Electromechanical Properties in Composites Based on Ferroelectrics (Engineering Materials and Processes)

Engineering Materials and Processes

Series Editor Professor Brian Derby, Professor of Materials Science Manchester Materials Science Centre, Grosvenor Street, Manchester, M1 7HS, UK

Other titles published in this series Fusion Bonding of Polymer Composites C. Ageorges and L. Ye

Materials for Information Technology E. Zschech, C. Whelan and T. Mikolajick

Composite Materials D.D.L. Chung

Fuel Cell Technology N. Sammes

Titanium G. Lütjering and J.C. Williams

Casting: An Analytical Approach A. Reikher and M.R. Barkhudarov

Corrosion of Metals H. Kaesche

Computational Quantum Mechanics for Materials Engineers L. Vitos

Corrosion and Protection E. Bardal Intelligent Macromolecules for Smart Devices L. Dai

Modelling of Powder Die Compaction P.R. Brewin, O. Coube, P. Doremus and J.H. Tweed

Microstructure of Steels and Cast Irons M. Durand-Charre

Silver Metallization D. Adams, T.L. Alford and J.W. Mayer

Phase Diagrams and Heterogeneous Equilibria B. Predel, M. Hoch and M. Pool

Microbiologically Influenced Corrosion R. Javaherdashti

Computational Mechanics of Composite Materials M. Kamiński

Modeling of Metal Forming and Machining Processes P.M. Dixit and U.S. Dixit

Gallium Nitride Processing for Electronics, Sensors and Spintronics S.J. Pearton, C.R. Abernathy and F. Ren

Modelling Stochastic Fibrous Materials with Mathematica® W.W. Sampson

Vitaly Yu. Topolov • Christopher R. Bowen

Electromechanical Properties in Composites Based on Ferroelectrics

123

Professor Vitaly Yu. Topolov Department of Physics Southern Federal University 5 Zorge street 344090 Rostov-on-Don Russia

Dr Christopher R. Bowen Department of Mechanical Engineering Faculty of Engineering and Design University of Bath Bath BA2 7AY United Kingdom

ISBN 978-1-84800-999-8

e-ISBN 978-1-84882-000-5

DOI 10.1007/978-1-84882-000-5 Engineering Materials and Processes ISSN 1619-0181 A catalogue record for this book is available from the British Library Library of Congress Control Number: 2008934905 © 2009 Springer-Verlag London Limited Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publishers. 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 laws and regulations and therefore free for general use. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. Cover design: eStudio Calamar S.L., Girona, Spain Printed on acid-free paper 9 8 7 6 5 4 3 2 1 springer.com

To our relatives, teachers and pupils

Preface Composite material, also called Composite, a solid material that results when two or more different substances, each with its own characteristics, are combined to create a new substance whose properties are superior to those of the original components in a specific application. Encyclopædia Britannica 2006 Ultimate Reference Suite ... functional composites make use of a number of underlying ideas including connectivity patterns leading to field and force concentration; the use of periodicity and scale in resonant structures; the symmetry of composite structures and its influence on physical properties; polychromatic percolation and coupled conduction paths; varistor action and other interfacial effects; sum, combination, and product properties; coupled phase transformation phenomena; and the important role that porosity and inner composites play in composite materials. These ideas provide a basic understanding of functional composites and have been discussed previously. R.E. Newnham

After the discoveries of piezoelectricity (1880) and ferroelectricity (1921), many attempts have been made to obtain new high-effective materials with predictable physical properties and to develop available technologies. Engineered composites based on ferroelectrics (i.e., composites with ferro-, piezo- and pyroelectric properties) have been manufactured and studied since the late 1970s. The 1980s and 1990s have been concerned with an active development of a previously untapped piezo-composite landscape in order to study interconnections between microstructure, composition and properties in novel piezo-active materials, especially those based on ferroelectrics that belong to so-called “smart materials”. The modern piezo-composite landscape is a specific area of research and innovations, and research developments in this area depend on multi-disciplinary efforts of scientists and engineers, such as solid state physics, physics of ferroelectrics, solid-state chemistry, materials science, mechanics of heterogeneous media, mechanics of electro-elastic media, mechanical engineering, etc. On studying the area of piezo-composites, it becomes obvious that knowledge of these materials and prediction of their properties cannot be attained without the good physico-mathematical basis and understanding of principles of modern materials science and mechanical engineering. To the best of our knowledge, there are few books concerned with the piezocomposites and their effective properties, and these problems were mainly considered in separate chapters or sections only. Earlier, two books devoted to methods for prediction of the effective electromechanical properties for some

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connectivity patterns of the two-component piezo-composites were written in Russian and published in the USSR (Khoroshun LP, Maslov BP, Leshchenko PV, (1989) Prediction of Effective Properties of Piezo-active Composite Materials. Naukova Dumka, Kiev) and Russia (Sokolkin YuV, Pan’kov AA, (2003) Electroelasticity of Piezo-composites with Irregular Structures. Fizmatlit, Moscow). In such a situation the present book aims to fill a gap in piezoelectric materials science and be an international edition devoted to different groups of piezo-composites and their effective properties. Our book is intended to discuss recent original theoretical and experimental results on effective electromechanical (piezoelectric, dielectric and elastic) properties in the piezo-composites based on ferroelectrics. For the last decades, single-crystal, bulk ceramic and thin-film ferroelectrics have found a number of applications as a result of their remarkable piezoelectric properties. Piezoelectricity as a physical phenomenon is concerned with an electromechanical interaction in acentric single crystals including ferroelectrics. Recent work performed in the rapidly growing field of smart materials demonstrates that both ferroelectricity and piezoelectricity represent an important link between solid-state science and engineering. The effective electromechanical properties associated with a complicated distribution of both the electric and elastic fields in piezo-composite samples depend on a number of factors, such as microstructure, connectivity, chemical composition and volume fraction of components, poling conditions for ferroelectric components, etc. Of particular interest are the non-monotonic dependences of the effective electromechanical properties and related parameters on a series of factors, such as the shape, spatial distribution and volume fraction of inclusions, porosity of a matrix, and poling degree of ferroelectric components. Knowledge of the extreme points of the effective parameters is important for manufacturing the highperformance piezo-composites for their particular piezotechnical applications. The characterisation and evaluation of smart materials in general requires a solid basis for the description of interconnections in the well-known dependency triangle of “composition – structure – properties”. For the piezo-composites based on ferroelectrics, this triangle becomes intricate as a result of electromechanical coupling, complex interrelations between the internal mechanical (tensor) and electric (vector) fields and a dependence of the properties in ferroelectric components on domain structure, microstructure, heterophase states, and external fields. Moreover, even in ferroelectric single crystals pertaining to one structural type, the piezoelectric and dielectric properties measured in weak electric fields vary significantly. The present book summarises and generalises a series of the authors’ works on the problem of prediction and non-monotonicity of the effective electromechanical properties in different two- and three-component composites based on the traditional ferroelectric ceramics and the newly available relaxor-ferroelectric single crystals. Therein we pay due attention to the analysis of interrelations between electromechanical constants of the components and to the description of different analytical schemes of averaging the properties of these materials having different connectivity and microgeometrical characteristics. Undoubtedly, this book is concerned with the important topic related to smart materials and

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optimisation of their effective properties. The main aim of the book is to show not only the advantages of different methods for predicting the electromechanical properties and choosing the optimum components, but to demonstrate the nontrivial behaviour of specific composite architectures and their parameters that are valuable for transducer, sensor, actuator, hydroacoustic, and other applications. This book has been written on the basis of the authors’ research results obtained at the Rostov State University (Russia, until December 2006), Southern Federal University (Russia, since December 2006), University of Bath (UK), and Karlsruhe Research Centre (Germany). The academic style of presentation of the results and the discussion about these results indicate that the book would be useful to engineers, postgraduate students, researchers, and lecturers working in the field of ferro-, piezoelectric and related materials, dealing with methods of mechanics of heterogeneous materials and with their different applications. This book will be of benefit to all specialists looking to understand the modelling and detailed behaviour of the electromechanical response of the piezo-composites and related smart materials. Some chapters and sections of the book could serve as a basis for a university course devoted to piezo-active materials and their properties. In this connection one can mention the well-known sentence “Knowledge itself is power” (F. Bacon). Based on our knowledge and experience, we hope that the 21st century, being called The Century of New Materials and Technologies, will favour the further fruitful development of the piezo-composite landscape and new scientific directions therein and in adjacent areas. Rostov-on-Don, Russia Bath, United Kingdom July 2008

Prof. Dr. Vitaly Yu. Topolov Dr. Christopher R. Bowen

Acknowledgements

The authors are grateful to Prof. Dr. A. V. Turik, Prof. Dr. A. E. Panich, Prof. Dr. V. G. Gavrilyachenko, Prof. Dr. A. A. Grekov, Dr. V. V. Eremkin, and Dr. V. G. Smotrakov (Southern Federal University, Russia), Prof. Dr. R. Stevens and Prof. Dr. A. Mileham (University of Bath, UK), Dr. S. A. Wilson (Cranfield University, UK), Prof. Dr. O. Kraft and Dr. M. Kamlah (Karlsruhe Research Centre, Germany), Dr. W. S. Kreher (Technical University of Dresden, Germany), Dr. H. Berger (University of Magdeburg, Germany), Prof. Dr. M. Lethiecq and Dr. F. Levassort (University of Tours, France), Prof. Dr. P. Bisegna (University of Rome “Tor Vergata”, Italy), Prof. Dr. H. L. W. Chan (The Hong Kong Polytechnic University, P.R. China), Prof. Dr. J. H. Huang (Feng Chia University, Taiwan, Republic of China), Prof. Dr. W. Cao (The Pennsylvania State University, USA), Prof. Dr. N. Kikuchi (The University of Michigan, USA), Prof. Dr. A. Safari and Dr. E. K. Akdogan (Rutgers – The State University of New Jersey, USA), Prof. Dr. E. C. N. Silva (University of São Paulo, Brazil), and Prof. Dr. J. S. Ono Fonseca (Federal University of Rio Grande do Sul, Brazil) for their interest in the research problems and for their interesting papers that have been cited in the book. Furthermore, the authors emphasise the vast geographic area wherein the piezocomposites are studied or manufactured for various applications. Dr. C. R. Bowen would also like to thank past supervisors including Prof. Dr. B. Derby (University of Manchester, UK), Prof. Dr. N. Claussen (Technical University of HamburgHarburg, Germany) and Prof. Dr. R. Stevens (University of Bath, UK). Special thanks are extended to Prof. Dr. R. E. Newnham and his co-workers (The Pennsylvania State University, USA) for the pioneering work that stimulated research activity in the field of piezo-composites and other smart materials in the last decades. The authors express much thanks to Dr. A. Doyle and Mr. S. Rees (Springer London Ltd, UK) for their effective and timely co-operation in the field of editing and producing this book. The authors sincerely thank Mrs. P. Carruthers and Mr. D. Barker (University of Bath, UK) for the technical help and Mrs. D. Stern (Karlsruhe Research Centre, Germany) for the bibliographic help. Copyright permissions obtained from Prof. Dr. P. Bisegna (University of Rome “Tor Vergata”, Italy), Springer (www.springer.com), Elsevier (www.elsevier.com), IOP Publishing (www.iop.org), AIP (www.aip.org), Taylor & Francis

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(www.informaworld.com), and Maney Publishing (www.maney.co.uk) are acknowledged with due attention and gratitude. Finally, financial support that promoted the fruitful research collaboration and writing this book is acknowledged with many thanks. Hereupon gratefully and proudly the authors mention the timely and effective support from the Royal Society (London, UK), Karlsruhe Research Centre and Helmholtz Society (Eggenstein-Leopoldshafen, Germany), EPSRC (Swindon, UK), National Physical Laboratory, QinetiQ (UK), Great Western Research (GWR, UK), University of Rome “Tor Vergata” (Rome, Italy), Rostov State University, and Southern Federal University (Rostov-on-Don, Russia).

Contents

1 From Smart Materials to Piezo-composites.................................................... 1 1.1 Piezo-composites as an Important Group of Smart Materials.................... 1 1.1.1 Smart Materials .............................................................................. 1 1.1.2 Composites ..................................................................................... 1 1.1.3 Piezo-active Composites ................................................................ 2 1.2 Classification of Composite Materials....................................................... 4 1.2.1 Connectivity ................................................................................... 4 1.2.2 Sizes and Shape .............................................................................. 4 1.2.3 Arrangement ................................................................................... 4 1.2.4 Classification, Connectivity Patterns and Specific Examples ........ 5 References........................................................................................................... 9 2 Effective Electromechanical Properties in Piezo-composites...................... 11 2.1 Piezoelectric Medium and Its Properties ................................................. 11 2.2 Sum, Combination and Product Properties .............................................. 22 2.2.1 Sum Properties.............................................................................. 22 2.2.2 Combination Properties ................................................................ 22 2.2.3 Product Properties ........................................................................ 24 2.3 Methods for Evaluation of Effective Parameters ..................................... 25 2.4 Evolution of D–E Connectivity Patterns .................................................. 30 2.5 Modelling of Effective Properties in Piezo-composites with Planar Interfaces ............................................................................. 32 2.6 Connectivity – Links – Properties............................................................ 35 References......................................................................................................... 35 3 Non-monotonic Volume-fraction Dependences of Effective Properties in D–E Ceramic / Polymer Piezo-composites..................................................... 43 3.1 Composites with 2–2 Connectivity, Manufacturing and Applications.... 43 3.1.1 Effective Electromechanical Constants of 2–2 Composites......... 46 3.1.2 Examples of Volume-fraction Dependences Predicted for 2–2 Composites....................................................................... 48 3.2 Composites with 1–3 Connectivity, Manufacturing and Applications .... 52

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3.2.1 Examples of Volume-fraction Dependences Predicted for 1–3 Composites........................................................................ 54 3.2.2 Diagrams of Changes in Piezoelectric Properties Predicted for 1–3 Composites........................................................................ 57 3.2.3 Maxima of Squared Figures of Merit of 1–3 Composites............. 61 3.3 Composites with 0–3 Connectivity, Manufacturing and Applications.... 62 3.3.1 Modelling of the Effective Properties in 0–3 Composites with Spheroidal Inclusions............................................................ 63 3.3.2 Piezoelectric Properties in 0–3 Composites Based on PbTiO3-type Ceramics................................................... 65 3.3.3 0–3 Composites with a Hierarchy of Inclusions........................... 75 3.3.4 0–3 Composites with Planar Microgeometry................................ 82 3.4 Composites with 3–3 Connectivity, Increasing the Piezoelectric Sensitivity and Applications.................................................................... 85 3.5 Connectivity – Non-monotonic Behaviour – Manufacturing................... 92 References......................................................................................................... 93 4 Piezoelectric Response of Porous Ceramic and Composite Materials Based on (Pb, Zr)TiO3 .................................................................................. 101 4.1 Porous Materials, General Characteristic and Manufacturing ............... 101 4.2 Volume-fraction Dependences of Parameters of Porous Piezo-active Materials ........................................................................... 103 4.3 Model of the Modified Layered Composite: Application to Porous Materials ............................................................ 110 4.4 Comparison of Calculated and Experimental Results............................ 111 4.5 From 3–3 Connectivity to Porous Materials.......................................... 116 References .......................................................................................................118 5 Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectric Single Crystals .......................................................... 123 5.1 Relaxor-ferroelectric Solid Solutions and Engineering ......................... 123 5.2 0–3 Single Crystal / Ceramic Composites.............................................. 125 5.3 0–1–3 Single Crystal / Ceramic / Polymer Composites..........................132 5.4 1–3 Single Crystal / Ceramic Composites.............................................. 142 5.5 1–0–3 Single Crystal / Polymer Composites ..........................................150 5.6 High Performance in Wide Ranges........................................................ 154 References .......................................................................................................155 6 Comparison of Results on Two-component Piezo-composites .................. 159 6.1 The Pros and Cons of Different Methods .............................................. 159 6.2 0–3 Versus 2–2 or 1–3............................................................................ 160 6.3 Data on 1–3 Composites.........................................................................165 6.3.1 Composites with Cylindrical Ceramic Rods................................ 165 6.3.2 Composites with Parallelepiped-shaped Ceramic Rods...............168 6.3.3 Variational Bounds for Effective Properties in 1–3 Composites with Cylindrical Ceramic Rods.................... 171

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6.3.4 Variational Bounds for Effective Properties in 1–3 Composites with Parallelepiped-shaped Ceramic Rods................................................................................173 6.4 Data on 0–3 Composites .........................................................................175 References ....................................................................................................... 182 7 Conclusions.................................................................................................... 185 7.1 From Components to Effective Electromechanical Properties .............. 185 References ....................................................................................................... 186 Appendix Formulae for Effective Electromechanical Constants of 2–2 and 1–3 Piezo-composites ....................................................................... 187 References .............................................................................................................189 List of Abbreviations.......................................................................................... 191 About the Authors .............................................................................................. 193 Index......................................................................................................................195

1 From Smart Materials to Piezo-composites

1.1 Piezo-composites as an Important Group of Smart Materials 1.1.1 Smart Materials In recent decades the structure and properties of smart materials (or intelligent functional materials) has represented a thriving research topic for academia and industry. This topic is transdisciplinary and attracts the attention of researchers in the fields of materials science, solid-state physics, mechanics, chemistry, crystallography, and mechanical engineering. Smart materials are able to adjust their behaviour to changes in external parameters and show clear analogies to a variety of biological systems [1, 2]. They can adapt functionality in an intelligent fashion and find applications as diverse as aeronautics to consumer products [2]. The main directions for the exploitation of smart materials are related to their performance for transducer, sensor or actuator applications. Advances in materials science and technology of the 21st century are inconceivable without smart materials and include ferroelectric, piezoelectric and pyroelectric ceramics and composites, electrostrictive and magnetostrictive materials, shape memory alloys, and relaxor-ferroelectrics [1, 2]. The physical properties of smart materials can be tailored [1, 3, 4] by understanding the “composition – structure – properties” relationships of the materials (Figure 1.1) that ultimately enable the optimisation of properties for specific applications. 1.1.2 Composites The majority of smart materials are solid-state heterogeneous or microinhomogeneous materials and include ceramics, polycrystalline films and composites. Composites can be defined as heterogeneous systems that consist of two or more components that differ in chemical composition and properties and are separated by distinct interfaces. Each composite material is characterised by the two following features [5, 6]. Firstly, the typical size of the separate structural elements are small in comparison with the whole composite sample (i.e., there is an element of micro-inhomogeneity). Secondly, the typical sizes of the structural

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Electromechanical Properties in Composites Based on Ferroelectrics

