = c +  η 0⋅  1 − 1 λ 1 γ ⋅ + ⋅ − ⋅   0   γ Φ m      Φ m      

n−1 a

  : model equation

320

Filled Polymers

A6.4.1 Effect of Filler Fraction (λ0 cst = 0.01 s, a = 2, σc = 200 Pa) Shear viscosity, Pa.s

1.106 1.105 1.104 1.103 100 1.10–3 0.01 0.1 1 10 100 1.103 Shear rate, 1/s Vol.Fract.=0.3 Vol.Fract.=0.1 Vol.Fract.=0.5

A6.4.2 Effect of Characteristic Time λ0 1.106 1.105

Shear viscosity, Pa.s

(Φ cst = 0.2, a = 2, σc = 200 Pa)

1.104 1.103 100 10 1.10–3 0.01 0.1 1 10 Shear rate, 1/s

100 1.103

Lambda = 0.05 Lambda = 0.01 Lambda = 0.6

321

Polymers and White Fillers

A6.4.3 Effect of Yasuda Exponent a (Φ cst = 0.3, λ0 cst = 0.01 s, σc = 200 Pa) Shear viscosity, Pa.s

1.106 1.105 1.104 1.103 100 1.10–3 0.01 0.1 1 10 Shear rate, 1/s a = 1.0 a = 0.5

100 1.103 a = 2.0

A6.4.4 Effect of Yield Stress σc (Φ cst = 0.3, λ0 cst = 0.01 s, a = 2) Shear viscosity, Pa.s

1.106 1.105 1.104 1.103 100 1.10–3 0.01 0.1 1 10 Shear rate, 1/s

100 1.103

Yield stress = 100 Pa Yield stress = 250 Pa Yield stress = 500 Pa

322

Filled Polymers

A6.4.5  Fitting Experimental Data for Filled Polymer Systems Fitting experimental data with the above model is not straightforward and requires both an extanded set of data and the appropriate fitting strategy, as demonstrated below with shear viscosity data on a series of TiO2 filled polystyrene at 180°C, as published by Minagawa and White [J. Appl. Polym. Sci, 20, 501–523, 1976]. In order to apply the model, TiO2 particles are considered as spheres of equal diameter in such a manner that the maximum packing fraction Φm can be considered equal to 0.64. virgin Polystyrene (PS)

PS + 4.8% TiO 2 Φ TiO = 0.012 2

Shear Rate s−1

 0.0027  0.00072   0.0258  0.0798   0.2470  0.4049  0.7646  Data1 : =  1.6629  3.6164   8.4407  18.3571  39.9234   93.1805  233.3986   440.7292

Shear Visco kPa.s 17.191  15.781   15.781  13.684   12.7743  11.450  8.922   6.028  4.374   2.656  1.731   1.089  0.638  0.361   0.219 

Shear Rate s-1

 0.0012  0.0027   0.0072  0.0258   0.0798  0.2470  0.4049   0.7646 Data2 : =  1.6629   3.6164  8.4407  18.3571   39.9234  93.1805   2333.3986  440.7292

Shear Visco kPa.s 22.701  21.750   19.544  18.860   18.200  15.781  13.684   9.581  Data3: = 6.708   5.044  3.289   1.997  1.212  0.736   0.416  0.253 

PS + 14.3% TiO 2 Φ TiO = 0.034 2

Shear Rate s−1

 0.0012  0.0027   0.0072  0.0258   0.0798  0.2470  0.4049   0.8806  1.6629   3.6164  8.4407  18.3571   39.9234  93.1805   233.3986  440.7292

Shear Visco kPa.s 53.022  41.314   32.191  27.914   24.205  19.544  16.354   11 1.450  8.308   5.417  3.532   2.303  1.501  0.849   0.4480  0.291 

323

Polymers and White Fillers

PS + 24.0% TiO 2

PS + 38.6% TiO 2

Φ TiO2 = 0.056

Φ TiO2 = 0.087

Shear Rate s -1

 0.0027 393.422   0.0072 178.145     0.0146 100.715    70.518   0.0258  0.0422 56.940    422.813   0.0798  0.1308 34.570     0.2470 27.914    0.4049 23.357  Data4 : =   0..8806 15.781    9.929   1.7846  3.6164 6.708     9.0585 4.07 73    2.752   18.3571  39.9234 1.859    1.051   93.1805  195.6 6190 0.594     440.7292 0.348 

Shear Visco kPa.s

Shear Rate s−1  0.8806  2.0553   3.6164   9.0585  18.3571 Data5 : =   39.9234  93.1805   195.1805   195.6190  440.77292

Shear Visco kPa.s 25.083 14.695 9.581 5.613 3.793 2.473 1.398 1.398 0.790 0.480

              

A6.4.6  Observations on Experimental Data Shear viscosity, kPa.s

1.103 100 10 1 0.1 1.10–3 0.01 0.1 1 10 100 1.103 Shear rate, 1/s Virgin PS 4.8% TiO2 14.3% 38.6% 18.0%

: Experimental shear viscosity functions

324

Filled Polymers

As can be seen, the shear viscosity function is well investigated over six decades of shear rate for the virgin polystyrene and the filled compounds up to 18% TiO2 content. Whilst the 38.6% compound is only documented in a shorter shear rate range (i.e., 1–1000 s−1), the occurrence of a yield stress behavior that depends on the filler level makes however no doubt. In the high shear region, the shear thinning behavior is not much affected by the presence of the filler, since the flow index n seems to be the same, whatever is the filler content. Presumably, the characteristic time λ0 is also not depending on the filler content. It follows that, for the filled compounds, only σc η0 and a must be fit to experimental data A6.4.7 Extracting and Arranging Shear Viscosity Data j: = 0 .. 4 γ 10 : = Data1< 0>

γ 11 : = Data2 < 0>

γ 12 : = Data3< 0>

γ 13 : = Data 4< 0>

γ 1 4 : = Data5< 0>

η0 : = Data1< 1>

η1 : = Data2 < 1>

η2 : = Data3< 1>

η3 : = Data 4< 1>

η4 : = Data5< 1>

n1: = length ( γ 1 0 )

n2 : = length ( γ 1 1 )

n1 = 15

n2 = 16





n3 : = length ( γ 1 2 )

n4 : = length ( γ 1 3 )





n3 = 16

n4 = 18

n5 : = length ( γ 1 4 ) n5 = 9

A6.4.8 Fitting the Virgin Polystyrene Data with the Carreau–Yasuda Model a η1( γ °,λ 0, a ) = η 0⋅ 1 + ( λ 0⋅ γ °)   

  (η ) + (η ) 0 0 0 2  2    1 C: =  γ 1°0 )round  n1 , 0 (   2    0.3   2 

            

n− 1 a



: Carreau–Yasuda equation

Guess parameters for nonlinear fitting algorithm (GenFit function)

n1: = length (η0) : number of data  < = extracting guess parameters from experimental data, i.e., -T  he average of two lowest shear rate viscosity data for η0 - The reverse of the mid shear rate range for λ0 - A common value for molten polymers for the flow index - a = 2 (Carreau model)

325

Polymers and White Fillers



Model equation and partial derivatives:

C2 − 1  C C0 ⋅ 1 + ( C1 ⋅ γ ° ) 3  C3      C2 − 1  1 + ( C1 ⋅ γ ° )C3  C3      C2 − C3 − 1    C0 C C   ⋅ ( C2 − 1) ⋅ ( C1 ⋅ γ ° ) 3 ⋅ 1 + ( C1 ⋅ γ ° ) 3  C3    C 1 F( γ °, C):=   C2 − 1  C0  C C ⋅ 1 + ( C1 ⋅ γ ° ) 3  C3 ⋅ ln 1 + ( C1 ⋅ γ ° ) 3       C 3    C3 ⋅ ( C1 ⋅ γ ° )C3 ⋅ ln ( C1 ⋅ γ ° )  …     C3 ⋅ 1 + ( C1 ⋅ γ ° )C3   C2 − 1     C3  C2 − 1    C3 ⋅  C0 ⋅  C  ⋅ 1 + ( C1 ⋅ γ ° )  C 3 1 + ( C1 ⋅ γ ° ) 3  ⋅ ln 1 + ( C1 ⋅ γ °     + −    C3 ⋅ 1 + ( C1 ⋅ γ ° )C3        

Res1: = GenFit ( γ 1°0, η0, C, F ) p : = 1..24

zl : = 0.001

: calling the nonlinear regression algorithm

γ s° p := 10log( zl) + p⋅0.25   : discrete shear rate range for calculations

 Res12 − 1   Res13 Res13   η0fp : =  Res10 ⋅ 1 + ( Res11 ⋅ γs° p )       Res12 − 1    Res1 r02 : = corr   Res10 ⋅ 1 + ( Res11 ⋅ γ 1°0 ) 3  Res13 , η0         100

: Fitting equation

: Correlation coefficient r2

Virgin PS

16.712   1.183  η0  10 Res1 =   0.277     η0fp   1  0.77  0.1 1.10–3 0.01 0.1

                    C3 )      

1 10 100 1.103 1.104 γ1°0, γs°p

  

r02 = 0.998

326

Filled Polymers



Assembling results R1:=



A0 ← 0 for i ∈ 0..3 A1 + i ← Res 1i A5 ← 0 A6 ← r 02 A

 0  166.712     1.183    R1 =  0.277   0.77     0   0.998   

Φ η0 λ0 n a σc r2

A6.4.9  Fitting the Filled Polystyrene Shear Viscosity Data Model equation: a −2 −2    σc   Φ  Φ      η( γ °,λ 0, a,σ c ) = +  η 0⋅ 1 − ⋅ γ °   ⋅ 1 +  λ 0⋅ 1 − γ °   Φ m  Φ m        

or:

a    Φm 2    σc  Φm 2 η( γ °,λ 0, a, σ c ) = +  η 0⋅ ⋅ γ °   ⋅ 1 +  λ 0⋅ γ °  ( Φ m− Φ )2    ( Φ m− Φ )2  

n− 1 a

n− 1 a

When compared with the virgin PS, the shear viscosity data for the PS + 4.8% TiO2 compound are essentially slightly shifted upwards with a possible modification of the curve in the transition region (which depends in fact on the value of the parameter a), and the yield stress behavior is barely visible at low shear rate. This suggests that the characteristic time λ0 and the flow index n obtained for the virgin polymer can be kept for the filled compounds, so that only the parameters are σc, η0 and a must be determined by nonlinear fitting. Φ m: = 0.64 : maximum packing fraction for a random arrangement of spheres (of equal diameter) PS + 4.8%TiO2

ΦTiO2 = 0.012

λ0: = Res11

n: = Res12

Φ1: = 10.012

Φ: = Φ1

: model parameters considered as constant (i.e., depending on the polymer matrix only)

327

Polymers and White Fillers

 0.001  D1 : =  Res10   < = guest parameters:  1.10−3  A low value for σ  Res13  c D1= 16.712  -T  he low shear viscosity  0.77  of the virgin PS for η0 -F  or a, the value obtained for the virgin PS  A = 1.039

A: =

Φ m2

( Φ m − Φ )2

        Model equation and partial                derivatives: n− 1  D0 D + ( D1 ⋅ A ) ⋅ 1 + ( A ⋅ λ 0⋅ γ  ) 2  D2    γ °   1  γ°  G(γ ° , D) : =   ( n − 1)   D   A ⋅ 1+ ( λ 0⋅ A ⋅ γ °) 2   D2       ( n − 1)    ( n − 1) ⋅ D1 ⋅ A  ln ( λ 0⋅ A ⋅ γ ° ) ⋅ ( λ 0⋅ A ⋅ γ ° D  ⋅ 1 + ( λ 0⋅ A ⋅ γ ° ) 2   D2  ⋅   D    D 1 + ( λ 0⋅ A ⋅ γ ° ) 2 2   

)D

2

          D2 ln 1 + ( λ 0⋅ A ⋅ γ ° )     −    D2 

Res2: = GenFit ( γ 1°1,η1, D1, G )   : calling the nonlinear regression algorithm  n− 1  Res2 Res2 2 Res2  0 2   η1fp : =  + ( Res2 1 ⋅ A ) ⋅ 1 + ( λ 0⋅ A ⋅ γ s° p )    : Fitting equation    γ s° p   n− 1    Res2 0 Res2  r12 := corr  + ( Res2 1 ⋅ A ) ⋅ 1 + ( λ 0⋅ A ⋅ γ 1°1 ) 2  Res22 , η1      γ 1°1   

100

PS + 4.8% TiO2

η1 10 η1fp

 3.285 ⋅ 10−3  Res2 =  19.469   0.772 

1

0.1 –3 1.10 0.01 0.1



  : Correlation coefficient r2

1 10 100 1.103 1.104 γ1°1, γs°p



r12 = 0.997

328

Filled Polymers



Assembling results R 2 : = A0 ← Φ A1 ← Res 2 1 for i ∈1.. 2 A1 + i ← Res 1i



 0.012  Φ  19.469  η   0  1.183  λ 0   R 2 =  0.277  n  0.772  a    3.285 ⋅ 10−3  σ 2  0.997  r 2  

A4 ← Res 2 2 A5 ← Res 2 0 A6 ← r 12 A

PS + 14.3% TiO2 λ0: = Res11

ΦTiO  = 0.034

Φ2: = 0.034

2

Φ: = Φ2

n: = Res12

 <= guest parameters:  Res2 0  -F  or σc, the value obtained for the lower filled compound D2 : =  Res2 1  above - For η0, the value obtained for the lower filled compound  Res2 2  above -F  or a, the value obtained for the lower filled compound above  3.285 ⋅ 10−3  D2 =  19.469   0.772  call G(γ°, D)

A: =

Φ m2

( Φ m − Φ )2



A = 1.115

: model equation and partial derivatives

Res3: = GenFit ( γ 1°2 , η2 , D2 ,G ) : calling the nonlinear regression algorithm  n− 1  Res3 Res32 Res3  0  2  η2fp : =  + ( Res31 ⋅ A ) ⋅ 1 + ( λ 0⋅ A ⋅ γ s° p )    γ s° p 

Fitting equation

329

Polymers and White Fillers

 n− 1    Res30 Res3  r22 : = corr  + ( Res31 ⋅ A ) ⋅ 1 + ( λ 0⋅ A ⋅ γ 1°2 ) 2  Res32 , η2        γ 1°2   

100 η2

PS + 14.3% TiO2

 0.026  Res3 =  31.734   0.526 

10

η2fp

1

0.1 1.10–3 0.01 0.1

1 10 100 1.103 1.104 γ1°2, γs°p





r22 = 0.999

Assembling results R 3 : = A0 ← Φ A1 ← Res 31 for i ∈1.. 2 A1 + i ← Res 1i



PS + 24.0% TiO2

Correlation coefficient r2

A4 ← Res 32 A5 ← Res 30 A6 ← r 22 A ΦTiO  = 0.056 2

 0.034  Φ  31.734  η   0  1.183  λ 0   R 3 =  0.277  n  0. 526  a    0.026  σ c  0.999  r 2  

Φ3: = 0.056

Φ: = Φ3

λ0: = Res11

n: = Res12



 <= guest parameters: - For σc, the value obtained for the lower filled compound above - For η0, the value obtained for the lower filled compound above - For a, the value obtained for the lower filled compound above

 Res30  D3 : =  Res31     Res32 

330

Filled Polymers

 0.026  D3  31.734   0.526 

A: =

call G (γ°,D)

: model equation and partial derivatives

Res4: = GenFit(γ 1°3,η3,D3 ,G)

: calling the nonlinear regression algorithm

Φ m2

( Φ m − Φ )2



A = 1.201

 n− 1  Res4 Res42 Res4  0  2 η3f p : =  + ( Res41 ⋅ A ) ⋅ 1 + ( λ 0⋅ A ⋅ γ s° p )    γ s° p   n− 1    Res40  1 + ( λ 0⋅ A ⋅ γ 13 )Res42  Res42  , η3  r32 : = corr  + Res4 ⋅ A ⋅ ( ) 1      γ 13   

Shear viscosity, kPa.s

1.103



  Correlation coefficient r2

PS + 24.0% TiO2

 0.939  Res4 =  54.889   0.386 

100 10 1 0.1 1.10–3 0.01 0.1 1 10 100 1.103 1.104 Shear rate, 1/s



r32 = 1

Assembling results R 4 := A0 ← Φ A1 ← Res 41



Fitting equation

for i ∈1.. 2 A1 + i ← Res 1i A4 ← Res 42 A5 ← Res 40 A6 ← r 32 A

 0.056   54.889     1.183    R 4 =  0.277   0. 386     0.939   1   

Φ η0 λ0 n a σc r2

331

Polymers and White Fillers

PS + 38.6% TiO2 λ0: = Res11

ΦTiO  = 0.087

Φ4: = 0.087

2

n: = Res12

Φ: = Φ4

a: = Res13

 <= guest parameters: -F  or σc, the value obtained for the lower filled compound  Res40    above D : =  Res41  4 - For η0, the value obtained for the lower filled compound  Res42  above - For a, the value obtained for the lower filled compound above  0.939  D4 =  54.889   0.386 

call G(γ°,D)

A: =

Φ m2

( Φ m − Φ )2



A = 1.339

: model equation and partial derivatives

Re s5: = GenFit ( γ 1° 4, η4, D4 ,G ) : calling the nonlinear regression algorithm  n− 1  Res5 Res52 Res5  0  2 η4fp : =  + ( Res51 ⋅ A ) ⋅ 1 + ( λ 0⋅ A ⋅ γ s° p )    γ s° p   n− 1    Res50 Res5  r42 : = corr  + ( Res51 ⋅ A ) ⋅ 1 + ( λ 0⋅ A ⋅ γ 1° 4 ) 2  Res52 , η4      γ 1° 4    

Fitting equation

  Correlation coefficient r2

1.104 η4 η4f

1.103

 11.514  Res5=  58.428   0.316 

100 10 1 0.1 1.10–3 0.01 0.1

1 10 100 1.103 1.104 γ1°4,γs°



r42 = 1

332

Filled Polymers



Assembling results R 5 : = A0 ← Φ A1 ← Res 51

 0.087   58.428     1.183    R 5 =  0.277   0.316     11.514   1   

for i ∈1.. 2 A1 + i ← Res 1i



A4 ← Res 52 A5 ← Res 50 A6 ← r 42 A

Φ η0 λ0 n a σc r2

A6.4.10 Assembling and Analyzing all Results AR: = augment(augment(augment(augment(R1, R2), R3), R4), R5)  0 16.712   1.183  AR =  0.277  0.77   0  0.998 

0.012 19.469 1.183 0.277 0.772 3.285 ⋅ 10−3 0.997

0.034 31.734 1.183 0.277 0.526 0.026 0.999

0.056 54.889 1.183 0.277 0.386 0.939 1

0.087  58.428  1.183   0.277  0.3166   11.514  1 

Φ η0 λ0 n a σc r2

Drawing fit curves: i: = 0 .. 4 AR 4,i −2   AR 5,i  AR 0,i     AR 0,i     ⋅ γs°  η(γs°,i): = +  AR 1,i ⋅  1 −  ⋅ 1 +  AR 2,i ⋅  1 −      γs° Φ m     Φm      −2

AR 3,i − 1 AR 4,i

333

Polymers and White Fillers

PS + TiO2 : Model curves

Shear Viscosity, kPa.s

1.104 1.103 100 10 1 0.1 1.10–3 0.01

1 10 100 1.103 1.104 Shear rate, 1/s virgin PS 14.3 % 4.8% TiO2 18.0% 38.6 %

PS + TiO2 : Experimental data

1.104

Shear viscosity, kPa.s

0.1

1.103 100 10 1 0.1 –3 1.10

0.01

0.1 1 10 Shear rate, 1/s virgin PS 4.8% TiO2 38.6% 18.0%

Effect of filler level on yield stress σc:

14.3%

100

Limit viscosity

10

Yield stress

1.103

Effect of filler level on limit viscosity η0:

15

5

0 0

100

0.05 Filler volume fraction

0.1

 

50

0 0

0.05 Filler volume fraction

0.1

334

Filled Polymers

Effect of filler level on Yasuda parameter a:

Parameter a

1

0

–1 0

0.05 Filler volume fraction

The variations of σc, η0, and a with the filler volume fraction are in agreement with the physics of the model. 0.1

Shear viscosity, kPa.s

1.104 1.103 100 10 1 0.1 1.10–3 0.01



1 10 100 1.103 1.104 Shear rate, 1/s virgin PS virgin PS fit 38.6% TiO2 38.6% TiO2 fit 0.1

Even for the highest TiO2 loading, for which experimental data are far to cover a sufficient shear rate range, the fitting strategy based on lower loaded compounds gives a reasonable curve, in agreement with the physics of the model.

335

Polymers and White Fillers

A6.5  Expanding the Krieger–Dougherty Relationship  : B “intrinsic viscosity,” should be equal − Φ ⋅B η(Φ) Φ  m  to 2.5 at very low volume fraction, in = η r (Φ) =  1 −  Φ m  η(0) order to match the Einstein equation : Φm maximum packing fraction Uniform spheres, cubic packing: Φm = 0.524 Uniform spheres, random packing: Φm = 0.621 − Φ ⋅η  Uniform spheres, hexagonal packing: Φ  m  Expanding  1 − :  Φ  = 0.741 m  Φm Fibers, perfectly aligned, max packing: Φm = 0.907 Prolate ellipsoids, random packing: Φm = 0.74 η r (Φ) = 1 + B ⋅ Φ + B ⋅

or:

( Φ m⋅ B + 1) ⋅ Φ 2 + B ⋅ ( Φ m⋅ B + 1) ⋅ ( Φ m⋅ B + 2 ) ⋅ Φ 3 … 2 !⋅ Φ m

3 !⋅ Φ m 2

+B ⋅

( Φ m⋅ B + 1) ⋅ ( Φ m⋅ B + 2 ) ⋅ ( Φ m⋅ B + 3) ⋅ Φ 4 …

+B ⋅

( Φ m⋅ B + 1) ⋅ ( Φ m⋅ B + 2 ) ⋅ ( Φ m⋅ B + 3) ⋅ ( Φ m⋅ B + 4) ⋅ Φ 5 …

+B ⋅

( Φ m⋅ B + 1) ⋅ ( Φ m⋅ B + 2 ) ⋅ ( Φ m⋅ B + 3) ⋅ ( Φ m⋅ B + 4) ⋅ ( Φ m⋅ B + 5) ⋅ Φ 6 + ...

