178 Advances in Polymer Science
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Polymeric and Inorganic Fibers
With contributions by J. J. M. Baltussen · P. den Decker · T. Ishikawa · M. G. Northolt · S. J. Picken · R. Schlatmann
2 3
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Library of Congress Control Number: 2004117780 ISSN 0065-3195 ISBN-10 3-540-24016-0 Springer Berlin Heidelberg New York ISBN-13 978-3-540-24016-7 Springer Berlin Heidelberg New York DOI 10.1007/b104206 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, re-printing, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: KünkelLopka GmbH, Heidelberg/design & production GmbH, Heidelberg Typesetting: Fotosatz-Service Köhler GmbH, Würzburg Printed on acid-free paper 02/3141 xv – 5 4 3 2 1 0
Editorial Board Prof. Akihiro Abe
Prof. Jean-François Joanny
Department of Industrial Chemistry Tokyo Institute of Polytechnics 1583 Iiyama, Atsugi-shi 243-02, Japan
[email protected] Physicochimie Curie Institut Curie section recherche 26 rue d’Ulm 75248 Paris cedex 05, France
[email protected] Prof. A.-C. Albertsson Department of Polymer Technology The Royal Institute of Technology S-10044 Stockholm, Sweden
[email protected] Prof. Ruth Duncan Welsh School of Pharmacy Cardiff University Redwood Building King Edward VII Avenue Cardiff CF 10 3XF United Kingdom
[email protected] Prof. Hans-Henning Kausch EPFL SB ISIC GGEC J2 492 Bâtiment CH Station 6 CH-1015 Lausanne, Switzerland
[email protected] Prof. S. Kobayashi Department of Materials Chemistry Graduate School of Engineering Kyoto University Kyoto 615-8510, Japan
[email protected] Prof. Karel Dušek
Prof. Kwang-Sup Lee
Institute of Macromolecular Chemistry Academy of Sciences of the Czech Republic Heyrovský Sq. 2 16206 Prague 6, Czech Republic
[email protected] Department of Polymer Science & Engineering Hannam University 133 Ojung-Dong Daejeon 306-791, Korea
[email protected] Prof. Dr. W. H. de Jeu
Prof. L. Leibler
FOM-Institute AMOLF Kruislaan 407 1098 SJ Amsterdam, The Netherlands
[email protected] and Dutch Polymer Institute Eindhoven University of Technology PO Box 513 5600 MB Eindhoven, The Netherlands
Matière Molle et Chimie Ecole Supérieure de Physique et Chimie Industrielles (ESPCI) 10 rue Vauquelin 75231 Paris Cedex 05, France
[email protected] Prof. Timothy E. Long Department of Chemistry and Research Institute Virginia Tech 2110 Hahn Hall (0344) Blacksburg, VA 24061, USA
[email protected] VI
Editorial Board
Prof. Ian Manners
Dr. E. M. Terentjev
Department of Chemistry University of Toronto 80 St. George St. M5S 3H6 Ontario, Canada
[email protected] Cavendish Laboratory Madingley Road Cambridge CB 3 OHE United Kingdom
[email protected] Prof. Dr. Martin Möller
Prof. Brigitte Voit
Deutsches Wollforschungsinstitut an der RWTH Aachen e.V. Pauwelsstraße 8 52056 Aachen, Germany
[email protected] Institut für Polymerforschung Dresden Hohe Straße 6 01069 Dresden, Germany
[email protected] Prof. Gerhard Wegner Prof. Oskar Nuyken Lehrstuhl für Makromolekulare Stoffe TU München Lichtenbergstr. 4 85747 Garching, Germany
[email protected] Max-Planck-Institut für Polymerforschung Ackermannweg 10 Postfach 3148 55128 Mainz, Germany
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Preface
Contents
The Tensile Strength of Polymer Fibres M. G. Northolt · P. den Decker · S. J. Picken · J. J. M. Baltussen · R. Schlatmann . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Advances in Inorganic Fibers T. Ishikawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Author Index Volumes 101–178 . . . . . . . . . . . . . . . . . . . . . . . 145 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Adv Polym Sci (2005) 178: 1– 108 DOI 10.1007/b104207 © Springer-Verlag Berlin Heidelberg 2005