Properties

Structure

Composition

Figure 1.1. Composition – structure – properties triangle

elements are greater than the size of their individual atoms or molecules so that each component is regarded as a continuous medium. As a result, to describe the composite properties it is possible to consider the physical laws and equations suitable for continuous media. Each component of the composite represents a set of the structural elements with equal physical constants. These elements can be distributed regularly (uniformly) or irregularly and can have a specific, random or variable shape. The composite as a whole can be described using a set of microstructural characteristics, for example, connectivity, volume fractions of the components, spatial distribution of the components, and other parameters [6–9]. 1.1.3 Piezo-active Composites The piezo-active composites (hereafter referred to as piezo-composites) are an important group of smart materials. The piezo-composite group is vast due to the large number of components that may be involved in the design of the composites. Among these components, ferroelectrics play a leading role [1, 3, 4] and modern processing technologies enable the synthesis of a variety of ferroelectric single crystals, ferroelectric ceramics (or polycrystalline materials), ferroelectric films, and nano-sized ferroelectrics. Ferroelectric properties are generally related to altering the spontaneous polarisation vector using an external electric field (domain switching) and such a process takes place in acentric (non-centrosymmetric) crystal structures. These crystal structures can be divided by symmetry criteria into 21 point groups [3, 4] and piezoelectric properties due to the appearance of positive and negative charges on the surfaces of a mechanically stressed sample are particular to the acentric single crystals belonging to the following 20 point groups [4, 10]: 1 (triclinic system), 2 and m (monoclinic system), 222 and mm2 (orthorhombic system), 3, 32 and 3m (trigonal or rhombohedral system), 4, 422, 4mm, 4 , and 4 2m (tetragonal system), 6, 622, 6 , and 6 m2 (hexagonal system),

and 23 and 4 3m (cubic system). The piezoelectric effect can also take place in

From Smart Materials to Piezo-composites

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acentric anisotropic media, so-called piezoelectric textures, belonging to one of the following Curie groups [11]: f, fmm or f/2. Ferroelectric ceramics exhibit piezoelectricity in the poled state and are piezoelectric textures with fmm symmetry. Among the ferroelectric ceramics, the perovskite-type compositions are of practical interest due to their excellent piezoelectric and dielectric properties, high electromechanical coupling and large piezoelectric anisotropy [3, 4, 6, 11–13]. A large number of ferroelectric ceramic compositions have been studied over the past 60 years, including BaTiO3, Pb(Zr1–xTix)O3 and (1 – x)Pb(A1/3Nb2/3)O3 – xPbTiO3, where A = Mg or Zn. The latter solid solutions exhibit relaxor-ferroelectric properties and various heterophase states near the morphotropic phase boundary (MPB) and exhibit very high piezoelectric activity in the single-crystal form with engineered domain structures [14, 15]. The piezoelectric properties of the perovskite-type and other ferroelectric ceramics are highly dependent on manufacturing and poling methods, temperature history, mechanical stress history, and microstructure [3, 4, 13]. In the perovskite-type ceramics the piezoelectric charge coefficient (that characterises a charge per unit force) d33 a 102 pC / N [3, 4, 12, 13] and is two orders of magnitude higher than the piezoelectric coefficients dij of quartz (d11 = –2.25 pC / N and d14 = 0.853 pC / N [11]), which is an example of a classic non-ferroelectric piezoelectric single crystal. In single crystals of (1 – x)Pb(A1/3Nb2/3)O3 – xPbTiO3 with compositions chosen near the MPB, typical experimental values are even higher with d33 a 103 pC / N [14, 15]. Piezo-composites based on ferroelectrics are the important group of smart materials due to the ability to vary the microgeometry, effective physical properties and their anisotropy across a wide range. These significant variations appear upon combining different components, for example, (i) ceramic and polymer, (ii) single crystal and ceramic or (iii) single crystal and polymer. The presence of one or more piezoelectric components enables a set of parameters or properties to be attained that are not particular to the separate components, and this synergy is important for many practical transducer applications [1, 4]. Piezoelectricity is an interaction between elastic variables (mechanical stress Vkl and mechanical strain [qr) and electric variables (electric field E and electric displacement D) [10, 11] and the interaction can be weak or strong in homogeneous piezoelectric materials, including single-domain ferroelectric single crystals. The same interaction becomes even more complicated in the piezocomposites where an electromechanical interaction between the components and a redistribution of internal electric and mechanical fields strongly depend on microgeometry and boundary conditions. In general, the piezo-composites based on ferroelectrics form the final link in hierarchy-of-properties chains of “singledomain single crystal ĺ polydomain single crystal ĺ ceramic ĺ composite” and “single-domain single crystal ĺ polydomain single crystal ĺ heterophase single crystal ĺ composite”. As a consequence, the study of piezo-composites and their effective properties are important in connection with the “structure–properties” and “composition–properties” sides of the fundamental triangle shown in Figure 1.1.

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Electromechanical Properties in Composites Based on Ferroelectrics

1.2 Classification of Composite Materials In order to consider the mechanics and physics of micro-inhomogeneous media it is first necessary to define the structure of the materials being analysed. In the case of composites (including piezo-composites), the structure criteria play an important role and the materials must be classified according to the concepts [6–9]: (i) connectivity of the structural elements, (ii) typical relative sizes and shape of structural elements and (iii) features of the arrangement of structural elements. These three concepts will now be described. 1.2.1 Connectivity

Based on knowledge of the connectivity of the structural elements it is possible to divide the composites into three large groups. The first group are the matrix composites in which the structural elements of the first component present a monolithic system (the matrix) and the structural elements of the remaining components are isolated and do not contact each other at small volume fractions. These remaining components form a system of isolated inclusions. The second group comprises composites in which each component presents a monolithic and interconnected framework. The corresponding composite structures are often called interpenetrated. In the composites of the third group, the structural elements of each component are not interpenetrated or isolated. In addition, it is possible to obtain some structures with the structural elements related to the three different groups. 1.2.2 Sizes and Shape

With regard to the sizes and shape of the structural elements it is necessary to consider granular, fibrous and laminated composites. The granular composites consist of the structural elements having linear dimensions li of one order-ofmagnitude in three co-ordinate directions (i = 1, 2 or 3). If l1 | l2 << l3 in one of the components, then the composites are characterised as fibrous. In laminated composites, a relation l1 | l2 >> l3 holds for one or more components. If the latter relation is not true for all the components, such a composite can be regarded as a matrix composite reinforced by plate-like inclusions of one or more components. Complicated structures that are combinations of granular, fibrous and laminated structures can also be attained in some composite systems. Typical examples of such materials are composites reinforced by grains and fibres, porous laminated and fibrous composites and laminated composites with layers reinforced by fibres. 1.2.3 Arrangement

Taking into consideration the arrangement of the structural elements it is possible to have many variants of a layout of each component. In this difficult situation,

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some degree of regularity of the composite structure becomes a pertinent criterion for subsequent classification. According to this criterion, composites can be regarded as materials with a regular (determined) or irregular (stochastic) structure. The structural elements in the composites with a regular structure are characterised by a relevant geometric shape (e.g., rectangular parallelepiped or ellipsoid) and a regular (equidistant and uniform) arrangement in the sample. In the case of the irregular structure, the arrangement of the structural elements is random, and their shape can be either appointed or arbitrary. The random character of this arrangement can change and influence connectivity of the structural elements in the sample. An additional factor that characterises the arrangement of the structural elements is related to their spatial orientation. This factor is also to be taken into account when analysing both microgeometry and connectivity of the composites, in particular fibrous or laminated composites. 1.2.4 Classification, Connectivity Patterns and Specific Examples

A widespread classification of the two-component (or diphasic1) composites with planar interfaces was first put forward by Newnham et al. [7]. This classification is based on the concept of connectivity of each component. Connectivity is regarded as one of the main characteristics of the microstructure and expressed by the numbers of dimensions (or co-ordinate axes) in which each component is continuously distributed between limiting surfaces of the composite. The distribution of a self-connected state of a component can take place along zero, one, two, or three co-ordinate axes, i.e., connectivity D = 0, 1, 2, or 3 for the component 1 and connectivity E = 0, 1, 2, or 3 for the component 2. The connectivity of a two-component composite is written in a general form [1, 7, 9] of D–E where the connectivity of the piezoelectric or most piezo-active component takes the first position (D). In the case of D d E, the n-component composites are described by (n + 3)! / (3! n!) connectivities [7, 9], for n = 2 the number of connectivities is 10. It is also possible to introduce 10 connectivities D–E at D t E, and a scheme of corresponding connectivity patterns is shown in Figure 1.2. Since D and E can independently vary from 0 to 3, the two-component composite is described in terms of 42 = 16 connectivities as follows: 0–0, 0–1, 0–2, 0–3, … , 3– 2, or 3–3. The scheme shown in Figure 1.2 is also useful for the analysis of connectivities and microstructural features in the presence of non-planar interfaces. For example, a composite consisting of a bulk continuous matrix with small isolated inclusions embedded in the matrix would be described by 3–0 or 0–3 connectivity. The term 3–0 connectivity would be used to describe ferroelectric ceramics with closed air pores or isolated piezo-passive inclusions, whereas 0–3 connectivity would be used to describe a ferroelectric ceramic / continuous polymer ____________ 1

Ferroelectric materials (single crystals and ceramics) can be heterophase, for example, close to the MPB [3, 13–15] in a wide temperature range and the term “diphasic” would be substituted by “two-component”. In this case it is possible to consider a two-component composite, each component of which could be heterophase, as in the single crystal / ceramic composite with compositions chosen near the MPB.

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Electromechanical Properties in Composites Based on Ferroelectrics

Figure 1.2. Scheme of D–E connectivity patterns for the two-component composites with D t E. (XYZ) is the rectangular co-ordinate system, the shaded component is characterised by D connectivity. Arrows at faces of cubes are used to indicate the connected directions parallel to the co-ordinate axes (reprinted from Newnham et al. [7], with permission from Elsevier)

matrix composite with isolated inclusions. If the composite contains three components, then its connectivity is denoted as D–E–J with D, E, J = 0, 1, 2, or 3. Some examples of the three-component composites based on ferroelectric ceramics are provided in ref. [1]. The concept of connectivity is fundamental in developing an understanding of the electromechanical interaction between components within piezo-composites, in the study of the distribution of internal electric and mechanical fields and in the interpretation of experimental or calculated data related to composites with specific connectivity. The connectivity of the piezo-composites is crucial in influencing the piezoelectric response and electromechanical coupling of these materials. Moreover, after publishing the pioneering paper [7], connectivities D–E, D–E–J, etc., or related connectivity patterns are almost always mentioned when the research is devoted to the composites based on ferroelectrics or related materials. Particular examples include piezo-composites with connectivities 0–3, 1–3, 2–2, 2– 3, 3–1, 3–2, and 3–3 that have been experimentally studied (see, e.g., refs. [7, 16– 21]). Examples of the D–E composites based on ferroelectric ceramics are shown in Figures 1.3–1.6.

From Smart Materials to Piezo-composites

7

Figure 1.3. Micrograph of the 1–3 ferroelectric ceramic / polymer composite. ɋeramic fibres are embedded in epoxy, diameter of each fibre is 8u10–4 m

Figure 1.4. Micrograph of the 0–3 PbTiO3 ceramic / epoxy composite with so-called pearl chains (reprinted from Wilson et al. [18], with permission from IOP Publishing)

The concept of connectivity and the related classification scheme link the microstructures and effective properties in the piezo-composites and, therefore, enable us to understand the underlying science of the complex electromechanical interactions between components connected in various ways. A number of examples of the effective electromechanical properties of piezo-composites and their features will be analysed and described in subsequent chapters of this book.

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Electromechanical Properties in Composites Based on Ferroelectrics

Figure 1.5. Micrograph of the 2–2 volume-fraction gradient composite that comprises PZT ceramic and Spurr epoxy layers (reprinted from Safari and Akdogan [20], with permission from Taylor & Francis Group)

a

b Figure 1.6. Micrographs of the 3–3 porous ferroelectric ceramics (specific case of composites). In figure (b) two ceramic layers (each 4 mm thick) are the main elements of the hydrophone

From Smart Materials to Piezo-composites

9

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

Newnham RE, (1997) Molecular mechanisms in smart materials. Materials Research Society Bulletin 22(5):20–34 Tani J, Takagi J, Qiu J, (1998) Intelligent materials systems: Application of functional materials. Applied Mechanics Review 51:505–521 Xu Y, (1991) Ferroelectric Materials and their Applications. North-Holland, Amsterdam London New York Toronto Uchino K, (1997) Piezoelectric Actuators and Ultrasonic Motors. Kluwer, Boston Dordrecht London Chistensen RM, (1979) Mechanics of Composite Materials. Wiley, New York Khoroshun LP, Maslov BP, Leshchenko PV, (1989) Prediction of Effective Properties of Piezo-active Composite Materials. Naukova Dumka, Kiev (in Russian) Newnham RE, Skinner DP, Cross LE, (1978) Connectivity and piezoelectricpyroelectric composites. Materials Research Bulletin 13:525–536 Sokolkin YuV, Pan’kov AA, (2003) Electroelasticity of Piezo-composites with Irregular Structures. Fizmatlit, Moscow (in Russian) Newnham RE, (1994) Nonmechanical properties of composites. In: Kelly A, Cahn RW, Bever MB (eds.) Concise Encyclopedia of Composite Materials. Elsevier, Oxford, pp 214–220 Ikeda T, (1990) Fundamentals of Piezoelectricity. Oxford University Press, Oxford New York Toronto Zheludev IS, (1971) Physics of Crystalline Dielectrics. Vol. 2: Electrical Properties. Plenum, New York Dantsiger AYa, Razumovskaya ON, Reznitchenko LA, Grineva LD, Devlikanova RU, Dudkina SI, Gavrilyatchenko SV, Dergunova NV, Klevtsov AN, (1994) Highly Effective Piezoceramic Materials (Handbook). Kniga, Rostov-on-Don (in Russian) Gorish AV, Dudkevich VP, Kupriyanov MF, Panich AE, Turik AV, (1999) Piezoelectric Device-Making. Vol. 1: Physics of Ferroelectric Ceramics. Radiotekhnika, Moscow (in Russian) Zhang R, Jiang B, Cao W, (2001) Elastic, piezoelectric, and dielectric properties of multidomain 0.67Pb(Mg1/3Nb2/3)O3 – 0.33PbTiO3 single crystals. Journal of Applied Physics 90:3471–3475 Yin J, Cao W, (2002) Effective macroscopic symmetries and materials properties of multidomain 0.955Pb(Zn1/3Nb2/3)O3 – 0.045PbTiO3 single crystals. Journal of Applied Physics 92:444– 4 48 Ritter T, Geng X, Shung KK, Lopath PD, Park S-E, Shrout TR, (2000) Single crystal PZN/PT – polymer composites for ultrasound transducer applications. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 47:792–800 Smay JE, Cesarano III J, Tuttle BA, Lewis JA, (2003) Piezoelectric properties of 3–X periodic Pb(ZrxTi1-x)O3 – polymer composites. Journal of Applied Physics 92:6119– 6127 Wilson SA, Maistros GM, Whatmore RW, (2005) Structure modification of 0–3 piezoelectric ceramic / polymer composites through dielectrophoresis. Journal of Physics D: Applied Physics 38:175–182 Akdogan EK, Allahverdi M, Safari A, (2005) Piezoelectric composites for sensor and actuator applications. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 52:746–775 Safari A, Akdogan EK, (2006) Rapid prototyping of novel piezoelectric composites. Ferroelectrics 331:153–179

10 Electromechanical Properties in Composites Based on Ferroelectrics [21] Wang F, He C, Tang Y, Zhao X, Luo H, (2007) Single-crystal 0.7Pb(Mg1/3Nb2/3)O3 – 0.3PbTiO3 / epoxy 1–3 piezoelectric composites prepared by the lamination technique. Materials Chemistry and Physics 105:273–277

2 Effective Electromechanical Properties in Piezo-composites

2.1 Piezoelectric Medium and Its Properties The appearance of an electric polarisation in quartz single crystals that were subjected to a mechanical load was first reported by brothers P. Curie and J. Curie in experimental work in 1880 [1]. The effect that links a mechanical action (stress or strain) with an electric response (electric field, displacement or polarisation) is called the piezoelectric effect or, more exactly, the direct piezoelectric effect. At a later date, it was experimentally established that the converse piezoelectric effect is also observed in acentric single crystals in that an external electric field generates a mechanical response, i.e., a stress or strain of the sample [2, 3], similar to the electrostriction of dielectrics [3, 4]. It is generally realised that the piezoelectric effect reflects a linear relationship between electric and mechanical variables and originates from the displacement of ions of an acentric single crystal under an applied electrical field [1–4]. This relationship enables a change in the sign of the effect by switching the external electric field, unlike electrostriction that always exhibits a quadratic effect [3, 4] and does not undergo switching. Despite the linear character, the piezoelectric response of a single crystal is often intricate owing to various interconnections between the piezoelectric and other properties such as the elastic, dielectric and thermal properties [3]. A general consideration of the piezoelectric effect in single crystals is undertaken using thermodynamic functions, such as Helmholtz free energy, Gibbs free energy, elastic Gibbs energy, and electric Gibbs energy [2–4]. Each of these functions has at least three arguments that characterise the mechanical, electric and thermal states of the single crystal. The first argument can be either mechanical stress Vkl or mechanical strain [jr, the second argument can be electric field E or electric displacement (electric flux density) D, and the third argument can be temperature T or entropy S. In addition, it is possible to develop additional arguments in terms of magnetic field H or magnetic induction B [5]. Based on knowledge of the thermodynamic functions and relations between the arguments of

12

Electromechanical Properties in Composites Based on Ferroelectrics

the mechanical, electric and thermal states, it is possible to describe the linked “response – actions” in terms of [jr, Ei and T as follows: E Vkl = ckljr [jr – eikl Ei + Ekl 'T

Di = eikl[kl +

H if[ Ef + pi 'T

'S = Ejr[jr + pi Ei + (U / T0) c 'T

(2.1) (2.2) (2.3)

In Equations 2.1–2.3 the constants written in the tensor form are elastic moduli at E constant electric field ( ckljr ), piezoelectric coefficients (eikl), coefficients of thermal stress (Ekl), dielectric permittivities at constant mechanical strain ( H if[ ), pyroelectric coefficients (pi), density (U), and specific heat (c). The increments 'T and 'S denote differences T – T0 and S – S0, respectively, where T0 is the initial temperature and S0 is the initial entropy of the single crystal. It is assumed that the linear relations in Equations 2.1 – 2.3 hold in the case of relatively weak external fields. For example, the dependence Vkl([jr) from Equation 2.1 obeys Hooke’s law in an anisotropic medium at small strains [jr. The dependence Di(Ef ) from Equation 2.2 is linear at relatively low levels of the electric field Ef applied to the single crystal. With regard to ferroelectric single crystals, similar requirements concerning the linear dependences Vkl([jr) and Di(Ef ) are possible, but the Ef range becomes narrower than that in acentric linear dielectric single crystals. It is well known that in the presence of a low electric field, the polarisation of ferroelectric single crystals linearly depends on field [3, 4] so that domain-wall displacements are reversible and no nuclei of reoriented domains will form. In this connection the electric field Ef is usually regarded to be a few times lower than the coercive field Ec which is determined from a ferroelectric hysteresis loop [3, 4, 6]. Equations 2.1 – 2.3 enable the determination of a set of isothermal constants for a single crystal. An analogous set of adiabatic constants can also be derived using the thermodynamic functions and a combination of three arguments including 'S (for example, [jr, Ei and 'S). According to experimental data, the difference between the related isothermic and adiabatic constants of the piezoelectric single crystals is approximately 1% or less [3, 5]. The effect of thermal fields on the elastic and electric responses of piezoelectrics is often neglected, and the description of the piezoelectric effect is carried out on the basis of Equations 2.1 and 2.2 at 'T = 0. These equations written in the matrix form then reduce to [2, 3] E Vp = c pq [q – efp Ef