4 !⋅ Φ m 3

5 !⋅ Φ m 4

6 !⋅ Φ m 5

   η r( Φ ) = 1 + B ⋅ Φ + B ⋅ 



1

 2  ( Φ m⋅ B + i )   2   3 ⋅Φ + B⋅  i = 1 ⋅Φ … 3! Φ m 2

∏ (Φ m⋅ B + i) i =1

2!Φ m

∏

 3   4  Φ + i B ⋅ ( ( Φ m⋅ B + i ) )   m   4   5 ⋅Φ + B⋅  i = 1 ⋅Φ +… +B ⋅  i = 1 4!Φ m 3 5!Φ m 4

∏

or abridged form:

∏

 a−1  ( Φ m⋅ B + i )  n   a η r (Φ) = 1 + B ⋅ Φ + B⋅  i = 1 ⋅Φ a−1 ! Φ a m a= 2

∑

∏

336

Filled Polymers

Numerical illustration Φm: = 0.74

ηr (Φ,n) := 1 + B ⋅ Φ +

n

∑

   B⋅ 

a−1

∏(Φ i=1

Φ   ηrKD(Φ ) : =  1 −  Φ m 

− Φ m⋅B

 ⋅ B + i )  a ⋅Φ

m

a! Φ m

a= 2

a−1



: polynomial equation (expanded Krieger– Dougherty)

: Krieger–Dougherty equation

ηGG( Φ ) : = 1 + 2.5 ⋅ Φ + 14.1 ⋅ Φ 2

: Guth and Gold equation

Φ: = 0.5 ηrKD(Φ) = 8.029 ηr(Φ, 6) = 6.549 40 ηr(Φ, 10) = 7.618 ηr(Φ, 20) = 8.017 ηr(Φ, 30) = 8.029 30  <= a 31 terms polynomial nearly per20 fectly macthes the Krieger–Dougherty equation 10 50

Relative viscosity



Φ: = 0, 0.01...Φm

B: = 2.5

0

0

0.25 0.5 Filler volume fraction Polynomial, 6 terms Polynomial, 12 terms Krieger–Dougherty Guth & Gold

0.75

337

Polymers and White Fillers

Effect of type of packing and maximum packing fraction Φ: = 0, 0.01...Φ m

η: = 2.5

  a−1 ( Φ m⋅ B + i )  n − Φ m⋅B  a  Φ   η rKD ( Φ, Φ m ) : =  1 − η r ( Φ, Φ m, n) : = 1 + B ⋅ Φ + B⋅  i = 1 ⋅Φ a−1  a Φ ! Φ m  m a = 2

∑

Relative viscosity

10

30

0.2 0.4 Filler volume fraction Polynomial, 3 terms Polynomial, 6 terms Krieger–Dougherty

20

10

0

0.6

0

    30

10

Polynomial, 3 terms Polynomial, 6 terms Krieger–Dougherty

0.6

Polynomial, 6 terms Krieger–Dougherty

Fibers, aligned, max packing

0.2 0.4 Filler volume fraction

0.2 0.4 Filler volume fraction Polynomial, 3 terms

20

0 0

Uniform spheres, random packing

30

20

0 0

Relative viscosity

Uniform spheres, cubic packing

Relative viscosity

Relative viscosity

30

∏

0.6

   

Prolate ellipsoids, random packing

20

10

0 0

0.2 0.4 Filler volume fraction Polynomial, 3 terms Polynomial, 6 terms Krieger–Dougherty

0.6

7 Polymers and Short Fibers

7.1  Generalities Short fibers and polymers are used to prepare composites essentially with respect to the large differences between certain mechanical properties of both. For instance, most (unfilled) vulcanized elastomers have tensile (Young) modulus and ultimate strength in the 20–50 kPa and 2–5 MPa ranges, respectively and most thermoplastics exhibit the same properties in the 1–2 GPa and 30–80 MPa ranges, respectively. Certain synthetic fibers have the same properties up to 500 GPa (modulus) and 5 GPa (strength) respectively. It means that fibers and elastomers tensile PRoperties differ by factors of 103–107, and fibers and plastics by factors of 102–104. Such huge differences explain why fibers reinforcing effects are always spectacular in polymers, depending however on fiber–matrix interfacial phenomena, fibers orientation and processing difficulties that must be mastered for optimal results. Controlling the orientation of short fibers in fabricated composite parts is a serious problem in most processing operations, while certain processes are more prone to improvement than others, depending on the complexity of the associated flow fields. At best, what can be achieved with short fiber-filled polymer systems is either random or preferred orientation, never perfect orientation. It follows that the actual reinforcing effects obtained when adding short fibers to a polymer can be considered as being bounded by two extreme (ideal) situations: either all fibers are perfectly aligned with respect to the main strain axis, or they are all perpendicular. Figure 7.1 illustrates this concept with an ideal system made of perfectly aligned (long) fibers in a matrix. The tensile moduli of the fibers and the matrix are Efib and Emat, respectively. If the fibers are all aligned in the direction of the strain, both the matrix and the fibers experience the same, uniform strain, and the composite modulus is the upper-bound, calculated as follows:

Eupper = Efib × Φ fib + Emat × ( 1 − Φ fib )

(7.1)

where Φfib is the volume fraction of fibers. This equation corresponds to the simple mixing rule and is sometimes referred to, in mechanics, as the Voigt 339

340

Filled Polymers

Matrix with modulus Emat

Fibers with modulus Efib Strain parallel to fibers

Modulus, GPa

80

Emat = 3 GPa

Strain perpendicular to fibers Efib = 73 GPa

60 40 20 00

Eupper Elower 0.2 0.4 0.6 0.8 Fibers volumic fraction Φfib

1

Figure 7.1 Upper and lower bounds in (long) fiber composites; the upper and lower bound moduli curves where calculated with the typical tensile modulus values for E-glass fibers (Efib = 73 GPa) and for polyamide 6 (Emat = 3 GPa).

average, with respect to the assumption made by this author that “in a multiphase body, the average strain of each phase is equal to the applied strain.”1 If the strain is applied perpendicularly to the fibers orientation, both the fibers and the matrix experience the same, uniform stress, and the composite modulus is the lower bound, i.e.:



Elower =

Φ fib Efib

Efib Emat 1 = 1 − Φ fib Emat Φ fib + Efib ( 1 − Φ fib ) + Emat

(7.2)

This equation corresponds to the harmonic mixing rule (or inversed mixing rule) and is also referred to as the Reuss average, with respect to the assumption made by this author that (in a multiphase body) “the average stress in each phase is equal to the applied stress.2”

341

Polymers and Short Fibers

The above equations correspond to the intuitive understanding that in the former case, the mechanical response of the laminar composite is essentially dominated by the performance of the fibers, thus giving the upper-bound reinforcing effect, while in the latter case, the matrix (and the fibers-to-matrix adhesion) is playing the key role and dictates the lower-bound reinforcement. As we will see, some micro-mechanical models for short fibers composites clearly emphasize such upper and lower limits, and it follows that what can really be achieved when manufacturing a short fiber-reinforced polymer object is indeed between those two extremes. In terms of scientific research, a large variety of fiber-polymer systems have been (and are still) studied, with new compositions constantly investigated. However, in terms of industrial realizations, the list of useful systems is reduced to a few tens, as described in Table 7.1. As one might expect, any fiber-polymer system is unique, not only because of the differences in the respective properties of the components, but also because the fiber surface properties and the polymer chemistry have important effects. Not all aspects are yet understood and any generalization would indeed be abusive. A clear demonstration of this fact is offered when comparing the effect of a given type of short glass fiber (E-glass), at constant loading (30%) in different thermoplastics (Table 7.2). As can be seen, the effect of the filler is strongly depending on both the polymer and the property considered. Generally, the ultimate tensile and flexural strengths increase upon addition of short glass fibers (SGF), while the elongation at break decreases. The effect of fibers on the impact resistance depends essentially on the semicrystalline or amorphous nature of the polymer (see Figure 7.2). A net increase is observed with the former, a loss with the latter. The loss is particularly impressive in the cases of polycarbonate and poly(phenylene oxide); which explains why these polymers are generally used alone in applications where impact resistance is essential, but rarely as components of composites. SGF bring a benefit on heat distortion temperature, but generally larger with semicrystalline polymers (Figure 7.3). Whatever is the polymer, fibers reduce the thermal dilation, which is one of the reasons in using fibers-filled ­composites in applications where tight dimensional tolerance of parts is Table 7.1 Most Frequent Polymer-Short Fiber Systems for Industrial Applications Polymers Polypropylene PP Polyamides PA Polyvinylchloride PVC Saturated polyesters PET, PBT Polyphenylene oxide PPO Polycarbonate PC Vulcanizable elastomers NR, SBR, BR

Short Fibers Synthetic: Short glass fibers, SGF Aramids (staple, pulp), SAF Chopped carbon fibers, SCF Natural: Various cellulose fibers, untreated or treated   Natural, e.g., cellulose   Synthetic, e.g., aramids

Unit

MPa % MPa J/cm °C °C–1

Property

Ultimate tensile strength Elongation at break Ultimate flexural strength Izod impact, notched HDT (1.8 MPa) Thermal dilatation coefficient ( × 106)

77 150 100 179 71 50

Pure

+98 –146 +145 +250 +182 –33

Gain/Loss in Cpd

PA-66

30 300 40 125 70 90

Pure +53 –297 +70 +179 +79 –60

Gain/Loss in Cpd

PP

Effects of 30%wt Short E-Glass Fiber in Different Thermoplastics

Table 7.2

60 110 87 2500 139 68

Pure +62 –106 +93 −2143 +7 –46

Gain/Loss in Cpd

PC

55 150 90 179 69 75

Pure

+78 –147 +90 +89 +144 –50

Gain/Loss in Cpd

PBT

55 50 89 893 120 50

Pure

+23 –30

+75 –46 +59 –536

Gain/Loss in Cpd

PPO

342 Filled Polymers

343

Polymers and Short Fibers

IZOD impact, notched, g/cm

Impact resistance 3000

Pure polymer Polymer + 30% GF

2500 2000 1500 1000 500 0

PA-66

PP

PC

PBT

PPO

Figure 7.2 Effect of 30% short E-glass fibers on the impact resistance of various thermoplastics. Heat resistance 300

Pure polymer Polymer + 30% GF

HDT (1.8 MPa),°C

250 200 150 100 50 0

PA-66

PP

PC

PBT

PPO

Figure 7.3 Effect of 30% short E-glass fibers on the heat distortion temperature of various thermoplastics.

an issue (e.g., connectors for electronic and various applications in electro mechanics). The obvious difference between particulate (e.g., carbon black, silica, and fine minerals) and fibrous fillers is the large aspect ratio of the latter, which makes them particularly sensitive to strong (i.e., orienting) flow in processing operations. In addition, one understands easily that most of the short fibers-related benefits described above also strongly depend on the average orientation of the fibers. If, for a given composition, a majority of the fibers are aligned in the stretching direction, then the tensile properties must necessarily be close to the upper bound limit. Conversely, for better flexural properties, most of the fibers must be oriented perpendicularly to the load application. Such remarks highlight the importance of mechanical models for fibers-filled composites, not only with respect to their expectedly predictive capabilities, but also as convenient tools for understanding the very origin of the benefits obtained, as well as the likely sources of failure or problems.

344

Filled Polymers

7.2 Micromechanic Models for Short Fibers-Filled Polymer Composites 7.2.1  Minimum Fiber Length Short-fibers reinforced composites have been developed, used and analyzed for long as follow-up of many successful engineering developments, most of them on a rather pragmatic basis (trial-and-error). In parallel, there as been a continuing effort to develop theoretical expressions that would allow the mechanical properties of a composite to be predicted, knowing only the properties of the polymer matrix and the fibers. Let us consider a homogeneous matrix in which short anisometrical particles (e.g., ellipsoids, rods, fibers) have been randomly dispersed. The rigidity of the dispersed inclusions is considerably larger than the one of the matrix. Then let us submit this composite material to a strain (whatever is the mode of deformation). It is commonsense that stress singularities will developed in the matrix neighboring regions of the particles. To calculate the overall nonuniform stress field of the composite would obviously be a formidable task, however not really necessary because the problem can be tackled either in terms of average stress and strain, or by considering the local situation and extending it to the whole composite. Such considerations are the essence of socalled micromechanical approaches. There are a many excellent reviews3,4 and a few textbooks5–7 dealing with micromechanic models whose objective is to predict the average elastic properties of composite materials. The following ­section will be limited to a short description of the most easy-to-handle models (in the author’s opinion and experience), leaving purposely aside those approaches that need extensive numerical simulation efforts to be applied. Nearly all micromechanics models for short-fibers filled systems consider the same basic assumptions: • Both the fibers and the matrix are linearly elastic; the matrix is isotropic and the fibers have constant properties along their length. • The fibers are axisymmetric with a narrow distribution of shape and size, so that they can be idealized as rods, essentially characterized by an (average) aspect ratio, i.e., the length-to-diameter ratio L/D. • There is perfect bonding between the fibers and the matrix, and it remains so during deformation of the composite. No such effects as interfacial slip, fiber-matrix decohesion or matrix microcracking are considered. • Fibers concentration is finite but not so large to have direct contact between them. In short-fibers composites, loads are transferred from the matrix to the fibers in a zone near the fiber end, so that how the stress is distributed around a fiber is an important aspect. A basic approach to this problem is the classical

345

Polymers and Short Fibers

“shear-lag analysis” by Rosen and Dow.8 Fiber-end geometry and adjacent fibers effects are ignored, so that the fiber stress is zero at the end and increases gradually as the load is transferred from the matrix to the fiber. The maximum reinforcing effect is then achieved when fibers are long enough for complete load transfer to occur. If both fibers and matrix are elastic bodies, there is a minimum fiber length Lmin for this optimal load transfer, i.e. Lmin = D



1 Efib 1 − Φ fib 2 Gmat Φ fib

(7.3)

where D is the fiber diameter, Efib and Gmat the fiber elastic and the matrix shear moduli respectively, and Φfib the volume fraction of fibers. The minimum fiber aspect ratio Lmin/D decreases thus slightly with the increasing fiber loading but is much depending on the fiber-to-matrix modulus ratio. As Efib/Gmat increases the minimum fiber aspect ratio becomes very large, for instance with Efib = 73 GPa and Gmat = 73 kPa, one gets Lmin/D = 786 with D = 0.5 μm (see Appendix 7.1 for a numerical illustration of Equation 7.3). But the shear lag analysis suffers in fact from a number of excessive hypotheses and, for instance, finite element analysis has proved that there is a strong interfacial stress concentration at fiber ends, which contributes of course to composite’s hysteresis. The demonstration provided by Equation 7.3 that there is a minimum fiber aspect ratio required for effective stress transfer remains however fully valid and is of importance with respect to the processing of fibers-filled composites. Indeed, fibers inevitably brake during processing operations, and the lower the fiber ratio, the lower the reinforcement. However, because the Efib/Gmat ratio is generally large, substantial reinforcement can still be obtained, even with reduced L/D ratio. 7.2.2  Halpin–Tsai Equations The Halpin–Tsai equations are a set of empirical relationships that have long been used to predict the mechanical properties of composite materials.9,10 They express the composite property with respect to those of the matrix and of the reinforcing material, their respective volume fraction and the filler geometry. Essentially, these equations give the (mechanical) property of a composite Pcpd in terms of the corresponding property of the matrix Pmat and the reinforcing phase Pfil through the following relationships:



Pcpd

 Pfil   P  − 1  1 + ζ µ Φ fil  mat = Pmat     with   µ =  1 − µ Φ fil   Pfil  + ζ  P  mat

(7.4)

The Halpin–Tsai equations were originally developed for continuous-fiber composites with respect to early self-consistent models for ideal systems

346

Filled Polymers

with perfectly aligned fibers. The mechanical properties that can typically be predicted using the above equations are the longitudinal and transverse tensile moduli, the shear modulus and the Poisson’s coefficient. Depending on the property and the geometry of the filler particles, different expressions must be used for ζ, as summarized in Table 7.3. The above equations are essentially empirical (i.e., the ζ term has a weak theoretical background) but their validity is confirmed by numerous experimental measurements, at least for moderate filler volume fractions. There were also found in excellent agreement with finite element calculations on idealized short fibers composites.4 For filler fractions larger than 0.4, Hewitt and de Malherbe11 have proposed to make the ζ term depending on the filler volume fraction by adding a component equal to 40 Φ10 (found by curve fitting). As demonstrated through calculation in Appendix 7.2, this extra term brings indeed negligible changes until Φ is larger than 0.4. Halpin and Kardos10 noted that ζ must lie between 0 and infinity. If ζ = 0, Equation 7.4 reduces to the harmonic mixing rule (Equation 7.2), and if ζ = ∞, the HalpinTsai form becomes the simple mixing rule (Equation 7.1). Choosing the appropriate value for ζ is in fact the most critical aspect in using the Halpai-Tsai equations, because the relationships quoted in Table 7.3 have been obtained by considering somewhat idealized systems, either with respect to the orientation of the filler (relevant for fibers, plates, and whiskers only) and/or with Table 7.3 Detailed Expressions for the ζ Parameter in Halpin–Tsai Equations Depending on Filler Particle’s Geometry Particle Geometry

Longitudinal Tensile Modulus E11

Spherical particle

ζ = 2

Oriented short fibers

ζ =2

Oriented plates

Oriented whiskers

Transverse Tensile Modulus E22

Shear Modulus G12

ζ = 2

ζ = 1

ζ = 2

ζ = 1

 L ζ=2  T

 L ζ=2  W

 L +W  ζ=  2 T 

 L ζ=2   D

ζ = 2

L ζ=   D

L D

1.73

1.73

Notes: • L and D are respectively, the length and diameter of fibers and whiskers. • The minimum effective length for short fibers can be calculated with Equation 7.3. • L, T and W are respectively, the length, thickness and width of plates. • Whatever the geometry of the filler particle, the Poisson’s ratio of the composite is νfil Φ fil + (1 − Φ fil ) νmat . • Whiskers are crystalline metallic tiny, filiform hairs that spontaneously grow from metallic surfaces. Whiskering is seen on elemental metals and on alloys. In the present context, whiskers must be seen as “curved” or “curled” filaments, in contrast with short fibers which are essentially rectilinear rods.

347

Polymers and Short Fibers

the implicit assumption that all particles are identical since their geometry is expressed through a set of single numbers. This latter assumption appears less critical than the actual fibers’ orientation. A numerical illustration of the mathematical virtues of the Halpin-Tsai equations is given in Appendix 7.2, as well as typical curves for the longitudinal, transverse and shear moduli, in the case of polypropylene-SGF composites. In short fibers-filled systems, the fiber aspect ratio plays a role only in the equation for the longitudinal modulus. Figure 7.4 shows typical curves as calculated with the Halpin-Tsai equations and a comparison of the calculated longitudinal and transverse moduli with (average) measured flexural modulus data on commercial PP-SGF systems. As can be seen, experimental data fall in between calculated E11 and E22 moduli likely because, in the tested commercial samples, fibers are oriented neither longitudinally nor transversally. For a complete random orientation of fibers, the overall modulus would be given by an appropriate fractional summation of longitudinal and transverse moduli, i.e.12 Erandom =



3 5 E11 + E22 8 8

(7.5)

It follows that, practically, the average fibers’ orientation could be expressed through an adjustable parameter X, such that Equation 7.5 can be rewritten as follows: Erandom (Φ) = X E11 (Φ) + (1 − X ) E22 (Φ)



20

Calculated longitudinal Calculated transverse Experimental

20

Maximum fiber packing : 0.9

Shear modulus, GPa

Tensile or flexural modulus, GPa

30

(7.6)

10

10

0

0

0.1 0.2 Fiber volume fraction

0.3

0

0

0.5 Fiber volume fraction

1

Figure 7.4 Typical curves as calculated with Halpin–Tsai equations and parameters for short glass fiber– polypropylene composites; parameters used in calculation: short glass fibers: Efib = 77.0 GPa, νfib = 0.20, L = 1 mm; D = 5 μm, i.e., a fiber aspect ratio of 200; polypropylene, Emat = 1.14 GPa, νmat = 0.43; experimental data are average flexural modulus data from various suppliers of PP-SGF composites. For both the fibers and the matrix, the shear moduli were calculated using the standard equation :G = E/[2(1 + ν)].

348

Filled Polymers

30 Longitudinal modulus Modulus, GPa

20 X=

4.5 8

10 Transverse modulus 0

0

0.1 0.2 Fiber volume fraction

0.3

Figure 7.5 Fitting experimental data with Halpin–Tsai equations and a parameter for average short fiber orientation; parameters used in calculation: short glass fibers: Efib = 77.0 GPa, νfib = 0.20, fiber aspect ratio of 200; polypropylene, Emat = 1.14 GPa, νmat = 0.43.

As shown in Figure 7.5, a better fit of experimental data is obtained with the appropriate value for the “orientation parameter” X. It is pretty obvious that fiber orientation distribution and how to control it during processing are amongst the most critical variables which affect the mechanical properties and hence the efficient use of short-fiber composites. Certain processing techniques allow some control of fiber orientation, essentially because important extensional flow fields and the associated flow anisotropy effects can be generated. It has been shown for instance that fiber aspect ratio, fiber-matrix interaction, processing tools geometry, shear rate, temperature, and fiber volume loading are among the very important parameters which control the final fiber orientation in processes such as extrusion, injection and transfer molding.13 It is difficult to assess the fiber orientation distribution. However E11 and E22 are easily calculated providing the matrix and fibers parameters and the fibers fraction are known. It follows that the average orientation of fibers in a given composite sample can be estimated from the measured modulus. Such an approach was indeed used by Leblanc et al.14 to study the mean fibers’ orientation in injection molded fatigue test specimens (ASTM D1708) with commercial short glass fiber composites with either polybutylene therephtalate (PBT) or copolyamide 6/6T (PA/PAT) as matrix material. Parallelepiped samples (3 × 5 × 10 mm) were cut out of the fatigue specimens, parallel to the cavity filling axis, and tensile Young modulus was measured (ASTM D638). Average short glass fibers dimensions were measured by electron microscopy and found to be L ≈ 300 µm and D ≈ 10 µm. Glass fibers were of the E-glass type (i.e., E11 = 73 GPa). The longitudinal and transverse moduli were

349

Polymers and Short Fibers

calculated using the Halpin–Tsai equations, then the orientation parameter was calculated using (see details in Appendix 7.2): X=



Emeas − E22 E11 − E22

(7.6a)

For randomly aligned fibers, X would be equal to 0.375. As seen in Table 7.4, all calculated X values are higher than 0.375 and clearly decrease with increasing fiber content. It follows that, as expected, fibers in the fatigue test specimens tend to be aligned along the longitudinal mold axis, but the higher the fiber content, the less effective the longitudinal flow induced orientation, probably due to fiber hindrance. Microphotographs of polished sections cut in the fatigue specimens essentially confirmed the general but not perfect orientation of fibers along the mold filling direction. Even for well aligned short fibers systems however, the Halpin–Tsai equations do not compare well with experimental results for large volume fractions and their theoretical maximum applicability limit is the maximum hexagonal packing of perfectly aligned fibers, i.e., π/(2 3 ) ≈ 0.907 . As shown in Figure 7.4, experimental data for current fiber loadings lie between the longitudinal and the transverse calculated tensile moduli. This corresponds of course to the fact that in real short fibers systems the alignment is neither perfect nor completely at random, as demonstrated above. For large fibers fraction systems (i.e., above 0.4–0.5) Nielsen and Lewis15,16 have suggested that the filler packing limit must be taken into consideration, as an upper-bound parameter, through the following modified form of the Halpin–Tsai equation, i.e.  1 + ζ µ Φ fil  Pcpd = Pmat   1 − F(Φ fil ) µ Φ fil 



(7.5)

Table 7.4 Calculating the Average Orientation from Measured Modulus and Halpin–Tsai Equations for PBT and PA/PAT Composites with Short Glass Fibers Material

ΦFiber

Modulus (GPa)

Orientation Parameter X

Virgin PBT PBT + 20% SGF PBT + 30% SGF PBT + 50% SGF Virgin PA/PAT PA/PAT + 25% SGF PA/PAT + 35% SGF PA/PAT + 50% SGF

– 0.1115 0.1765 0.3288 – 0.1308 0.1938 0.2962

2.600 6.475 8.445 14.265 3.200 8.431 10.771 14.930

– 0.606 0.587 0.576 – 0.697 0.657 0.636

350

Filled Polymers

with various explicit forms for the functional F(Φfil), for instance:



 1 − Φ max  F(Φ fil ) = 1 +   Φ fil 2  Φ max

(7.5a)

or

F(Φ fil ) =

1   Φ Φ 1 − exp  fil max  Φ fil − Φ max Φ fil 

   

(7.5b)

The functional F(Φfil) must be such that the product F(Φfil) × Φfil fulfills the following boundary conditions:



F(Φfil) × Φfil = 0   at   Φfil = 0,

d [ F(Φ fil ) × Φ fil ] = 1   at   Φfil = 0 d Φ fil

and   F(Φfil) × Φfil = 1   at   Φfil = Φmax.