The Tensile Strength of Polymer Fibres M. G. Northolt ( R. Schlatmann 1
2 3
4
) · P. den Decker · S. J. Picken · J. J. M. Baltussen ·
Magellan Systems International, 8310 Shell Road, Richmond VA 23237, USA
[email protected] Teijin Twaron Research, P.O. Box 9600, 6800 TC Arnhem, The Netherlands Delft University of Technology, Polymer Materials & Engineering, Julianalaan 136, 2628 BL Delft, The Netherlands
[email protected] Akzo Nobel Chemicals Research Arnhem, P.O. Box 9300, 6800 SB Arnhem, The Netherlands
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Fracture of a Polymer Fibre . . . . . . . . . . . . . . . . . . . . . . . . Fracture Envelope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Shear Stress and Critical Shear Strain . . . . . . . . . . . . . . Distribution of the Strain Energy in a Fibre . . . . . . . . . . . . . . . Effect of Cracks on the Strength and the Relation with the Chain Orientation . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Polymer Fibre Considered as a Molecular Composite . . . . . . . . 2.5.1 Ultimate Strength of a Fibre . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Estimates of the Shear Modulus, Shear Strength and Ultimate Strength of Cellulose II, PpPTA and PIPD-HT Fibres . . . . . . . . . . . . . . . . 2.5.3 Theoretical and Experimental Relations between Strength and Modulus of PpPTA, PBO, PIPD-HT, POK, Cellulose II and PET Fibres . . . . . . . 2.6 Relation between the Concentration of the Spinning Solution and the Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1 2.2 2.3 2.4
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14 14 17 21
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23 27 29
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35
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43
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3.1 3.1.1 3.1.2 3.1.3 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.4 3.5
Relation between the Ultimate Strength and the Chain Length Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Forces Acting on the Chain . . . . . . . . . . . . . . . . . . . Force in the Cross Section of the Fibre . . . . . . . . . . . . . Ultimate Strength for Various Chain Length Distributions . . . Monodisperse Distribution . . . . . . . . . . . . . . . . . . . Flory Distribution . . . . . . . . . . . . . . . . . . . . . . . . Half-Gauss Distribution . . . . . . . . . . . . . . . . . . . . . Uniform Distribution . . . . . . . . . . . . . . . . . . . . . . Effect of Low Molecular Weight Fraction on Fibre Strength . . Theoretical Effect of Molecular Weight on the Modulus . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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46 46 46 48 53 55 55 58 60 61 63 67 70
4 4.1
Dependence of the Fibre Strength on the Time and the Temperature . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72 72
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2 4.2
M. G. Northolt et al.
. . .
73
4.4 4.5
Simple Derivation of the Lifetime of a Polymer Fibre under Constant Load . . . . . . . . . . . . . . . . . . . . . . . . . . . Relationship Between Strength and Load Rate Derived from the Eyring Reduced Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . Lifetime Relationship Derived from the Eyring Reduced Time Model . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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80 86 90
5 5.1 5.2 5.3
Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . Can Post-drawing Improve the Fibre Strength? . . . . . . . . . Effect of Residual Stress, Chemical Impurities and Degradation
. . . .
4.3
References
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. . . .