Dk = ekl[l +

[

H kr Er

(2.4) (2.5)

and represents the first pair of piezoelectric equations linking the variables of strain

Effective Electromechanical Properties in Piezo-composites

13

and field ([, E). The first term in the right part of Equation 2.5, Pk = ekl[l, describes the piezoelectric polarisation produced by an external strain as a result of the direct piezoelectric effect. In addition, the converse piezoelectric effect is described by the efp Ef term in Equation 2.4. From the thermodynamic treatment of the interrelations between the electric and elastic fields [2, 3], the converse and direct piezoelectric effects can also be described by the three following pairs of equations: E [p = s pq Vq + dfp Ef

V

(2.6)

H kr Er

(2.7)

D [p = s pq Vq + gfp Df

(2.8)

Ek = –gkl Vl + E krV Dr

(2.9)

Dk = dkl Vl + in variables (V, E),

in variables (V, D) and D Vp = c pq [q – hfp Df

(2.10)

Ek = –hkl [l + E kr[ Dr

(2.11)

in variables ([, D). Superscripts V and D denote measurement conditions at V = const and D = const, respectively, constants E krV and E kr[ are determined using equalities

E D or c pq and elastic E krV H rvV = Gkv and E kr[ H rv[ = Gkv, and elastic moduli c pq

E D E D compliances s pq or s pq are related by conditions c pq sqrE = Gpr and c pq sqrD = Gpr,

where Gkv is the Kronecker symbol. It should be noted that Equations 2.4–2.11 contain four types of piezoelectric coefficients, namely ekl, dkl, gkl, and hkl. The piezoelectric coefficient ekl (stress coefficient) relates the electric field to the mechanical stress (Equation 2.4) and the mechanical strain to the electric displacement (Equation 2.5). The piezoelectric coefficient dkl (charge coefficient) characterises the relation between the electric field and the mechanical strain (Equation 2.6) and the relation between the mechanical stress and the electric displacement (Equation 2.7). The piezoelectric coefficient gkl (voltage coefficient) is introduced to relate the electric displacement to the mechanical strain (Equation 2.8) and the mechanical stress to the electric field (Equation 2.9). Finally, the piezoelectric coefficient hkl (strain coefficient) relates the electric displacement to the mechanical stress (Equation 2.10) and the mechanical strain to the electric field (Equation 2.11). In general, each piezoelectric coefficient characterises the relationship between components of two fields that are described by first- and second-rank tensors, and all the piezoelectric coefficients introduced in Equations 2.4–2.11 are defined to be components of the third-rank tensors, but written in the

14

Electromechanical Properties in Composites Based on Ferroelectrics

matrix form (with two subscripts) [2–6] only. These individual piezoelectric coefficients can be used to aid in the selection of piezoelectric materials for specific applications. For example, for an actuator application [4] it may be necessary to select materials with a high strain per unit applied electric field (i.e., a high dkl). For a pressure sensor, the electric field generated per unit mechanical stress is likely to be important (i.e., a high gkl). The interrelationships of the piezoelectric coefficients follow from the thermodynamic description [2, 3] and are expressed by a set of equalities E dfp = H Vfk gkp = efq sqp

(2.12)

E efp = H [fk hkp = dfq cqp

(2.13)

D gfp = E Vfk dkp = hfq sqp

(2.14)

D hfp = E [fk ekp = gfq cqp

(2.15)

All four types of piezoelectric coefficients from Equations 2.12–2.15 are also involved in relations [2] between dielectric or elastic constants measured on different conditions:

H krV – H kr[ = dkf erf

(2.16)

E kr[ – E krV = gkf erf

(2.17)

c

D pq –

c

E pq =

efp hfq

(2.18)

D E – s pq = dfp gfq s pq

(2.19)

The close connections between the elastic, piezoelectric and dielectric constants of a single crystal (see Equations 2.12–2.19) or a piezoelectric medium enable them to be collectively described as electromechanical constants. These constants characterise electromechanical properties in a variety of piezo-active materials, such as piezoelectric or ferroelectric single crystals, poled ferroelectric ceramics, thin ferroelectric films, piezo-composites, etc. It follows from Equations 2.1, 2.2 and 2.4–2.11 that the piezoelectric properties are described by third-rank tensors and are represented in the matrix form. Tables of matrices of the piezoelectric coefficients of the different symmetry classes have been collected in a series of monographs [2–5]. It should be noted that the matrices of the piezoelectric coefficients || d || and || g || differ from the matrices of the piezoelectric coefficients || e || and || h || for 3, 32, 3m, 6 , and 6 m2 classes [2, 3]. The conditions for measuring the piezoelectric coefficients are concerned with the electric and mechanical variables that are fixed for the direct or converse piezoelectric effect. For example, if the direct piezoelectric effect is considered, then the corresponding piezoelectric coefficients are measured under the following conditions [3]: [ = 0 and E = 0 (for dkl), [ = 0 and D = 0 (for gkl), V = 0 and D = 0 (for hkl), and V = 0 and E = 0 (for ekl). In the case of the converse piezoelectric

Effective Electromechanical Properties in Piezo-composites

15

effect, the following conditions should be used: V = 0 and E = const (for dkl), V = 0 and D = const (for gkl), [ = 0 and D = const (for hkl), and [ = 0 and E = const (for ekl). The above-given conditions are determined on the basis of Equations 2.4–2.11 for the four types of piezoelectric coefficients. It has been shown that there are close connections between the variety of electromechanical constants, and two groups of these constants (i.e., piezoelectric / dielectric and elastic / piezoelectric) are described by Equations 2.12–2.19. The elastic, piezoelectric and dielectric constants are factors that ultimately measure the effectiveness of the conversion of electric energy into mechanical energy and vice versa. From the analysis of energy conversion [6], the effectiveness strongly depends on differences between dielectric permittivities from Equation 2.16 or differences between elastic compliances from Equation 2.19, and these differences stem from the piezoelectric properties in the medium. In a general form, the effectiveness of the energy conversion is described by an electromechanical coupling factor [3, 5, 6] k = wpiezo / wel wmech

(2.20)

and is expressed in terms of volume densities of piezoelectric (or mutual) energy V wpiezo = Vi dmi Em / 2, electric energy wel = Ek H kr Er / 2, and mechanical energy wmech E = Vp s pq Vq / 2. The k2 value determined using Equation 2.20 characterises [2–6] a

ratio of stored mechanical energy to electric energy input or a ratio of stored electric energy to mechanical energy input. The absolute value of the electromechanical coupling factor depends [2, 4, 6] on the vibration mode, measurement methodology, sample shape, and the electromechanical constants of the piezoelectric material. To date, we have considered the constitutive equations and the electromechanical properties in linear piezo-active media. Equations 2.4–2.11 are often used to describe the linear behaviour of ferroelectric single crystals (singledomain, polydomain or unipolar) [3, 4] poled ferroelectric ceramics (piezoceramics) [6, 7] and piezo-active composites [5] under weak electric and mechanical fields. As discussed by Hall [7], the linear description provides a reasonable approximation of the functional characteristics of piezo-ceramics at low levels of the applied electric field and stress, but becomes increasingly inaccurate as these levels increase. Non-linear behaviour of the electromechanical properties in various perovskite-type ferroelectric ceramics is observed above the threshold values [7] of the electric field Ethr and mechanical stress Vthr. According to data [7], the typical values equal Ethr | 10 kV / m for “soft” PZT ceramics (i.e., Pb(Zr, Ti)O3-type ceramics with low coercive field Ec) and Ethr | 300 kV / m for wellaged “hard” PZT ceramics (i.e., having high Ec values). Similarly, non-linearity of mechanical properties in both the “soft” and “hard” PZT ceramics becomes evident as mechanical stress exceeds Vthr | 20 MPa [7]. The hysteresis and non-linear behaviour of electromechanical properties in ferroelectric single crystals and ceramics are associated with the reorientation of the spontaneous polarisation vectors Ps of domains under the external electric field

16

Electromechanical Properties in Composites Based on Ferroelectrics

[3, 4, 6, 7]. The domain structure of ferroelectrics is of interest due to the significant influence of both the domain arrangement and reorientations on the electromechanical properties in these materials. Ferroelectric single crystals with the perovskite-type structure represent the vast group of multiaxial ferroelectrics and have various 180q and non-180q domain structures in ferroelectric phases [3, 4]. Many examples of complicated domain structures have also been studied [4] in related perovskite-type ferroelectric ceramics. An effect of the domain structure on the electromechanical properties in the well-known perovskite-type single crystals has been studied for cases of the 180q (BaTiO3 [8]) and 90q (BaTiO3 [9], PbTiO3 [10]) domains in the ferroelectric 4mm phase. The domain structures are characterised [8–10] as regular laminar structures (Figure 2.1) with planar domain walls that are motionless under weak external fields. An orientation contribution caused by the elastic (or reversible) 90q domainwall displacements under applied electric fields has been determined [11] for BaTiO3 single crystals separated into 90q domains. The contribution of domain orientation to the electromechanical constants of a single crystal strongly depends on the mobility of the 90q domain walls within the material. The important role of the 60q, 90q and 120q domain-wall displacements on the piezoelectric coefficient d33 of the polydomain KNbO3 single crystal (mm2 phase) has been analysed in ref. [12]. The contributions of domain-wall displacements to the d33 under an electric field correlate with some crystallographic characteristics of the non-180q domain structures. Full sets of the electromechanical constants of polydomain single crystals are used to study the relationships between electromechanical properties in a single grain and a poled ceramic [10, 13–15] consisting of a large number of grains with different orientations of main crystallographic axes. This study is based on solving a problem [13, 14] of the electromechanical interaction between a spherical piezoactive grain and a piezo-ceramic medium surrounding this grain. Such an interaction is considered within the framework of the effective medium method [13, 14] that takes into account not only the piezoelectric effect but also the anisotropy of the piezoelectric properties in the grain and the ceramic medium. This anisotropy can be considerable, particularly in the presence of non-180q domain reorientations in the grains [16]. Modelling of the 180q and 90q reorientations under an electric field results in the full set of the electromechanical constants, calculated for the BaTiO3 ceramic [16] at various values of Pr / Ps, where Pr is the remanent polarisation of the ceramic and Ps is the spontaneous polarisation of the separate domain. In addition, the effect of the 90q domain-wall displacements within the grains on the electromechanical properties in poled BaTiO3 ceramic was studied in [11, 17, 18] where contributions from these domain-wall displacements to a variety of constants for the ceramic were evaluated. Of particular interest is the significant contribution of domain-wall displacements (37–72%) to the piezoelectric coefficients d3j [17, 18]. The E analogous contribution to the elastic compliances s pq is evaluated to be in a range from 11 to 43 % [18]. Differences between the experimental [6] and averaged [13] constants of the BaTiO3 ceramic were accounted for by the evaluated contributions [11, 17, 18] from the 90q domain-wall displacements.

Effective Electromechanical Properties in Piezo-composites

a

17

b

Figure 2.1. Schematic representation of the polydomain single crystal with the 180q (a) or 90q (b) domain structure. Spontaneous polarisation vectors of the domains are shown by arrows

In contrast to BaTiO3 ceramics, the electromechanical properties in the PbTiO3 ceramic are determined [10, 15] in the absence of 90q domain-wall displacements within the grains. According to results [15, 19], the 90q domain structure of the grains has a strong influence on the piezoelectric coefficients dfp and their anisotropy d33 / d31 in the PbTiO3-type ceramics. In particular, it is possible to attain large piezoelectric anisotropy (i.e., d33 / | d31| >> 1) by doping the ceramic [20], varying temperature [15] and by changing the volume fractions of the 90q domains [15] within the ceramic grains. A comparative analysis of the linear and non-linear effective properties in BaTiO3 ceramics was carried out within the framework of the model of a polydomain grain in an effective medium [21, 22]. Results on the modelling of the piezoelectric behaviour of the ceramic and on the electromechanical properties determined using the effective medium method demonstrates that the non-linear behaviour is a result of 90q domain-wall displacements in the grains. These displacements in the presence of an externally applied field become an important extrinsic factor that influences the piezoelectric coefficients dfp of the ceramic at different poling states. For the fully poled BaTiO3 ceramic, the aforementioned contributions from the 90q domain-wall displacements to dfp attain 68 % and 64 % [22] for fp = 31 and 33, respectively. Therefore, subsequent averaging procedures facilitate the consideration of the hierarchy-of-properties chain of “single-domain single crystal – polydomain single crystal – ceramic” in the presence of the 90q domain-wall displacements (BaTiO3) or on condition of the motionless 90q domain walls (PbTiO3) [10, 13, 15, 19, 20]. The main aims are to analyse factors that influence the piezoelectric properties in the related ferroelectric materials and to study the correlation between the microstructure, domain-wall displacements and electromechanical properties in the ferroelectric ceramic. Knowledge of the full sets of the electromechanical constants of ferroelectric ceramics enables appropriate material selection for various piezotechnical applications and offers a route to using the ceramics as piezo-active components for composite materials.

18

Electromechanical Properties in Composites Based on Ferroelectrics

At present, there is a limited number of full sets of electromechanical constants available in the literature for poled ferroelectric ceramics, primarily due to the effort involved in manufacturing and poling a variety of test samples. Electromechanical constants are often measured using standard methods [23, 24] at room temperature and with weak electric fields. The majority of the ferroelectric ceramics are manufactured as perovskite-type solid solutions [3–5, 25–27] with compositions varied and tailored to improve specific properties. Attempts to classify the ferroelectric ceramics have been made in different papers [25–27]. From the classification scheme [25] and by considering the conventional perovskite-type ceramics based on Pb(Zr1–xTix)O3, it is possible to identify materials that are stable with regard to electric and mechanical actions, materials V with high dielectric permittivity H 33 , materials having high sensitivity with respect to mechanical stress fields (high piezoelectric coefficients gfp), materials with the large piezoelectric anisotropy d33 / d31, materials with high stability to the V resonance frequency, materials with low dielectric permittivity H 33 , and hightemperature materials (high Curie temperature TC). From the classification method proposed in monograph [26], the Pb(Zr1–xTix)O3-based ceramics are divided into four groups with specific performances. These groups are related to molar concentrations of Ti from ranges 0 d x d 0.1, 0.1 d x d 0.4, 0.4 d x d 0.6, and 0.6 d x d 1, and some advantages of each group are mentioned [26] in connection with a variety of piezotechnical applications. Examples of the ferroelectric ceramics and full sets of their room-temperature electromechanical constants used in Equations 2.4 and 2.5 are listed in Table 2.1. Analogous sets related to the advanced relaxor-ferroelectric single crystals with engineered domain structures are also shown in Table 2.2 for comparison. Differences between the individual constants of the materials with the same composition (e.g., BaTiO3 and (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 ceramics from Table 2.1 or the 0.70Pb(Mg1/3Nb2/3)O3 – 0.30PbTiO3 single crystals from Table 2.2) are the result of technological, microstructural and poling factors, the selection of a composition in the proximity of the MPB, fluctuations of the composition over samples and a variable mobility of the domain walls. These factors can influence the domain structures [44], heterophase states [45, 46] and electromechanical properties [47]. For example, changes in volume fractions of the 90q domains or in the mobility of the 90q domain walls in the grains of the PbTiO3-type ceramics affect their piezoelectric and other properties [15, 19, 26, 47]. Furthermore, in the presence of an external electric field E, displacements of interfaces separating morphotropic phases in the 0.92Pb(Zn1/3Nb2/3)O3 – 0.08PbTiO3 single crystal promote an appreciable contribution to the piezoelectric coefficient d33 [46]. This contribution can exceed 50 % in a wide E range. The full sets of experimental electromechanical constants from Tables 2.1 and 2.2 have been used in our study on the piezo-composites, and we will analyse and discuss the results in Chapters 3–6. In the next section we consider some of the physical factors that influence the effective properties of two-component composites based on ferroelectrics.

Effective Electromechanical Properties in Piezo-composites

19

E Table 2.1. Experimental room-temperature values of elastic moduli c pq (in 1010 Pa),a

piezoelectric coefficients efp (in C / m2) and dielectric permittivities H [ff of ferroelectric ceramics BaTiO3 [28] b

BaTiO3 [6] b

(Ba0.917Ca0.083). . TiO3 [6, 29] 15.8

(Pb0.9625La0.025). (Ti0.99Mn0.01)O3 (I) [30] b 14.33

(Pb0.855Nd0.11). . (Ti0.94Mn0.02. . In0.04)O3 [31] 17.3

c11E

16.6

15.0

c12E

7.66

E 13

6.6

6.91

3.220

5.31

E 33

7.75

6.6

6.75

2.413

5.68

E 44

16.2

14.6

15.0

13.16

16.6

4.29

4.4

4.51

5.587

6.09

e31 e33 e15

H 11[ /H0

–4.34 18.6 11.6 1270

–4.35 17.5 11.4 1115

–3.14 13.5 10.9 1000

0.4584 6.499 5.923 210

2.00 8.81 4.84 270

H 33[ /H0

1420

1260

910

140

198

c

c

c

.

(Pb0.9625La0.025) (Ti0.99Mn0.01)O3 (II) [31] b 15.8

.

.

.

(Pb0.85Nd0.10) (Ti0.99Mn0.01)O3 [31] 17.5

(Pb, Ca)TiO3 [32]

TBK-3 [5] c

TBKS [5] d

.