The two first conditions are imposed by the fact that as the filler fraction goes to zero, one must recover the Einstein equation. Equations 7.5a and 7.5b were selected by Lewis and Nielsen as amongst the simplest ones fulfilling such conditions, but otherwise have no theoretical justification. Equation 7.5a is the equation of a straight line and Equation 7.5b has a maximum above 0.75 × Φfil. When multiplied by Φfil, both functions nearly superimpose up to Φfil = 0.12 and exhibit minor differences above this level. In fact, as shown in Appendix 7.3 (Section A7.3.2), such modifications of the Halpin-Tsai equations do not bring much changes in the fiber fraction range of practical interest (0–0.25). For short-fiber composites, it is clear that the above equations are expected to give good predictions only if the fibers are somewhat aligned. As the fiber aspect ratio L/D increases, the longitudinal modulus of the composite increases and tends to the limit Efib × Φfib of long-fiber composites. As we have seen with Equation 7.3, the fiber aspect ratio must be larger with higher Efib/Gmat ratio. It follows that reinforcement with short fibers is more efficient for relatively small Efib/Gmat ratio. For a given polymer matrix, with respect to the range of practical fiber aspect ratio (e.g., 200–300), at loadings that are compatible with processing constrains and requirements, the longitudinal tensile modulus of composites reach an asymptotic limit, and using short fibers with larger tensile modulus brings only minor changes. The transverse and the shear moduli however are independent of the fiber aspect ratio, and if the fiber modulus is large compared to the matrix modulus, Ey and G

351

Polymers and Short Fibers

become also independent of the fiber modulus, in which case, the following simple relationships can be considered:

 1 + 2 Φ fib  Ey = Emat    1 − Φ fib 

(7.6a)



 1 + Φ fib  Gcpd = Gmat    1 − Φ fib 

(7.6b)

7.2.3  Mori–Tanaka’s Averaging Hypothesis and Derived Models Mori and Tanaka17 considered likely interactions between inhomogeneities in complex systems (explicitly metallic alloys in their paper) and introduced the concept of an average field that would include the inhomogeneities and the surrounding matrix. Essentially the Mori–Tanaka hypothesis can be stated as follows: within a heterogeneous concentrated composite submitted to a strain, each particle “feels” a far-field strain that is equal to the average strain in the matrix. Alternatively this average stress/strain concept could be defined as follows: when submitted to a given applied stress, the average stress in the matrix differ from the applied one due to the presence of inclusions; however, the (volume) average of the affected parts in both the inclusions and the surrounding matrix must vanish in order to satisfy the equilibrium condition. As described in their paper, such a hypothesis allows to calculate the average internal stress (upon strain) in a matrix containing inclusions. The approach is not easy and consists of complex mathematical manipulations of field variables with respect to the concept of equivalent inclusions. The Mori and Tanaka hypothesis was however considered by several authors, namely Benveniste18 who provided a much simpler description of the average strain concept. Benveniste’s explanation prompted several authors to formulate simple expressions for moduli of materials, when some simplifying assumptions can be made. Otherwise numerical methods are needed to use the so-called Mori–Tanaka model.4 Tandon and Weng19 have derived a complete set of explicit relationships for moduli of a composite model in which randomly distributed ellipsoidal particles (i.e., short-fiber like) are unidirectionally aligned. The Mori– Tanaka’s average strain concept is also used as the transformation tensor described by Eshelby20 when he solved the problem of an ellipsoidal inclusion in an elastic field. The Eshelby’s transformation tensor is a 4th order tensor whose components depend only the aspect ratio of the inclusion and the elastic moduli of the matrix. Some of the equations obtained by Tandon and Weng must however be solved iteratively and handling them implies calculating at first quite a impressive number of “constants,” in fact various

352

Filled Polymers

combinations of matrix and inclusions moduli and Poisson’s ratio, and the appropriate nonzero components of the Eshelby’s tensor with respect to the case considered. Explicit formulas were given in the Tandom and Weng’s paper and are reproduced in Appendix 7.4 (Sections A7.4.1 through A7.4.3), with a numerical illustration of their virtues in predicting the tensile and the shear moduli of short-fibers filled composites. With respect to fiber volume fraction, all the calculated moduli fall between the lower and upper bounds (Equations 7.1 and 7.2) and are relatively insensitive to the fiber aspect ratio, except the longitudinal (tensile) modulus, (see Figure 7.6). As the aspect ratio decreases, the modeled E11 becomes close to the lower bound prediction, but when the L/D ratio is larger than 100, no difference is seen between the Mori–Tanaka’s approach and the upper bound prediction. As shown in Figure 7.6, one can hardly see a difference in the modeled shear modulus when the fiber aspect ratio varies from two to several hundreds. As the predicted shear modulus remains close to the lower bound whatever the fiber volume fraction, it is quite clear that the average strain approach is far to correspond to common experimental observations which reveal a steeper variation of the shear modulus when the short fiber content increases. If the fiber aspect ratio is set to one (i.e., spherical particles), one notes that there is a mathematical singularity in the Eshelby tensor and the Tandon and Weng explicit equations for short fibers cannot be used. However, in such a case, there are some simplifications in the tensor and particularly simple

40

L = 200 D 50

Shear modulus, GPa

Tensile modulus, GPa

100

20 2

0 0

0.5 Fiber volume fraction Φ

1

Emat = 1.19 GPa νmat = 0.35 Gmat = 0.44 GPa Efib = 77.0 GPa

νfib = 0.20

Gfib = 32.1 GPa

30

20

10

0 0

L = 200, 20, 2 D 0.5 Fiber volume fraction Φ Upper bound modulus Lower bound modulus

Figure 7.6 Mori-Tanaka’s average strain approach; sensitivity to fiber aspect ratio.

1

Polymers and Short Fibers

353

expressions are obtained (see Appendix 7.4, Section A7.4.9), for instance for the shear modulus:



Gcpd

    Gfill − Gmat )  Φ fill  (  = Gm 1 +   ( 1 − Φ fill )   2 ( 4 − 5 νmat )    G G + G − ( ) mat mat   15 (1 − ν )  fill    mat

(7.7)

where Gcpd, Gmat and Gfill are the moduli of the composite, the matrix and the filler respectively, and νmat the Poisson’s ratio of the matrix. In the particular case of polymer films containing “liquid fillers,” Gao and Tsou21 have extended the Tandon and Weng’s work to derive easy-to-handle expressions for the moduli and the Poisson’s ratio. Through a comparison with finite element calculations on idealized arrays of fibers, Tucker and Liang4 have evaluated models derived from the Mori–Tanaka’s average field concept, the (easier-to-handle) Halpin–Tsai equations and other models. They found that the Halpin–Tsai equations give reasonable estimation for stiffness but the best predictions (of finite element calculations) were obtained with the Mori–Tanaka model. The evaluation by Tucker and Liang is however somewhat artificial since they considered (for their Finite Element (FE) calculations) an ideal short fibers composite, with a model array (i.e., arrangement) of the fibers. A more pertinent evaluation is made with respect to measured data (see Appendix 7.4, Section A7.4.8 for details). Figure 7.7 compares the prediction of the Mori–Tanaka’s average strain approach with experimental data on commercial short glass fibers-filled polypropylene composites. The following data were used in calculating the model curves: short glass fibers, Efib = 77.0 GPa, νfib = 0.20, L = 1 mm; D = 50 μm, i.e., a fiber aspect ratio of 20; polypropylene, Emat = 1.14 GPa, νmat = 0.43. For both the fibers and the matrix, the shear modulus was calculated using the standard equation: G = E/2(1 + ν). Essentially the E11 longitudinal modulus was calculated, to be compared with both the flexural and tensile moduli (which are generally equal for isotropic systems and appear to be also equivalent for filled thermoplastics; see Figure 7.8). For the sake of comparison, the much simpler modified Guth and Golf equation, i.e., Ecpd = Emat (1 + 2.5f Φ + 14.1f 2 Φ2) with f an (empirical) anisometry factor, was used to draw other model curves. As can be seen the Mori–Tanaka’s approach only allows experimental data to be met at low fiber volume fraction. Above Φ ≈ 0.10, experimental moduli are systematically higher than the model and the simpler modified Guth–Gold equation with f = 4.2 gives a better fit. 7.2.4  Shear Lag Models So-called shear lag models are really micromechanical by nature as they consist in considering the behavior of the matrix near the fiber surface. Such

354

Filled Polymers

20

Tensile modulus, GPa

Flexural modulus, GPa

20

10

0

10

0

0.25

0.05 0.15 0.10 0.20 Fiber volume fraction Φ

0

0

0.05 0.10 0.15 0.20 Fiber volume fraction Φ

Average data, various suppliers Single supplier's data

0.25

Mori–Tanaka's averaged strain Modified Guth & Gold

Figure 7.7 Comparing the Mori–Tanaka’s average strain model with experimental data on commercial short glass fiber filled polypropylene composites.

Flexural modulus, GPa

20

5

18

4

16

3 2

14

1

12

0

8

PP + CaCO3

[D.W. Van Krevelen Properties of Polymers, 3rd Ed., Elsevier (2003)] [Table 6.7]

6

PP + Talc

[Table 6.8]

PP + Mica

[Table 6.9]

4

PP + SGF

[Table 7.6]

PA6 + SGF

[Table 7.7]

PA11 + SGF

[Table 7.9]

10

0

1

2

2 0

0

5

3

4

5

Pure polymers

10 15 Tensile modulus, GPa

20

Figure 7.8 Comparing flexural and tensile moduli for thermoplastic polymer systems.

355

Polymers and Short Fibers

models are algebraically simple and have an obvious ­physical ­significance. The shear lag analysis made by Cox22 focuses on a single fiber of length L and radius R, embedded in a concentric cylindrical shell of matrix with radius r; see Figure 7.9 (note: the matrix shell thickness is thus r–R). Essentially, the model intends to predict the longitudinal (tensile) modulus E11, so only the axial stress and strain are of interest. Poisson’s ratios effects are neglected and it is considered that when the composite is submitted to a stress, there is a difference in displacement between the fiber surface and the outer surface of the cylindrical matrix shell. The key hypothesis of the shear lag model is that the axial shear stress at the fiber surface is proportional to this difference. With the boundary condition of zero stress at fiber’s ends, an expression for the average fiber stress is obtained, which combined with a corresponding average fiber strain through a so-called “efficiency factor,” allows to derive an equation for the longitudinal modulus. The model is then completed by combining the average fiber stress with an average matrix stress in order to obtain a modified rule of mixture for the axial modulus, i.e. E11 = nL Φ fib Efib + (1 − Φ fib ) Emat



(7.8)

 βL    tanh  2   where nL is the “efficiency factor,” i.e., nL = 1 −  βL     2 β=



2 Gmat  r 2 R Efib ln    R 2

3

Fiber

Figure 7.9 Shear lag model.

Mesophase thickness micron

150

r

(7.8b)

1

Matrix shell

R

(7.8a)

Hexagonal fiber packing π 2

100

L

3

50 0 0

0.2

0.4 0.6 0.8 Fiber volume fraction

1

356

Filled Polymers

Efib and Emat are the tensile moduli of the fiber and the matrix respectively, Gmat the shear modulus of the matrix, L and R the length and radius of the fiber. In using the above equations for a real composites, i.e., many fibers (all perfectly aligned) with average length L and radius R, the remaining problem is to have a reasonable estimation for the matrix shell radius r. It is quite obvious that the shell thickness must decrease with increasing fiber volume fraction (see the inset in Figure 7.6). Several choices are possible for r, all depending on the fibers spatial arrangement, and are conveniently expressed through the following equality: r=R



Kr Φ fib

(7.8c)

Kr is a constant (in fact the packing ratio of the fibers in the composite) that depends on the assumption made about the fibers packing mode. Cox considered an hexagonal packing and chose r as the distance between the centers of two neighbor fibers (see Table 7.5). This choice is not really realistic however since, for touching fibers, r would be equal to 2 R with no more matrix layer between the fibers. Another apparently easy choice is Kr = 1 so that the fiber and its matrix shell have together the same volume fraction as the fibers in the composite.23,24 However we do not consider this choice as realistic since Kr cannot be larger than π/2 3 ≈ 0.907 , i.e., the maximum hexagonal packing fraction for fibers of equal diameter. Other authors have considered a square array of fibers, with r as half the distance between centers of nearest fibers. In such as case, Kr is close to experimental values for the maximum fibers ­packing fraction. Except the Cox choice, all the others give nearly identical

Table 7.5 Shear Lag Model; Values of Kr Fibers Arrangement r

r

Matrix Shell Radius

Kr

Cox

2π = 3.628 3

Hexagonal

π = 0.907 2 3

Square

π = 0.785 4

r

357

Polymers and Short Fibers

smooth decreases of the matrix shell thickness with larger fibers fraction. It is clear however that each assumption gives a somewhat different dependence of the efficiency factor Φfib and hence (slightly) different values for the predicted modulus. Larger values of Kr lead to lower values of E11. Whatever is the arrangement of the fibers, the mathematical form of Equation 7.8c dictates a smooth decrease of the shell thickness (r–R) as the fiber content increases, with a significant difference between the Cox arrangement and the other arrangements (hexagonal and cube), owing to the particular choice for Kr (see upper left graph in Figure 7.6). The key factor in the shear lag model remains however the fiber aspect ratio, as easily demonstrated when comparing calculated E11 values with experimental data on short fibers-filled composites (Figure 7.10; see calculation details in Appendix 7.5). Depending on the fibers arrangement selected, an adequate choice for the fiber ratio allows measured data to be well fitted, up to fiber fractions of around 0.18, essentially because the model exhibits a slight curvature with respect to Φfib. For commercial PP–SGF composites, the fibers ratio is likely to be between 20 (hexagonal and cube arrangement) and 30 (Cox arrangement), if one considers the shear lag model with confidence.

Kr =

0 0

2π

Longitudinal/flexural modulus, GPa

π 2

3

Kr =

Kr =

π 2

10

0 0

50 30 20 10

0.1 0.2 Fiber volume fraction

0.3

L/D 50 30 20 10

0 0

1

L/D 3

2π 3

10

π 4

0.5 Fiber volume fraction

30

20

20

3 Kr =

200

Kr =

Longitudinal/flexural modulus, GPa

400

30

0.1 0.2 Fiber volume fraction

30 Longitudinal/flexural modulus, GPa

Matrix shell thickness, micron

600

20

Kr = π 4

10

0 0

0.3

L/D 50 30 20 10

0.1 0.2 Fiber volume fraction

0.3

Figure 7.10 Typical features of the shear lag model; effect of fiber aspect ratio on calculated modulus, compared with experimental data; material parameters: Short Glass Fibers, Efib = 77.0 GPa, νfib = 0.20, L = 1 mm; D = variable; polypropylene, Emat = 1.14 GPa, νmat = 0.43.

358

Filled Polymers

When applying the above model to experimental data, one considers implicitly that the L/D ratio is an average value, whatever is the actual distribution of fiber lengths. Would this distribution be known, one could obviously calculate a distribution of efficiency factors. Numerical handling of the above equations with respect to fictitious fiber length distributions demonstrates very little benefit in this approach with respect to a simple L/D ratio average. A more significant concern is that, in the shear lag model, the fibers are considered as perfectly aligned. In practice this is never the case and Equation 7.8 could be rewritten as:

E11 = O f nL Φ fib Efib + (1 − Φ fib ) Emat

(7.9)

where Of is a coefficient that depends of the distribution of fibers orientation. For a random 2D distribution of fibers orientation, Of = 1/3, and for a random 3D distribution, Of = 1/6. It is fairly obvious that, with all the other parameters being kept constant, a random orientation of the fibers gives a lower longitudinal modulus. Through numerical simulation, some authors25 have however concluded that shear-lag type models do not apply to random fiber networks, because most of the axial stress is transferred directly from fiber to fiber rather than through an intermediate shear-loaded matrix layer. It is worth noting that the shear lag model is in complete contradiction with the average stress and strain field concepts of Mori–Tanaka’s and followers, but is in line with the concept of mesophase introduced by Theocaris26 (see Chapter 5, Section 5.1.9).

7.3  Thermoplastics and Short Glass Fibers Lightweight reinforced thermoplastics materials with great strength and stiffness are needed in a number of so-called technical applications. Therefore short glass fibers are used to stiffen thermoplastics, for instance polypropylene, polyamides and also more technical polymers such as polybutylene terephthalate. Thermoplastic processors do not compound themselves their materials and consequently there is quite a large choice of ready-to-be­processed SGF reinforced thermoplastic composites. Table 7.6 gives the average properties of typical commercial PP–SGF composites, as compiled from manufacturers’ data sheets (when available). As can be seen, the hardness, the tensile and flexure properties and the thermal resistance (HDT, Vicat) generally increase with higher fiber loading, with however a large scatter that likely reflects the various grades of polypropylene used, as well as (unknown) differences in compounding processes. E-glass fibers are the most common type but the actual fiber dimensions are

3

Calculated density (g/cm ) Volume fraction from calculated density: Volume Fract from given density: Property Unit Density g/cm3 Melt flow g/10 min Hardness, rockwell R Hardness, shore D Tensile strength, ultimate MPa Tensile strength, yield MPa Elongation at break % Elongation at yield % Tensile modulus GPa Flexural modulus GPa Flexural yield strength MPa Izod impact, unnotched J/cm Charpy impact, notched J/cm2 Gardner impact J Izod impact, notched J/cm Deflection temperature at 0.46 MPa, °C Deflection temperature at 1.8 MPa, °C Vicat softening point °C

Short Glass Fibers (%wt)

0.90 ± 0.02 28 ± 18 63 ± 37 60 ± 18 26.6 ± 7.0 26.2 ± 8.2 169 ± 83 12 ± 11 1.19 ± 0.35 1.15 ± 0.42 30 ± 11 9.97 ± 8.56 – 12 ± 11 2.34 ± 3.83 90 ± 25 52 ± 6 108 ± 50

0

0.95 ± 0.04 4.6 ± 3.9 84 ± 11 51 ± 26 34.3 ± 12.7 32.0 ± 15.6 11.7 ± 11.8 3.6 ± 0.8 2.46 ± 0.67 1.87 ± 0.6 49 ± 15 4.01 ± 1.17 0.23 0.68 ± 0.48 0.44 ± 0.21 117 ± 45 86 ± 37 104 ± 12

0.962 0.0339 0.0335

10

Commercial PP–SGF Composites; Average Suppliers’ Data

Table 7.6

1.15 ± 0.212 9.3 ± 8.1 110 73 49.9 ± 8.3 60 3.3 ± 1.1 5 3.48 ± 0.05 3.19 ± 0.61 79 ± 14 4.54 ± 1.13 0.23 4.51 1.31 ± 1.17 174 ± 40 129 ± 30 110 ± 42

0.989 0.0500 0.0581

15

1.11 ± 0.24 11.8 ± 15.0 95 ± 20 73 ± 3 22.9 ± 16.8 57.6 ± 29.7 4.6 ± 2.8 5.7 ± 4.7 4.66 ± 2.32 3.34 ± 2 71 ± 43 3.76 ± 2.65 0.53 1.04 ± 0.94 1.59 ± 1.96 139 ± 29 120 ± 42 132 ± 30

1.015 0.0656 0.0717

20

1.21 ± 0.2 8.5 ± 11.3 98 ± 17 75 ± 1 69.4 ± 55.6 73.1 ± 34.8 5.5 ± 7.1 4.5 ± 3.1 6.02 ± 3.02 5.36 ± 3.42 119 ± 81 4.13 ± 3.42 1.27 5.00 ± 7.46 1.39 ± 1.5 150 ± 17 123 ± 42 131 ± 35

1.064 0.0952 0.1082

30

1.27 ± 0.2 5.1 ± 4.9 101 ± 17 79 ± 6 98.3 ± 52.7 87.8 ± 38.4 4 ± 4.1 3.8 ± 2.9 8.06 ± 3.12 7.1 ± 2.8 156 ± 90 4.61 ± 3.47 2.04 0.57 ± 0.52 2.75 ± 3.00 156 ± 9 142 ± 25 134 ± 4

1.111 0.123 0.1406

40

1.33 – – – 63.0 – 1 – 11.72 8.90 98 1.87 – – 0.53 – 121 –

1.155 0.1492 0.1718

50

– – 3.78 ± 1.43 – 318 –

1.49 ± 0.04 – – – – 164 ± 34 1.4 ± 0.5 – 19.00 ± 6.05 18.07 ± 3.61 328 ± 95

1.196 0.1738 0.2166

60

Polymers and Short Fibers 359

360

Filled Polymers

18 Flexural modulus, GPa

16

20

PP + SGF, various suppliers

18

PP + SGF , single supplier

Tensile modulus , GPa

20

14 12 10 8 6 4

14 12 10 8 6 4 2

2 0 0.00

16

0.05 0.10 0.15 Filler volume fraction

0.20

0 0.00

0.05 0.10 0.15 Filler volume fraction

0.20

Figure 7.11 Mechanical properties of commercial polypropylene–short glass fibers composites; squares are averaged suppliers data; the vertical bars indicate the standard deviations; shaded diamonds are data from one single manufacturer; the curves were calculated with the modified Guth and Gold equation, using an anisometry factor equal to 4.5.

not disclosed (if known) by the suppliers. The trends observed are therefore quite remarkable and very significant where tensile and flexural moduli are concerned (see Figure 7.11). A simple modified Guth and Gold equation with an anisometry factor equal to 4.5 fits the data well. Polyamides are particularly versatile polymers, most widely used as engineering thermoplastic materials. They are melt processable and offer a unique combination of high mechanical strength, low wear and abrasion properties together with good chemical resistance. Their semicrystalline nature results in an excellent combination of properties, which can be further reinforced when preparing composites with short glass fibers. Various types of polyamide are available commercially today including polyamide 6 (polycaprolactame), polyamide 66 (polyhexamethylene diamine adipamide), polyamide 11 (polyaminoundecanoic acid) and aromatic polyamides, which can be modified in a variety of ways, through compounding with various additives including flame retardants, plasticisers, stabilizers, lubricants, nucleants, mineral fillers, and short glass fibers. The filling of aliphatic polyamides by short glass fibers is widely used to improve their physicomechanical and antifrictional properties. Properly dispersed, glass fibers significantly reinforce the polymer matrix since they form an internal structure which acts as a load-carrying frame and supports the main part of the load applied to the sample; therefore higher stiffness, strength, and heat distortion temperature are obtained, as illustrated by average properties of typical commercial polyamide–SGF composites. Tables 7.7 through 7.9 were compiled from available manufacturers’ data sheets for composites based on polyamide 6, 66, and 11, respectively.

361

Polymers and Short Fibers

10 8 6 4 2

PA6 + SGF PA66 +SGF

0 0.00 0.05 0.10 0.15 0.20 0.25 Fibers volume fraction

Charpy impact, notched, J/cm2

Charpy impact, unnotched, J/cm2

12

2.4 2.2 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.00 0.05 0.10 0.15 0.20 0.25 Fibers volume fraction

Figure 7.12 Impact resistance of commercial short glass fibers-filled polyamides.

Despite the inevitable scatter due to the diversity of the sources, clear trends can be seen: mechanical (tensile and flexural) properties, impact resistance as well as heat distortion temperature increase with higher fibers content. The improvement in impact resistance seems however to be maximum at around 0.18–0.20 fibers fraction, and damaged samples (i.e., notched) are considerably more fragile than intact ones Figure 7.12). Figures 7.12 through 7.14 are plots of flexural and tensile moduli data for commercial SGF-filled composites based on polyamides 6, 66, and 11, respectively. It is interesting to compare such data with the predictions of micromechanical models. With respect to the discussion in Section 7.2, the Halpin–Tsai model is the most interesting one and the easiest to implement because all parameters are relatively accessible. Suppliers do not give information about the type of glass fibers they used but E-glass type, whose tensile and flexural moduli are in the 75 GPa range, is the most common one. The overall fiber orientation in the samples, whose (averaged) properties are compiled in Tables 7.7 through 7.9, is obviously unknown but because injection molding is a standard practice in preparing test samples, near longitudinal orientation is a reasonable hypothesis. This commands to use Equation 7.4 with ξ = 2(L/D). The fiber aspect ratio is however another missing information but would the initial L/D ratio be known, compounding and processing operations inevitably break the fibers, in a random and unknown manner, as documented by some authors.27–29 It follows that the fiber aspect ratio has to be considered as a fitting parameter whose value is granted by the best superposition of model curves with experimental data. Curves in Figures 7.13 through 7.15 were consequently calculated with Equation 7.4, Efib = 77 GPa, either the tensile or flexural moduli of the polyamides given in Tables  7.7 through 7.9 and the best fit for L/D. The upper bound composite moduli were also calculated with Equation 7.1 for comparison.

362

Filled Polymers

PA 6 + SGF

Halpin–Tsai (L/D=50) 20

18

18

16

16

Tensile modulus, GPa

22

20 Flexural modulus, GPa

22

14

Upper bound

14 12

12

10

10 8 6

8 6

4

4

2

2

0 0.00 0.05 0.10 0.15 0.20 0.25 Fiber volume fraction

0 0.00 0.05 0.10 0.15 0.20 0.25 Fiber volume fraction

Figure 7.13 Flexural and tensile moduli of commercial SGF-filled polyamide 6 composites compared with the predictions of the Halpai–Tsai model.

PA66 + SGF

Halpin–Tsai (L/D=20)

18

18

16

16

14 12 10 8 6

Tensile modulus, GPa

20

Flexural modulus, GPa

20

Upper bound

14 12 10 8 6

4

4

2

2

0 0.00 0.05 0.10 0.15 0.20 0.25 Fiber volume fraction

0 0.00 0.05 0.10 0.15 0.20 0.25 Fiber volume fraction

Figure 7.14 Flexural and tensile moduli of commercial SGF-filled polyamide 66 composites compared with the predictions of the Halpai–Tsai model.

363

Polymers and Short Fibers

Halpin–Tsai (L/D=20)

20

20

18

18

16

16

14 12 10 8 6

Tensile modulus, GPa

Flexural modulus, GPa

PA11 + SGF

Upper bound

14 12 10 8 6

4

4

2

2

0 0.00 0.05 0.10 0.15 0.20 0.25 Fiber volume fraction

0 0.00 0.05 0.10 0.15 0.20 0.25 Fiber volume fraction

Figure 7.15 Flexural and tensile moduli of commercial SGF-filled polyamide 11 composites compared with the predictions of the Halpai–Tsai model.