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. 91 . 91 . 99 . 103
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Abstract A theory of the tensile strength of oriented polymer fibres is presented. From an analysis of the observed fracture envelope it is shown that the criterion for fracture of the fibre is either a critical shear stress or a critical shear strain. Owing to the chain orientation distribution in the fibre, the initiation of fracture is likely to occur in domains whose symmetry axes have orientation angles in the tail of this distribution. By considering the fibre as a molecular composite, the tensile strength is calculated as a function of the modulus. The results are compared to the observed values of PET, POK, cellulose II, PpPTA, PBO and PIPD fibres. In addition, the relation between the ultimate strength and the chain length distribution is investigated. By using the critical shear strain as a fracture criterion in the Eyring reduced time model, relations are derived for the fibre strength as a function of the load rate, as well as for the lifetime under constant load. Moreover, this model predicts the dependence of the strength on the temperature. The theoretical relations are compared to the experimental results on PpPTA fibres. Keywords Polymer fibre · Strength · Chain length distribution · Creep fracture · Lifetime · Poly(p-phenylene terephthalamide) Abbreviations and Symbols A Cross-sectional area c Concentration dc Interplanar spacing d.r. Draw ratio D Diameter of the fibre DABT Poly(p-benzanilide terephthalamide) DP Degree of polymerisation ec Chain modulus e1 Modulus transverse to the chain axis esd Estimated standard deviation E Fibre modulus ERT Eyring reduced time f(z) Chain length distribution fw(z) Molecular weight distribution g Shear modulus of the domain gv Apparent shear modulus G Torsional modulus of the filament
The Tensile Strength of Polymer Fibres HT h(z) I(U) j(t) kB K Ld LC LG LP m Mn Mw Mz m.u. NA p PAN PBO PE PET PIPD POK PpPTA P(s) P2 PD q r RH s.s. t tb T Tg Tni T0 u ua uc u0 U UHMW V Vc Vcell W Wa Wb WC WS
3
High tenacity Crossing length distribution Transition density distribution Creep compliance Boltzmann constant Kelvin Contour projection length of the chain Contour length Griffith crack length Persistence length of the chain Weibull modulus Number-average molecular weight Weight-average molecular weight Z-average molecular weight Monomeric unit Avogrado’s number Distance between periodic force centres Polyacrylonitrile Poly(p-phenylene benzobisoxazole) Polyethylene Poly(p-ethylene terephthalate) Poly({2,6-diimidazole[4,5-b:4¢,5¢-e]pyridinylene-1,4(2,5-dihydroxy)phenylene}) Polyetherketone Poly(p-phenylene terephthalamide) Cumulative failure probability Internal order parameter Order parameter of the directors Crack size Radius of chain cross section Relative humidity Spinning speed Time Lifetime Temperature Glass transition temperature Nematic–isotropic transition temperature Reference or Vogel temperature Chain length Average chain length Bonded chain length Monomer length Activation energy Ultra-high molecular weight Volume Chain volume fraction Unit cell volume Strain energy Activation energy of creep Fracture energy Strain energy of the chain Shear energy
4 WmS W0S wg z zc zn zw
M. G. Northolt et al. Maximum shear energy Shear energy of fracture Surface energy of a crack Chain length in monomeric units Bonded chain length in m.u. Number-average chain length in m.u. Weight-average chain length in m.u.