15.05

15.7

14.6

E 12

c

4.20

5.49

3.68

6.3

5.18

c13E

4.55

5.65

3.09

6.2

5.06

E 33

15.2

16.5

12.78

15.3

14.2

E 44

6.32

6.26

1.20

4.4

4.9

e31 e33 e15

H 11[ /H0

1.32 6.80 3.35 203

1.80 8.83 5.41 265

1.71 8.8 0.33 170

–3.2 12.5 9.1 950

–0.68 7.71 4.56 410

H 33[ /H0

133

193

140

950

336

c11E

c

c

c11E

NBS-1 [5] e 11.0

ZTBS-3 [5] f 15.1

ZTS-19 [5] g 10.9

PZT-4 [6] g 13.9

PZT-5 [6] 12.1

PZT-5A [32] 12.04

c12E

4.22

7.9

6.1

7.78

7.54

7.52

E 13

4.12

8.0

5.4

7.43

7.52

7.51

E 33

10.0

13.6

9.3

11.5

11.1

11.09

E 44

3.45

2.9

2.4

2.56

2.11

2.3

e31 e33 e15

H 11[ /H0

–3.3 14.8 7.7 900

–7.9 17.7 15.4 1610

–4.9 14.9 10.6 820

–5.2 15.1 12.7 730

–5.4 15.8 12.3 916

–5.4 15.8 12.3 540

H 33[ /H0

890

1280

840

635

830

830

c

c

c

20

Electromechanical Properties in Composites Based on Ferroelectrics Table 2.1 (continued)

c11E

PZ 27 [33] 14.28

PZ 34 [33] 18.84

PCR-1 [25] h 16.0

PCR-7 [25] 12.5

PCR-7M [25] 13.3

PCR-8 [25] 14.4

c12E

9.98

8.02

10.1

8.4

9.2

8.5

E 13

c

9.22

8.23

9.6

8.1

9.1

7.7

c33E

11.45

12.72

13.3

12.1

12.5

11.6

E 44

2.28

5.89

2.58

2.36

2.28

2.83

e31 e33 e15 H 11[ /H0

–3.71 15.77 11.74 1121

3.81 6.87 2.31 181

–3.7 11.1 10.8 570

–9.0 28.3 17.9 1430

–9.5 31.1 20.0 1980

–7.5 13.6 11.6 830

H 33[ /H0

878

154

300

1350

1810

740

c11E

PCR-13 [25] 11.9

PCR-21 [25] 13.4

PCR-62 [25] 12.7

PCR-63 [25] 14.1

PCR-73 [25] 17.3

c12E

4.8

7.3

4.5

6.6

13.3

E 13

3.1

6.2

0.8

5.6

12.5

E 33

10.0

11.8

11.3

11.2

14.4

E 44

3.50

2.45

4.70

4.15

2.29

e31 e33 e15 H 11[ /H0

–6.0 10.4 7.1 710

–7.3 16.2 12.0 890

–6.6 8.9 6.0 560

–4.6 11.8 6.9 830

–9.1 29.2 22.4 2220

H 33[ /H0

710

890

590

1000

2380

c

c c

c

Notes to Table 2.1 a. Ferroelectric ceramics poled along the OX3 axis are characterised [2–6] by fmm symmetry and a relation c66 = ( c11E – c12E ) / 2 b. Different processing conditions lead to different sets of the electromechanical constants as shown for the BaTiO3 and (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 ceramics c. Based on (BaxCa1–x)TiO3 d. Based on (BaxCa1–x–yPby)TiO3 e. Based on (Ba1–xPbx)Nb2O6 f. Based on (Ba1–xPbx)(Zr1–yTiy)O3 g. Based on Pb(Zr1–xTix)O3 h. PCR is the abbreviation for the group “piezoelectric ceramics from Rostov-on-Don”. These materials based mainly on Pb(Zr1–xTix)O3 are manufactured [25] by the conventional method (PRC-13 and PCR-62) or by means of hot pressing (PCR-1, PCR-7, PCR-7M, PCR8, PCR-21, PCR-63, and PCR-73)

Effective Electromechanical Properties in Piezo-composites

21

E Table 2.2. Experimental room-temperature values of elastic moduli c pq (in 1010 Pa), [ piezoelectric coefficients efp (in C / m2) and dielectric permittivities H ff of (1 –

x)Pb(A1/3Nb2/3)O3 – xPbTiO3 single crystals (4mm symmetry)

c11E

A= Mg, x= 0.30 [34] 11.7

A= Mg, x= 0.30 [35] 16.04

A= Mg, x= 0.33 [36] 11.5

A= Mg, x= 0.38 [37] 21.25

A= Mg, x= 0.42 [38] 17.51

A= Zn, x= 0.045 [39] 11.1

A= Zn, x= 0.07 [40] 11.3

c12E

10.3

14.96

10.3

14.33

8.51

10.2

10.3

c13E

10.1

7.51

10.2

13.51

8.3

10.1

10.5

E 33

10.8

12.0

10.3

9.92 a

10.5

10.5

10.91

E c44

7.1

5.38

6.9

5.56

2.85

6.4

6.30

c66E

6.6

2.87

6.6

6.95

8.0

6.3

7.10

–2.4 27.1 13.6 3307

–5.22 30.4 31.84 4963

–3.9 20.3 10.1 1434

–3.26 11.46 21.12 2171 b

–2.1 12.2 37.50 c 3054

–3.7 15.0 8.9 3000

–2.3 15.1 11.1 2779

1242

1386

680

255

259

1000

823

c

e31 e33 e15 H11[ /H0 H 33[ /H0

c11E

A = Zn, x = 0.08 [41] 11.5

A = Zn, x = 0.08 [42] 11.1

A = Zn, x = 0.08 [43] 11.47

c12E

10.5

10.2

10.30

E 13

10.9

9.97

10.37

E 33

11.51

10.2

10.71

E 44

6.34

6.1

6.70

c c

c c

E 66

e31 e33 e15 H11[ /H0 H 33[ /H0

6.50

6.56

3.05

–5.1 15.4 10.1 2720

–9.15 12.2 9.39 2848

–5.83 12.60 6.22 1415

984

620

560

Notes to Table 2.2 E E E a. According to our calculations based on formula [2] c pq sqr = Gpr and values s11 = 11.73, s12E = –2.71, s13E = –8.17, and s33E = 21.53 (in 10–12 Pa–1) [37], it would be c33E = 14.89u1010 Pa instead of 9.92u1010 Pa [ V b. According to our calculations based on formula [2] H kr = H kr (1 – k152) and experimental

V [ values H 11 / H0 = 4301 and k15 = 0.459 [37], it would be H 11 / H0 = 3395 instead of 2171. The [

V

value H 11 / H0 = 3395 would also be attained using Equation 2.16, H 11 / H0 = 4301, d15 = 380 pC / N, and e15 = 21.12 C / m2 from ref. [37] c. According to our calculations based on Equation 2.13 and experimental values d15 = 131 E pC / N and c44 = 2.85u1010 Pa [38], it would be e15 = 3.73 C / m2 instead of 37.50 C / m2

22

Electromechanical Properties in Composites Based on Ferroelectrics

2.2 Sum, Combination and Product Properties Ferroelectrics and related piezo-active materials are characterised by the full sets of electromechanical constants that are listed in Tables 2.1 and 2.2. Ferroelectric ceramics and single crystals are used as components of piezo-composites and actively influence the effective properties of the composites. Each physical property of a piezo-composite is analytically determined using specific formulae or procedures of averaging (see, e.g., [5, 32, 33]). The physical property can also be studied by experimental methods and characterisation. In the case of piezocomposites, results on averaging depend on the volume fractions of components and their properties, microstructure and connectivity [5, 33, 48, 49]. Numerous theoretical and experimental results point out that there are three main effects [48, 49] related to forming the effective properties in the piezo-composites, namely (i) the effect of the addition of the properties (forming the sum properties), (ii) the effect of the combination of the properties (forming the combination properties) and (iii) the effect of the generation of the properties (forming the product properties). 2.2.1 Sum Properties For example, the effective property Y* of a two-component piezo-composite1 obeys a condition Y(1) < Y*< Y(2), where Y(n) is the analogous property of the nth component, n = 1 and 2. Y* belongs to the sum properties when it is characterised by a monotonic dependence (Figure 2.2a) on the volume fraction m of one of the components. It is the simplest example of the volume-fraction dependence of the effective property, that is determined by a general formula Y*(m) = Y(1) m + Y(2) (1 – m) + 'Y, where 'Y depends on Y(n) and m and characterises deviations of Y*(m) from the linear dependence (compare solid and dotted lines in Figure 2.2a). Such *E behaviour is typical of elastic moduli cab , piezoelectric coefficients d ij* and dielectric permittivities H *prV of 0–3 or 1–3 composites [5, 50–54] comprising a PZT-type ceramic and a piezo-passive polymer. 2.2.2 Combination Properties In this case, the properties Y(n) and Z(n) of the components (n = 1 and 2) form the effective properties Y*(m) and Z*(m) that are monotonic dependences (Figures 2.2a and b). It is assumed that )(n) is expressed in terms of Y(n) and Z(n) and also ____________ 1

Hereafter, we denote all of the effective constants and parameters of composites by an asterisk. Following Equations 2.6 and 2.7, the full set of effective constants is written as s*pqE , *V *V d *fp and H kr . The similar set of the constants s *pqD , g *fp and E kr is derived from Equations 2.8

and 2.9. Finally, the full set of the effective constants c*pqD , h*fp and E kr*[ can be determined using Equations 2.10 and 2.11.

Effective Electromechanical Properties in Piezo-composites

23

Figure 2.2. Sum properties Y*(m) and Z*(m) (a and b) and combinations properties )*(m) (c–e) in two-component composites. m is the volume fraction of the first component. In graphs a and b, possible versions of the volume-fraction dependence are shown with dotted lines

characterises some property of the nth component. The monotonic character of Y*(m) and Z*(m) does not imply a monotonic volume-fraction behaviour of the effective property )*. In this case a non-monotonic )*(m) dependence (often with values |)*(m) | > |) (n) |, Figures 2.2c and d) is concerned with the effect of combination of the properties. This effect is important, for example, when *V [ considering the piezoelectric coefficients g 3* j = d 3* j / H 33 and h3* j = e3* j / H 33 (j = 1 and 3) of various piezo-composites [50, 52, 54] (see also Sections 3.1–3.3 and 5.4).

24

Electromechanical Properties in Composites Based on Ferroelectrics

Furthermore, the combination property )*(m) can be described by a monotonic volume-fraction dependence (Figure 2.2e), but a configuration of the )*(m) curve differs from both the Y*(m) and Z*(m) curves. The effect of combination of three properties (i.e., elastic, piezoelectric and dielectric properties) is concerned with the volume-fraction dependence of electromechanical coupling factors of the 1–3 ceramic / polymer composites [32, 51]. 2.2.3 Product Properties The effect of generation of properties in a two-component composite is described as follows. It is assumed that a response V1 of the first component is the result of an external action X. Simultaneously, the second component does not show a response on the action X, but generates only a response V2 in response to V1. This implies that the internal action V1 from the first component due to the external action X is the reason for the new response V2. As a result, the formation of product properties is associated with two different responses, V1 and V2. Such an effect takes place, for example, in composites [55, 56] comprising one magnetostrictive (CoFe2O4 or other ferrites) and one piezoelectric (BaTiO3, PbTiO3 or PZT-type) ceramics. In the presence of an external magnetic field, a mechanical strain is generated in the magnetostrictive ceramic. This strain is transferred into the piezoelectric component of the composite that generates an electric field in the piezoelectric ceramic. Thus, the magnetic field applied to the composite sample generates an electric field due to the combination of the magnetostrictive (but piezo-passive) and piezo-active (but showing no magnetostriction) components. In this composite the role of both the electromechanical and electromagnetic interactions between the components is obviously of importance. In general, the product property appears due to the interaction of two components that have different properties being mutually complementary. O ther examples of the product properties [49] in the composites based on ferroelectrics are concerned with pyroelectricity (taking into account piezoelectricity of the first component and thermal expansion of the second component) and photostriction (taking into account the photovoltaic effect in the first component and electrostriction in the second component). Recent work [57–59] has been devoted to the study of more complicated effects in the piezo-composites and disordered piezo-active heterogeneous materials. Taking into account electric conductivities of the components, it is possible to achieve giant dielectric permittivity [57], giant piezoelectric and dielectric enhancement [58]. Our further consideration of the effective electromechanical properties is closely connected with examples of non-monotonic dependences of these properties that are often caused by the effect of the combination of properties. A transition to an analysis of the properties would be impossible without a characterisation of methods for determination of the effective properties in the piezo-composites with different connectivity patterns and components. These will now be described.

Effective Electromechanical Properties in Piezo-composites

25

2.3 Methods for Evaluation of Effective Parameters It is well known [2–6, 26, 29] that Equations 2.4–2.19 are valid not only for piezoelectric single crystals, but also for poled ferroelectric polydomain single crystals, ferroelectric ceramics and composites containing at least one piezo-active component. The applicability of Equations 2.4–2.19 to describe the electromechanical properties of piezo-composites is reasonable in cases when these properties can be regarded as effective (or overall), i.e., characterising the response of the composite sample as a whole. The effective electromechanical properties and related parameters enable us to move away from the properties of the individual components, the specifics of microgeometry and other features of heterogeneous materials and to estimate the potential advantages and applications of these materials. The problem of prediction of the effective physical properties in advanced composites is of interest to many specialists undertaking theoretical and experimental studies of these heterogeneous materials. A formulation of the related problem on the effective electromechanical properties in the piezo-composite is given in ref. [5]. Equations 2.4 and 2.5 are examples of constitutive equations that describe the behaviour of the piezo-active material under electric and mechanical fields. If averaged components of the electric field <Ef > and the mechanical strain <[q> in a macroscopic region of the piezo-composite are independent of coordinates xj (j = 1, 2 and 3), then Equations 2.4 and 2.5 can be rewritten as follows: = c*pqE <[q> – e*fp <Ef >

(2.21)

= e <[l> + H <Er >

(2.22)

* kl

*[ kr

In Equations 2.21 and 2.22 elastic moduli c*pqE , piezoelectric coefficients e*fp and dielectric permittivities H kr*[ constitute the full set of effective electromechanical constants of the piezo-composite. These constants are determined [5] taking into account equations of electric and mechanical equilibrium and boundary conditions at the surface of the macroscopic region in the heterogeneous medium. Calculations of the effective electromechanical constants of the piezo-composite are closely connected with averaging a series of vector and tensor components of the electric and mechanical fields, for example, [q, Ef , Vp, and Dk (see Equations 2.4–2.11). This averaging is performed on volume fractions of the composite components2 (n = 1, 2, …) for which the full sets of electromechanical constants, such as c (pqn ),E , e (nfp ) and H kr(n ),[ or other constants involved in Equations 2.6–2.11, are known. The formulation of the problem allows the development of a variety of methodologies for the determination of the full sets of electromechanical constants and various methods of numerical realisation for specific connectivity patterns and arrangement, etc. [5, 32, 33, 51–54, 60–63]. ____________ 2

In order to avoid confusion owing to vector (tensor) components and composite components, the latter are also called constituents.

26

Electromechanical Properties in Composites Based on Ferroelectrics

The central aspect of this problem is a method of homogenisation of the electromechanical properties [33, 63]. It is clear that the effective properties are to be regarded as properties in a material at distances exceeding the linear dimensions of an individual inclusion or the thickness of a separate layer in a composite sample. However, the same inclusions can be inhomogeneous (for example, ferroelectric ceramic particles that are split into domains or are heterophase). It is therefore necessary to attain an additional homogenisation of the properties in these inclusions. Such a situation needs a compromise. For example, the effective electromechanical properties in the 2–2 piezo-composite are determined [64, 65] by assuming that their components are characterised by an equivalent homogeneity [66]. The authors of refs. [67, 68] have considered a variety of applications of the approach developed with attention to piezo-composite and biological materials. It is also noted that homogenisation methods are applicable [67] to media with periodic and quasi-periodic structures. Based on the detailed analysis of results concerning composite materials and their effective properties, Khoroshun et al. [5] placed emphasis on two main scientific directions in modern mechanics of composites. The first direction represents the mechanics of the composites having regular (periodic) structures. The second direction is related to the mechanics of stochastically heterogeneous materials, including the composites with arbitrary connectivity, shape and distribution of structural elements. The first direction deals with periodic solutions of equations of mechanics of microinhomogeneous media in which the regular distribution of the structural elements is present. As a rule, such solutions are applicable to the matrix piezocomposites with inclusions having the appropriate shape, e.g., spheroids. The accompanying method, called the method of regularisation of structure [5], is applied to the composites [32, 33, 60, 63, 69] with a certain disorder in the shape and distribution of the structural elements. This method makes it possible to substitute the real composite by a model material with regularly distributed structural elements having the appropriate shape only. In such a model material, a periodic representative volume element (similar to unit cells in single crystals, as shown, for example, in the Banno cube model [69]) is chosen and analysed using micromechanical methods including the finite-element method (FEM). A representative volume element used for a prediction of effective parameters of the porous piezo-ceramic [70] with 3–3 connectivity is shown in Figure 2.3. The second direction implies using random functions to describe the properties in the composite with a stochastic structure. These properties are regarded as randomly varied ones over the composite sample, and the homogenisation can be performed with regard to mathematical features of the random functions. This approach is suitable for the prediction of the effective properties in the piezocomposites without restrictions on the shape and distribution of inclusions and connectivity. A considerable amount of research is devoted to the 0–3 piezo-composites [5, 33, 50, 53, 60–62] that contain a system of isolated inclusions embedded in a matrix. The inclusion is often made from a ferroelectric ceramic, and the matrix is regarded as a polymer medium being either piezoelectric or piezo-passive. Such piezo-composites are manufactured using a variety of techniques (see, e.g., reviews

Effective Electromechanical Properties in Piezo-composites

a

27

b

Figure 2.3. Micrograph of the porous PZT ceramic (a) and example of the representative FEM image (b) that is used for calculations of the effective electromechanical properties. x, y and z are co-ordinate axes (reprinted from Ramesh et al. [70], with permission from Elsevier)

[48, 71, 72]) and studied using a series of experimental methods [73–79]. Theoretical studies of 0–3 piezo-composites are focused on the effective electromechanical properties that are strongly dependent on the properties and volume fractions of the components [5, 33, 50, 61], on the shape of the inclusions and features of their distribution in the matrix [5, 60, 80] as well as on poling conditions [75, 77] for the composite sample as a whole. In these studies, it is often assumed that the inclusions have the shape of a spheroid [53, 81] or a rectangular parallelepiped [33] and aligned regularly in the matrix. An electromechanical interaction between each inclusion and the surrounding matrix is taken into account [5, 60, 61] in connection with internal electric and mechanical fields and boundary conditions for the specific shape of the inclusion. An approach developed by Banno [69, 82, 83] stems from concepts [84] initially applied to 2–2 composites. Based on these concepts, the 0–3 composite is represented [83] by the Banno unit cell (or the representative volume element) that comprises the parallelepiped-like inclusion and the surrounding matrix. The effective electromechanical properties in this composite are determined by direct averaging of the properties in the components. In this averaging procedure, the shape of the parallelepiped (i.e., a ratio of their edges), but not the interaction between the components plays an important role. In an alternative matrix method [33] proposed for the prediction of the electromechanical properties in the 0–3 composite, the theoretical description is based on the matrix approach [64, 85] previously applied to the 2–2 composite. This approach enabled the authors of ref. [33] to perform the averaging procedure for the 0–3 composite with a regular distribution of the parallelepiped-like inclusions. The method developed by Levassort et al. [33] has been extended to 3– 3 and 1–3 ceramic / polymer piezo-composites [86, 87] with planar interfaces. In refs. [33, 86, 87], the electromechanical constants of the components c (pqn ),E , e (nfp ) and H kr(n ),[ are involved in matrices of the properties (9 u 9) and boundary conditions. To avoid mathematical complications concerned with the presence of the matrix elements with strongly differing degrees of values (e.g., c (pqn ),E a (1010 – 1011) Pa and e (nfp ) a (1–10) C / m2 for the nth component), the matrix approach has been modified and successfully applied to the 1–E piezo-composites with E = 1, 2