As can be seen, experimental data are always below, but not far from, the upper bound curves. This justifies the choice of the Halpin–Tsai equation to calculate the longitudinal tensile modulus E11 for short fiber-filled composites. Depending on the polyamide matrix, different values for the (average) fibers aspect ratio must be used however for the best superposition of calculated curves on experimental data, i.e., L/D = 50, 20 and 20 for PA6, PA66 and PA11, respectively. Assuming that fibers had initially the same aspect ratio before compounding, this would suggest that more breakage occurs in PA 66 and 11 than in PA 6. Filling polyamides with short glass fibers appears to be a convenient manner to tailor a number of interesting properties for engineering applications. There are however some substantial disadvantages. For instance SGF-filled polyamides exhibit quite complex rheological properties, with strong flowinduced orientation effects and the associated important stress-overshoot phenomena in processes, such as injection molding.30 Obviously glass-filled polyamides have a higher abrasivity not only in the molten state which hinders their processing and gives rise to increased wear of processing equipment, but also in the solid state which produces severer wear of adjacent metal parts in contact with SGF-polyamide joints of friction. Such effects are common to most SGF-filled polymers but likely exacerbated in the case of polyamides by the very low viscosity in the molten state. Residual stresses in molded SGF–polyamide composites are generally higher and whilst a higher impact resistance is imparted by the fibers (up to a certain level however), a

Calculated density (g/cm3) Volume fraction from calculated density: Volume fraction from given density: Property Unit Density g/cm3 Tensile MPa strength, ultimate Tensile MPa strength, yield Elongation at % break Modulus of GPa elasticity Flexural GPa modulus Flexural yield MPa strength

Short Glass Fibers (%wt)

0.0606

0.0622

1.23 116 ± 7

116 ± 7

3.3 ± 0.5

5.9 ± 0.2

5.1 ± 0.7

163 ± 6

0.0000

0.0000

1.11 69 ± 21

69 ± 21

–

2.5 ± 0.6

2.4 ± 0.6

100 ± 21

15

1.20

1.11

0

20

4.5 7.5 – –

6.90

6.80

–

140

140

4

1.27 140

0.082

0.0792

1.23

1.24 140

0.0733

0.0719

1.22

18

Commercial PA6-SGF Composites; Average Suppliers’ Data

Table 7.7

25

215

7.4

8.3 ± 0.3

3.4 ± 0.2

155 ± 7

1.31 155 ± 7

0.1016

0.0971

1.25

30

235

8.7 ± 0.4

9.5 ± 0.1

3.6 ± 0.2

180 ± 9

1.35 180 ± 9

0.1208

0.1143

1.28

35

255

10.2 ± 0.4

10.9 ± 0.4

3.1 ± 0.4

193 ± 5

1.39 192 ± 5

0.1397

0.1308

1.30

40

275

10.9 ± 0.3

12.8 ± 0.3

3

202 ± 6

1.45 202 ± 6

0.1606

0.1468

1.33

45

330 ± 32

12.5

14.3 ± 0.4

3

210 ± 7

1.51 210 ± 7

0.1816

0.1621

1.35

50

310

13.9 ± 0.6

16.0 ± 0.9

2.5 ± 0.4

226 ± 5

1.56 226 ± 5

0.2016

0.177

1.37

60

340 ± 17

16.5

20

2.2

235

1.66 235

0.2413

0.2052

1.41

364 Filled Polymers

Charpy impact, notched Charpy impact, unnotched Izod impact, notched (ISO) HDT at 0.46 MPa HDT at 1.8 MPa Vicat softening point

–

73 ± 9

108 ± 50

°C

°C

°C

104 ± 12

191 ± 8

215 195

205

215

–

8 ± 0

134 ± 4

211 ± 6

220

14

9.2 ± 0.4

–

210

220

15

9.7 ± 0.56

–

210

220

–

–

210

220

–

9.67 ± 0.29 10

–

213 ± 3

220

17

–

210

220

–

9.68 ± 0.83 9

1.15 ± 0.21 1.40 ± 0.10 1.60 ± 0.07 1.77 ± 0.06 1.95 ± 0.07 1.73 ± 0.26 1.5

110 ± 42 132 ± 30 131 ± 35

180

–

215

–

8

5

6.8 ± 2

kJ/m2

–

–

J/cm2

1.5

3.68 ± 0.79 5.5

0.85 ± 0.23 0.64 ± 0.08 0.7

J/cm2

Polymers and Short Fibers 365

Calculated density (g/cm3) Volume fraction from calculated density: Volume fraction from given density: Property Unit Density g/cm3 Tensile strength, ultimate MPa Tensile strength, yield MPa Elongation at break % Elongation at yield % Modulus of elasticity GPa Flexural modulus GPa Flexural yield strength MPa Charpy impact, notched J/cm2 Charpy impact, unnotched J/cm2 Izod impact, notched (ISO) kJ/m2 HDT at 1.8 MPa °C

Short Glass Fibers (%wt)

4.5 85

1.14 – 90 – – 3.1 3.0 125 0.4

1.14

0

1.23 – 100 – 4 5.5 – – 0.6 6.5 8 228

0.0548

0.0426

1.21 105 105 4 – 5.0 4.5 – 0.5 3.5 5 232

0.0543

0.0423

13 1.22

1.20

10

Commercial PA66-SGF Composites; Average Suppliers’ Data

Table 7.8 15

1.24 125 125 4 – 6.2 5.3 – 0.7 4.1 6 245

0.0627

0.0622

1.23

20

1.29 145 – 4 – 7.4 6.4 – 0.8 4.8 7 250

0.0833

0.0812

1.26

25

1.32 170 – 3 – 8.4 7.3 280 0.9 5.4 8.3 255

0.1023

0.0995

1.28

30

1.36 190 – 3 – 9.7 9.1 280 1 8 10 253

0.1216

0.1171

1.31

33

1.39 130 – 4 – 8.5 7.8 – 1.4 8 15 248

0.1337

0.1272

1.32

35

1.41 210 – 3 – 11.7 9.5 – 1.4 9.5 13 255

0.1417

0.1340

1.33

50

1.57 240 – 2 – 16.2 13.5 – 1.6 9.5 14.5 255

0.2028

0.1810

1.40

366 Filled Polymers

Calculated density Volume fraction from calculated density Volume fraction from given density Property Unit Density g/cm3 Tensile strength, ultimate MPa Elongation at break % Tensile modulus GPa Flexural modulus GPa Flexural yield strength MPa Izod impact, unnotched J/cm Izod impact, notched J/cm HDT, at 0.46 MPa °C HDT, at 1.8 MPa °C

Short Glass Fibers (%wt)

1.11 82.7 4.0 2.76 2.76 96.5 5.87 0.747 163 154

1.06 38.0 50.0 1.30 1.90 –

– 135 47

1.12 0.0395 0.0391

10

1.06 0.0000 0.0000

0

Commercial PA11–SGF Composites; Suppliers’ Data

Table 7.9

1.13 68.9 4.5 3.79 3.03 96.5 5.34 0.534 178 170

1.15 0.0580 0.0571

15

1.50 110.0 5.0 6.69 5.52 117.0 6.41 1.070 179 174

1.17 0.0759 0.0969

20

1.22 110.0 5.0 7.69 – – – – – –

1.19 0.0863 0.0884

23

1.26 89.6 5.3 7.45 5.52 131.0 7.47 – – 182

1.23 0.1097 0.1127

30

1.35 82.7 3.5 8.27 6.89 124.0 4.54 1.120 188 182

1.27 0.1412 0.1495

40

– – –

1.42 146.0 4.0 12.40 – –

1.29 0.1501 0.1655

43

Polymers and Short Fibers 367

368

Filled Polymers

lower toughness is observed in impacted regions due to damages arising around the fibers in the matrix . These disadvantages can be somewhat compensated by the introduction of various modifying ingredients into glassfilled polyamides.

7.4 Typical Rheological Aspect of Short Fiber-filled Thermoplastic Melts As we have seen, short glass fiber-thermoplastic polymers offer improved stiffness, strength, and heat distortion temperature with respect to the unfilled polymer. But the overall properties of the fabricated parts strongly depend on the interrelation between macroscopically observable rheological properties, the local fiber orientation in the processing fields, and the resulting mechanical and thermal properties in the solid state. As clearly demonstrated by the mechanical models discussed above, the optimum reinforcement is only obtained when the fibers are properly oriented. How fibers orient themselves during flow is a key factor, obviously depending of the matrix flow properties, its shear and extensional viscosities and its shear thinning character, and the complex flow fields of the final processing operations (i.e., before the cooling down and the associated recrystallization step, if any), notwithstanding of course the elasticity of the fluid which surely has an influence on the orientation of the fibers. In contrast with the industrial importance of short fiber-filled thermoplastics, there is a rather limited scientific literature on the rheological properties and behavior of such systems and only a few key rheological experiments were performed with model systems. Amongst the likely reasons, there are the tremendous experimental difficulties in correctly assessing the flow properties of molten short fiber–polymer composites, due to a number of singular effects, e.g., nonlinear Bagley plots in capillary rheometry,31 nonrepeatability of steady shear experiments with cone-and-plate, rheometers, 32 orientation of the fibers during rheometrical test, nonlinear effects, 33 etc., Around 30 years ago, Chan et al. reported data on short glass fiber filled polyethylene and polystyrene melts34,35 and Knutson et al.36 investigated the extrusion behavior of SGF-filled polycarbonate. The orientation of the fibers under various flow conditions was investigated by various authors, namely Laun,30 Crowson and Folkes,31,37 and S. Kenig,38 who proposed a qualitative model as well as an identification of the various flow mechanisms that likely contribute to fiber orientation during injection molding, i.e., spreading radial flow in the vicinity of the cavity gate, fountain flow in the advancing front, converging type flow in the melt front zone, and shear flow at some distance from the wall. As a result of these mechanisms a layered structure is formed, having distinct fiber orientations. Other authors

Polymers and Short Fibers

369

reached similar conclusions, for instance Vincent and Agassant39 with the injection molding of a SGF–polyamide composite in center gated discs and Larsen40 with the injection molding of a rectangular box using short fiber reinforced polypropylene. In steady slit extrusion of short fiber reinforced thermoplastic composites, the formation of a skin-core of fibers oriented in the thickness direction has been observed, with fibers axis along the flow direction and parallel to the walls in the skin region, irrespective of the entrance geometry. Different fiber orientation distributions in the core region are however obtained by using different entrance geometries.41 During processing, fibers move, tumble, and rotate with the flow of the polymer matrix, which inevitably changes their orientation. Providing, the loading is sufficiently low for no or minimum interactions to take place between neighboring fibers, it is easy to understand the basic difference in short fibers motion in either extensional or shear flows. In (simple) shear flow, an isolated fiber tumbles over and over as it follows the mainstream, as foreseen by Jeffrey.42 In extensional flow, the same fiber rotates until its long axis is aligned with the main extensional axis and then stays there.43–45 In addition, a fiber in a shear flow has a minimum effect on the flow, when its axis is approximately parallel to the streamlines, and therefore, whilst tumbling, tends to stay in this position. In contrast, a fiber in extensional flow field is permanently in the position for which it has the maximum effect on the flow. Such behavioral differences explain why dilute suspensions of rod like particles and fibers exhibit substantial deviations from the Trouton ratio of 3. It must be noted that most published studies have been performed on model systems, with for instance short fibers in either Newtonian fluids or viscoelastic polymer solutions. One presumes that the behavior of the fibers in molten polymers is similar. Generally, converging flow results in high fiber alignment along the flow direction, whereas diverging flow causes the fibers to align at 90° to the major flow direction. Simple shear flow produces a decrease in alignment parallel to the flow direction and the effect is more pronounced at low flow rates. The net overall fiber orientation in a given process therefore results from the particular combination of converging-diverging and shear flow fields prevailing in the final shaping steps, i.e., the die in extrusion or the gate and the cavity in injection molding, before the melt solidifies. In addition to flow induced orientation effects, that are particularly strong in extensional flow fields, another but less understood mechanism is associated with fiberto-fiber interaction. Indeed, in many practical applications, the fiber volume fraction is sufficiently high for fibers to flow in close proximity to each other. In such a case, various types of fiber interactions may come into play, for instance excluded-volume (due to fiber tumbling and/or rotation), and friction or mechanical interactions between fibers, in addition to usual hydrodynamic forces effects. A recent work by Guo et al.29 on E-glass fibers-filled (up to 35%) linear low-density polyethylene melts generated experimental

370

Filled Polymers

results that clearly demonstrate the effects of fiber–fiber interactions and the coupling between fiber orientation and polymer chains conformation on the rheological properties of molten composites.

7.5  Thermoplastics and Short Fibers of Natural Origin Certain natural fibers such as jute, hemp, flax, sisal, etc., have relatively high strengths that make them attractive for preparing composites for various applications. Like any other fibers, aspect ratio, loading, and orientation are the first parameters that control the properties of the natural fibers composites and, providing differences in mechanical properties (e.g., moduli) between the fibers and the polymer matrix are sufficiently large, similar reinforcing effects can be expected as with their synthetic counterparts. However the efficiency and the level of reinforcement may be different, due to a few specific aspects. Firstly, natural fibers often exhibit quite large degrees of nonuniformity in most characteristics, i.e., chemical composition, surface aspect and properties, length and diameter, cross-sectional shape, strength, and stiffness.46 In addition, most cellulose based materials are prone to moisture absorption and therefore the durability of the composite may be questionable. Most polymers and hence polymer composites absorb moisture in humid atmosphere or when immersed in water. Natural fibers absorb of course more water than synthetic fibers, which results in degradation of mechanical properties such as tensile strength. Moisture diffusion in a composite depends on various factors, with fibers volume fraction, void volume, fibers treatment and additives, humidity and temperature the most important ones. Preparing polymer composites with lignocellulosic fibers is not just a novelty. Contemporary environmental concerns and the growing concern for sustainable development have renewed the interest in materials that offer both economic and environmental benefits. Applications considered are in the automotive, construction, furniture and packaging industries. Chopped natural fibers, with controlled aspect ratios and tailored specific properties, could be used for higher value applications than typical wood polymer composites. An attractive aspect is that compared to short glass fibers, natural fibers are nonabrasive and consequently higher fiber content in plastics could be considered, without excessive wear and damage of compounding and processing equipment. The high moisture sensitivity of natural fibers and their low microbial resistance are however disadvantages that needs to be considered, namely by insuring that they are properly encapsulated in the polymer with good fiber-matrix bonding, either using the appropriate additives or through suitable preliminary treatments, for instance through acetylation of hydroxyl groups present in the fiber.47

371

Polymers and Short Fibers

Like wood flour, the main limitation of the use of natural fibers is the lower permissible processing temperature, in order to avoid fiber degradation and / or emission of volatiles. The processing temperatures are thus limited to 180–200°C, although it is possible to use higher temperatures for very short periods. Of course, processing equipment must be properly designed in order to avoid any possibility of stagnation or low flow zones. Practically, this limits the type of polymers that can be used to commodity thermoplastics such as PE, PP, PVC and PS. The main component of natural fibers is cellulose that, owing to its crystalline structure, has one of the highest tensile modulus exhibited by polymer materials, around 136 GPa compared with 77 GPa for E-glass fibers. Inside a natural fiber, cellulose chains are assembled in microfibrils which are packed in several layers. The high stiffness of natural fibers is due to the helicoidal supramolecular organization of crystalline cellulose which, during stretching however brings a torsion/traction coupling that can modify the fiber– matrix interface and therefore affects the deformation behavior and the ultimate fracture mechanisms. There are also hydrogen bonding between cellulose chains which explain the excellent properties of cellulose based fibers.48,49 All these characteristics explain the large diversity in mechanical and physical properties of natural fibers. Table 7.10 somewhat positions typical natural fibers such as jute and kenaf vs. glass fiber, when used at equal level in polypropylene. Tensile, flexural, and impact properties, along with water absorption and specific gravities were measured on injection molded samples. As can be seen, natural fibers composites are quite challenging the glass fibers-filled composite with however a significant difference in moisture sensitivity. So-called WPCs are thermoplastic composites that combine wood flour with plastics, intended to offer the advantages of wood-like materials with the processing and part-design capabilities of thermoplastic polymers. Table 7.10 Commercial Fibers-Filled Polypropylene Composites PP-fibers Fibers content Fibers volume fraction Specific gravity Tensile modulus Tensile strength Elongation at break Flexural modulus Flexural strength Water absorption Linear mold shrinkage

%wt g/cm3 GPa MPa % GPa MPa %/24 h cm/cm

Virgin PP

E-Glass

0 0 0.90 1.7 30 170 1.2 40 0.02 0.028

50 0.15 1.33 11.5 88 1.4 8.9 98 0.05 0.004

Jute 50 0.39 1.08 8.1 69 2.3 7.3 97 1.13 0.003

Kenaf 50 0.39 1.07 7.9 58 2.2 7.3 98 1.05 0.003

372

Filled Polymers

Whilst finely grinding wood material may considerably hide the fiber aspect of wood particles, WPC exhibit properties which are typical of systems filled with short fibers of natural origin. As WPC is a rich subject in itself, only a few basic aspects will be discussed hereafter. It is obvious that not all thermoplastics can be used to prepare WPC. Indeed the polymer component melts or at least softens at or below the degradation point of the wood component, normally 200–220°C but must be rigid at normal usage temperatures (i.e., up to around 65°C). Practically, this reduces the choice to polypropylene, lowand high-density polyethylenes, polystyrene, and vinylics (essentially PVC). As such wood flour, used as a filler in thermoplastic composites, offers only modest, if any, reinforcement, but wood fibers can lead to superior composite properties and act more as reinforcing filler. Commercial wood flour is a by-product of the wood industry, often mechanically processed from waste materials such as planer shavings, chips, and sawdust, which are reduced to fine powders, with various grades available depending upon the particle size and the wood species. Wood (cellulose) fibers are produced through more or less complex defibrillation techniques, using raw materials from both virgin and recycled resources, and are different from natural fibers, such as jute, hemp, or sisal. Composites with wood material and polymer can be prepared in two ways. In the first, the wood fiber/flour is a reinforcing agent or a filler in a continuous thermoplastic matrix. In the second, the thermoplastic is a binder to the majority wood component. Only the first approach is discussed hereafter since a continuous thermoplastic matrix determines the processability of the composite material and is the necessary condition for processing WPC with conventional thermoplastic processing techniques, whilst certain adaptation of the equipment and operation may be required. Wood-filled thermoplastic composites were introduced on the market in the early 1990s, essentially in North America. By far the biggest sector of the WPC industry is the fabrication of deck board and railing, fencing, door and window frames for the residential construction market. Deck board, alone, accounts for about 60% of overall WPC volume, with a steady growth of nearly 14%/year from 1996 to 2006, despite the 10–20% higher price compared to treated wood. Obviously in the long run, WPC need less maintenance than natural wood and the initial higher investment is recovered in a few years. The second sector is the fabrication of wood-filled thermoplastic composite lumbers, with a similar growth rate. It is worth noting that all such products are made through continuous process, with extrusion the major technique. Natural orientation of wood fibers along the main flow lines in suitably designed extrusion dies is likely the reason as it will optimize the mechanical performance (in terms of flexural and tensile modulus, for instance) and aesthetics (surface aspect). Injection molding and compression molding together account for less than 10% of the WPC market, with more developments in Europe (Austria and Germany) than in North America. Injection molding is a challenging technique for WPC because wood begins

Polymers and Short Fibers

373

to degrade at temperatures above 200°C and flow easiness dramatically decreases with increasing wood flour/fiber content. WPC cannot be considered only as mere substitutes to natural wood because using the resources of thermoplastic processing techniques, products can be fabricated that would be impossible or at least difficult and costly to make with wood. Indeed extrusion allows long, hollow profiles with any cross-sectional design to be made. The resources of coextrusion and other specific techniques can also be exploited, for instance by introducing a foaming agent into the molten wood/plastic matrix, so that lighter products with an internal rigid foamy structure are obtained. A basic problem in the production and processing of wood polymer composites is the moisture content and sensitivity of natural materials, which generally require a thorough drying before compounding and a control of the storage conditions of the compounds before the ultimate shaping/processing steps. It is generally considered that the maximum moisture content for easy processing is around 10–12% for extrusion or calendering operations, and less than 3–4% for injection molding. Generally speaking, a thumb rule is the drier, the better. When developing WPC, a number of technical problems must be solved, for instance fiber-to-matrix bonding, fiber damage during compounding and processing, fiber orientation in finished part, providing obviously that the correct fiber-matrix selection has been made. Basic research works to tackle one or several of these issues have been initiated around 40 years ago, with varied industrial or scientific successes, and still go on owing to the great number of likely interesting systems and the recently renewed interest in materials of natural origin. Ageing is an issue in WPC technology and various additives are used to protect the wood flour or fibers against moisture, light, mold, mildew, insects, and other pests. But the best technique is a full encapsulation of the wood particles or fibers in the polymer matrix. Virgin and recycled PE (both high- and low-density) remains the most common thermoplastic used in the WPC industry. PP offers enhanced properties (higher stiffness, less creep, higher HDT) but its higher melt temperature may pose some processing challenges. The use of PVC continues to grow as more structurally enhanced products emerge. Although most WPCs still incorporate wood flour, derived from reclaimed pine, maple and oak sawdust, planer shavings, sanding dust, and milled scraps, there are today alternatives that permit more targeted products in terms of performance, for instance pulp cellulose fiber, which is generally reclaimed from postconsumer waste paper.50 Pulp cellulose fiber has zero coefficient of thermal expansion and, because it contains no lignin, it does not photo-degrade like wood flour. Natural fibers are also an alternative but large scale uses require a regularity in large volume availability, at constant quality. The potential for natural fiber is likely as a reinforcing component with wood floor the major filler. Flax, jute, hemp, and kenaf fibers can be produced with millimeter range lengths and, properly oriented in the material would give enhanced

374

Filled Polymers

Polypropylene/wood flour

Tensile or flexural modulus, GPa

5

Tensile

4

Flexural

3 2

Part.size, µm 282

1 0

24

0

10

20 30 40 Wood flour content, %

50

Tensile or flexural strength, MPa

flexural properties. Compared to short glass fibers, natural fibers have higher strength-to-weight ratios, similar surface functionality and handling characteristics as wood, and can be easily incorporated into a wood composites.51 Today a wood–plastic composite is quite a complex formulation. A typical WPC formulation would consist for instance of 30–35%wt polymer, 50–55% wood flour, 5% talc, 4% lubricant, 2% colorant, 2–3% other additives such as coupling agents and/or mold and tannin inhibitors. Additives play a key role in enhancing product characteristics, particularly with respect to weatherability. Additives are lubricants, UV stabilizers, flame retardants, antimicrobials, color concentrates, coupling agents, foaming agents, dispersion agents, mineral fillers, antioxidants, and compatibilizers. Maleic anhydride-based coupling agents, for instance MA-grafted polyolefins, improve dispersion and wetting of the lignocellulosic materials by the polymer, with a direct effect on product durability, because the rate of moisture absorption is reduced, while strength properties are improved.52 Lubricants improve surface appearance and processability, and reduced manufacturing cost through increased throughput. Zinc borate is quite a common additive to WPCs to ward off fungal decay and inhibit the growth of mold and mildew. A number of published research works allow to understand some basic scientific aspects of the WPC technology.53–55 The effect of wood flour particle size is qualitatively the same as with other particulate fillers, except that the fiber nature seems to play in role in what the ultimate properties are concerned, as illustrated in Figure 7.16 with results on (pine) wood flourfilled polypropylene composites.56 As can be seen, tensile and flexural moduli increase with increasing wood flour content (left graph); whilst flexural strength is practically not affected, the tensile strength steadily decreases with increasing filler content (right graph). In the range considered, 24–282 µm, particle size has practically no effect.

50 40 30

Flexural Tensile

20 10 0

0

15 25 40 Wood flour content, %

Figure 7.16 Effect of (pine) wood flour on the mechanical properties of PP based composites. (Data from J.C. Caraschi, A.Lopes Leão. Mat. Res., São Carlos, Brazil, Oct./Dec.5 (4), 405–409, 2002.)