Greek symbols b Critical shear strain in tensor notation g Shear strain in engineering notation d Relaxation time e Strain eb Strain at fracture e bs Shear strain at fracture ef Fibre strain e0 Ultimate strain at fracture efy Yield strain of the fibre e13 Shear strain in tensor notation v e13 Viscoelastic shear strain of a domain e13y Shear yield strain in tensor notation z Strength of orienting nematic potential h Viscosity q Orientation angle at stress s qb Orientation angle at fracture Q Orientation angle in the unloaded state l Load rate n Frequency Ç(q) Orientation distribution of the chains s Stress sb Tensile strength s bs Fibre strength based on shear deformation only scomp Strength of a macrocomposite s0 Ultimate strength sL Longitudinal strength sT Transverse strength sy Yield stress t Shear stress tb Shear strength tm Maximum shear stress tn Normalised shear stress ty Shear yield stress t0 Ultimate shear strength c Euler’s constant w Angular frequency W Activation volume
The Tensile Strength of Polymer Fibres
5
1 Introduction Organic polymer fibres offer an impressive range of mechanical properties. The tensile modulus of these fibres varies between 5 and 330 GPa, with a tensile strength up to 7 GPa, a compressive strength up to 1.7 GPa, and a temperature resistance up to 400 °C. The tensile curves of these fibres for temperatures below the glass transition temperature, including the yield phenomenon, are well described by the continuous chain model [1–10]. Considerable attention has been given in the literature to the relation between the tensile strength and the chain length distribution [11–14]. As will be shown here, there are also other factors of similar importance which determine the strength of a polymer fibre. In this report a relationship is derived describing the fibre strength as a function of the orientation distribution of the chains and the intrinsic mechanical properties, such as the elastic modulus of the polymer chain and the modulus for shear between the chains. In addition, a modified version of Yoon’s model for the description of the relation between the strength and the chain length distribution is presented. Finally, a model is proposed for the dependence of the fibre strength on the time and the temperature. Before embarking on the discussion of these intrinsic factors determining the strength of polymer fibres, the effect of structural and morphological imperfections on the fibre strength are briefly discussed. During the manufacturing process of polymer fibres all kinds of imperfections are introduced, like structural inhomogeneities, impurities and voids. These so-called extrinsic factors result in an imperfect bonding between the chains and may give rise to stress concentrations, which after a catastrophic growth of pre-existing cracks can lead to fracture. These imperfections cause the size effects, viz. the transverse effect or the dependence of the strength on the fibre diameter, and the longitudinal effect or the dependence of the strength on the test length [15–17]. Two different approaches can be recognised for the description of the size effects. The first is based on Griffith’s theory of crack propagation, which considers the energy balance between the external work, the surface energy of the crack and the elastic energy of the material [18, 19]. This theory is based on the elastic theory of infinitesimal deformations, and so does not apply to highly deformable materials. It can be applied to the transverse effects and leads to the semi-empirical equation for the strength of a material 1 1 D 4 = 4 + K · kl sb s0
(1)
where sb is the actual strength of the fibre, s0 the strength of the flawless fibre or the ultimate strength, K a constant and D the diameter of the fibre [20]. It was later shown by Penning et al. that the scaling of the tensile strength with D–0.5 can be derived from geometrical considerations as well [16]. An example of this relation is presented in Fig. 1, where the yarn strength of poly(p-phenyl-
6
M. G. Northolt et al.
Fig. 1 The inverse of the observed strength of PpPTA yarns versus the square root of the diameter of the filaments. Linear regression yields sb–1=0.063(16)+0.0643(27) D0.5 (GPa)–1 with s0=16(4) GPa, and estimated standard deviations in parentheses
ene terephthalamide) or PpPTA is plotted versus the filament diameter [21]. Apparently, for the strength of a flawless PpPTA filament the extrapolation yields s0=16±4 GPa. As will be shown in this report, this value is too large. Whereas in the second approach of the size effects it is also assumed that fracture is controlled by defects, the strength is now considered a statistically distributed parameter rather than a physical property characterised by a single value. The statistical distribution of fibre strength is usually described by the Weibull model [22, 23]. In this weakest-link model the strength distribution of a series arrangement of units of length L0 is given by
L s P(s) = 1 – exp – 4 4 L0 sp
m
(2)
where P(s) is the cumulative failure probability at a stress s, sp a scaling parameter and m the Weibull modulus. To make a so-called Weibull plot of a yarn P(s) is approximated by ni P =8 n+1
(3)
where ni is the number of filaments that have broken at or below a stress s and n is the total number of filaments tested. The length dependence is expressed through the test length L and can be written as log [–log (1 – P)] – log L + log L0 = mlog s – mlog sp
(4)
The Tensile Strength of Polymer Fibres
7
Fig. 2 Weibull plot of the filament strength for a test length of 10 cm of a PpPTA yarn yielding a Weibull modulus of 20.6. The average filament strength is 3.2 GPa
Thus, given a Weibull distribution of the filament strength, a plot of log [–log(1–P)] versus logs results in a straight line with a slope m. For the range 5<m