28

Electromechanical Properties in Composites Based on Ferroelectrics

and 3 [88, 89]. The matrices describing the electromechanical properties in the components comprise elastic compliances s (pqn),E a (10–11 – 10–10) Pa-1, piezoelectric ) coefficients d (n a (10–11 – 10–9) pC / N and dielectric permittivities H kr(n ),V a (10–11 – fp

10–8) F / m, and use of such a set of electromechanical constants [88, 89] avoids the aforementioned significant differences in the degrees of the values of the electromechanical constants. Many publications on 0–3 composites are concerned with a piezoelectric ellipsoidal inclusion problem [5, 53, 60, 90–95]. This problem stems from the classical Eshelby analysis [96, 97] of the stress and strain fields in and around an elastic ellipsoidal inclusion. Results [96, 97] on the uniform stress and strain fields in such inclusions and the equivalent inclusion method stimulated intensive investigations into heterogeneous media and their effective physical properties [98]. Work on the coupled electric and mechanical fields in heterogeneous materials implied the application of the equivalent inclusion method [96, 97] for the description of the piezo-composites, poled ferroelectric ceramics and piezoactive materials with pores or cracks [94, 95, 99–101]. The inclusion can be either piezo-active or piezo-passive, and no restrictions on its symmetry are introduced. The matrix can be either piezo-active (transversely isotropic) or piezo-passive (isotropic) [91–95, 99, 101]. The analysis of the coupled electromechanical behaviour of an “inclusions – matrix” system is based on knowledge of the Eshelby tensors. These tensors represent the strain and electric fields that are induced in the inclusion due to a constraint of the matrix in the external fields. The Eshelby tensors are expressed in the explicit form for the following inclusion shapes [101, 102]: spheroid, elliptic cylinder, rod, penny, and ribbon-like. All of the inclusions are assumed to be aligned along the axis of anisotropy, i.e., along the poling axis in the transversely isotropic medium. As in the work on elastic inclusions in an elastic matrix [96, 97], the Eshelby tensors depend on the shape of the inclusion and on the properties in the matrix. Further averaging of the electromechanical properties is often carried out according to a self-consistent scheme. The main variants of this scheme are concerned with the effective medium method [94, 103, 104] and the effective field method (or the mean field method) [53, 94, 95, 104–106]. In the effective medium method, the inclusion is regarded as a region in a continuous medium (i.e., in a matrix containing an ensemble of similar inclusions). The effective properties in this medium are found using the Eshelby tensors for the specified shape of the inclusions, and calculations are performed within the framework of iterative (self-consistent) procedures. The effective field method (EFM) stems from the Mori – Tanaka method [107] based on the concept of an average stress in the matrix containing the inclusions with so-called eigenstrains. The eigenstrains represent the strains appearing in a stress-free region, similar to the spontaneous strains in ferroelectric or ferroelastic phases. In the EFM, the inclusion interacts with the matrix subjected to an action of an external electric and mechanical field. Due to the presence of the ensemble of similar inclusions in the matrix, the field can be regarded as the effective field

Effective Electromechanical Properties in Piezo-composites

29

that acts on each inclusion therein. This means that the effective (or average) electric and mechanical fields in the composite sample (see Equations 2.21 and 2.22) play a significant role in forming the overall effective electromechanical properties. The EFM is one of the most powerful and most suitable methods [105] for studying the overall behaviour of piezo-composites and for predicting their effective electromechanical properties. Moreover, the EFM is known to produce physically realistic results with regard to the effective properties in the piezocomposites [53, 104, 106] in a wide range of inclusion volume fractions. This method has been applied to determine the effective electromechanical properties in the 0–3 ferroelectric ceramic / polymer [53, 80], ceramic / ceramic [61, 105] and single crystal / ceramic [108] composites. In addition to the above approaches, the effective electromechanical properties in a piezo-composite with spheroidal inclusions have also been determined using the dilute concentration limit (or dilute solution) [94, 106, 109], the differential scheme [94] and the conditional moments method [5]. These methods are based on different averaging procedures and take into account features of the arrangement of the inclusions and their electromechanical interaction. In addition to the piezoelectric effect in the composites studied, the piezomagnetic and magnetoelectric effects have also been examined in 0–3 composite systems [110–112]. These coupled effects are of interest due to the involvement of three subsystems (e.g., electric, magnetic and mechanical) in materials whereby an interaction between piezoelectric and piezomagnetic components [110, 111] occur. The study of these effects is based on knowing the magneto-electro-elastic Eshelby tensors for spheroidal inclusions [112]. The authors of refs. [110–112] generalised earlier results [101] for the case of a 0–3 piezoelectric–piezomagnetic composite. The determination of the effective properties in piezo-composites with arbitrary connectivity is often carried out by means of the FEM [63, 113–118]. This method employs numerical modelling techniques that take into account boundary conditions for electric and mechanical fields in a composite system. The model of the composite is based on the unit cell being the periodic representative volume element, and this element is assumed to be characteristic of the whole composite structure. An important advantage of the FEM is that there are no restrictions on connectivity of the composite, sizes of the inclusions and their spatial distribution, the number of the components and their properties. Interesting examples of the numerical realisation of the FEM for evaluating the effective electromechanical constants are given in refs. [116, 117] (0–3 connectivity), [63, 68, 113, 118] (1–3 connectivity) and [114, 115] (3–0, 3–3 and mixed connectivity patterns). Trends in the study of the piezo-composites and other piezo-active materials show that the FEM and related finite-element software becomes increasingly more widespread in mechanical-engineering, smart-materials and piezotechnical applications (transducers, sensors, actuators, hydrophones, acoustic antennae, etc.). However, advances in numerical techniques do not exclude the development of analytical methods for connectivity patterns such as 0–3, 1–3 and 2–2.

30

Electromechanical Properties in Composites Based on Ferroelectrics

2.4 Evolution of D–E Connectivity Patterns An example of the evolution of the connectivity patterns in the two-component composites becomes clear with reference to the problem of a piezo-composite with ellipsoidal inclusions (see Section 2.3). In a general case, a composite containing isolated ellipsoidal inclusion is characterised by 0–3 or 3–0 connectivity, where the index 3 refers to the number of co-ordinate axes along which the matrix of the composite is distributed continuously. In the case of 0–3 connectivity, this corresponds to a ferroelectric ceramic / polymer composite with ceramic inclusions. Infinite elongation of one of the axes of the ellipsoid enables us to consider the inclusions having the shape of the elliptic cylinder or a lengthy rod, and this microgeometry results in 1–3 or 3–1 connectivity of the composite. However, the structure of the 0–3 composite containing penny-shaped inclusions approaches a structure of the 2–2 composite. These changes in connectivity are accompanied by changes in the Eshelby tensors [101] so that they can be utilised to predict the effective properties in the 1–3, 3–1 or 2–2 composites with one or two piezoelectric components. Another example of the evolution of the connectivity patterns is associated with various modifications of the 2–2 composite structure. According to Levassort et al. [33, 86, 87], the modified patterns are described by 0–3, 1–3 or 3–3 connectivity. The effective properties in the corresponding piezo-composites are determined by means of the matrix method [33, 64]. It can be seen from the examples above that some connectivities of the twocomponent composite can be associated with slight modifications of the initial connectivity pattern. In this section, we consider the entire complex of the D–E connectivity patterns (Figure 1.1), their evolution and interconnections between them [89] in the two-component composites with planar interfaces. It is assumed that there is a group of junction connectivity patterns and that their combination can form new connectivity patterns. Each composite with a regular arrangement of components is represented by the Banno unit cell (or representative volume element) [82, 83]. The presence of planar interfaces and their certain hierarchy inherent in the D–E composite suggest that the 2–2 connectivity pattern (i.e., a system of series or parallel layers) may be regarded as one of the junction connectivity patterns. For example, a simple multiplication of the 2–2 connectivity pattern results in the 1–1 connectivity pattern [88], see Figure 2.4 and the connection between the 2–2 and 1–1 structure. A further simplification of the 1–1 connectivity pattern enables us to obtain the 1–3 or 3–1 connectivity pattern.3 It should be noted that this simplification has similarities to the formation or rearrangement of ferroelectric domain structures, in which not all domain types are present even though these types are predicted [119] in accordance with symmetry changes at phase transitions. ____________ 3

It is seen from Figure 2.4 that the above-mentioned 2–2, 1–1, 1–3, and 3–1 connectivity patterns make up the full set of the junction connectivity patterns. These patterns show numerous interconnections between the structures of the two-component composites from Figure 1.1.

Effective Electromechanical Properties in Piezo-composites

31

Figure 2.4. Links between the D–E connectivity patterns. C1 and C2 denote components 1 and 2, respectively. Double lines connecting the 2–2 and 1–1 connectivity patterns as well as the 1–1 and 0–0 connectivity patterns denote a multiplication of structural elements at transformations 2–2 o 1–1 and 1–1 o 0–0. Dashed lines connecting the 1–1 and 1–3 connectivity patterns as well as the 1–1 and 3–1 connectivity patterns denote a simplification of the 1–1 structure at transformations 1–1 o 3–1 and 1–1 o 1–3 (reprinted from Topolov and Glushanin [89], with permission from IOP Publishing)

The main interconnections between the D–E connectivity patterns are shown in the schematic in Figure 2.4 that contains links between the junction connectivity patterns and other connectivity patterns in the D–E composites. All the D–E composite structures can be generated by systems of the layers. These layers are either homogeneous (see components C1 and C2 at the base of Figure 2.4) or heterogeneous. The heterogeneous layers are characterised by the junction connectivity pattern, by the combination of the junction connectivity patterns or by the combination of the junction connectivity pattern and one of the components. The evolution of the connectivity patterns demonstrates the important role of the junction connectivity patterns. The main features of this evolution are described as follows. 1. From the symmetrical arrangement of the junction connectivity patterns and other elements in the scheme (Figure 2.4), it can be concluded that the evolution of the connectivity patterns is primarily caused by a motion along the symmetry axis (i.e., the line that mentally connects the connectivity patterns with D = E) and adjacent branches. 2. The structures of the interfaces are rather complicated for 0–1 and 1–2 connectivity patterns (Figure 1.1) that are generated by three or two junction connectivity patterns. The 0–0 connectivity pattern is situated at a dead-end

32

Electromechanical Properties in Composites Based on Ferroelectrics

between the 1–1 connectivity pattern and the component Ci (i = 1 or 2). As a consequence, the 0–0 connectivity pattern does not influence the formation of the 0–E structures with E = 1, 2 and 3. 3. The maximum distance between the 3–3 connectivity pattern and the components Ci, along with the lack of direct links between them are also typical of the evolution of the connectivity patterns. It is shown in Figure 2.4 that the remaining connectivity patterns are located in between 3–3 and Ci. The continuous distribution of the C1 or C2 component or each component in the 3–3 connectivity pattern along three co-ordinate axes represents two extreme cases of the arrangement in the scheme (see upper and bottom parts of Figure 2.4). 4. The number of systems of the interfaces decreases at D = 3 or E = 3. Such a decrease is a result of a continuous distribution of one (at D = 3 z E or E = 3 z D) or two (at D = E = 3) components along three co-ordinate axes. In fact, any introduction of an additional type of interface would destroy this continuity. Knowledge of the evolution of the connectivity patterns enables the analysis of the D–E piezo-composites with planar interfaces as well as understanding the resulting effective electromechanical properties in these heterogeneous materials. The concepts developed in this section can ease both the modelling and optimisation of the effective electromechanical properties, especially in the piezocomposites with complicated connectivity patterns. An example of the 1–2 composite and a series of parameters introduced for determination of its effective electromechanical properties are considered below.

2.5 Modelling of Effective Properties in Piezo-composites with Planar Interfaces An important step following the analysis of the evolution of the connectivity patterns is the synthesis, i.e., a combination of parts to form a connected composite. An approach developed in ref. [89] is based on the assumption that different D–E composite structures with planar interfaces can be generated using a system of layers (Figure 2.5). It is assumed that the layers with specified connectivity regularly alternate along the OX3 axis that represents the poling direction. The C1 component is shown as the dotted area in all cross-sections (Figure 2.5) of the D–E connectivity patterns (D ” E) and in the schematic arrangement of the 1–2 composite (Figure 2.5, see bottom right). In order to determine the effective electromechanical properties of the 1–2 composite, one can introduce a series of concentration parameters [89], for example, t = |AC| / |AE|, n = |FH| / |FL|, tc = |AB| / |AC|, nc = |FG| / |FH|, and v = |MN| / |MP|. These parameters4 characterise volume ____________ 4 The concentration parameters t, n and tc, which are related to the 0–0 and 0–1 connectivity patterns (Figure 2.5), are introduced by analogy with the concentration parameters related to the 1–2 composite. The D–E connectivity patterns obeying the condition D • E are depicted by the similar schemes wherein the dotted areas shown in Figure 2.5 become white and the white areas become dotted.

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Figure 2.5. Systems of layers forming the D–E connectivity patterns (reprinted from Topolov and Glushanin [89], with permission from IOP Publishing)

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fractions of the C1 and C2 components in the 1–2 composite and are assumed to be constant over the composite sample. Any variations of the concentration parameters mean that relative widths (or lengths) of several parallelepipeds are simultaneously varied in all the layers of the composite. The layers forming the D– E composites are parallel to the (X1OX2) plane as shown for the 1–2 composite (Figure 2.5, see bottom right). The calculation of the effective electromechanical constants of the D–E composites can be performed by means of the matrix method [64] (as shown for the 1–1, 1–2 and 1–3 ferroelectric ceramic / polymer composites [88, 89]) or the FEM. In these methods the electromechanical interaction between the adjacent regions is taken into account in averaging procedures. For this calculation, full sets of electromechanical constants of both components are required. The effective properties in the aforementioned 1–E composites with E = 1, 2 and 3 are determined [88, 89], for example, using elastic compliances s (pqn),E , piezoelectric ) coefficients d (n and dielectric permittivities H kk(n ),V of the ferroelectric ceramic (n = fp

1) and polymer (n = 2). In a general case, the matrix method [64] is applied to predict the effective electromechanical properties in the periodically layered 2–2 composite or the ferroelectric single crystal with a laminar domain structure [120, 121]. This approach can be applied, providing the wavelength of an incident acoustic field is much longer that the thickness of the separate layer. In a range of low frequencies Z of the AC electric field, the condition ZW << 1 holds (where W is the Maxwell– Wagner relaxation time) and the electric conductivities of the layers are to be taken into account [57] for the correct prediction of the effective electromechanical properties in the 2–2-type piezo-composite. The matrix method [64], which does not take into account the electric conductivities, can be applied at frequencies that are considerably more than those where the Maxwell–Wagner relaxation is in progress. At the same time, the frequency of the electric field should be well below a piezoelectric resonant frequency of the composite sample for specific oscillation modes. The modelling of the effective properties in the piezo-composites is closely connected with their potential applications. Clearly, for successful and optimum piezotechnical applications, the piezo-composites with different connectivity patterns, volume fractions of the components and other parameters are required. The optimisation of the effective properties and related parameters as well as manufacturing the appropriate piezo-composites are of independent interest. It should be noted that structure – processing – property relations in the ferroelectric ceramic / polymer composites are described using connectivity as the unifying central concept [71]. Computer-aided design-based fabrication methods have advanced in recent years [72]. In particular, novel processing technologies make it possible to attain unprecedented structural complexity [71] as compared to the traditional methods [48, 69] for manufacturing the ceramic / polymer composites.

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2.6 Connectivity – Links – Properties Chapter 2 has been devoted to the description of the electromechanical properties in piezo-active materials. An emphasis has been placed on the piezo-composites, their connectivity and trends in forming their effective electromechanical properties. Three types of effective properties (i.e., sum, combination and product properties) have been considered and applied to the two-component composites based on ferroelectrics. Results on the evolution of the D–E connectivity patterns and the role of systems of planar interfaces inherent in the D–E piezo-composites have been discussed. A scheme demonstrating the links between the D–E connectivity patterns has been put forward for the modelling of the effective properties in the piezo-composites with planar interfaces. In general, Chapter 2 can be regarded as a transition from the connectivity concept to determination of the volume-fraction dependences of the effective properties in different groups of piezo-composites, such as ferroelectric ceramic / polymer, ferroelectric single crystal / polymer, ferroelectric single crystal / ferroelectric ceramic, with two or more piezoelectric components, etc. Further discussion will examine the determination and interpretation of calculated data relating to specific connectivity patterns and the aforementioned groups of the piezo-composites. In Chapter 3 we will consider the most important D–E connectivity patterns as well as features of the effective electromechanical properties and applications of the two-component composites based on ferroelectric ceramics.