Polymers and Short Fibers

375

There are generally large differences in surface polarity between wood particles or fibers and polyolefin matrices, which lead to poor wetting, hence dispersion mixing difficulties and eventually limited mechanically properties. Various coupling agents have been studied in the literature with some advantages assigned to functionalized polymers.57 Maleic anhydride-grafted PP, PE-PP and (in a lower extent) PE are the most commonly used, likely owing to their wide commercial availability, e.g., Exxelor® VA1801, MA-gEPR, 1.21%MA (Exxon Mobil Chemical Co) or Epolene® G2608, MA-g-PE, acid number: 8 mgKOH/g (Eastman Chemical Co). Such compatibilizers readily enhance the mechanical properties of composites, especially tensile and flexural strength, likely because on one hand either polar interaction or covalent links are obtained between the anhydride carbonyl and hydroxyl groups of wood surfaces, and on the other hand because of good compatibility with polyolefins.58,59 Stark and Rowlands60 studied for instance the effects of 40%wt wood flour and fiber of different particles size (aspect ratio), compounded in polypropylene, without or with maleated PP as coupling agent. Tensile and flexural strength and modulus were found to increase with the fiber aspect ratio. Notched impact energy increases with higher particle size, whereas unnotched impact energy exhibits the reverse trend. Wood fibers give generally higher strengths than wood flour, but the higher aspect ratio had little effect on impact energy. The MA-grafted PP coupling agent gives higher strength and the effect is larger with wood fibers than with wood flour. The coupling agent does not significantly affect tensile or flexural moduli. Dányádi et al.61,62 studied the effect of MA-grafted PP ­molecular weight and content on the properties of wood flour filled polypropylene composites. They found that stiffness increases with wood flour content, with very little effect from the type or the amount of the functionalized polymer additive used. However, ultimate properties are strongly influenced by the amount and properties of MA-g-PP, with larger molecular weight and smaller functionality having beneficial effects on both tensile strength and impact resistance. Indeed, the functionalized agent increases considerably the interfacial adhesion between the wood particles and the polymer matrix, which completely changes the deformation mechanism; the fracture of wood particles becomes the dominating factor because the matrix polymer essentially deforms by shear yielding. The optimum MA-g-PP/wood flour ratio is around 0.05.

7.6  Elastomers and Short Fibers Filling an elastomer with short fibers essentially involves combining the viscoelastic behavior of a rubber matrix with the strength and stiffness of the fiber, in order to obtain useful engineering materials. Reinforcing elastomers

376

Filled Polymers

with long fibers, textiles, or fabric is a long established practice in the rubber industry, namely in tires, belts, hoses, and other demanding applications, but long fiber rubber composites are outside the scope of this book. It is in principle possible to prepare short fiber-filled rubber composites by exploiting the resources of both the rubber latex and the regenerated cellulose technologies. Research results have been published that described such materials.63–65 For instance, short-fiber filled compositions were prepared by coprecipitating mixtures of natural rubber or nitrile rubber latex with cellulose xanthate. Compounds with cellulose fiber content up to 30 phr were obtained that exhibit increased tensile modulus and strength and decreased elongation at break as fiber level increases, as well as some (minor) effects on curing properties. It is clear that such systems are worth consideration with respect to their potential for sustainable development, because they partly use renewable components. However, despite encouraging laboratory results, there has been so far no industrial application (to the author’s knowledge). Only more practical rubber compounds with short fibers added during mixing operations are discussed hereafter. Because the dispersed fibers do flow with the rubber matrix during the processing operations, quite intricate parts can be shaped through techniques such as extrusion, calendering, and the various molding techniques (compression, transfer and injection), then vulcanized. There are however a few constraints for fiber-reinforcement to be used in rubber technology: first an adequate dispersion of fibers must be achieved, with limited fiber breakage, second a good adhesion between fibers and rubber must be obtained, generally by using either a suitable pretreatment of the fibers or an appropriate adhesion-promoting bonding system/agent. In contrast with fibersfilled thermoplastics for which a good wetting of the fibers by the polymer is generally sufficient because only (small strain) elastic properties are concerned, most rubber parts must support larger strain, in the several 100% range. Rubber-fiber bonding is therefore essential, and can only be achieved by chemical means. Since the early works in the late 1970s by Coran et al.,66–68 short natural fiber–rubber composites have received a considerable attention in scientific literature and a few systems have achieved a significant importance in certain engineering and consumer goods applications, because of their high strength-to-weight ratio, manufacturing flexibility, and ease of processing. Particularly, these authors demonstrated that in sulfur-cured SBR compounds, unregenerated (hardwood) cellulose fibers, properly bonded to the matrix (by an undisclosed system), gives more that 60% of the tensile modulus that would be obtained with the same volume fraction of short glass fibers. They also concluded that, with respect to the benefits offered by cellulose fibers, not much advantages would be obtained by using fibers with modulus higher than around 25–40 GPa. Such results prompted the development of unregenerated cellulose grades especially for use in elastomer reinforcement,69 which were commercialized under the trade name

Polymers and Short Fibers

377

Santoweb® by Monsanto Co. (now Flexsys). Today, four grades of treated cellulose fiber product are commercially available for use in NR, SBR, BR, and CR compounds (Santoweb® D), or EPDM and IIR elastomers (Santoweb® H). It seems that the treatment consists in adding the correct amount of resorcinol-formaldehyde resin, i.e., a methylene receptor, so that the use of a methylene donor chemical, for instance hexa(methoxymethyl)melamine, HMMM, is required. Maybe the most significant industrial achievements in rubber technology with such treated cellulose fibers are based on the fact that, during processing, fibers tend to become oriented in the direction of the flow. Special designed extruder dies (so-called expanding mandrel extrusion dies) can therefore be used to provide control of the average fiber orientation, such that hoses can be made with fibers oriented in the circumferential direction. Extensive reviews on short fibers-reinforced elastomers have been published by Goettler and Shen,70 and more recently by Rajeev.71 Table 7.11 is a list of selected published works on natural fiber-filled rubber composites, sorted by rubber type. Only vulcanizable rubbers were considered in preparing the table; thermoplastic rubber and rubber-plastics blends were omitted. As can be seen most grades of conventional elastomers have been considered with quite a large variety of natural fibers. Natural rubber, SBR, and EPDM compounds have received much attention, as expected with respect to their industrial importance. A bonding system is used in most cases, with resin types (i.e. resorcinol-formaldehyde, resorcinol-hexamethylenetetramine, silica-resorcinol-hexamethylenetetramine) the most frequent ones. The predominance of resin bonding is likely related to the thorough understanding of the chemistry involved when using resin formers to obtained fiber-to-rubber adhesion. The development of adhesion essentially occurs In situ, during vulcanization, when resin forms in the boundary fiber–matrix regions. As demonstrated by Morita,72 typical Mooney scorch curves are readily observed when resin formation and rubber vulcanization are made to occur subsequently through adequate choice in the formulation. Recommended formulations are however such that both events occur simultaneously during curing. Resorcinol or resorcinol derivatives are common resin formers, playing the role of “methylene acceptors,” while hexamethylenetetramine (HEXA) or HMM act as “methylene donors.” Silica has been found to enhance the adhesion significantly. As mentioned above, such resins systems were the background of patents obtained by Monsanto Co more than 30 years ago. It is pretty obvious that a pretreatment of fibers with resorcinol or resorcinol-formaldehyde will favor fiber dispersion because it will reduce hydrogen bonding between fibers and therefore give an easier separation of the single fibers from fiber bundles, with a minimum force in such a manner that fiber breakage is limited during mixing. Industrial usages over the last 20 years have largely demonstrated that resin systems are likely the best choice for cellulose-lignin fibers, i.e., most natural fibers. Resin systems are not very effective with polyester fiber, not effective at all with

378

Filled Polymers

Table 7.11 Selected Published Works on Natural Fiber Filled Rubber Composites (Vulcanizable Rubbers only) Rubber(s) CR CR CR CR, EPDM, PUR EPDM EPDM EPDM EVA NBR NR NR NR NR NR NR NR, NBR NR, SBR, EPDM NR + LDPE + Liquid NR PU, NR latex SBR SBR SBR SBR, BR XNBR

Fiber(s) Cotton, polyester, Cellulose PET, PA Silk PET PET Melamine PAN, Aramid PAN, Carbon Jute Jute Silk Polyester PA Grass Hemp Glass, PA, carbon, aramid, cellulose Glass, aramid, cellulose Kenaf PA, aramid, HMW-PE Jute, glass Glass, PA, carbon, cellulose, polyester Sisal Cellulose Jute

Bonding System

Reference

None

88

R-F* S-R-H* None 1,4-carboxysulphonyl-diazide S-R-H None None S-R-H S-R-H S-R-H None Undisclosed/none Silane (TESPT)* None S-R-H

89 75 90 91

None

104

Silane, PPgMA Corona, γ-irradiation

105 106

R-H* R-F

107 108

R-H R-H S-R-H

109 84 88

92,81 93 94 95 96, 87, 97 98 99 100, 101, 105 102 103 74

* R-F: resorcinol–formaldehyde; R-H resorcinol-hexamethylenetetramine; S-R-H: silica-­ resorcinol-hexamethylenetetramine; TESPT: bis(3-triethoxysilylpropyl)tetrasulfane.

glass fiber and mixed results were reported with silk, amide and aramide fibers in various elastomer systems.73–75 Special fiber–elastomer composites have been reported to exhibit good properties without the need for bonding systems, e.g., polyurethane and aramid short fibers.76 It is clear that the possibility to fine-tune a short-fiber filled rubber compounds through (proprietary) adjustments of the formulation suits well the practice of the rubber industry. Wennekes et al.77 have recently published quite an extended review of the various treatment/additives that can be used to promote fiber-rubber

Polymers and Short Fibers

379

adhesion. They described namely other alternatives to the well established resorcinol-HEXA system, such as fiber surface roughening (through aggressive chemical treatment), adhesion promoter additives, impregnated fibers, and plasma treatment. Whatever are the natural fiber and the rubber type, qualitatively the same results are always reported when increasing the short fiber content: • Reinforcement of the tensile properties, i.e., increased modulus and tensile strength, but lower elongation at break • Increased hardness and stiffness • Improvements in resistance to cuts, tears and punctures • Enhanced resistance to solvent swelling (constraining effect of fibers) • Lower extrudate swell (essentially reflecting fiber alignment in the flow direction) • Overall benefits increase when an appropriate bonding system is used. Expectedly quantitative results vary considerably from one fiber–rubber system to another, and other compounding ingredients may induce additional effects (positive or negative). The qualitative effects are completely in line with the expected role of short fibers, in agreement with micromechanic considerations (see Section 7.2). Certain authors have used well established micromechanic approaches, e.g., Voigt and Reuss averages (Equations 7.1 and 7.2), and Halpin–Tsai equations (Equation 7.5) to consider the effects on short natural fibers in rubber compounds.78–80 The trade literature shows that certain short natural fiber–rubber composites have achieved significant importance in a number of applications. This is certainly true for fiber materials that are commercially available and that meet the requirements in product quality and consistency for engineering applications, e.g., the Santoweb® treated cellulose fibers grades from Flexsys, the cotton flocks W200 and D220 supplied by IFC, International Fiber Corp., North Tonawanda, NY, the “jet process” milled cellulose fiber Interfibe® ­supplied by Interfibe LP, Solon, OH and the nitrile rubber coated cellulose fiber Nicote® supplied by Vellumoid Inc., Worchester, MA. All those commercially available products are fibers of 10–25 µm diameter and average length in the 250–600 µm range. Short fiber reinforced rubber composites have found well established usages in hose, belt, tires and other automotives applications, generally because either they offer specific properties that cannot be obtained with more conventional fillers only (e.g., carbon black or silica) or they are a cheaper alternative at equivalent properties. It is obviously the capability of short fibers to be properly oriented during the (adapted) fabrication process and therefore to achieve well-controlled anisotropic mechanical properties, that is the most

380

Filled Polymers

critical reason for using such systems. Providing the fiber aspect ratio is larger than 100, an adequate mechanical anisotropy can be imparted to the rubber composite through the processing operation. Indeed, with high aspect ratio treated cellulose fibers, large difference in tensile properties are achieved between the processing flow direction and the cross-machine direction in carefully milled sheets (up to 10 times in term of tensile modulus).55 Short fibers, well dispersed in a viscoelastic matrix at concentrations above 6%, can therefore be very efficiently oriented in certain type of flow fields where the elongational component is dominant. Goettler and Lambright developed a proper design of a converging–diverging extruder die such that average fiber deviation from the flow axis is substantially different.81,82 As illustrated in the upper part of Figure 7.17, the right combination of (wall) shear forces with the extensional flow occurring between the screw head and the orifice of a rubber hose die cause the fibers to become aligned parallel to the flow direction. Because, at some intermediate point, the restriction is followed by a specific type of expansion, a somewhat controllable transverse orientation pattern of the fibers is obtained. As illustrated in the lower part of the figure, this technique led to the so-called “extrusion moving die” technology to produce curved hoses.83 By moving the inner or the outer portions of the die out of concentricity in a programmed sequence, the opening between inner and outer dies is varied and the extrusion direction can be made to deviate from the machine axis. S-bend hoses can thus be produced, whose performances (i.e., burst pressure) are similar to cord reinforced coolant hoses. Die component motions and timing sequences must be controlled in a very tight manner through computer control. This special extrusion technique was commercially available by the mid 1980s (namely from the now defunct Iddon Brothers Ltd Co) and is in widespread use today. This unique process for producing bend hoses is however contingent upon the use of short fiber reinforcement, since well dispersed fibers must be present in the compound before it is extruded. It suits obviously well treated cellulose fibers like Santoweb®. As reported by Goettler,84 the technology is not restricted to rubber materials but can also be applied with soft thermoplastics, such as plasticized PVC. In such a case an isocyanate bonding agent is recommended.85 Of all the fibers that have (and are still) evaluated in rubber compounds over the last 30 years, only treated unregenerated cellulose fibers and chopped aramid fibers have really achieved a certain industrial importance. The current research efforts being made, essentially by university groups in the producing countries, on natural fibers such as jute, sisal, kenaf, etc., are not (yet) leading to documented industrial applications, at least to the author’s knowledge. Actual reasons for this situation are unclear, except maybe that industrial applications for treated unregenerated cellulose fibers were inherited from the important preliminary works by former Monsanto’s scientists, prior to the commercial development of Santoweb® fibers. When

381

Polymers and Short Fibers

Converging mandrel die

Circumferential fiber orientation

Converging–expanding die

Mixed fiber orientation

Moving extrusion die technology

Mandrel axis Outer die axis

Lateral translation of outer die produces eccentricity then hose curvature Figure 7.17 Controlling short fiber orientation in extruded rubber hoses with converging–expanding dies; application in the moving die technology to produce curve hoses.

comparing the effects of various type of fibers, i.e., hardwood cellulose, acetate, nylon 66, PET, acrylic and glass, on the mechanical properties on a carbon black filled SBR compound, Coran et al.54 came to the conclusion that properly bonded and treated cellulose fibers gave the best balance of performances, namely tensile properties not so far to what is obtained with short glass fibers. In fact, when normalizing their results for changes in fiber concentration, orientation, geometry and matrix modulus, they clearly demonstrated that not much is gained with fibers whose tensile modulus is higher than about 27–40 GPa. Glass fibers exhibit typically tensile modulus in the 70–90 GPa range but suffer from severe breakage during mixing

382

Filled Polymers

operations because (like carbon fibers) they have low ­bending strength. Ductile fibers, such as cellulose and synthetic fibers, are more flexible and resistant to bending, the former because they are generally ribbon shaped. Therefore such fibers have the capability to preserve aspect ratios largely above the critical value for an effective stress transfer from the matrix to the fibers. A number of factors must be taken into consideration when comparing the potential of different fibers for (sizeable) industrial applications. Regular large volume availability with minimum variation in properties is obviously required and is quite a challenging constraint for all materials of natural origin. In addition, there are some important technical constraints to consider, i.e., to name a few: • Fiber size and aspect ratio (in the composite, after the mixing operations) • Fiber–polymer matrix interactions, with respect to the fiber pretreatment and/or the use of bonding agents, if necessary • Fiber mechanical properties, at first tensile and flexural moduli • Fiber structure, either external, i.e., linear or branched geometry, fibrillar, bundles, etc., or internal, i.e., hollow or plain fiber • Fiber surface (smooth, rugged, …) and surface chemistry Except for synthetic fibers, not all such information are easily available, and as a matter of fact are missing for most natural fibers even for those which have achieved a significant industrial success (e.g., Santoweb® brand). Such a ­situation obviously limits scientific investigation on such systems, namely considerations based on well established micromechanic models for (synthetic) fiber filled thermoplastic systems. What is nevertheless well established in what short fiber-filled rubber composites are concerned is that an average length (or length-to-diameter ratio) above a critical value must be kept to achieve an effective polymer-to-fiber load transfer, plus an excellent bonding between fibers and polymer matrix. It seems that, where treated cellulose fibers are concerned, this is effectively the case because the minimum critical aspect ratio is relatively low and silica-resorcinol-­hexamethylenetetramine systems appear as optimal in terms of cost/performance ratio. Murty and De86 on Natural Rubber (NR)/jute systems, and Chakraborty et al.87 on carboxylated nitrile rubber (XNBR)/jute composites found for instance that an aspect ratio as low as forty still gives a good reinforcement, likely because an excellent bonding is achieved between the fibers and the rubber matrix. Fiber orientation is also an important aspect and it is well understood that below an aspect ratio of around 100, and providing the fiber content is not excessively high to avoid bundling, beneficial orientation effects can be obtained through adequate processing (as previously explained). Such considerations allow

Polymers and Short Fibers

383

to somewhat understand why timing V-belts and water coolant hoses have become quite traditional applications for treated cellulose–rubber systems.

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75. L. Ibarra Rueda, C. Chamorro Antón, M.C. Tabernero Rodriguez. Mechanics of short fibers in filled styrene-butadiene rubber (SBR) composites. Polym. Compos., 9 (3), 198–203, 1988. 76. C. Vajrasthira, T. Amornsakchai, S. B.-Limcharoen. Fiber-matrix interactions in aramid-short-fiber-reinforced thermoplastic polyurethane composites. J. Appl. Polym. Sci., 87, 1059–1067, 2003. 77. W.B. Wennekes, R.N. Datta, J.W. Noordermeer, F. Elkink. Fiber adhesion to rubber compounds. Rubb. Chem. Technol., 81, 523–540, 2008. 78. A.Y. Coran, K. Boustany, P. Hamed. Unidirectional fiber-polymer composites: swelling and modulus anisotropy. J. Appl. Polymer Sci., 15, 2471, 1971. 79. W. Guo, M. Ashida. Mechanical properties of PET short fiber-polyester thermoplastic elastomer composites. J. Appl. Polym. Sci., 49, 1081–1091, 1993. 80. R.S. Rajeev, A.K. Bhowmick, S.K. De, G.J.P. Kao, S. Bandyopadhyay. New composites based on short melamine fiber reinforced EPDM rubber. Polym. Compos., 23 (4), 574–591, 2002. 81. L.A. Goettler, A.J. Lambright (to Monsanto Co). Process for controlling orientation of discontinuous finer in a fiber reinforced product formed by extrusion. U.S. Patent 4,056,591, 1977. 82. L.A. Goettler, A.J. Lambright, R.I. Leib, P.J. DiMauro. Extrusion-shaping of curved hose reinforced with short cellulose fibers. Rubb. Chem. Technol., 54, 277– 301, 1981. 83. L.A. Goettler, J. Sezna, P.J. DiMauro. Short fiber reinforcement of extruded rubber profiles. Rubb. World, 187, 33–42, 1982. 84. L.A. Goettler. The extrusion and performance of plasticized Poly(viny1 chloride) hose reinforced with short cellulose fibers. Polym. Compos., 4 (4), 249–255, 1983. 85. L.A. Goettler (to Monsanto Co). Treated fibers and bonded composites of cellulose fibers in vinyl chloride polymer characterized by an isocyanate bonding agent. U.S. Patent 4,376,144, 1981. 86. V.M. Murty and S.K. De. Short jute fiber reinforced rubber composites. Rubb. Chem. Technol., 55, 287–308, 1982. 87. S.K. Chakraborty, D.K. Setua, S.K. De. Short jute fiber reinforced carboxylated nitrile rubber. Rubb. Chem. Technol., 55, 1286–1307, 1982. 88. J.W. Rogers. The use of fibers in V-belt Compounds. Rubb. World, 183 (6), 27–31, 1981. 89. M. Ashida, T. Noguchi, S. Mashimo. Dynamic moduli for short Fiber-CR ­composites. J. Appl. Polym. Sci., 29, 661–670, 1984. 90. M. Ashida, T. Noguchi, S. Mashimo. Effect of matrix's type on the dynamic properties for short fiber-elastomer composite. J. Appl. Polym. Sci., 30, 1011–1021, 1985. 91. L. Ibarra. Dynamic properties of short fiber-EPDM matrix composites as a function of strain amplitude. J. Appl. Polym. Sci., 54, 1721–1730, 1994. 92. R.S. Rajeev, S.K. De, A.K. Bhowmick, G.J.P. Kao, S. Bandyopadhyay. Atomic force microscopy studies of short melamine fiber reinforced EPDM rubber. J. Mater. Sci., 36, 2621–2632, 2001. 93. S. Jin, Y. Zheng, G. Gao, Z. Jin. Effect of polyacrylonitrile (PAN) short fiber on the mechanical properties of PAN/EPDM thermal insulating composites. Mater. Sci. Eng. A, 483–484, 322–324, 2008.

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94. N.C. Das, T.K. Chaki, D. Khastgir, A. Electromagnetic interference shielding effectiveness of ethylene vinyl acetate based conductive composites containing carbon fillers. Chakraborty. J. Appl. Polym. Sci., 80, 1601–1608, 2001. 95. S.S. Bhagawan, D.K. Tripathy, S.K. De. Stress relaxation in short jute fiber­reinforced nitrile rubber composites. J. Appl. Polym. Sci., 33, 1623–1639, 1987. 96. V.M. Murty, S.K. De. Effect of particulate fillers on short jute fiber-reinforced natural rubber composites. J. Appl. Polym. Sci., 27, 4611–4622, 1982 97. V.M. Murty, S.K. De, S.S. Bhagawan, R. Sivaramakrishnan, S.K. Athithan. Viscoelastic properties of short-fiber-reinforced rubber composites and the role of adhesion. J. Appl. Polym. Sci., 28, 3485–3495, 1983. 98. D.K. Setua, S.K. De. Short silk fiber-reinforced Natural Rubber composites. Rubb. Chem. Technol., 56, 808–826, 1983. 99. A.F. Younan, M.N. Ismail, A.I. Khalaf. Thermal stability of natural rubber­polyester short fiber composites. Polym. Degradation Stability, 48, 103–109, 1995. 100. N. Kikuchi. Tires made of short fiber reinforced rubber. Rubb. World, 214 (3), 31–32, 1996. 101. T.D. Sreeja. Cure characteristics and mechanical properties of Natural Rubber— short Nylon fiber composites. J. Elast. Plast., 33 (3), 225–238, 2001. 102. D. De, D. De, B. Adhikari. The effect of grass fiber filler on curing characteristics and mechanical properties of natural rubber. Polym. Adv. Technol., 15 (12), 708–715, 2004. 103. E. Osabohien, S.H.O. Egboh. Utilization of bowstring hemp fiber as a filler in natural rubber compounds. J. Appl. Polym. Sci., 107, 210–214, 2008. 104. M.A. López Manchado, M. Arroyo. Short fibers as reinforcement of rubber ­compounds. Polym. Compos., 23 (4), 666–673, 2002. 105. H. Anuar, W.N. Wan Busu, S.H. Ahmad, R. Rasid. Reinforced thermoplastic Natural Rubber hybrid composites with Hibiscus Cannabinus, L and short glass fiber—part I: processing parameters and tensile properties. J. Compos. Mater., 42, 1075–1087, 2008. 106. M. Epstein, R.L. Shishoo. Measurement of adhesion between fibers and fast curing elastomer resin. J. Appl. Polym. Sci., 50, 863–874, 1993. 107 V.M. Murty, S.K. De. De. Short-fiber-reinforced styrene-butadiene rubber ­composites. J. Appl. Polym. Sci., 29, 1355–1368, 1984. 108 L. Ibarra Rueda, C. Chamorro Antón, M.C. Tabernero Rodriguez. Short-fiberreinforced styrene-butadiene rubber composites. Polym. Compos., 9 (3), 198–203, 1988. 109. R.P. Kumar, M.L.G. Amma, S. Thomas. Short sisal fiber reinforced styrenebutadiene rubber composites. J. Appl. Polym. Sci., 58, 597–612, 1995.

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Appendix 7

A7.1 Short Fiber-Reinforced Composites: Minimum Fiber Aspect Ratio The shear-lag analysis of fiber-reinforced composites considers that below a certain length, a fiber is no longer effective in supporting the transfer of load from the matrix. There is thus a minimum effective fiber length Lmin that can be assessed as :

Lmin = D ⋅

D: fiber diameter Efib: tensile modulus of the fiber Gmat: shear modulus of the matrix Φfib: fiber volume fraction

1 Efib 1 − Φ fib ⋅ ⋅ 2 Gmat Φ fib

Numerical illustration: D: = 0.5 · 10−6·m

Φfib: = 0,0.01..0.50   : fiber volume fraction range

Efib: = 73 · 109·Pa

 = 73 GPa: typical tensile modulus for E-glass fiber

Emat: = 3 · 10 ·Pa

 = 3 GPa: tensile modulus range for poly amide 66

9

For most thermoplastics, the shear modulus is consideraly smaller than the tensile modulus, let’s say 1000 times smaller:



Gmat : =

Emat    Gmat = 3 · 106 ·Pa 1000

A7.1.1 Effect of Volume Fraction on Effective Fiber Length Lmin ( Φ fib ) : = D ⋅

1 . Efib 1 − Φ fib    ⋅ 2 Gmat Φ fib

Fiber/matrix moduli ratio

Efib = 2.433 ⋅ 10 4 Gmat

390

Filled Polymers

Effective fiber length, m

2.10–4 1.5.10–4 1.10–4 5.10–5 0

0

0.2 0.4 Fiber volume fraction

0.6

Above 10% fibers loading, the effective fiber length slightly decreases with increasing fibers content



Lmin ( 0.15 ) = 6.937 ⋅ 10−5 m

Lmin ( 0.15 ) = 138.735 D



Lmin ( 0.20 ) = 6.132 ⋅ 10−5 m

Lmin ( 0.20 ) = 122.633 D



Lmin ( 0.30 ) = 5.012 ⋅ 10−5 m

Lmin ( 0.30 ) = 100.232 D

A7.1.2 Effect of Matrix Modulus on Effective Fiber Length Gmat: = 0.1 ⋅ 106 ⋅ Pa, 0.2 ⋅ 106 ⋅ Pa.3 ⋅ 106 ⋅ Pa   Constant fiber loading: Φfib: = 0.2 Lmin (Gmat ) : = D ⋅

Effective fiber length, m





1 Efib 1 − Φ fib ⋅ ⋅ 2 Gmat Φ fib

4.10–4

2.10–4

0

0

5.105

1.106 15.106 2.106 2.5 .106 Shear modulus of matrix, Pa

3.106

391

Polymers and Short Fibers

Gmat i : =

i: = 0 .. 3

7.3 · 106 · Pa

7.3 · 10  · Pa 5

7.3 · 104 · Pa 7.3 · 103 · Pa

Efib = Gmat i

Lmin (Gmat i )m =

1 · 104

3.931 · 10−5

1 · 105



1.243 · 10

−4

Lmin (Gmat i ) = D

78.615 248.603

1 · 106

3.931 · 10−4

786.151

1 · 107

1.243 · 10−3

2.486 · 103

A7.1.3 Effect of Fiber-to-Matrix Modulus Ratio on Effective Fiber Length/Diameter Ratio EGrat: = 0 .. 2.105    Φfib: = 0.2



LD min ( EG rat ) :=



1 1 − Φ fib ⋅ EG rat ⋅ 2 Φ fib

Fiber length/diameter ratio

400 4 . 104

200

0

   ⇐ LD min ( 4 ⋅ 10 4 ) = 157.23

0

5 . 104 1 . 105 1.5 . 105 Fiber-to-matrix modulus ratio

2 . 105

A7.2 Halpin–Tsai Equations for Short Fibers Filled Systems: Numerical Illustration Materials’ data:  µm: = 10−6 ⋅ m  nm: = 10−9 ⋅ m  GPa : = 109 ⋅ Pa : defining units

Polymer matrix (e.g., PP): Emat: = 1.14 ⋅ GPa   νmat: = 0.43  Gmat : =

Emat   Gmat: = 0.399 °GPa 2 ( 1 + νmat )

392

Filled Polymers

Fibers (e.g., E-glass; typical diameter 10 μm): Efib: = 77.0⋅GPa   νfib: = 0.20   Gfib : =

Efib    Gfib: = 32.083 °GPa 2 ( 1 + νfib )

L: = 1 ⋅ mm   D: = 5 ⋅ 10−3 ⋅ mm   α : =

L : aspect ratio    α = 200 D

Φ: = 0, 0.01 .. 1  : fiber volume ratio range A7.2.1 Longitudinal (Tensile) Modulus E 11 ζL(Φ) := 2 ⋅

L D

0  10   15  20 Data: =  30   40   50  60

 1 + ζL ( Φ ) ⋅ µ(Φ) ⋅ Φ  Ex cpd (Φ) := Emat ⋅    1 − µ (Φ ) ⋅ Φ

Longitudinal/flexural modulus, GPa

Longitudinal/flexural modulus, GPa

0.9

50

0 0

0.5 Fiber volume fraction Model

PP+SGF

1

 

Average data various suppliers SGF % wt

 Efib   E  − 1 mat µ(Φ) :=  Efib   E  + ζL(Φ) mat

100

Flexural modulus (GPa) for short glass fibers filled PP

Φ % wt 0 0.0337 0.05405 0.06865 0.1017 0.1318 0.1605 0.1952

Flex Mod GPa 1.152   1.87  3.19   3.34  5.36   7.1   8.964  18.07 

30

20

10

0 0

0.1 0.2 Fiber volume fraction Model

0.3

PP+SGF

The Halpin–Tsai equation predicts a linear variation of the longitudinal ­modulus with increasing fiber volume fraction; such a behavior is not exhibited by commercial PP + SGF composites likely because fibers are far to be perfectly aligned in such materials.

393

Polymers and Short Fibers

A7.2.2 Transversal (Tensile) Modulus E 22

A7.2.3 Shear Modulus G12

ζT(Φ): = 2 + 40 ⋅ Φ10

ζG(Φ): = 1 + 40 ⋅ Φ10

40

π 2.

3

20

0 0

0.5 Φ

ζT(Φ)



1

ζG(Φ)

 0.1   2   1   0.2   2   1         0.3   2    1  For fiber fractions       below 0.4, the 40 Φ10 φ : =  0.4    ζT(φ) =  2.004    ζG(φ) =  1.004  term is negligible  0.5   2.039   1.039         0.6   2.242  1.242   0.7   3.13   2.13       



 Efib   E  − 1 mat µT (Φ) :=  Efib   E  + ζT (Φ) mat

 1 + ζT (Φ) ⋅ µT (Φ) ⋅ Φ  Ey cpd (Φ) : = Emat ⋅    1 − µT (Φ) ⋅ Φ

 Gfib   G  − 1 mat µG(Φ) : =  Gfib   G  + ζG(Φ) mat

 1 + ζG(Φ) ⋅ µG(Φ) ⋅ Φ  Gcpd (Φ) := Gmat ⋅    1 − µG(Φ) ⋅ Φ

394

Filled Polymers

40 0.9

0.9 Shear modulus, GPa

Transverse modulus, GPa

100

50

0 0

20

0.5 Fiber volume fraction

A7.2.4 Modulus for Random Fiber Orientation

1

0

 

0

0.5 Fiber volume fraction

1

A7.2.5 Fiber Orientation as an Adjustable Parameter x : = 0.560 Eracpd(Φ) : = x ⋅ Excpd(Φ) + (1−x) ⋅ Eycpd(Φ)

30

30

20

20

Modulus, GPa

Modulus, GPa

3 5 Ercpd (Φ) := ⋅ Ex cpd (Φ) + ⋅ Ey cpd (Φ) 8 8

10

10

0 0

0.1 0.2 Fiber volume fraction Model Ex PP+SGF Model Ey Random orientation

0 0

0.1 0.2 Fiber volume fraction Model Ex PP+SGF Model Ey 0.56

A7.2.6 Average Orientation Parameters from Halpin–Tsai Equations for Short Fibers Filled Systems Materials’ data:  µm: = 10−6 ⋅ m  nm: = 10−9 ⋅ m  GPa: = 109 ⋅  Pa: defining units SGF: Efib: = 73.0 ⋅ GPa   L: = 300 ⋅ µm  D: = 10 ⋅ µm  α: =

L : aspect ratio   α = 30 D

395

Polymers and Short Fibers

Polymer matrix and Composites: Injection molded fatigue test samples (ASTM D1708) Tensile measurements (ASTMD638) on specimens (3 × 5 × 10 mm) cut out of fatigue samples PBT + SGF

Φ Fiber

 0  0.115 Data1: =   0.1735   0.3288

Emeas

PA/PAT+SGF

GPa

Φ Fiber

 0  0.1308 Data2: =   0.1938   0.2962

2.6  6.475     8.445   14.265 

Emeas GPa

3.2  8.431  10.771   14.930 



E1mat : = ( Data1<1> )0 ⋅ GPa

E2 mat : = ( Data2 <1> )0 ⋅ GPa



Φ1 : = Data1< 0>

Φ2 : = Data2 < 0>

i: = 0..3  E1meas. : = ( Data1<1> )i ⋅ GPa

E2 meas. : = ( Data2 <1> )i ⋅ GPa

i

i

A7.2.6.1  Longitudinal (Tensile) Modulus E11 ζL: = 2 ⋅



L D

 Efib   E1  − 1 mat µL1: = µL2: =  Efib  + L ζ  E1    µL1 = 0.307    mat

Ex cpd1i : = E1mat ⋅

1 + ζL ⋅ µL1 ⋅ Φ1i 1 − µL1 ⋅ Φ1i

 Efib   E2  − 1 mat  Efib   E2  + ζL   µL2 = 0.263 mat

Ex cpd2i : = E2 mat ⋅

1 + ζL ⋅ µL2 ⋅ Φ 2 i 1 − µL2 ⋅ Φ 2 i

396

Filled Polymers

A7.2.6.2  Transversal (Tensile) Modulus E22



ζT: = 2  Efib   E1  − 1 mat µT 1: =  Efib   E1  + ζT mat

 Efib   E2  − 1 mat µT 2: =  Efib   E2  + ζT mat

 1 + ζT ⋅ µT 1 ⋅ Φ1i  Ey cpd1i : = E1mat ⋅   1 − µT 1 ⋅ Φ1i 

 1 + ζT ⋅ µT 2 ⋅ Φ 2 i  Ey cpd2i : = E2 mat ⋅   1 − µT 2 ⋅ Φ 2 i 

A7.2.6.3  Orientation Parameter X

X 1i : =

E1measi − Ey cpd1i Ex cpd1i − Ey cpd1i 0 X1 =

0.606 0.587 0.576

Orientation parameter X



X2 i : =

E2 measi − Ey cpd2i Ex cpd2i − Ey cpd2i 0

0.8 X2 = 0.6

0.4 0

0.697 0.657 0.636

0.2 0.4 Fiber volume fraction PBT+SGF PA/PAT+SGF

A7.3 Nielsen Modification of Halpin–Tsai Equations with Respect to the Maximum Packing Fraction: Numerical Illustration Materials’ data:  µm: = 10−6⋅m  nm: = 10−9⋅m  GPa: = 109 ⋅ Pa:defining units Polymer matrix (e.g. PP): Emat: = 1.14⋅GPa   νmat: = 0.43   Gmat : =

Emat    Gmat: = 0.399 °GPa 2 ( 1 + νmat )

397

Polymers and Short Fibers

Fibers (e.g. E-glass; typical diameter 10 μm): Efib: = 77.0⋅GPa   νfib: = 0.20   Gfib : =

Efib    Gfib: = 32.083 °GPa 2 ( 1 + νfib ) L D

L: = 1⋅mm   D: = 8 ⋅ 10−3 ⋅ mm   α : = Φmax: = 0.6 Φ: = 0,0.01..Φmax :fiber volume ratio range

: aspect ratio    α = 125 Flexural modulus (GPa) for short glass fibers filled PP Average data various suppliers SGE %wt

A7.3.1  Maximum Packing Functions [Lewis and Nielsen, J. Appl. Polym. Sci., 14, 1449, 1970]

0  10   15  20 Data: =   30   40  50   60

 1 − Φ max  F1(Φ) : = 1 +  ⋅Φ  Φ max 2 

F2(Φ) :=

1   − Φ ⋅ Φ max   ⋅  1 − exp   Φ max − Φ   Φ 

0 0.0337 0.05405 0.06865 0.1017 0.1318 0.1605 0.1952

1.152  1.87  3.19   3.34  5.36   7.1  8.964   18.07 

1

2

0.5

1.5

1 0

Flex Mod GPA

Φ %wt

0.2

Φ F1(Φ)

0

0.4 F 2(Φ)

  

0

0.2 F1 (Φ).Φ

Φ

0.4 F 2 (Φ).Φ

Φ

Φ: = 0.75⋅Φmax  Φ2c: = Maximize (F2,Φ)  Φ2 c := 0.496 ⇐ position of maximum

of F 2 function

398

Filled Polymers

A7.3.2 Longitudinal (Tensile) Modulus E 11

Φ: = 0,0.01…Φmax

L ζL := 2 ⋅ D

 Efib   E  − 1 mat µ:=  Efib   E  + ζL mat

 1 + ζL ⋅ µ ⋅ Φ  Ex1cpd (Φ) := Emat ⋅   1 − µ ⋅ F 1(Φ) ⋅ Φ 

 1 + ζL ⋅ µ ⋅ Φ  Ex2 cpd (Φ) := Emat ⋅   1 − µ ⋅ F 2(Φ) ⋅ Φ 



30 Longitudinal/flexural modulus, GPa

Longitudinal/flexural modulus, GPa

100

Φmax

50

0 0

0.5 Fiber volume fraction F1(phi)

F2(phi)

A7.3.3 Transverse (Tensile) Modulus Ey

20

10

0

1 PP+SGF

0.12

 

0

0.1 0.2 Fiber volume fraction F1(phi)

F2(phi)

A7.3.4  Shear Modulus G



ζT(Φ) : = 2 + 40 ⋅ Φ10

ζG(Φ) : = 1 + 40 ⋅ Φ10



 Efib   E  − 1 mat µT (Φ) :=  Efib   E  + ζT (Φ) mat

 Gfib   G  − 1 mat µG(Φ) :=  Gfib   G  + ζG(Φ) mat

 1 + ζT ( Φ ) ⋅ µT ( Φ ) .Φ  Ey cpd (Φ) : = E mat ⋅   1 − F 1( Φ ) ⋅ µT ( Φ ) ⋅ Φ 

PP+SGF

 1 + ζG(Φ) ⋅ µG(Φ).Φ  Gcpd (Φ) : = Gmat ⋅   1 − F 1(Φ) ⋅ µG(Φ) ⋅ Φ 

399

Polymers and Short Fibers

30 Shear modulus, GPa

Transverse modulus, GPa

100

20

50

10

0

0

0.2 0.4 Fiber volume fraction

0

0.6

0

0.2 0.4 Fiber volume fraction

 

0.6

A7.4 Mori–Tanaka’s Average Stress Concept: Tandon–Weng Expressions for Randomly Distributed Ellipsoidal (Fiber-like) Particles: Numerical Illustration Materials’ data:  µm: = 10−6 ⋅ m  nm: = 10−9 ⋅ m  GPa: = 109 ⋅ Pa: defining units Polymer matrix (e.g., PP): Emat: = 1.14 ⋅ GPa  νmat: = 0.43   Gmat : =

Emat   Gmat: = 0.399 °GPa 2 ( 1 + νmat )

Fibers (e.g., E-glass; typical diameter 10 μm): Efib: = 77.0 ⋅ GPa   νfib: = 0.20   Gfib : =

L: = 1 ⋅ mm   D: = 50 ⋅ 10−3   α : =

Efib    Gfib: = 32.083 °GPa 2 ( 1 + νfib ) L D

: aspect ratio    α = 20

Φ: = 0,0.01..1   : fiber volume ratio range

A7.4.1 Eshelby’s Tensor (Depending on Matrix Poisson’s Ratio and Fibers Aspect Ratio only)

fα : =

(α

α 2

− 1)

3

⋅ α ⋅

(α 2 − 1) − ac osh (α )

fα = 0.993

400





Filled Polymers

S1111 : =

S 2222 : =

1 3 ⋅ α2 − 1  3 ⋅ α2    ⋅ fα  ⋅ 1 − 2 ⋅ νmat + 2 −  1 − 2 ⋅ νmat + 2 α −1  2 ⋅ ( 1 − νmat )  α − 1  

  3 ⋅ α2 1 9 ⋅ fα + ⋅ 1 − 2 ⋅ νmat − 8 ⋅ ( 1 − νmat ) ⋅ ( α 2 − 1) 4 ⋅ ( 1 − νmat )  4 ⋅ ( α 2 − 1)  S 3333 : = S 2222



S 2233 : =

    α2 1 3 ⋅ fα  ⋅ − 1 − 2 ⋅ νmat +  2 2 4 ⋅ ( 1 − νmat )  2 ⋅ ( α − 1)  4 ⋅ ( α − 1)   S 3322 : = S 2233



S 2211 : =

1 −α 2  3 ⋅ α2  − ( 1 − 2 ⋅ νmat ) ⋅ fα + ⋅ 2 ⋅ ( 1 − νmat ) ⋅ ( α 2 − 1) 4 ⋅ ( 1 − νmat )  α 2 − 1  S 3311 : = S 2211



S 1122 : =

  −1 1  1 3  1 − 2 ⋅ νmat + 2 + ⋅ 1 − 2 ⋅ νmat + ⋅ fα 2 ⋅ (1 − νmat )  α − 1  2 ⋅ (1 − νmat )  2 ⋅ (α 2 − 1)  S1133 : = S 1122



S 2323 : =

    α2 1 3 ⋅ fα  ⋅ + 1 − 2 ⋅ νmat −  2 2 4 ⋅ ( 1 − νmat )  2 ⋅ ( α − 1)  4 ⋅ ( α − 1)   S 3232 : = S 2323



S1212 : =

3 ⋅ ( α 2 + 1)    α2 + 1 1  1 ⋅ fα  ⋅ 1 − 2 ⋅ νmat − 2 − ⋅ 1 − 2 ⋅ νmat −  2 α −1  α −1 2  4 ⋅ ( 1 − νmat )   S1313 : = S1212

A7.4.2 Materials’ Constants (i.e., Not Depending on Fiber Volume Fraction)



D1 := 1 + 2 ⋅

Gfib − Gmat νfib ⋅ Efib νmat ⋅ Emat − (1 + νfib ) ⋅ (1 − 2 ⋅ νfib ) (1 + νmat ) ⋅ (1 − 2 ⋅ νmat )

D1 = 4.346

401

Polymers and Short Fibers

νmat ⋅ Emat

(1 + νmat ) ⋅ (1 − 2 ⋅ νmat )

+ 2 ⋅ Gmat



D2 : =



νmat ⋅ Emat (1 + νmat ) ⋅ (1 − 2 ⋅ νmat ) D3 : = νmat ⋅ Emat νfib ⋅ Efib − 1 ν ⋅ 1 − 2 ⋅ ν 1 ν + + ( fib ) ( ( mat ) ⋅ (1 − 2 ⋅ νmat ) fib )

νmat ⋅ Emat νfib ⋅ Efib − (1 + νfib ) ⋅ (1 − 2 ⋅ νfib ) (1 + νmat ) ⋅ (1 − 2 ⋅ νmat )

D2 = 0.171

D3 = 0.129

A7.4.3 Materials and Volume Fraction Depending Constants

B1 (Φ): = Φ ⋅ D1 + D2 + ( 1 − Φ ) ⋅ ( D1 ⋅ S1111 + 2 ⋅ S2211 )



B2 (Φ) : = Φ + D3 + ( 1 − Φ ) ⋅ ( D1 ⋅ S1122 + S2222 + S2233 )



B3 (Φ) := Φ + D3 + ( 1 − Φ ) ⋅ [ S1111 + ( 1 + D1 ) ⋅ S2211 ]



B4 (Φ) := Φ ⋅ D1 + D2 + ( 1 − Φ ) ⋅ ( S1122 + D1 ⋅ S2222 + S2233 )



B5 (Φ) := Φ + D3 + ( 1 − Φ ) ⋅ ( S1122 + S2222 + D1 ⋅ S2233 )



A1 (Φ) := D1 ⋅ ( B4 (Φ) + B5 (Φ)) − 2 ⋅ B2 (Φ)



A2 (Φ) := ( 1 + D1 ) ⋅ B2 (Φ) − ( B4 (Φ) + B5 (Φ))



A3 (Φ) := B1 (Φ) − D1 ⋅ B3 (Φ)



A4 (Φ) := ( 1 + D1 ) ⋅ B1 (Φ) − 2 ⋅ B3 (Φ)



A5 (Φ) :=



A(Φ) := 2. B2 (Φ) ⋅ B3 (Φ) − B1 (Φ) ⋅ ( B4 (Φ) + B5 (Φ))

(1 − D1 )

B4 (Φ) − B5 (Φ)

402

Filled Polymers

A7.4.4  Calculating the Longitudinal (Tensile) Modulus E11 E11 (Φ) :=

Emat ( A (Φ) + 2 ⋅ νmat ⋅ A2 (Φ)) 1+ Φ⋅ 1 A(Φ)

  

E11 ( 0.2 ) = 9.286  GPa

upper bound: Eup (Φ) := Emat ⋅ ( 1 − Φ ) + Efib ⋅ Φ

Fiber aspect ratio : α = 20

lower bound : Elow ( Φ ) :=



Emat ⋅ Efib Efib ⋅ ( 1 − Φ ) + Emat ⋅ Φ

100

50

0

E11 is the most sensitive elastic constant to the fiber aspect ratio; when the aspect ratio becomes larger than 50, the Mori–Tanaka’s average stress approach gives E11 not much different from the upper bound prediction 0

0.5 Φ

1

E11(Φ).GPa–1

Elow(Φ).GPa–1 Eup(Φ).GPa–1

A7.4.5  Calculating the Transverse (Tensile) Modulus E22

E22 (Φ) :=

Emat

[ −2 ⋅ νmat ⋅ A3 (Φ) + (1 − νmat ) ⋅ A4 (Φ) + (1 + νmat ) ⋅ A5 (Φ) ⋅ A(Φ)] 1+ Φ⋅ 2 A(Φ)

E22(0.2) = 1.907 °GPa

403

Polymers and Short Fibers

100

50

00

0.5 Φ E22(Φ).GPa–1 Elow(Φ).GPa–1 Eup(Φ).GPa–1

1

The transverse modulus if not really sensitive to fiber aspect ratio

A7.4.6  Calculating the (In-Plane) Shear Modulus G12     Φ  G12 (Φ) := Gmat ⋅ 1 + Gmat  + 2 ⋅ (1 − Φ ) ⋅ S1212   Gfib − Gmat 

s1212 = 0.247

G12 ( 0.2 ) = 0.594° GPa

upper bound: Gup (Φ) := Gmat ⋅ (1 − Φ ) + Gfib ⋅ Φ



lower bound : Glow (Φ) :=



Gmat ⋅ Gfib Gfib ⋅ ( 1 − Φ ) + Gmat ⋅ Φ

40 30 20 10 0

0

0.5 Φ G12(Φ).GPa–1 Glow(Φ).GPa–1 Gup(Φ).GPa–1

1

The shear modulus if not really sensitive to fiber aspect ratio

404

Filled Polymers

A7.4.7  Calculating the (Out-Plane) Shear Modulus G23

  Φ 1 +  G mat G23 (Φ) : = Gmat ⋅  + 2 ⋅ (1 − Φ ) ⋅ S2323    Gfib − Gmat  

s2323 = 0.28

G23 ( 0.2 ) = 0.572° GPa

upper bound: Gup (Φ) := Gmat ⋅ (1 − Φ ) + Gfib ⋅ Φ



lower bound : Glow (Φ) : =



Gmat ⋅ Gfib Gfib ⋅ ( 1 − Φ ) + Gmat ⋅ Φ

40

30

20

10

0

0

0.5 Φ

1

G12(Φ).GPa–1 Glow(Φ).GPa–1 Gup(Φ).GPa–1

A7.4.8  Comparing with Experimental Data Flexural modulus (GPa) for short glass fibers filled PP

Rem: flexural modulus = tensile modulus × (gyration radius)2

405

Polymers and Short Fibers

  Average data various suppliers

Φ % wt

SGF %wt 0  10   15  20 Data1: =   30   400  50   60

0 0.0337 0.05405 0.06865 0.1017 0.1318 0.1605 0.1952

Flex Mod GPa 1.152  1.87  3.19   3.34  5.36   7.1  8.964   18.07 

Data single supplier

SGF %wt 0  10   13 Data2: =   20  30   40

Φ % wt

0 0.03375 0.04335 0.0656 0.0959 0.126

Flex Mod GPa 1.1303  2.309  2.757   3.654  5..329   6.756 

20

10

0 0

0.1 0.2 Φ, Datal<1>, Data2<1> E11(Φ).GPa–1

Datal<2> <2>

Datal

The model meets reasonnably well experimental data up to around 0.10 volume fraction and with a fiber ratio of around 20

406

Filled Polymers

A7.4.9 Tandon–Weng Expressions for Randomly Distributed Spherical Particles: Numerical illustration Materials’ data:   nm: = 10−9⋅m   GPa: = 109⋅Pa   : defining units Polymer matrix: Emat: = 2.76⋅GPa   νmat: = 0.35   Gmat : =

E mat    Gmat = 1.022 °GPa 2 ( 1 + νmat )

Filler: Efill: = 72.4⋅GPa   νfill: = 0.20   Gfill : =

E fill    Gfill = 30.167  °GPa 2 ( 1 + νfill )