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3 Non-monotonic Volume-fraction Dependences of Effective Properties in D–E Ceramic / Polymer Piezo-composites

3.1 Composites with 2–2 Connectivity, Manufacturing and Applications Since the 1970s, there has been significant progress in the development and application of piezo-composites consisting of ferroelectric ceramic and polymer components [1–6]. The D–E ferroelectric ceramic / polymer composites combine high piezoelectric activity of a ceramic with the beneficial mechanical properties of a polymer. A polymer is often chosen as the piezo-passive material and is characterised by low dielectric permittivity and low elastic modulus. By combining the physical properties of the ceramic and polymer components and tailoring the microgeometrical features (see, e.g., Figure 1.2) significant advantages can be achieved over the performance of the individual piezo-active component. Some advantages of the D–E piezo-composites are concerned with a variety of non-monotonic dependences of the effective electromechanical properties on the volume fractions of the ceramic components. Many examples of such dependences are known from work on the 2–2 ceramic / polymer composites with the series connection (Figure 3.1a) and the parallel connection (Figure 3.1b) of layers. The 2–2 series-connected composites based on the Pb(Zr, Ti)O3-type ceramic are characterised by several types of volume-fraction dependence of the thickness * * D *[ 1/2 electromechanical coupling factor kt* = e33 / ( c33 H 33 ) [7, 8] and elastic *E * modulus c11 measured at an electric field E = const [9], where e33 is the *D is the elastic modulus (at electric displacement D = piezoelectric coefficient, c33 *[ is the dielectric permittivity (at mechanical strain [ = const). The const) and H 33 * and calculated volume-fraction dependences of the piezoelectric coefficient e31

*V dielectric permittivity H 11 show maxima, and their values are highly dependent on the ratio between the elastic moduli of the individual components [9]. The

44

Electromechanical Properties in Composites Based on Ferroelectrics

Figure 3.1. Schematic drawing of the 2–2 series-connected (a) and parallel-connected (b) composites. m and 1 – m are the volume fractions of components C1 (n = 1) and C2 (n = 2). (X1X2X3) is the rectangular co-ordinate system * piezoelectric coefficient e31 of the 2–2 composite [10] can pass through the zero

* > 0, although both the components (n = 1 and 2) have value and reach max e31 (n) . A squared strain–voltage negative values of the piezoelectric coefficient e31

Non-monotonic Volume-fraction Dependences of Effective Properties

45

* 2 figure of merit (Q33 ) = d33* g 33* also exhibits a maxima [11] in the 2–2 composites based on the Pb(Zr, Ti)O3-type ceramic. In comparison to the series-connected composite in Figure 3.1a, the parallelconnected composite (Figure 3.1b) contains layers alternating in the OX1 direction. This structure is characterised by an anisotropy of the electromechanical properties in the (X1OX2) plane. The corresponding 2–2 composite based on the Pb(Zr, Ti)O3type ceramic also exhibits the non-monotonic behaviour of a number of parameters. For example, the thickness electromechanical coupling factor kt* and * * * + d 32 + d 31 and the hydrostatic (or hydrophone) piezoelectric coefficients d h* = d 33 * * * * gh = g 33 + g 32 + g 31 pass through maxima [12, 13] on varying the volume fraction

of the ceramic component. The simplicity of the 2–2 structure (Figure 1.2) and the possibility of varying and tailoring the volume fractions of the components (Figure 3.1) in a wide range, enable the manufacture and design of composites with extreme values of parameters. Methods suitable for manufacturing the 2–2 ferroelectric ceramic / polymer composites are discussed in a series of review papers (see, e.g., [4–6]). The simplest method is the dice-and-fill process [4, 5] whereby a series of parallel cuts is made in a ceramic bulk sample (block). The cuts attached to the ceramic base are backfilled with polymer, and the resulting filled composite consisting of a system of alternating ceramic and polymer layers is finally removed from the ceramic base. In the injection-moulding method [4, 5, 14] preforms containing a number of thin ceramic sheets are initially prepared and then heat treated. The heat treatment enables the removal of the binder and subsequent sintering. Final filling of the structure with a polymer results in a fine-scale 2–2 composite. The tapelamination method [4, 5, 15] utilises thin ceramic sheets that are either laminated with a passive material or stacked with a spacer, and then filled with polymer. Jet machining [4, 5, 16] is applied to a green ceramic tape that contains the powder and binder. The main stages of this method are lamination with a photoresist, exposing through a photomask, the development of the photoresist, and jet machining. This method is often used to manufacture the 2–2 ceramic / polymer composite with a system of annular rings [4, 5]. The co-extrusion method is also suitable for manufacturing the 2–2 composites [4, 5]. The initial stage of coextrusion involves mixing ceramic powder with a thermoplastic polymer and then extruded the mixture to form a sheet. This sheet is then laminated with a carbonfilled thermoplastic sheet. Finally, the co-extrusion of these laminates is implemented and results in the formation of the 2–2 composite. Recently, rapid prototyping methods (or solid freeform fabrication) [6] have become attractive and enable the manufacture of fine-scale 2–2 and other piezocomposites using computer-aided design. Among these methods, those suitable for the fabrication of separate polymer and ceramic parts [5, 6] with specified microgeometry are of interest. The main stages of manufacturing are the fused deposition of the ceramic, filament fabrication, the build process (using computeraided design), and the post-processing stage. As a result, a number of 2–2 piezocomposites [6, 7, 17, 18] have been manufactured, namely, composites with curved

46

Electromechanical Properties in Composites Based on Ferroelectrics

ceramic skeletons, composites with volume-fraction gradient, radial composites and oriented composites. The 2–2 piezo-composites find various applications, mainly as active elements of sensors, actuators and transducers for medical-imaging applications [5, 6]. In addition, the transducer can perform various functions via conversion of acoustic energy into the electric energy or vice versa. The transducer can also combine passive and active functions in, for example, the pulse–echo mode. The 2–2 piezocomposites based on the Pb(Zr, Ti)O3-type ceramic are often used as cantilevers or multilayers [19]. The 2–2 ceramic / polymer composites that exploit the transverse piezoelectric mode for hydroacoustic applications are also of interest [19, 20]. In the 1990s, novel 2–2-type structures became widespread due to the development of “moonies” and “cymbals” [19, 20] that exhibit high actuator performance owing to the combination of ferroelectric ceramic layers and metal elements as strain transformers. These novel transducers provide both a considerable displacement and a large generative force for hydrophone, actuator and transceiver applications. 3.1.1 Effective Electromechanical Constants of 2–2 Composites In the following section we will consider the effective electromechanical properties in simple series- and parallel-connected composites (Figure 3.1) that are termed 2– 2 ceramic / polymer composites. The orientation of the interfaces between the layers with respect to the poling direction influences these properties. Alternating the ceramic (C1) and polymer (C2) layers along the poling direction OX3 (Figure 3.1a) or along the OX1 direction, perpendicular to the poling direction (Figure 3.1b), results in different boundary conditions [13, 21] for the internal electric and mechanical fields and, as a consequence, influences electromechanical coupling in the 2–2 structures. It is assumed that the layers of the 2–2 composite have a large length in the OX1 and OX2 directions (Figure 3.1a) or in the OX2 and OX3 directions (Figure 3.1b). The interfaces separating these layers represent planes, and perfect bonding between the adjacent layers is assumed. To determine the full set of effective electromechanical constants of the 2–2 composite, a matrix method is used [21, 22]. The electromechanical constants of the components are given by the 9 u 9 matrices

§ || s ( n ), E || || d ( n ) ||t · ¸ || C || = ¨¨ ( n ) ( n ),V ¸ H || || d || || ¹ © (n)

(3.1)

where || s(n),E || is the 6 u 6 matrix of elastic compliances, || d(n) || is the 6 u 3 matrix of piezoelectric coefficients and || H(n),V || is the 3 u 3 matrix of dielectric permittivities of the ceramic (n = 1) and polymer (n = 2) components, and superscript t denotes the transposition. A matrix || C* || of the effective electromechanical properties in the 2–2 composite is written [21, 22] in the form || C* || = [|| C(1) ||.|| M || m + || C(2) || (1 – m)] [|| M || m + || I || (1 – m)]–1 (3.2)

Non-monotonic Volume-fraction Dependences of Effective Properties

47

The structure of the matrix || C* || is similar to that shown in Equation 3.1. The effective properties of the composite are determined using Equation 3.2 in a longwave approximation [21], i.e., in a case when the wavelength of an external acoustic field is much longer than the thickness of the separate layer (C1 or C2, see Figure 3.1) in the composite. In Equation 3.2 || M || is the matrix related to the boundary conditions at interfaces xi = const (i = 1 or 3), and || I || is the identity matrix. The boundary conditions, for example, at x1 = const (Figure 3.1b) imply a continuity of components of mechanical stress V11, V12 and V13, strain [22, [23 and [33, electric displacement D1, and electric field E2 and E3. By taking into account the transversal isotropy of the poled ceramic component and that the polymer component can be isotropic (piezo-passive) or transversely isotropic (piezo-active) [10, 12, 13], the matrix || M || for interfaces x1 = const is || M || = || m1 ||–1|| m2 || where

§1 0 0 ¨ ¨ ¨ ( n ), E ( n ), E ( n ), E s11 s13 ¨ s12 ¨ ¨ s ( n ), E s ( n ), E s ( n ), E 13 33 ¨ 13 ¨ ¨0 0 0 ¨ ¨ || m n || ¨ 0 0 0 ¨ ¨0 0 0 ¨ ¨ ¨0 0 0 ¨ ¨ 0 0 ¨0 ¨ ¨¨ 0 0 0 ©

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

( n ), E s 44

0

0

0

d 15( n )

0

1

0

0

0

0

0

1

0

0

0

d 15( u )

0 0

0 H 11( n ), V

0

0

0

0

1

0

0

0

0

0 ·¸ ¸ (n) ¸ d 31 ¸ ¸ d 33( n ) ¸ ¸ ¸ 0 ¸ ¸ ¸ 0 ¸ ¸ 0 ¸ ¸ ¸ 0 ¸ ¸ ¸ 0 ¸ ¸ 1 ¸¸ ¹

is written in terms of the electromechanical constants of the components (n = 1 or 2). * Taking the matrix elements Cab from Equation 3.2 and using Equations 2.12– 2.19, it is possible to determine the effective piezoelectric coefficients d kl* (m),

ekl* (m), g kl* (m), and hkl* (m) for the 2–2 composites (Figure 3.1) where the volume fraction m is varied from 0 to 1.

48

Electromechanical Properties in Composites Based on Ferroelectrics

3.1.2 Examples of Volume-fraction Dependences Predicted for 2–2 Composites In this section we consider the piezoelectric properties of 2–2 composites based on the Pb(Zr, Ti)O3- or PbTiO3-type ceramics. As can be seen in Table 2.1, these ferroelectric ceramics are distinguished by a piezoelectric anisotropy, dielectric (1) properties and signs of the piezoelectric coefficients e31 . Calculations are performed using the room-temperature experimental data on typical ferroelectric ceramics (see Table 2.1) and polymer (araldite from ref. [23]). Of particular interest are the volume-fraction dependences of the four types of piezoelectric coefficients ĭ3l* ( ĭ = d, e, g, or h) that characterise an electric response of the composites in the poling direction OX3 when an external mechanical field is applied along one of the co-ordinate axes OXl (Figure 3.1). The 2–2 composites examined are characterised by fmm (series connection) or mm2 * * * (parallel connection) symmetry. A relation ĭ31 = ĭ32 z ĭ33 holds for the ceramic / polymer series-connected composites with interfaces x3 = const (Figure 3.1a). In this case there is no distinction between the response along the OX1 and OX2 axes. The orientation of interfaces x1 = const (Figure 3.1b) results in a reduction of the * * * macroscopic symmetry of the composite so that a relation ĭ31 z ĭ32 z ĭ33 is held. The series-connected composites are characterised by relatively small values of e and d 3l* at volume fractions 0 < m < 0.9 due to the poor orientation of the layers (Figure 3.1a) with respect to the poling direction. In the external electric field E || OX3 a considerable depolarisation field appears in the ceramic layers C1, and these layers provide a small contribution to the piezoelectric response of the composite as a whole. For example, at m = 0.5 the PZT-5 ceramic / araldite composite is * (1) * (1) characterised by the following values: e31 = 1.07u10–2 e31 , e33 = 5.27u10–4 e31 , * 3l

* (1) * (1) = 4.69u10–3 d 31 , and d 33 = 3.59u10–3 d 33 . In addition, the studied composite d 31 * (1) * (1) * (1) = 0.956 g 31 , g 33 = 0.729 g 33 and h31 = 1.07 h31 . For the attains values g 31 (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic / araldite composite at m = 0.5 the * (1) * (1) * (1) * values e33 = 2.69u10–3 e33 , d 33 = 2.53u10–2 d 33 , g 31 = 0.957 g 31 , and g 33 = (1) are predicted. A relatively slow increase in the dielectric permittivities 0.543 g 33

H 33*V (m) and H 33*[ (m) of the composites leads to relatively large values of the piezoelectric coefficients *V (m) g 3l* (m) = d 3l* (m) / H 33

(3.3)

*[ (m) h3l* (m) = e3l* (m) / H 33

(3.4)

in comparison with d 3l* (m) and e3l* (m) (see Equations 2.12–2.15). The volume-fraction dependence of the piezoelectric coefficients of the parallel-connected composites (Tables 3.1 and 3.2) indicates that such an arrangement of the layers C1 and C2 (see Figure 3.1b) leads to a significant

Non-monotonic Volume-fraction Dependences of Effective Properties

49

Table 3.1. Volume-fraction dependences of the piezoelectric coefficients e3l* (in C / m2), 8 d 3l* (in pC / N), g 3l* (in mV m / N), and h3l* (in 10 V / m), which have been calculated for the 2–2 PZT-5 ceramic / araldite parallel-connected composite m

* e31

0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m

* a g 31

0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

* e32

* e33

* d 31

* d 32

* d 33

0

0

–0.0187 –0.0394 –0.0878 –0.149 –0.228 –0.335 –0.487 –0.721 –1.13 –2.01 –5.40

–0.112 –0.225 –0.456 –0.695 –0.945 –1.21 –1.51 –1.86 –2.31 –3.07 –5.40

0 0.944 1.89 3.78 5.66 7.53 9.38 11.2 13.0 14.6 16.0 15.8

0

0

–43.7 –67.0 –93.1 –109 –121 –131 –140 –148 –155 –163 –170

–61.2 –92.4 –124 –140 –149 –156 –160 –163 –167 –169 –170

0 136 205 273 307 328 342 352 359 365 369 373

* b g 32

* c g 33

* h31

* h32

* d h33

0

0

–79.4 –55.7 –35.2 –26.3 –21.3 –18.1 –15.9 –14.3 –13.1 –12.1 –11.3

–111 –76.8 –46.9 –33.7 –26.2 –21.5 –18.2 –15.8 –13.9 –12.5 –11.3

0 247 170 103 74.0 57.6 47.2 40.0 34.6 30.6 27.3 24.7

0

0

–0.452 –0.497 –0.568 –0.647 –0.747 –0.880 –1.07 –1.36 –1.87 –2.98 –7.34

–2.70 –2.85 –2.95 –3.02 –3.10 –3.19 –3.32 –3.51 –3.83 –4.55 –7.34

0 22.9 23.9 24.5 24.6 24.7 24.7 24.6 24.5 24.3 23.7 21.5

Notes to Table 3.1 * = * (0.016) = –102 mV m / N a. min g 31 g 31 * = * (0.016) = –143 mV m / N b. min g 32 g 32 * * (0.015) = 320 mV m / N c. max g 33 = g 33

* ( * (m) | 2.47u109 V / m) is attained at 0.334 d m d 0.546 d. A diffuse maximum of h33 h33

* * increase in e33 (m), d 33 (m) and h3l* (m) on increasing the volume fraction m of the ceramic layers. In both composites the piezoelectric coefficients g 3l* (m) show non-

monotonic behaviour at fairly small m values. This behaviour is due to the relatively rapid increase in the piezoelectric coefficients d 3l* (m) at small dielectric *V (m) in a range of 0 < m << 1 (see Equation 3.3). In contrast to the permittivity H 33 * * (m) in the composite based peaks of g 3l (m), a diffuse maximum is peculiar to h33 *[ (m) cannot on the PZT-5 ceramic (see Table 3.1): in this case a slow increase in H 33

50

Electromechanical Properties in Composites Based on Ferroelectrics

Table 3.2. Volume-fraction dependences of the piezoelectric coefficients e3l* (in C / m2),

d 3l* (in pC / N), g 3l* (in mV m / N), and h3l* (in 108 V / m), which have been calculated for the 2–2 (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic / araldite parallel-connected composite m

* e31

* e32

* e33

* a d 31

0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 m

0 0.00134 0.00283 0.00631 0.0107 0.0164 0.0242 0.0354 0.0528 0.0837 0.153 0.458

0 0.0185 0.0370 0.0741 0.111 0.149 0.187 0.226 0.266 0.308 0.359 0.458

0 0.322 0.644 1.29 1.93 2.58 3.22 3.86 4.51 5.16 5.81 6.50

* c g 31

* d g 32

* e g 33

0

0

–116 –88.0 –52.8 –35.1 –24.7 –17.8 –13.1 –9.50 –6.76 –4.58 –2.81

–49.1 –29.8 –15.3 –10.0 –7.40 –5.84 –4.81 –4.09 –3.55 –3.14 –2.81

0 270 205 131 95.4 74.9 61.6 52.3 45.4 40.1 35.9 32.6

0 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

* b d 32

* d 33

0

0

–12.3 –15.9 –17.6 –17.1 –15.8 –14.2 –12.4 –10.5 –8.50 –6.47 –4.40

–5.22 –5.39 –5.11 –4.89 –4.75 –4.64 –4.57 –4.51 –4.47 –4.43 –4.40

0 28.8 37.2 43.8 46.5 48.0 49.0 49.6 50.1 50.5 50.8 51.0

* h31

* h32

* h33

0 0.138 0.179 0.227 0.269 0.317 0.379 0.467 0.601 0.837 1.37 3.70

0 1.90 2.35 2.67 2.80 2.87 2.93 2.97 3.02 3.09 3.21 3.70

0 33.1 40.9 46.3 48.5 49.7 50.4 50.9 51.3 51.6 51.9 52.5

Notes to Table 3.2 * * a. min d 31 = d 31 (0.207) = –17.6 pC / N * * b. min d 32 = d 32 (0.085) = –5.41 pC / N * * c. min g 31 = g 31 (0.034) = –121 mV m / N * * d. min g 32 = g 32 (0.021) = –62.8 mV m / N * * = g 33 (0.031) = 285 mV m / N e. max g 33 * compete with a rapid increase in e33 (m) in the wide m range (see Equation 3.4). It

is seen that both the composites show the large anisotropy of e3l* (m) and h3l* (m) at various m values, i.e., conditions * * / | e3* f | >> 1 and h33 / | h3* f | >> 1 e33

(3.5)

Non-monotonic Volume-fraction Dependences of Effective Properties

51

hold, where f = 1 and 2. The composite based on the PbTiO3-type ceramic is also of interest due to positive values of the piezoelectric coefficients e3l* (m) and h3l* (m) in the complete

m range (l = 1, 2 and 3, see Table 3.2). At the same time, the piezoelectric coefficients d 3l* (m) have different signs. According to Equation 2.12, these piezoelectric coefficients in the parallel-connected composite (mm2 symmetry) are interrelated by formulae *E *E *E * * * * = e31 + e32 + e33 s11 s12 s13 d 31

(3.6)

*E *E *E * * * * = e31 + e32 + e33 s12 s22 s23 d 32

(3.7)

*E *E *E * * * * = e31 + e32 + e33 s13 s23 s33 d 33

(3.8)

An analysis of Equations 3.6–3.8 in connection with data from Table 3.2 and * (m) is primarily conditions 3.5 suggests that the volume-fraction dependence d 31 *E

* due to the item e33 s13 < 0 and closely connected with a competition between

*E

*E

*E

* items e32 s22 > 0 and e33* s23 < 0. In addition, an item e33* s33 > 0 plays a * (m). It is obvious that the large significant role in forming the dependence d 33

(1) / piezoelectric anisotropy, peculiar to the PbTiO3-type ceramic layers (wherein d 33

(1) * * * * | > 10), cannot affect both the d 33 / d 31 and d 33 / d 32 ratios in the wide m | d 31

*E

range (Table 3.2) due to changes in elastic compliances sab from Equations 3.6 – 3.8. Undoubtedly, these changes are associated with the parallel-connected layers (Figure 3.1b) that promote the appreciable elastic anisotropy at various volume fractions m. It should be added that analytical formulae are known for the effective electromechanical constants of both the series-connected [13] and parallelconnected [12, 13] piezo-composites (Figure 3.1) with a regular arrangement of the layers C1 and C2. A comparison of data from Tables 3.1 and 3.2 with the results obtained using formulae [12, 13] shows that a difference between the similar constants does not exceed 1% in the range of 0 < m < 1. A further comparison of the volume-fraction dependences of the electromechanical properties in the seriesconnected composites can be made in order to account for [24–26] the electric conductivity of the layers. The volume-fraction dependences of the effective elastic properties determined using Equation 3.2 coincide with those predicted [24] taking into account the electric conductivity of the layers. A discrepancy between the *V (m)) effective properties (for example, volume-fraction dependences d 3l* (m) or H 33 predicted using the matrix approach [21, 22] and the same properties determined using formulae [24, 26] is accounted for by the distribution of internal electric and mechanical fields in the layered structure with an element of conductivity of the

52

Electromechanical Properties in Composites Based on Ferroelectrics

components. A possible “giant” increase in the dielectric and piezoelectric properties in the series-connected composites is expected [24, 26] at low electric field frequencies, i.e., at the condition ZW << 1 where Z is the circular frequency of the electric field E and W is the Maxwell–Wagner relaxation time. In this case, the role of the electric conductivity of the layers of the composite and the redistribution of electric charges near the interfaces separating these layers and oriented perpendicular to the electric field E is of importance. Such behaviour of the effective properties in the 2–2 series-connected piezo-composites could lead to potentially new applications for these composite materials in the infralowfrequency range.