Φ: = 0,0.01 .. 1   : filler volume ratio range

A7.4.9.1  Eshelby’s Tensor (Depending on Matrix Poisson’s Ratio Only)

S1111 :=



5 νmat ⋅ −1 4 − 5 ⋅ νmat 7 − 5 ⋅ νmat    S1122 : =    S1212 : = ⋅ − ⋅ ( 1 − νmat ) 15 1 ν 15 ( mat ) 15 ⋅ ( 1 − νmat )

s2222 : = s1111

s1313 : = s1212

s3333 : = s1111

s2323 : = s1212

s1133 : = s1122

s3322 : = s2233

s2233 : = s1122

s3232 : = s2323

s3311 : = s1122

s3131 : = s1212

s2211 : = s3311

A7.4.9.2 Materials’ Constants (i.e., Not Depending on Filler Volume Fraction)

D1 : = 1 + 2 ⋅

Gfill − Gmat νfill ⋅ Efill νmat ⋅ Emat − 1 ν ⋅ 1 − 2 ⋅ ν ν + + 1 ( fill ) ( ( mat ) ⋅ (1 − 2 ⋅ νmat ) fill )

D1 = 4.288

407

Polymers and Short Fibers

νmat ⋅ Emat

D2 : =

(1 + νmat ) ⋅ (1 − 2 ⋅ νmat )

+ 2 ⋅ Gmat

νfill ⋅ Efill νmat ⋅ Emat − (1 + νfill ) ⋅ (1 − 2 ⋅ νfill ) (1 + νmat ) ⋅ (1 − 2 ⋅ νmat )

νmat ⋅ Emat (1 + νmat ) ⋅ (1 − 2 ⋅ νmat ) D3 : = νmat ⋅ Emat νfill ⋅ Efill − 1 + ν ⋅ 1 − 2 ⋅ ν 1 ν + ( fill ) ( ( mat ) ⋅ (1 − 2 ⋅ νmat ) fill )

D2 = 0.25

D3 = 0.135

A7.4.9.3  Materials and Volume Fraction Depending Constants

B1 (Φ): = Φ ⋅ D1 + D2 + ( 1 − Φ ) ⋅ ( D1 ⋅ S1111 + 2 ⋅ S2211 )



B2 (Φ) := Φ + D3 + ( 1 − Φ ) ⋅ ( D1 ⋅ S1122 + S2222 + S2233 )



B3 (Φ) := Φ + D3 + ( 1 − Φ ) ⋅ [ S1111 + ( 1 + D1 ) ⋅ S2211 ]



B4 (Φ) := Φ ⋅ D1 + D2 + ( 1 − Φ ) ⋅ ( S1122 + D1 ⋅ S2222 + S2233 )



B5 (Φ) := Φ + D3 + ( 1 − Φ ) ⋅ ( S1122 + S2222 + D1 ⋅ S2233 )



A1 (Φ) := D1 ⋅ ( B4 (Φ) + B5 (Φ)) − 2 ⋅ B2 (Φ)



A2 (Φ) := ( 1 + D1 ) ⋅ B2 (Φ) − ( B4 (Φ) + B5 (Φ))



A3 (Φ) := B1 (Φ) − D1 ⋅ B3 (Φ)



A4 (Φ) : = ( 1 + D1 ) ⋅ B1 (Φ) − 2 ⋅ B3 (Φ)    A5 (Φ) :=



A ( Φ ) := 2.B2 ( Φ ) ⋅ B3 ( Φ ) − B1 ( Φ ) ⋅ ( B4 ( Φ ) + B5 ( Φ ))

(1 − D1 ) B4 (Φ) − B5 (Φ)

408

Filled Polymers

A7.4.9.4  Calculating the Tensile Modulus E

E (Φ) :=

Emat

( A (Φ) + 2 ⋅ νmat ⋅ A2 (Φ)) 1+ Φ⋅ 1

   Guth and Gold : EGG (Φ) := Emat ⋅ (1 + 2.5 ⋅ Φ + 14.1 ⋅ Φ 2 )

A(Φ)

E ( 0.2 ) = 3.777 GPa    lower bound :

Elow (Φ) :=

Emat ⋅ Efill Efill ⋅ ( 1 − Φ ) + Emat ⋅ Φ

Tensile modulus, GPa

80

60

40

20

0 0

0.5 Filler volume fraction

1

Rem: the longitudinal and transverse (tensile) moduli E11 and E22 are identical when particles are spherical

Mori-Tanaka's average stress Lower bound modulus Guth & Gold equation

A7.4.9.5  Calculating the Shear Modulus G     Φ    G(Φ) : = Gmat ⋅ 1 + Gmat  + 2 ⋅ ( 1 − Φ ) ⋅ S1212   Gfill − Gmat 

G ( 0.2 ) = 1.528 GPa   lower bound :

Guth and Gold : GGG (Φ) := Gmat ⋅ ( 1 + 2.5 ⋅ Φ + 14.1 ⋅ Φ 2 )

Glow (Φ) :=

Gmat ⋅ Gfill Gfill ⋅ ( 1 − Φ ) + Gmat ⋅ Φ

409

Polymers and Short Fibers

Shear modulus, GPa

40

30

20

Rem: the in-plane and outplane shear moduli G12 and G23 are identical when particles are spherical

10

0

0

0.5 Filler volume fraction

1

Mori-Tanaka's average stress Lower Bound modulus Guth & Gold equation

A7.5  Shear Lag Model: Numerical illustration Materials’ data:  µm: = 10−6⋅m  nm: = 10−9⋅m  GPa: = 109⋅Pa: defining units Polymer matrix (e.g. PP): Emat: = 1.14⋅GPa  νmat: = 0.43  Gmat : =

Emat   Gmat: = 0.399 °GPa 2 ( 1 + νmat )

Fibers (e.g. E-glass; typical diameter 10 μm): Efib: = 77.0⋅GPa  νfib: = 0.20  Gfib : =

L: = 1⋅mm  D: = 50⋅10−3⋅mm  α : =

L D

Efib   Gfib: = 32.083 °GPa 2 ( 1 + νfib )

: aspect ratio   α = 20  R : =

D 2

410

Filled Polymers

Fibers’ arrangement Kr : =

π    2⋅ 3

2⋅π [ Cox ]     3

π 2⋅ 3

[ Hexagonal ]   

Φ: = 0,0.01...09  :fiber volume ratio range

: matrix shell radius



SGF %wt 0   10  15  Data: =  20  30   40   50  60

200

100

0

β(Φ) :=

Average data various suppliers

300 Mesophase thickness, microns

Mesophase thickness r(Φ) – R:

Kr Φ

0

[ Cube]

Flexural modulus (GPa) for short glass fibers filled PP

Longitudinal (tensile) modulus E11   r(Φ) := R ⋅

π 4

0.5 Fiber volume fraction

Φ %wt 0 0.0337 0.05405 0.06865 0.1017 0.1318 0.1605 0.1952

Flex Mod GPa 1.152   1.887  3.19   3.34  5.36   7.1   8.964  18.07 

1

 β(Φ) ⋅ L    2 ⋅ Gmat  tanh  2    r(Φ)     nL (Φ) : = 1 −  : efficiency factor R 2 ⋅ Efib ⋅ ln  β(Φ) ⋅ L    R    2

Shear lag model equation:  E11(Φ): = nL(Φ)⋅Φ⋅Efib + (1 − Φ)⋅Emat Longitudinal/flexural modulus, GPa

Longitudinal or flexural modulus, GPa

30

Shear lag model equation :

100

20

50

0

10

0

0.5 Fiber volume fraction

1

 

0 0

0.1 0.2 Fiber volume fraction

0.3

Index A Acetylene black, 24 Agglomerates voids, degree, 42 Aggregates, 21, 25–26 comparison, 41 diameter, size, 37 elementary particles, assessing, 40 flocculation process, 159–160 Medalia, concept of, 37–38 Medalia, occluded rubber, 42–43 solid fraction, 218 structure, 17 Aggregates flocculation/entanglement model, 218–221 aggregate, solid fraction of, 219–220 filler volume fraction, modulus, 220–221 strain dependence, modulus, 221–222 Alkoxy derivatives, 261 Aluminium tri-hydrate (ATH), 263 American Society for Testing Materials (ASTM), 27, 29–30, 35 Anchoring knots, 133 Antimony tri-oxide, 263 ASTM, see American Society for Testing Materials (ASTM) ATH, see Aluminium tri-hydrate (ATH) Attraction–repulsion mechanism, 154 B Bagley method, 108 Barrier properties, 292 BdR, see Bound rubber (BdR) BET method, see Brunauer, Emmet, Teller (BET) method Bingham fluid and Herschel–Bulkley equation, 101 Bis(triethoxysilylpropyl)tetrasulfane (TESPT), 236, 240 Blanc fixe, 56

Bound rubber (BdR) absolute value of, 116 concept, 108 content from thermogravimetry analysis (TGA), 118 toluene extraction kinetic method, 117 3D structure of, 109–110 effect of carbon black size on, 109–110 elastomer and storage on, 119 extraction kinetic method for assessment, 116 extraction time, 116–117 factors affecting, 114–121 kinetic aspects in formation of, 118 and NMR results, 110–111 50 phr Carbon Black SBR 1500 Compounds, 115 singular flow properties, 112 BR, see Butadiene rubber (BR) Brunauer, Emmet, Teller (BET) method, 17, 28 nitrogen adsorption, 238 Butadiene rubber (BR), 45 C Calcite, 11 Calcium carbonate, 285, 287–288 ground natural (GCC), 280 precipitated (PCC), 280, 290 Calcium oxide, 56 white minerals used as fillers for polymers, 57 Capillary rheometer, 98 Carbonates dolomite, 56 GCC and PCC, 55 grades, (PCC), 55 Carbon black (CB) aggregates comparison, 41 411

412

diameter, size, 37 elementary particles, assessing, 40 Medalia, concept of, 37–38 Medalia, occluded rubber, 42–43 ASTM designation vs. characterization data, 31–32, 35 characterization data, relationships, 84 data, source, 82–83 DBPA of, 129 di-butyl phthalate (DBP) absorption numbers (cDBPA), 103 dispersion on dynamic properties, 145–148 dynamic moduli, 141 dynamic properties, 133 EBA/CB composites, 177 effect on type G′ and tan δ, 144 electrical conductivity of, 176 fabrication processes aggregates, 21 for furnace black, 22 lamp black, manufacturing process, 23 methods, 22 smoke, 23–24 thermal black process, 24 thermo-oxidative processes, 21 filled compounds extraction kinetic data on N330, 117 morphology and nonlinear flow properties, 113 stress overshoot experiments on, 106 tridimensional representation of morphology of, 111 Filled SBR 1500 Compound, 92 Gent and Park equation, 136 Guth, Gold and Simha equation, 125 interactions between rubber segments and, 169 inter-aggregates distances face-centered cubic lattice model, 93–94 for optimal reinforcement, 94 junction distance, 135–136

Index

level and postextrusion swelling, 107 temperature on dynamic properties, 143 manufacturing process for gas black, 24 mass fractal dimension, 30, 33 aggregate, volume of particle, 39 and aggregates, 33 connectivity exponent, 36 critical filler level, 43–44 DPBA, 39–40 Ethylene-Propylene-Diene Monomer rubber (EPDM) compounds, 38 fractal geometry, 36–37, 41 fractal nature, 43 geometry description of aggregates, 34 HAF grade, 40 and mean DPA absorption number, 35 Medalia aggregates, concept of, 37–38 Medalia floc simulation approach, 38–39 Medalia occluded rubber, 42–43 pellets and agglomerates, 42 pellets mixing operations, 43 TEM/AIA study, 37 void ratio, 39 volume fraction, 41–42 well dispersed state, 43 X-ray scattering experiments, 35 mechanical properties, effect on, 125 Medalia’s floc simulation data, 85 Mooney and Rivlin equation, 130–131 optimum dispersion, 92–93 particles/aggregate, 89 postextrusion swelling bound rubber (BdR), 109–110 entrance pressure drop, 108 level and, 107 production properties and process, 25 properties of, 91 on rheological properties, effects of, 95

Index

rubber–CB interactions, 119 rubber–filler interactions and, 103 rubber reinforcement by, 148–151 shear viscosity complex cosmetic material, 98–99 filled BR compound, 98 function, effect on, 95 Herschel–Bulkley equation, 100 plots and level, 97 power law model with yield stress, 99–100 silica comparison, low strain dynamic properties, 239 comparison, surface chemistry, 236 tensile properties, 238 spatial organization and reinforcing character of, 26 specific gravity, 92 Stokes diameter, 93 strain amplification factor, 131 strain sweep tests, 178 structural aspects and characterization aggregates, 25–26 ASTM classification of, 29–30 Brunauer, Emmet, Teller (BET) method, 28 cetyltrimethylammonium bromide CTAB adsorption methods, 27–28 dibutylphthalate (DBP) method, 28 DPA absorption method, 28 elementary analysis, 26 elementary particles and structure of aggregates, size, 28–29 rubber reinforcement, 24–25 shapes of, 25 standard characterization methods, 27 tread and carcass tire applications, 30 surface energy aspects energetic sites, 47–48 esterification of, 48

413

lamp and gas, chemical functions detection, 44 new equilibrium gas adsorption techniques, 46–47 percolation level, 46 probe, 46 rubber–filler interactions, 45 rubber grade, 44 surface activity, 44 ToF-SIMS and XPS, 45 Toom model, 48 surface energy components for, 36 temperature effect and tensile stress at break, 127 theoretical concept for, 162 type and temperature on damping properties of SSBR compounds, 145 tan δ of SSBR compounds, 142 usages, 21–22 viscosity and levels of, 121–122 volume fraction, 41–42, 92, 134 and yield stress, 105 Young’s modulus, 130 Carbon black pellets, 42 Carbon fibers aramid fibers polypara-phenylene terephtalamide fibers, 71–72 tensile strength and modulus, 74 PAN-fibers and MPP-fibers, 71 Carbon fibers reinforced epoxy resins, 1 Carbon nanotubes, 5 Carboxylated nitrile rubber (XNBR), 382 Carcass tire applications, 30 Carreau equation for shear viscosity function of polymer melts, 278 Carreau–Yasuda model, 268–269, 278 virgin polystyrene (PS) data, fitting, 324, 326–330 assembly result, 329–332 Casson model, 101 for yield stress fluids, 102 Cathetometer, 96 CCA, see Cluster-cluster-aggregation (CCA) model

414

cDBP absorption test maximum volume fraction, 188 Cellulose derivatives, 13 Cetyltriethylammonium bromide (CTAB), 17 adsorption methods, 27–28 Characteristic time, effect of, 320 Chlorosilanes, 247 Chlorosulfonated polyethylene compounds, 260 Chopped natural fibers, 370 Clay/polymer nanocomposites, 4 Clay(s), 300 and chemical grafting of, 50–51 and elastomers grades of, 257–259 rubber extrusion and calendering, 257 exfoliated, 301 minerals, 11 one-step and two-step grafting technique, 50 mechanisms, 51 physical adsorption of functional polymers, 50 Cluster-cluster-aggregation (CCA) model, 158, 218–221 aggregate, solid fraction of, 219–220 filler volume fraction, modulus, 220–221 strain dependence, modulus, 221–222 Cohen–Addad percolation model maximum adsorbed polymer at saturation, 251 for silica-PDMS, 120 Coloring carbon blacks, 173 Commercial filled polypropylene compounds volume fraction, effect of mica, 300 talc, 295–296 Commercial PP–mica composites data, 299 Complex modulus, 255 Concept of maximum volume fraction, 266 Conductive carbon blacks, 175

Index

Connective filaments between rubber–filler aggregates, 112 Connectivity exponent, 274 Contour length, 246 Controlled stress rheometers, 98 Creep phenomena, 125 Creep resistance, 292 Critical shear rates, 269 Crushed dibutyl phtalate adsorption test (ASTM D-3493), 133 CTAB, see Cetyltriethylammonium bromide (CTAB) D DBP, see Di-butylphthalate (DBP) Deagglomeration-reagglomeration of filler aggregates, Kraus model, 155 Degussa Si264 TCPTS, 240 Si69 TESPT, 240 De Mattia fatigue tests, 260 Deutsches Institute für Normung (DIN), 27 Diamond, 265 Di-butylphtalate absorption (DPBA), 39–40, 43 Di-butyl phthalate (DBP), 17, 35, 37, 39 absorption numbers (cDBPA), 103 method, 28 Diene elastomers carbon black and (untreated) silica, comparison, 237–239 reinforcement, 239–246 DIN, see Deutsches Institute für Normung (DIN) Dolomite, 56 DPBA, see Di-butylphtalate absorption (DPBA) Dynamic moduli variation with strain amplitude, 141–142 with temperature, 142–144 Dynamic strain softening (DSS) fractional linear solid model, 222–223 Lion model, 167 mathematical aspects, 226 modeling, 223–225

Index

Maier and Göritz model, 227–232 percolation theory, 153 Dynamic stress softening (DSS), 141 effect of mixing duration on magnitude, 148 as filler network effect, 152 as filler–polymer network effect, 168 nonlinear viscoelastic properties, 151 origins of, 151–152 Payne effect, 151 stress/strain proportionality, 151 E Effective fiber length effect fiber-to-matrix modulus ratio, 391 matrix modulus, 390–391 volume fraction, 389–390 Elasticity dissipation structure, 114 Elastic modulus, 166, 212, 223, 230 effect of carbon black specific area on, 144 temperature effect, 143 Elastomers carbon black filled SBR compound, mechanical properties, 381 cellulose fibers and chopped aramid fibers industrial importance, 380–381 extrusion moving die technology, 380 fiber orientation, 382–383 glass fibers exhibit, 381–382 hexa(methoxymethyl)melamine (HMMM), 377 inter-aggregates distances face-centered cubic lattice model, 93–94 for optimal reinforcement, 94 natural fiber filled rubber composites, selected published works, 378 optimum dispersion, 92–93 properties of, 91 reinforcing elastomers, 375–376 resorcinol-HEXA system, 379 rubber-fiber bonding, 376

415

rubber matrix, 375–376 short fiber reinforced rubber composites, 379–380 in extruded rubber hoses, controlling, 381 special fiber–elastomer composites, 378–379 specific gravity, 92 Stokes diameter, 93 strain amplification in, 130 styrene butadiene copolymer (SBR), 376–377 and talc fractured surfaces, 260 ground talc, 260 technical constraints, 382 thermoplastic systems micromechanic models for (synthetic) fiber, 382 volume fraction of, 92 Elastomers and white fillers, 235 elastomers and clays, 257–260 elastomers and silica, 235 diene elastomers carbon black, comparison, 237–239 polydimethylsiloxane, 246–257 silanisation, reinforcement diene, 239–245 surface chemistry of, 235–237 elastomers and talc, 260–266 Engineering plastics, 301 ENR, see Epoxidized natural rubber (ENR) EPDM, see Ethylene-Propylene-Diene Monomer rubber (EPDM) compounds Epoxidized natural rubber (ENR), 45 EPR, see Ethylene-Propylene rubber (EPR) Equivalent circuit model, 172 Eshelby’s tensor, 399–400, 406 transformation tensor, 351–352 Ethylene-Propylene-Diene Monomer rubber (EPDM) compounds, 38 Ethylene-Propylene rubber (EPR), 45 Evonik, see Degussa Experimental results calculated data, comparison, 191–192

416

Extraction kinetic method for bound rubber assessment, 116 Extrudate swell, see Postextrusion swelling F Fabrication processes for carbon black (CB) aggregates, 21 for furnace black, 22 lamp black, manufacturing process, 23 methods, 22 smoke, 23–24 thermal black process, 24 thermo-oxidative processes, 21 Face-centered cubic lattice model, 93 FE, see Finite element (FE) calculations Fiber-polymer systems, 341 Fiber-reinforced composites shear-lag analysis, 389 Fiber-to-matrix modulus ratio, effect of, 391 Fibrous fillers, 343 Filled/composite polymer systems classification, 7 Filled compounds with carbon black (CB) extraction kinetic data on N330, 117 morphology and nonlinear flow properties, 113 stress overshoot experiments on, 106 tridimensional representation of morphology of, 111 Filled polymers, 1 fitting experimental data, 322–324, 326 observations, 323 and polymer nanocomposites, 3 preparing and using, 2 shear viscosity function, model, 319–321 filler fraction, effect of, 320 Filled rubber compounds, flow anisotropy effects in, 115 Filler–filler network considerations, 171 Filler network effect, 152; see also Dynamic stress softening (DSS)

Index

Fillers classification based on fabrication process and reinforcing activity, 12 particle sizes, 14 effect of viscosity, 123–124, 333 Yasuda parameter, 334 yield stress, 333 fraction effect, 320 incorporation effect, 125 loading and vulcanization, expected variation of modulus function during, 126 and pigment, distinguishing between, 13 refractive indices, 12 shapes, 11–12 shear viscosity of, 95 structures, 12 types, 11 use as color modifier, 12 Filler–thermoplastic systems, 267 Filler volume fraction modulus, function, 220–221 Finite element (FE) calculations, 353 Fitting data for filled polymer systems, 322–323 filled polystyrene (PS) shear viscosity, 326 on PDMS/silica compound, 309–310 silica weight fraction compound, 311–312 virgin polystyrene (PS) data with Carreau–Yasuda model, 324 FKM, see Fluoroelastomers (FKM) Flexural modulus, 404–405 Floc simulation, 137 approach, 38–39 Flow anisotropy effects, 114 in filled rubber compounds, 115 Flow properties, 263 Flow zone, 125 Fluoroelastomers (FKM), 52 Fluorohydric acid (HF), 56 Fractal geometry solid volume fraction, 41 Fractal scaling law, 33 Fractional linear solid model, 222–223

417

Index

Fumed silica combustion process, 58 density, 58 pyrolitic process, 56 G Gas black process, 23 Gaussian peaks, 47 Gaussian statistics, 251 GCC, see Ground calcium carbonate (GCC) General purpose (GP) resins, 4 GenFit function, 324 Glass fibers, 69 types of, 71 Glass fibers reinforced polyesters, 1 Grain effect, 114 Green tire, 235 Ground calcium carbonate (GCC), 55 Ground talc, 260 Guth, Gold, and Simha equation, 122 Guth and Gold approach, 138 H HAF, see High abrasion furnace (HAF) Halpin–Tsai equations curves calculations, 347 fiber aspect ratio, 350–351 fiber orientation distribution, 348 filler particle’s geometry, ζ parameter expressions for, 346 fitting experimental data with, 348 glass fibers, 348–349 mechanical properties, 346 Nielsen modification of Halpin, 396 longitudinal (tensile) modulus, 398 maximum packing functions, 397 transverse (tensile) modulus and shear modulus, 398–399 PBT and PA/PAT composites with short glass fibers, 349 short fibers filled systems, 347, 391–392 average orientation parameters from, 394–396

longitudinal (tensile) modulus, 392 random fiber orientation, modulus and adjustable parameter, 394 transversal (tensile) and shear modulus, 393 Halpin–Tsai model, 361 commercial SGF-filled polyamide 6 and 66 composites, flexural and tensile moduli of, 362 commercial SGF-filled polyamide 11 composites, flexural and tensile moduli of, 363 Hard clay, 257 Hard fibers, 72 Henry’s law, 46 Herschel–Bulkley equation, 270 Herschel–Bulkley model for yield stress fluids, 100–101 HEXA, see Hexamethylenetetramine (HEXA) Hexa(methoxymethyl)melamine (HMMM), 377 Hexamethylenetetramine (HEXA), 377 HF, see Fluorohydric acid (HF) High abrasion furnace (HAF), 28–29 grade, 40 High structure silica, 61 HMMM, see Hexa(methoxymethyl) melamine (HMMM) Hydrophilic fillers, 263 Hydrophilic polymers, 4 Hydrous kaolin, 49 I IGC, see Inverse gas chromatography (IGC) Injection molded fatigue test samples (ASTM D1708), 395 in situ polimerization, 4 Inter-aggregates distances face-centered cubic lattice model, 93–94 for optimal reinforcement, 94 Intermediate super abrasion furnace carbon black (ISAF), 45

418

International Organization for Standardization (ISO), 27 Intrinsic viscosity, 272, 335 Inverse gas chromatography (IGC), 46 ISAF, see Intermediate super abrasion furnace carbon black (ISAF) ISO, see International Organization for Standardization (ISO) Isolator–conductor transition, 172 J Junction gap width, 189 in CB, 136 Junction rubber, 133 K Kaolin, 257 grades, 245, 258–259 Kaolins, see Clay(s) Kraus deagglomeration– reagglomeration model dynamic strain softening (DSS), 196–202, 209 complex modulus, 202 modeling, 197–200 soft spheres interactions, 196 Kraus model deagglomeration–reagglomeration of filler aggregates, 155 G′ and G″ data, 157 mathematical aspects, 204–205 rate equilibrium between dislocated and flocculated aggregates, 163 SBR/carbon black compounds, 158 Ulmer modification, dynamic strain softening (DSS), 212–215 modeling G′, strain, 213–215 Kraus–Ulmer equations, 253, 255 Krieger–Dougherty equation, 272, 274–275 polynomial, relative viscosity variations, 276 Krieger–Dougherty relationship expansion, 335 numerical illustration, 336 Kuhn length, 246

Index

L Ladouce–Stelandre model, 171 Lamp black process, 23 Langmuir’s theory, 168 Lattice gas model, 172 Layered silicates, 3, 5 Lennard-Jones potential, 154 Liege, 12 Lignin, 75, 77 Lion model dynamic strain softening (DSS), 223–225 Local drag flow mechanisms of rubber–aggregate flow units, 114 Longitudinal (tensile) modulus, 392, 395, 398, 409 calculation, 402 Low strain amplitude dynamic properties silica reinforcement, 238 M Magnesium hydroxide, 263 Magnesium oxide fluorohydric acid (HF), 56 Maier and Göritz model, 170, 227 development, 227–228 experimental data, fitting, 230 mathematical aspects, 229 modeling G′, strain, 231 modeling G″, strain, 232–233 Mass fractal dimension of carbon black (CB), 30, 33 aggregate, volume of particle, 39 and aggregates, 33 connectivity exponent, 36 critical filler level, 43–44 DPBA, 39–40 ethylene-propylene-diene monomer rubber (EPDM) compounds, 38 fractal geometry, 36–37, 41 fractal nature, 43 geometry description of aggregates, 34 HAF grade, 40

Index

Medalia, floc simulation approach, 38–39 Medalia aggregates, concept of, 37–38 Medalia occluded rubber, 42–43 pellets and agglomerates, 42 pellets mixing operations, 43 TEM/AIA study, 37 void ratio, 39 volume fraction, 41–42 well dispersed state, 43 X-ray scattering experiments, 35 Mastics, 247 Matrix modulus, effect of, 390–391 Maximum packing fraction, 315 effect of, 317, 337 Medalia’s aggregate morphology approach, 86 void ratio and DPB absorption, relationship, 87–88 Medalia’s floc simulation, 193 data, 85 Mesophase concept, 150 Mesophase-pitch-precursor (MPPfibers), 71 Metamorphic rocks, 54 Methoxy group, 243 Mica filled PP compounds, application, 298 and thermoplastics, 297 volume fraction, effect of, 300 Michelin formulation, 246 Mineral fillers cost of, 2 industrial use, 264 Model equation, 326 Modeling G′, strain, 213–215, 231–232 Modeling G″, strain, 215–218, 232–233 Modeling, polymer systems shear viscosity function, 312–315 Modeling tan δ, γ0, 200–202 Models aggregates flocculation/ entanglement model, 218–221 aggregate, solid fraction of, 219–220 filler volume fraction, modulus, 220–221 strain dependence, modulus, 221–222

419

Carreau–Yasuda model, 268–269, 278 Casson model, 101 for yield stress fluids, 102 cluster-cluster-aggregation (CCA) model, 158, 218–221 aggregate, solid fraction of, 219–220 filler volume fraction, modulus, 220–221 strain dependence, modulus, 221–222 Cohen–Addad percolation model maximum adsorbed polymer at saturation, 251 for silica-PDMS, 120 comparison sharp variation, 317 smooth variation, 317 equivalent circuit model, 172 face-centered cubic lattice model, 93–94 fractional linear solid model, 222–223 Halpin–Tsai model, 361 commercial SGF-filled polyamide 6 and 66 composites, flexural and tensile moduli of, 362 commercial SGF-filled polyamide 11 composites, flexural and tensile moduli of, 363 Herschel–Bulkley model for yield stress fluids, 100–101 Kraus deagglomeration– reagglomeration model dynamic strain softening (DSS), 196–202, 209 Kraus model deagglomeration– reagglomeration of filler aggregates, 155 G′ and G″ data, 157 mathematical aspects, 204–205 rate equilibrium between dislocated and flocculated aggregates, 163 SBR/carbon black compounds, 158 Ulmer modification, dynamic strain softening (DSS), 212–215

420

Ladouce–Stelandre model, 171 lattice gas model, 172 Lion model dynamic strain softening (DSS), 223–225 Maier and Göritz model, 170, 227–232 development, 227–228 experimental data, fitting, 230 mathematical aspects, 229 modeling G′, strain, 231 modeling G″, strain, 232–233 Mori–Tanaka’s averaging hypothesis and derived models average strain approach, 352 fiber aspect ratio, 352–353 finite element (FE) calculations, 353 liquid fillers, 353 Poisson’s ratio, 352 thermoplastic polymer systems, flexural and tensile moduli comparison, 353–354 network junction (NJ) model, 134 shear lag models, 353, 355 efficiency factor, 355 fiber aspect ratio, 357 fiber lengths, distribution of, 358 packing ratio, 356 shear viscosity function model, 319–321 Toom model, 48 Vand cubic model, 122 van de Walle, Tricot and Gerspacher (VTG) model, 165 White–Wang model, 101 for carbon black filled compounds, 103 CB filled rubber compounds, 102 Mohs hardness, 265 gypsum, 265 scale, 266 soft minerals, 265 wollastonite, 265 Molten polymer, 3 Monte Carlo method, 167 Montmorillonite, 50 Mooney and Rivlin equation, 130–131 Mooney functional analysis, 272

Index

Mooney viscometer, 98 Mori–Tanaka’s average stress concept Eshelby’s tensor, 399–400 experimental data, comparison, 404–405 longitudinal (tensile) modulus calculation, 402 materials and volume fraction depending constants, 401 materials constants, 400–401 shear modulus calculation, 403–404 transversal (tensile) modulus calculation, 402–403 Mori–Tanaka’s averaging hypothesis and derived models average strain approach, 352 fiber aspect ratio, 352–353 finite element (FE) calculations, 353 liquid fillers, 353 Poisson’s ratio, 352 thermoplastic polymer systems, flexural and tensile moduli comparison, 353–354 Mullins effect, 127–128 N Nanocomposites commercial demand, 5 fundamental research on, 6 Hamed’s proposal, 6–7 Nanometer-size materials, 4–5 Natural fiber filled rubber composites selected published works, 378 Natural fibers cellulose structure, 74, 77 composition, 72 lignin, 75, 77 polymers matrix interactions, chemical approaches, 78 potential fillers for, 76 SEM microphotographs, 78 synthetic fibers, properties of, 72–73 Natural rubber (NR), 45, 382 pseudo-Newtonian plateau, 95

421

Index

Natural rubber (RSS1) compounds, 237 Natural silica neuburg silica, 52 quartz, 52 siliceous fillers, 53 diatomaceous earths, 53 white calcium silicate (CaSiO3), 52–53 Network junction (NJ) model, 134, 187 junction gap width, 189 theory, strain amplification factor, 190–194 typical calculations, 188 strain amplification factor experimental data, comparison, 194–196 Network junction (NJ) theory, 133 model development, 185–189 absorbed DBP, 185 NJ model, 185–187 strain amplification factor and, 139 Neuburg silica, 52 Newtonian plateau, 267 Nielsen modification of Halpin, 396 longitudinal (tensile) modulus, 398 maximum packing functions, 397 transverse (tensile) modulus and shear modulus, 398–399 Nonlinear fitting algorithm, 309, 311 NR, see Natural Rubber (NR); Natural rubber (NR) N330 SBR compound, 93 Nuclear magnetic resonance (NMR), 248 O OESBR/BR tread compound, 261 Organic ammonium salts, 4 Organic fillers chemical synthesis by, 12–13 of natural origin, 12 Organo-clays, 50–51 Organophilic clays, 4 Organophilicity, degree of, 237 Organo-silanes, 236, 243, 261 Bis(3-triethoxysilylpropyl) tetrasulfane (TESPT), 240

silanization efficiency alkoxy groups, effect of, 243 3-thiocyanatopropyl-triethoxy silane (TCPTS), 240 Orientation parameter, 396 P Packing fraction, 337 Packing of objects, 270 Parallelepiped samples, 348 Particle/matrix adhesion, 284 Payne effect, 141 PBT, see Polybutylene therephtalate (PBT) PCC, Precipitated Calcium Carbonate (PCC) PDMS, see Polydimethylsiloxane (PDMS) Percolation level, 46 Percolation theory, 153; see also Dynamic stress softening (DSS) Phyllosilicate, 3 Plan-plan rheometer, 96 Plastics organoclay additives, 3 Platy fillers, oriented, 260 Polyacrylonitrile (PAN-fibers), 71 Polyamides, 4, 360 Polybutylene therephtalate (PBT), 348 Polydimethylsiloxane (PDMS), 237, 246–247 adsorption kinetics of, 250 chains, 248 chlorosilanes, hydrolysis of, 247 gel, 252 Polyethylene, 17–18 Polymers adsorption kinetics, model, 308 cost of, 2 molecular weight, effect of, 308–310 PDMS/silica compound, fitting data, 309 natural fibers matrix interactions, chemical approaches, 78 potential fillers for, 76 polymer–glass fiber systems, 150

422

polymer-short fiber systems for industrial applications, 341 and short fibers elastomers tensile properties difference, 339 fiber-polymer systems, 341 fibers-to-matrix adhesion role, 341 fibrous fillers, 343 harmonic mixing rule, 340–341 upper and lower bound moduli curves, 340 Voigt average, 339–340 silicone polymer molecular weight, 308–310 silica adsorption kinetics of, 308–312 silica weight fraction, 310–312 and short fibers filler, 341 Polynomial equation, 275 Polyolefins, 2 Polyorganosiloxanes, 246 Polypropylene (PP), 287, 298 filler compounds, effect of talc volume fraction, 295–296 Polyvinyl chloride (PVC), 2, 18 Postextrusion swelling, 106 and entrance pressure drop in CB, 108 PP–CaCO3 composites, 281–282, 286–287 mechanical properties, 284, 289 Precipitated calcium carbonate (PCC), 55 Precipitated silica, 58 amorphous, grades of, 58 dispersion of, 61 high structure silica, 61 manufacturing process of, 58–60 properties of, 60 synthetic, types of, 62 Probe, 46 PVC, see Polyvinyl chloride (PVC) Q Quartz BET, specific surface area, 52

Index

R Reduced segmental mobility of rigid body with interactions, 150 Reinforcement fillers, 15 structure of, 17 polyethylene or polypropylene, 18 promoters, 240 relative variation of rubber compound properties, 15–16 thermoplastics, 17–18 Reinforcing elastomers, 375–376 Relative viscosity of suspensions comparing model equations, 273 Relaxation modulus function, 125 Rheology of suspensions of rigid particles, 315 Rigid particles suspensions rheology, models, 315–319 criteria, 315–316 Rigid PVC CaCO3 particle size effect, 290 Rigid spherical particles models, 125 Rubber, 2 dynamic properties as tire technology requirements, 140 elastic behavior of modeling, 190–191 matrix CB aggregates, 186 well dispersed state, 43 reinforcement, 24–25 silica silanisation reaction, type on, 244 technology, 91 Rubber–carbon black interaction chemical reticulation, 169 hard and soft regions, 129 stress–strain behavior, 131 as topological constraints effect, 120 Rubbery plateau, 125 S SAXS, see Small-angle X-ray scattering (SAXS) SBR, see Styrene-butadiene rubber (SBR)

Index

Scanning tunneling microscopy (STM), 35 Sealants, 247 Sedimentary rocks, 54 SEM microphotographs, 78 SGF, see Short glass fibers (SGF) Shape factor, 271 Shear lag analysis, 389 models, 171, 409 analysis, 344–345 efficiency factor, 355 fiber aspect ratio, 357 fiber lengths, distribution of, 358 longitudinal (tensile) modulus E11, 409 packing ratio, 356 Shear modulus, 393, 398–399 calculation, 403–404, 408–409 Shear viscosity complex cosmetic material, 98–99 filled BR compound, 98 function, effect on, 95 Herschel–Bulkley equation, 100 plots and level, 97 power law model with yield stress, 99–100 Shear viscosity function capabilities, model filled polymers, 319–321 critical shear rate and viscosity, 314 effect on carbon black (CB), 95 modeling, polymer systems, 312–315 mathematical aspects, 313 Short E-glass fiber in different thermoplastics, effects of, 342 on heat distortion temperature of various thermoplastics, effects of, 343 on impact resistance of various thermoplastics, effects of, 343 Short fibers carbon black filled SBR compound, mechanical properties, 381 cellulose fibers and chopped aramid fibers industrial importance, 380–381

423

extrusion moving die technology, 380 fiber orientation, 382–383 glass fibers exhibit, 381–382 hexa(methoxymethyl)melamine (HMMM), 377 natural fiber filled rubber composites, selected published works, 378 natural origin commercial fibers-filled polypropylene composites, 371 wood flour and high stiffness of, 371 wood–polymer composites (WPC), 371–374 and polymers elastomers tensile properties difference, 339 fiber-polymer systems, 341 fibers-to-matrix adhesion role, 341 fibrous fillers, 343 filler, 341 harmonic mixing rule, 340–341 upper and lower bound moduli curves, 340 Voigt average, 339–340 reinforcing elastomers, 375–376 resorcinol-HEXA system, 379 rubber-fiber bonding, 376 rubber matrix, 375–376 short fiber reinforced rubber composites, 379–380 in extruded rubber hoses, controlling, 381 special fiber–elastomer composites, 378–379 styrene butadiene copolymer (SBR), 376–377 technical constraints, 382 thermoplastic systems micromechanic models for (synthetic) fiber, 382 Short fibers-filled polymer composites, micromechanic models Halpin–Tsai equations, 345 curves calculations, 347 fiber aspect ratio, 350–351

424

fiber orientation distribution, 348 filler particle’s geometry, ξ parameter expressions for, 346 fitting experimental data with, 348 glass fibers, 348–349 mechanical properties, 346 PBT and PA/PAT composites with short glass fibers, 349 short fibers-filled systems, 347 minimum fiber length homogeneous matrix, 344 load transfer, 344–345 shear-lag analysis, 345 Mori–Tanaka’s averaging hypothesis and derived models average strain approach, 352 average stress/strain concept, 351 Eshelby’s transformation tensor, 351–352 fiber aspect ratio, 352–353 finite element (FE) calculations, 353 liquid fillers, 353 moduli composite model, explicit relationships, 351 Poisson’s ratio, 352 shear lag models, 353, 355 efficiency factor, 355 fiber aspect ratio, 357 fiber lengths, distribution of, 358 packing ratio, 356 Short fibers-filled systems, 347, 391–392, 394–396 average orientation parameters longitudinal (tensile) modulus, 392, 395 orientation parameter, 396 transversal (tensile) modulus, 393, 396 random fiber orientation, modulus and adjustable parameter, 394 Short glass fibers (SGF), 341 aliphatic polyamides, filling, 360 commercial composites flexural and tensile moduli data for, 361 PA6, average suppliers’ data, 364–365

Index

PA11, average suppliers’ data, 367 PA66, average suppliers’ data, 366 polyamide 6 and 66 composites, flexural and tensile moduli of, 362 polyamide 11 composites, flexural and tensile moduli of, 362 residual stresses, 363, 368 commercial polypropylene, mechanical properties of, 360 E-glass fibers, 358, 360 filling polyamides with, 363–364 impact resistance, 361 thermoplastic polymers Bagley plots in capillary rheometry, 368 high fiber alignment, converging flow results in, 369–370 injection molding, 368–369 slit extrusion of, 369 Short synthetic fibers carbon fibers aramid fibers, 71–72 PAN-fibers and MPP-fibers, 71 glass fibers, 69 types of, 71 Silanes as coagents, benefits, 240 Silanisation, 262 efficiency, organo-silanes alkoxy groups, 243 reinforcing properties, effect of NR compounds, 244 silica, 239 silica, reaction rubber type, effect, 244 in situ, 241, 243, 245 ethanol formation, 243 problems, 241 steps silane condensation reaction, 242 silanol, alkoxy reaction, 242 Silanols, 245 Silica carbon black comparable series, 237 tensile properties, 238

Index

carbon black, effect of low strain dynamic properties, 239 fabrication processes fumed silica, 56, 58 precipitated silica, 58–62 modification, bi-functional organosilane, 241–242 network, 238 peculiar dynamic properties, 235 reinforcement low strain amplitude dynamic properties, 238 silanisation, 239–240, 243, 246 silanes, as coagents, 240 silica/PDMS systems, 247–249 adsorption, kinetics properties, 247 dynamic strain softening effect, modeling, 254–255 low strain dynamic properties, variation of, 256 silica/polysiloxane system, 253 surface chemistry of, 235–237, 249 carbon black, comparison, 236 surface energy aspects chemistry of, 68–69 components for, 70 dispersive and polar components, 69 particles, 69 weight fraction, effect of, 310–312 fitting data, 311–312 Silicates calcined clays calcination steps, 51 clay and chemical grafting of, 50–51 China clay, 49–50 hard and soft clays, distinction of, 49 montmorillonite, 50 one-step and two-step grafting technique, mechanism for, 51 physical adsorption of functional polymers, 50 polymer nanocomposites, 50 mica color, 52 muscovite, 51–52

425

talc lamellar surface, 51 Silicone polymers silica, adsorption kinetics of, 308–312 polymer molecular weight, 308–310 silica weight fraction, 310–312 Siloxane group, 236 chains, adsorption empirical model, 248 Simulation algorithms, 277 Small-angle x-ray scattering (SAXS), 35 Soft clays, 257 Soft spheres interactions, 196 Spring-and-dashpot system, 164 S-SBR, see Styrene butadiene copolymer (S-SBR) Stiffness (elasticity modulus), 292 STM, see Scanning tunneling microscopy (STM) Stokes diameter, 93 Storage effects, 118 Storage modulus strain dependence, 221–222 Strain amplification concept of Mullins and Tobin, 132 Strain amplification factor, 193–194 NJ theory, 190–194 elastic behavior of a rubber, modeling, 190–191 Strain crystallization effect of NR, 127 Strain sweep experiments, 213, 230 on SSBR compounds, 141 Strain sweep tests, 178 Stress overshoot experiments on carbon black filled compounds, 106 stress overshoot effect, 105 Structural aspects and characterization of CB aggregates, 25–26 ASTM classification of, 29–30 Brunauer, Emmet, Teller (BET) method, 28 cetyltrimethylammonium bromide CTAB adsorption methods, 27–28 dibutylphthalate (DBP) method, 28 DPA absorption method, 28

426

elementary analysis, 26 elementary particles and structure of aggregates, size, 28–29 rubber reinforcement, 24–25 shapes of, 25 standard characterization methods, 27 tread and carcass tire applications, 30 Styrene-butadience rubber (SBR), 45 Styrene butadiene copolymer (S-SBR) anionic polymerization, production, 246 Styrene-butadiene rubber (SBR) formulation, 91 Surface energy aspects of carbon black (CB) energetic sites, 47–48 esterification of, 48 lamp and gas, chemical functions detection, 44 new equilibrium gas adsorption techniques, 46–47 percolation level, 46 probe, 46 rubber–filler interactions, 45 rubber grade, 44 surface activity, 44 ToF-SIMS and XPS, 45 Toom model, 48 Surface properties, 263 Swollen rubber–filler gel, 118 Synthetic resins, 13 Synthetic silica, 53 ASTM classification, 64 characterization and structural aspects of, 62–63 IGC techniques, 64 precipitated silicic acids (silica) and active silicates, properties of, 54 property ranges for, 63 silicates, 64 suppliers data, 65–67 usage in, 54 T Talc, 260, 291 chemical modification, 261

Index

coarse grades, 262 and elastomers fractured surfaces, 260 ground talc, 260 magnesium silicate, crystalline form, 291 minerals, 11 myriad products, 292 platelets, basal surfaces, 261 PP Composites, data, 292–294 properties, 292 thermoplastics, application, 297 volume fraction, effect of, 295–296 Tandon–Weng expressions for randomly distributed spherical particles Eshelby’s tensor, 406 materials and volume fraction depending constants, 407 materials constants, 406–407 shear modulus, calculation, 408–409 tensile modulus, calculation, 408 Tan δ variation, with strain amplitude and temperature, 142 TCPTS, see 3-Thiocyanatopropyltriethoxy silane (TCPTS) TEM/AIA, see Transmission electronmicroscopy/automatedimage-analysis (TEM/ AIA) study Tensile measurements (ASTMD638), 395 Tensile modulus, 126 Tensile modulus calculation, 408 Tensile stress softening (TSS), 127–128 TESPT, see Bis(triethoxysilylpropyl) tetrasulfane (TESPT) Tetra organic phosphonium solutions, 4 Theoretical model approximate fitted equation, comparison, 192 Thermal black process, 24 Thermo-activated reactions, 243 Thermo-oxidative processes, 21

427

Index

Thermoplastics, 172 effect of CB on electrical conductivity, 175–177 rheological properties of, 173–175 materials polyamides, 360 processors, 358–359 natural origin commercial fibers-filled polypropylene composites, 371 lignocellulosic fibers, polymer composites preparation, 370 moisture diffusion, 370 wood flour and high stiffness of, 371 wood–polymer composites (WPC), 371–374 thermoplastic polymer systems flexural and tensile moduli, comparison, 353–354 Thermoplastics and white fillers, 262 and calcium carbonates, 280 and clay(s), 300–301 and mica, 297–300 talc, 291 application, 297 properties, 292 3-Thiocyanatopropyl-triethoxy silane (TCPTS), 240 Tightly BdR, 112 Time-of-flight secondary ion mass spectrometry (ToF-SIMS), 45 Tire technology requirements for rubber dynamic properties, 140 Titanium dioxide, 288 ToF-SIMS, see Time-of-flight secondary ion mass spectrometry (ToF-SIMS) Toom model, 48 Transmission electron-microscopy/ automated-image-analysis (TEM/AIA) study, 37 Transversal (tensile) modulus, 393, 396 calculation, 402–403 Transverse (tensile) modulus, 398–399 Tread tire applications, 30 TSS, see Tensile stress softening (TSS)

Tunnel effects, 173 Two-roll laboratory mill, 260 U Unbound rubber, 112 Upper and lower bound moduli curves, 340 V Vand cubic model, 122 Van der Waals forces, 152 van de Walle, Tricot and Gerspacher (VTG) model, 165 Vapor-grown carbon fibers (VGCF), 176–177 Vegetal fibers, 12 VGCF, see Vapor-grown carbon fibers (VGCF) Vicinal silanols, 236, 243 condensation, 236 Vinyl-based polymer, 3 Viscous modulus, 166, 207, 212, 224, 230 Void ratio, 39 Medalia classification and assumption, 43 Volume Packing Fraction, 277 VTG, see van de Walle, Tricot and Gerspacher (VTG) model Vulcanizable elastomers, 247 Vulcanization, 56 silica–rubber bonding, 241–242 system, 240 Vulcanized PDMS/silica systems dynamic properties, 253 W Western Europe consumption of rubbers, 2 White fillers carbonates dolomite, 56 GCC and PCC, 55 grades, PCC, 55 concept of maximum volume fraction, 266–268

428

and elastomers, 235 mineral fillers barium sulfate, 56 calcium oxide, 56–57 zinc and magnesium oxide, 56 natural silica neuburg silica, 52 quartz, 52 siliceous fillers, 53 white calcium silicate (CaSiO3), 52–53 silica, surface energy aspects chemistry of, 68–69 components for, 70 dispersive and polar components, 69 particles, 69 silica fabrication processes fumed silica, 56, 58 precipitated silica, 58–62 silicates calcined clays and talc, 51 clays, 49–51 mica, 51–52 synthetic silica, 53 characterization and structural aspects of, 62–67 precipitated silicic acids (silica) and active silicates, properties of, 54 usage in, 54 thermoplastics properties, affect, 263 White–Wang model, 101 for carbon black filled compounds, 103 CB filled rubber compounds, 102 Wollastonite filled compounds, 53 fluoroelastomers (FKM), 52 grades, 52

Index

Wood flour, 12 Wood–polymer composites (WPC), 371 flexural properties, 373–374 formulation, 374 injection molding and compression molding, 372–373 maleic anhydride-grafted, 375 market amounts in North America, 11 PP based composites, (pine) wood flour effect of, 374 production and processing, problems, 373 wood-filled thermoplastic composites, 372 WPC, see Wood-polymer composites (WPC) X XNBR, see Carboxylated nitrile rubber (XNBR) XPS, see X-ray photoelectron spectroscopy (XPS) X-ray photoelectron spectroscopy (XPS), 45 Y Yield stress data for filled rubber compounds, 104 Z Zero-shear viscosity, 278 Zinc oxide, 56 as weight predispersion in EVA, 92