3.2 Composites with 1–3 Connectivity, Manufacturing and Applications In the 1–3 ceramic / polymer composite system, the ceramic component is selfconnected in one dimension (often along the poling axis) and the polymer component is self-connected in three dimensions. The 1–3 composites (Figures 1.2 and 3.2) are widespread due to the ease of poling and a variety of advantages over the poled monolithic ferroelectric ceramics [3–6, 27–31]. The ceramic component can be represented by a continuous row of particles, relatively long rods or fibres, or a series of discs. The combination of the relatively low dielectric permittivity and high piezoelectric activity at small volume fractions of the ceramic component (less than 10%) results in significant piezoelectric sensitivity of the 1–3 ferroelectric ceramic / polymer composite [32, 33]. These particular composite properties stimulated intensive research into the development and manufacture of 1–3 ceramic / polymer composites in the last decades (see, e.g., [4–6, 27–29, 34– 36]). Simultaneously, much research has been devoted to the analysis of the nonmonotonic dependence of the 1–3 composite’s parameters on the volume fraction of the ceramic component. For example, a number of papers report on extreme * * * [23, 38, 40, 41], g 33 [33, 42], and h33 piezoelectric coefficients d 3* j [27, 37–39], e33 [33], hydrostatic piezoelectric coefficients d h* [27, 29, 32, 34] and g h* [27, 32, 34, 43], the thickness electromechanical coupling factor kt* [23, 28, 29], and squared hydrostatic figure of merit (Qh* ) 2 = d h* g h* [30, 34, 43, 44]. Due to the rod shape of the highly piezo-active inclusions (Figure 3.2) oriented along the poling field the piezoelectric anisotropy values * * ]e = e33 / e31 * * ]d = d 33 / d 31

are varied over wide ranges [45].

(3.9) (3.10)

Non-monotonic Volume-fraction Dependences of Effective Properties

53

Figure 3.2. Schematic drawing of the 1–3 composite with cylindrical rods. m and 1 – m are volume fractions of the rods (component 1) and the matrix (component 2). (X1X2X3) is the rectangular co-ordinate system

Various methods for manufacturing 1–3 ferroelectric ceramic / polymer composites have been developed. The first attempts to create 1–3 composites were associated with using a coarse PZT powder (i.e., a composition of the Pb(Zr, Ti)O3 type) and polyurethane [3]. The average particle size approached the thickness of the composite sample and promoted saturation poling under an external electric field. The 1–3 composites were also made of PZT spheres or discs embedded in polymers [3]. Klicker et al. [30] proposed embedding PZT rods, aligned in a fixture, in a polymer matrix. A further development of this processing resulted in a rod placement (or pick-and-place) method [4, 5]. In this method, which is appropriate for a wide ceramic volume-fraction range, different microgeometrical features such as arrangement of rods, rod size and shape of the base of the rod, etc. are considered. The methods for manufacturing the 2–2 ceramic / polymer composites (see Section 3.1) also became popular for the 1–3 composite system. These methods include dice-and-fill, injection moulding, co-extrusion, tape-lamination, and rapid prototyping methods [3–6, 31]. The lost-mould method for manufacturing the 1–3 composites [4] uses a plastic mould that has been pre-prepared. The mould contains a negative structure relative to an intended composite structure and the negative mould is filled with ceramic / polymer slurry and subjected to a heat treatment. During the heat treatment, the binder is burned out and the ceramic structure is sintered. After sintering, this structure is filled with the polymer component, i.e., the 1–3 composite is formed. Dielectrophoresis [4, 46] is also suitable for manufacturing the 1–3 ceramic / polymer composites. This method provides an ordering of a colloidal suspension of ceramic particles in a liquid polymer under an external electric field. Due to the non-uniform electric field in the medium, the ceramic particles attract each other and form a chain-like structure that can be continuously distributed between electrodes. The 1–3 ferroelectric ceramic / polymer composites that exhibit high performance find various applications, for example, as active elements of hydrophones, actuators, ultrasonic transducers operating in the pulse-echo mode, medical-imaging devices [3–5, 28, 31]. In a high-frequency range (a 106 – 107 Hz), the 1–3 composites are exploited for non-destructive testing and biomedical diagnostics [4, 28, 31]. The 1–3 piezo-composites represent interesting examples of volume fraction dependencies of the effective piezoelectric properties. In the following section we

54

Electromechanical Properties in Composites Based on Ferroelectrics

consider two typical cases of volume-fraction behaviour of these properties in the composites based on the Pb(Zr, Ti)O3 or PbTiO3 ceramics. 3.2.1 Examples of Volume-fraction Dependences Predicted for 1–3 Composites During modelling of 1–3 composite system it is assumed that the composite consists of very long ceramic rods (component 1) that are regularly distributed in the continuous polymer matrix (component 2). The rods are considered to be in the form of circular cylinders aligned parallel to the poling OX3 axis and continuous in the same direction. In the (X1OX2) plane a square arrangement of the cylinders is assumed. Both the components can be piezo-active in a general case. The 1–3 piezo-composite as a whole is regarded as a transversely isotropic material. Electromechanical properties of its components are described by sets of the elastic ),[ moduli cij( n ),E , piezoelectric coefficients ekl(n ) and dielectric permittivities H (n pp where n = 1 and 2. The volume fractions of the ceramic and polymer components are m and 1 – m, respectively. Examples of calculated volume-fraction dependences of the four types of effective piezoelectric coefficients of the 1–3 ceramic / polymer composites are shown in Figures 3.3 and 3.4. The components are identical to those chosen for the analysis of the piezoelectric properties of the 2–2 composites (see Section 3.1.2). Calculations were performed on the basis of the EFM that enables analytical expressions to be obtained [32] for ekl* , d kl* , cij*E , and H *ppV . A simple comparison of results from Figures 3.3 and 3.4 suggests that, independently of sgn ekl(n ) , conditions 3.3 hold over the range 0 < m d 0.6. It is obvious that the vertical arrangement of the piezo-active rods in the 1–3 structure promotes the large anisotropy values of ]e from Equation 3.9. The piezoelectric coefficients d 3l* are determined using Equations 3.6 – 3.8 for the composite with fmm symmetry and it follows that: *E *E *E * * * * = d 32 = e31 ( s11 + s12 ) + e33 s13 d 31

(3.11)

*E *E * * * = 2 e31 + e33 s13 s33 d 33

(3.12)

In Equation 3.11 both items connected with e3l* are negative at 0 < m d 0.6 for the PZT-5-based composite. In the composite based on the PbTiO3-type ceramic a competition between the items from Equation 3.11 (i.e., positive and negative * * contributions into d 31 ) results in a non-monotonic dependence d 31 (m), as shown in * (m) is Figure 3.4. In both of the composites studied here the dependence d 33 primarily caused by the second item in Equation 3.12. As follows from the data in Figures 3.3 and 3.4, the piezoelectric anisotropy ]d from Equation 3.10 varies in a more restricted range compared with ]e.

Non-monotonic Volume-fraction Dependences of Effective Properties

55

a

b Figure 3.3. Volume-fraction dependences of the piezoelectric coefficients e3l* (in C / m2),

d 3l* (in pC / N), g 3l* (in mV m / N), and h3l* (in 108 V / m), which have been calculated for the 1–3 PZT-5 ceramic / araldite composite: (a) l = 3 and (b) l = 1

56

Electromechanical Properties in Composites Based on Ferroelectrics

a

b Figure 3.4. Volume-fraction dependences of the piezoelectric coefficients e3l* (in C / m2),

d 3l* (in pC / N), g 3l* (in mV m / N), and h3l* (in 108 V / m), which have been calculated for the 1–3 (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic / araldite composite: (a) l = 3 and (b) l = 1

Non-monotonic Volume-fraction Dependences of Effective Properties

57

As with the piezoelectric coefficients g 3l* (m) of the 2–2 parallel-connected * (m) and composites (see Tables 3.1 and 3.2), non-monotonic behaviour of g 31 * (m) in the 1–3 composites occurs at relatively small m values (see Figures 3.3 g 33 and 3.4), in accordance with Equation 3.3. In this volume-fraction range the piezoelectric coefficients d 3l* (m) change relatively quickly in comparison with *V (m), and this results in the considerable piezoelectric dielectric permittivity H 33 sensitivity at 0 < m < 0.1.

3.2.2 Diagrams of Changes in Piezoelectric Properties Predicted for 1–3 Composites The determination of the effective electromechanical properties of the 1–3 composite is usually carried out [10, 27, 29, 32, 33, 42, 47] in terms of two sets of constants of components. An approach proposed in ref. [35] is intended to analyse the effective properties using specific ratios of the component constants. This approach enables the derivation of all the volume-fraction dependences of the effective properties via a limited number of the basic constants of one of the ( 2) and components. Hereafter, the elastic modulus c33( 2),E , piezoelectric coefficient e33 dielectric permittivity H 33( 2 ),V of the matrix (component 2) are chosen as the basic constants. The volume-fraction dependences of the electromechanical properties calculated using the EFM1 [32] are given by ( 2) ( 2 ), E , eij* (m) = eij0 (m) e33 and H *ppV (m) = H 0pp,V (m) H 33( 2 ),V (3.13) c *fgE (m) = c 0fg, E (m) c33

In Equations 3.13 c 0fg, E (m), eij0 (m) and H 0pp,V (m) are dimensionless coefficients that depend on the volume fraction m of the rods. To diversify the volume-fraction dependences that are calculated on the basis of formulae in ref. [32], we consider the possibility of varying the properties of the components. This variation is implemented on changing the ratios [35] between the electromechanical constants of the ceramic rod and the polymer matrix,

rc,fg = c (fg1),E / c (fg2),E , re,ij = eij(1) / eij( 2) and rH,pp = H 33(1),V / H 33( 2 ),V

(3.14)

An effect of varying rc,fg and re,3j from Equations 3.14 on the piezoelectric properties and their anisotropy in the 1–3 composite with the piezo-active polyvinylidene fluoride (PVDF) matrix is shown in Figures 3.5 and 3.6. The ____________ 1

It should be noted ref. [32] are almost cylinder [48] or to between the related fraction range.

that the electromechanical constants calculated using the formulae in equal to those calculated using the Eshelby tensors for the circular those calculated according to formulae in ref. [29]: the difference constants at m = const does not exceed 1% over the wide volume-

58

Electromechanical Properties in Composites Based on Ferroelectrics

Figure 3.5. Graphic presentation of changes in volume-fraction dependences of the normalised piezoelectric coefficients e30 j that have been calculated for the 1–3 composite with the PVDF matrix (reprinted from Topolov and Turik [35], with permission from the American Institute of Physics)

Non-monotonic Volume-fraction Dependences of Effective Properties

59

Figure 3.6. Graphic presentation of changes in volume-fraction dependences of the normalised piezoelectric coefficients d 30 j that have been calculated for the 1–3 composite with the PVDF matrix (reprinted from Topolov and Turik [35], with permission from the American Institute of Physics)

60

Electromechanical Properties in Composites Based on Ferroelectrics

electromechanical properties of PVDF are taken from experimental data [10]. It is assumed that the components have almost equal anisotropy of the (n) ( n) electromechanical properties and piezoelectric coefficients e31 < 0 and e33 > 0. Figures 3.5 and 3.6 contain a system of graphs in cells related to fixed values of ratios from Equations 3.14. In Figure 3.5 these ratios are re,3j = 0.01, rc,fg = 0.01, 0.1, 1, 10, and 100 for cells 1, 2, 3, 4, and 5, respectively; re,3j = 0.1, kc,fg = 0.01, 0.1, 1, 10, and 100 for cells 6, 7, 8, 9, and 10, respectively; re,3j = 1, kc,fg > 0 for cell 11; re,3j = 10, rc,fg = 0.01, 0.1, 1, 10, and 100 for cells 12, 13, 14, 15, and 16, respectively. The piezoelectric coefficients e3* j (m) of the 1–3 composite are expressed in terms of the basic constants from Equations 3.13 as e3* j (m) ( 2) . = e30 j (m) e33

In Figure 3.6 for cells 1 – 10, the same re,3j and rc,fg values are assumed as those related to cells 1 – 10 of Figure 3.5; re,3j = 1, rc,fg = 0.01, 0.1, 1, 10, and 100 for cells 11, 12, 13, 14, and 15, respectively; re,3j = 10, rc,fg = 0.01, 0.1, 1, 10, and 100 for cells 16, 17, 18, 19, and 20, respectively. The piezoelectric coefficients d 3* j (m) of the 1–3 composite are expressed in terms of the basic constants from Equations ( 2) / c33( 2),E . 3.13 as d3* j (m) = d30j (m) e33 The graphs shown in cells of Figures 3.5 and 3.6 have been built for a series of hypothetical 1–3 composites in which the electromechanical properties are varied within a few orders of magnitude. Such large variations enable different combinations of the components to be considered, for example, almost piezopassive or with high piezoelectric activity, elastically soft or hard, etc. Changes in the piezoelectric coefficients e3* j (m) show that weak min e330 (m) < 0 or max e330 (m) > 0 can be observed at rc,fg > 1 (see cells 4, 5, 9, 10, 15, and 16 in Figure 3.5). The volume-fraction behaviour of the anisotropy value ]e(m) from Equation 3.9 is concerned with a convexity of the e30 j (m) curves. This behaviour undergoes changes at rc,fg = 1 or re,3j = 1 (see cells 2 – 4, 7 – 9, 13 – 15, or a sequence of cells 7, 11, 15, etc., in Figure 3.5). As follows from the graphs shown in Figure 3.6, the d30j (m) curves do not undergo significant changes on increasing rc,fg or re,3j. Contrary to ]e(m), changes in the anisotropy value ]d(m) from Equation 3.10 are observed in a fairly narrow range (Figure 3.6). Such behaviour is mainly associated with the relatively small ]e(0) and ]d(0) values of the piezo-active PVDF [10]. It is important to note that the changes in volume-fraction dependences e30 j (m), d30j (m), ]e(m), and ]d(m) (Figures 3.5 and 3.6) do not depend on any rH,pp variations

related to the dielectric properties of the individual components. This independence is caused by the 1–3 structure (see Figure 3.2) and boundary conditions for the electric-field vector components in the composite poled along the OX3 axis. 0 0 Points of weak max e33 (m) or min e33 (m) appear at rc,fg > 1, re,3j = const and almost equal values of the volume fraction m (Figure 3.5). Such extreme points are closely connected with the elastic moduli c (fgn ),E and piezoelectric coefficients e3( nj ) :

Non-monotonic Volume-fraction Dependences of Effective Properties

61

these electromechanical constants are characterised by the equal anisotropy in both components (see Equations 3.12). Any changes in the elastic anisotropy of one of 0 the components at re,3j = const result in a shifting of the extreme point of e33 (m) or can even result in the lack of an extreme point in the range of 0 < m < 1 [35]. 0 Changes in the volume-fraction behaviour of e33 (m) are associated with changes in ratios from Equations 3.14. These latter changes result in a redistribution of the mechanical field (see vertical directions in Figure 3.5) or the electric field (see horizontal directions in Figure 3.5) in the 1–3 composite along the poling axis OX3. 0 The same redistribution has a small influence on the e31 (m) dependence: the corresponding curves shown in Figure 3.5 do not undergo considerable changes at rc,fg > 1. As follows from results in ref. [35], inequalities re,3j < 1 and rc,fg > 1 are 0 necessary for reaching min e33 (m), and inequalities re,3j > 1 and rc,fg > 1 are 0 (m). An effect of the elastically hard rods on the necessary for reaching max e33 0 (m) dependence demonstrates the important role of the internal stress field in e33 forming the non-monotonic piezoelectric properties in the 1–3 composite. It is * noteworthy that max e33 (m) also takes place in the presence of the piezo-passive polymer matrix. A similar maximum was analysed in studies on the 1–E ceramic / polymer composites with planar interfaces [38, 41] where E = 1, 2 or 3. Thus, the approach developed in ref. [35] can be applied for prediction of the effective electromechanical properties in 1–3 piezo-composites and the interpretation of the non-monotonic volume-fraction dependence of different effective parameters, especially when the properties in each component can be varied in certain ranges. Particular cases for the application of this approach to the two-component piezo-composites are considered in ref. [45].

3.2.3 Maxima of Squared Figures of Merit of 1–3 Composites An interesting example of the volume-fraction behaviour of the effective * 2 parameters is concerned with maxima of squared figures of merit (Q33 ) and (Qh* ) 2 in the 1–3 composites based on the Pb(Zr, Ti)O3-type ceramics. These parameters are used [3, 20] to describe the sensor signal-to-noise ratio of the piezoelectric element and to characterise its piezoelectric sensitivity. For the 1–3 composite (Figure 3.2) poled along the OX3 axis and being transversely isotropic, the squared figures of merit are given by * 2 * * * 2 *V = ( d 33 ) / H 33 (Q33 ) = d 33 g 33

(3.15)

2 (Qh* ) 2 = d h* g h* = ( d h* ) / H 33*V

(3.16)

* * * * where d h* = d 33 + 2 d 31 and g h* = g 33 + 2 g 31 = d h* / H 33*V are the hydrostatic piezoelectric coefficients [32, 34, 44]. It is well known [34, 43, 44] that values of * max [ Q33 (m)]2 and max[ Qh* (m)]2 in the 1–3 composite, based on the Pb(Zr, Ti)O3-

62

Electromechanical Properties in Composites Based on Ferroelectrics

* type ceramic, are several tens of times larger than the [ Q33 (1)]2 and [ Qh* (1)]2

values related to the monolithic ceramic. * As a rule, for various 1–3 ceramic / polymer composites, max [ Q33 (m)]2 and max [ Qh* (m)]2 are attained at relatively small volume fractions of the Pb(Zr, Ti)O3– type ceramic, i.e., at 0 < m < 0.1 [32, 34, 43]. In the same m range, the 1–3 composite is characterised by large anisotropy of the piezoelectric coefficients e3* j ,

i.e., | ]e | >> 1. Taking into account these facts, one can rewrite Equations (3.15) and (3.16) as follows [44]: * * (1) *E 2 *V ( * / c33 ) / H 33 | K elas [m e33 / (m c33(1),E + c33( 2),E )]2 / (m H 33(1),V ) (Qh* ) 2 |K elas ,h e33 ,h * 2 * / * ) * 2 and (Q33 ) | (K elas , 33 K elas ,h (Qh )

(3.17)

* * where K elas = {[1 + ( E12* )–1 – 2( E13* )–1] / [1 + ( E12* )–1 – 2( E13* )–1( E 33 / E13* )]}2, ,h * * * *E = {[1 + ( E12* )–1] / [1 + ( E12* )–1 – 2( E13* )–1( E 33 / E13* )]}2 and E ab = c11*E / cab K elas ,33 (1) >> depend on the elastic properties in the composite only, and an inequality e33 ( 2) e33 holds. Both the squared figures of merit from Equations 3.17 depend on a ratio

J = m e33(1) / (m c33(1),E + c33( 2),E ) so that a higher e33(1) / c33(1),E ratio leads to higher (Qh* ) 2 and * 2 (Q33 ) values for 0 < m < 0.1. It is worth noting that among the perovskite-type (1) / c33(1),E ratio ferroelectric ceramics listed in Table 2.1, PCR-7M has the highest e33 * (m)]2 at room temperature. For example, max[ Qh* (m)]2 = 6240 (PPa)–1 and max[ Q33

= 6.24u105 (PPa)–1 are attained in the 1–3 PCR-7M / elastomer composite [44]. The max [ Qh* (m)]2 value is about 3 times larger than that of the typical 1–3 piezocomposites [5]. Moreover, the same max[ Qh* (m)]2 value is approximately 1.5 times larger than the ( Qh* )2 value typical of the 1–3–0 Pb(Zr, Ti)O3-based composite [5] with a porous polymer matrix. These findings could promote new hydroacoustic, medical and other piezotechnical applications of the studied 1–3 and related composites.

3.3 Composites with 0–3 Connectivity, Manufacturing and Applications The 0–3 ferroelectric ceramic / polymer composite system is one of the most common composite types used for sensor applications. The 0–3 composite (Figure 1.4) consists of the three-dimensionally connected polymer matrix reinforced by a system of discrete (isolated) ceramic inclusions. The most common ceramic components are perovskite-type compositions based on solid solutions of Pb(Zr, Ti)O3 (e.g., PZT, PZ, Navy, ZTS, PKI, APC, and PCR) [2, 3, 10, 20, 40, 49] or Pb(Ca, Ti)O3 [50–52]. The polymer components can be either piezo-active or

Non-monotonic Volume-fraction Dependences of Effective Properties

63

piezo-passive. As the effective parameters of the 0–3 piezo-composites are highly dependent [2, 3, 50–53] on their microstructure, properties and volume fractions of the components and poling conditions a large research effort has studied the interrelations between microgeometry and the effective electromechanical properties. The 0–3 ceramic / polymer composite can be made in a variety of forms [3] with a range of mechanical flexibility (e.g., piezorubber [2, 19, 20]). These performances differ from those of the 2–2, 1–3 and 3–3 composites. Technological aspects of processing and manufacture of 0–3 piezo-composites have been studied [53–58]. As with 1–3 composites, dielectrophoresis can also be used to manufacture 0–3 composites. A variety of chain-like structures with 0–3 connectivity are formed in a colloidal suspension of the ceramic particles under the electric field [58]. These structures contain prolate ceramic inclusions formed due to the attraction of the particles and mutual dielectrophoresis. Some aspects related to the poling procedure and choosing the filler materials for the 0–3 composites have been discussed in ref. [3]. Various 0–3 composite films and cables are manufactured by hot-rolling [3, 54] the ceramic particles into the rubber matrix. These highly effective materials find applications as active elements for flexible hydrophones, keyboards, bloodpressure cuffs and sensor applications [20]. 3.3.1 Modelling of the Effective Properties in 0–3 Composites with Spheroidal Inclusions The electromechanical properties of the 0–3 piezo-composites can be predicted by a variety of methods and averaging procedures [10, 59–68]. The most popular models for predicting the properties consider regularly shaped inclusions (for example, sphere [63, 66, 67], spheroid [60, 67, 68] or parallelepiped [38, 59, 64]) or the hybrid inclusions [51, 52] that form the Banno unit cell [2, 54]. An approach developed in monographs [10, 40] also enables the consideration of 0–3 piezocomposites with an irregular arrangement of piezoelectric inclusions in the matrix. However, most research deals with a regular distribution of inclusions with a specific geometric shape, such as spheroid or parallelepiped. Of practical interest is the averaging procedure [60, 62, 68–71] that allows for an electromechanical interaction between the piezo-active inclusions in the matrix of the 0–3 composite. This procedure is based on the EFM, i.e., the known Mori – Tanaka method [72] generalised for heterogeneous piezoelectric media [69]. In this method, an individual inclusion within the matrix, which is under an external electroelastic field, is subjected to the action of an unknown average electroelastic field. This average (or effective) field is a result of the external field applied to the matrix (and, therefore, to the composite sample as a whole) and by the interaction between the inclusions therein. Using the equivalent inclusion approach, the coupled electroelastic behaviour of a 0–3 piezo-composite was modelled by Dunn and Taya [73]. This modelling can be performed for the composite with aligned ellipsoidal (spheroidal) piezo-active inclusions [73] that are isolated (not touching) and characterised by a volume fraction 0 < m d 0.5. The EFM can also be applied

64

Electromechanical Properties in Composites Based on Ferroelectrics

in the range of 0.5 d m < 1 [74], but model predictions are approximate only since an error due to near-touching or touching of inclusions would increase as m o 1. In the model [68] of the 0–3 ceramic / polymer composite with spheroidal inclusions (Figure 3.7), their shape is described by the equation (x1 / a1)2 + (x2 / a1)2 + (x3 / a3)2 = 1

(3.18)

relative to the axes of the rectangular co-ordinate system (X1X2X3). In Equation 3.18 the semi-axes of the spheroid are a1 = a2 and a3. The aspect ratio of the spheroid is defined as U = a1 / a3. The spheroidal ceramic inclusions are uniformly aligned in the continuous polymer matrix, so that the remanent polarisation vector of each inclusion is oriented along the OX3 axis, which is the poling axis for the sample as a whole. It is assumed that centres of symmetry of the inclusions occupy sites of a simple tetragonal lattice with unit-cell vectors parallel to the OXj axes. For improved poling the composite sample, the ceramic (n = 1) and polymer (n = 2) components obey an inequality Ȗ(2) • Ȗ(1) for electrical conductivities Ȗ(n). It is also assumed that surface charges appearing owing to the ferroelectric polarisation of the inclusions are to be fully screened by free charge carriers that would flow to interfaces separating the components. The effective electromechanical constants of the 0–3 composite are written in a matrix form as

§||c E || || C* || = ¨ ¨ ©||e ||

t || e || ·¸  ||İ ȟ || ¸¹

(3.19)

where the superscript t denotes the transposition and the matrices || c*E ||, || e* || and || İ*ȟ || contain the constants from Equations 2.21 and 2.22. Taking into account the electromechanical interaction between the piezo-active inclusions, the effective properties in the 0–3 ceramic / polymer composite are determined by means of the EFM [62, 68–70]. The corresponding 9 u 9 matrix of the effective constants is given by

Figure 3.7. Schematic of the 0–3 composite with spheroidal inclusions. The polarisation vector P of the inclusion is denoted by an arrow. (X1X2X3) is the rectangular co-ordinate system, ai are semi-axes of the spheroid, m is the inclusion volume fraction, and 1 – m is the matrix volume fraction

Non-monotonic Volume-fraction Dependences of Effective Properties

65

|| C* || = || C(2) || + m (|| C(1) || – || C(2) ||) [|| I || + (1 – m) || S || || C(2) ||–1 (|| C(1) || – || C(2) ||)]–1 (3.20) In Equation 3.20 the matrices of the electromechanical constants || C(1) || (ceramic) and || C(2) || (polymer) have a form similar to that shown in Equation 3.19, m is the volume fraction of the ceramic, || I || is the identity matrix, and || S || is the matrix that contains the Eshelby tensor components [48] depending on the elements of || C(2) || and the aspect ratio U. The polymer medium surrounding the ceramic inclusions can be either piezo-passive (isotropic) or piezo-active (transversely isotropic). The 0–3 composite as a whole (see Figure 3.7) is described by fmm symmetry. Based on the effective electromechanical constants from Equation 3.20 and taking into account Equations 2.12–2.15, it is possible to represent the piezoelectric coefficients in the matrix form as follows: || d* || = || e* ||˜|| c*E ||–1, || g* || = || d* ||˜|| İ*ı ||–1 and || h* || = || e* ||˜|| İ*ȟ ||–1 (3.21) In Equations 3.21 the piezoelectric coefficients || d* ||, || e* ||, || g* ||, and || h* || are written as 3 u 6 matrices and determined as functions of m and U. The electromechanical constants of the 0–3 Pb(Zr, Ti)O3-type ceramic / polymer composite that are calculated using Equations 3.20 and 3.21 are in full agreement with the similar constants evaluated in ref. [60]. Despite the different formulae for the Eshelby tensor components used in refs. [48, 68] and [60], the difference between the constants remains less than 1% over a wide range of m and U. A small difference is observed in the limiting case of U = 0, as the composite shown in Figure 3.7 approaches that of a 1–3 geometry and the piezo-active inclusion is elongated in the poling axis (see Figure 3.2) due to a3 o f. The corresponding electromechanical constants, evaluated for the 1–3 Pb(Zr, Ti)O3type ceramic / polymer composite by means of Equations 3.20 and 3.21 and the Eshelby tensor [48] for the rod-shaped inclusion are more or less equal to constants calculated using formulae from refs. [29], [32] or [40] over a wide range of m: the difference between the related constants is less than 0.5%. Another limiting case of U o f (i.e., the piezo-active inclusion is elongated normal to the poling axis) enables one to attain a good correlation between the electromechanical constants, calculated using Equations 3.20 and 3.21 and the Eshelby tensor [48] for the penny-shaped inclusions, and the constants evaluated using formulae from ref. [13] for the series-connected 2–2 composite. For U > 102, a difference between the values of the related constants becomes less than 1% over the range of 0 < m < 1. Further examples of comparison of the volume-fraction dependences of the effective parameters of the 0–3 composites are considered in Chapter 6. 3.3.2 Piezoelectric Properties in 0–3 Composites Based on PbTiO3-type Ceramics As is seen from Table 2.1, the PbTiO3-type ceramics have the piezoelectric coefficients [10, 75, 76]

66

Electromechanical Properties in Composites Based on Ferroelectrics (1) (1) > 0, e33 > 0 and e15(1) > 0 e31

(3.22)

and exhibit large anisotropy of the piezoelectric coefficients

d3(1j)

and

electromechanical coupling factors k : across a wide temperature range, (1) 3j

inequalities d33(1) / | d31(1) | >> 1 and k33(1) / | k31(1) | >> 1 hold [77]. Important examples of the large piezoelectric anisotropy were considered in work on the (Pb1–xMex)TiO3 ceramics (Me = Ba, Ca or Sr) [78–80]. In this section we analyse how the highly anisotropic piezoelectric component influences the piezoelectric properties in the 0–3 ceramic / polymer composite with spheroidal inclusions. We will focus our attention on two examples of the PbTiO3-based composites. In the first example, the ceramics of the (Pb, A)(Ti, B)O3 type are involved, where both the Pb and Ti ions are substituted by specified ions A and B (or even Bc and Bs) in order to vary (1) (1) / e31 ratio (Table 2.1). the electromechanical properties and, in particular, the e33 The second example is related to the (Pb1–xCax)TiO3 solid solutions (Table 3.3) that are characterised by the non-monotonic behaviour of both the piezoelectric (1) (1) anisotropy values, e33 / e31 and d33(1) / d31(1) , at 0.10 d x d 0.30. The effective piezoelectric coefficients in the 0–3 composites are calculated using Equations 3.20 and 3.21 in specific ranges of m and U. In the case of 0 < U d 1, the volume fraction m of the inclusions is varied from 0 to 0.5 at the upper bound mu = S / 6 for the spherical inclusions (U = 1) in a primitive cubic cell. The calculated volume-fraction dependences of four types of effective piezoelectric coefficients are shown in Figure 3.8. The piezoelectric response of * (m, ȡ)|ȡ = const (see, the composite studied is characterised by the presence of min d 31 for example, curve 3 in Figure 3.8a) and the corresponding appearance of * * min g31 (m, ȡ)|ȡ = const and max g 33 (m, ȡ)|ȡ = const (curves 5 and 6 in Figures 3.8a and b) where ȡ is taken from a range of aspect ratios 0 < ȡ < ȡ*. It can be seen that for various combinations of the PbTiO3-type ceramics from Table 2.1 and piezopassive polymers, the ȡ* value belongs to a range [0.11; 0.16] and increases ( 2) of polymers. The mf values that slightly with an increase in the elastic moduli cab * * * * (m2, ȡ) = g31 and g 33 (m3, ȡ) = g33 are almost equal, and satisfy conditions g31 ,m ,m * * (m, ȡ), d33 (m, ȡ) and an inequality | m2 – m3 | ” 0.01 holds. The calculated e33

g3* j (m, ȡ) dependences (curves 2, 4–6 in Figure 3.8) are similar to those determined [59, 60, 63, 67] for various 0–3 composites based on the perovskite (1) (1) = –sgn e33 < 0. ceramics with sgn e31 * values calculated for ȡ << 1 (i.e., elongated inclusions along the The g33 ,m

poling axis) are highly dependent on the elastic moduli of the polymer component. For example, at ȡ = 0.01 in the (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II) / araldite * composite, a ratio g33 / (1) = 9.13 is observed, and at ȡ = 0.01 in the , m g 33 (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II) / polyethylene composite, the same ratio increases to 24.6. Moreover, in these composites at 0 < ȡ < ȡ*, the piezoelectric

Non-monotonic Volume-fraction Dependences of Effective Properties

67

(1), E Table 3.3. Calculated values of elastic moduli cab (in 1010 Pa),a piezoelectric coefficients

2 (1) (in eij(1) (in C / m ), relative dielectric permittivities H (pp1),[ / H0, piezoelectric coefficient g 33 (1) (1) / d 31 of the (Pb1–xCax)TiO3 ceramic mV m / N), and piezoelectric anisotropy value d 33 (reprinted from Glushanin et al. [68], with permission from Elsevier)

x

c11(1),E

c12(1),E

c13(1),E

(1), E c33

(1), E c44

0.10 0.15 0.20 0.23 0.24 0.25 0.26 0.30 x

19.1 18.8 18.5 18.7 18.4 18.9 18.9 18.6

5.22 5.09 4.76 5.03 4.72 5.19 5.14 4.89

e15(1)

18.6 18.3 18.0 18.2 17.9 18.3 18.3 18.1 H 33(1),[ / H0

6.85 6.76 6.79 6.76 6.76 6.76 6.77 6.77

(1) e33

5.15 5.01 4.68 4.93 4.61 5.06 5.02 4.80 H 11(1),[ / H0

0.10 0.15 0.20 0.23 0.24 0.25 0.26 0.30

3.90 4.15 4.31 4.55 4.69 5.09 5.30 5.48

1.62 1.73 1.77 1.80 1.87 1.95 2.03 2.27

111 118 126 137 145 152 161 202

109 115 123 133 141 147 156 197

21.1 21.4 20.6 19.7 19.4 19.5 19.1 20.2

(1) g 33

b

(1) e31

0.676 0.751 0.844 1.06 1.07 1.33 1.36 1.08 d 33(1) / d 31(1) –11.7 –13.0 –18.7 –30.3 –39.8 –71.9 –65.2 –17.6

Notes to Table 3.3 (1) a. For the poled ceramic with ’mm symmetry, c66 = ( c11(1),E – c12(1),E ) / 2 according to formulae in refs. [10, 40] b. The electromechanical constants have been calculated by the self-consistent (effective medium) method [79, 80]. Spherical grains of the ceramic sample are assumed to be split into 90° lamellar domains of two types with equal volume fractions and these domains are separated by planar 90° domain walls. These domain walls are assumed to be practically motionless so that a contribution from their displacements under external electric or mechanical fields into the electromechanical constants of the ceramic approaches zero [79, 80]. The electromechanical constants of the single-domain crystal sample have been evaluated by the method from ref. [79] on the basis of experimental room-temperature data on the (Pb1–xCax)TiO3 solid solutions

anisotropy is characterised by values ]e >> 1 and ]d § 3–5. The distinctions between the effective piezoelectric coefficients e3* j and d 3* j demonstrate the important role of the elastic properties of the polymer component in forming the piezoelectric response of the 0–3 composite with highly prolate ceramic inclusions (ȡ << 1) with the piezoelectric coefficients e3(1j) that obey conditions 3.22. * (m, ȡ) at In contrast to the above-considered PbTiO3-based composites, min d 31

ȡ << 1 is not observed in the Pb(Zr, Ti)O3- or BaTiO3-based 0–3 composites [59, * 60, 70]. The piezoelectric coefficient d 31 of the 0–3 composites is expressed in *E terms of the piezoelectric coefficients e3* j and elastic compliances sab in

68

Electromechanical Properties in Composites Based on Ferroelectrics

* accordance with Equation 3.11. It is known that d 31 < 0 is observed within a wide range of m and ȡ values (see, for example, curve 3 in Figure 3.8), and as a consequence, an inequality ]e > –( s11*E + s12*E ) / s13*E is true for the composites with

e3* j > 0. *E The analogous inequality is written using the elastic moduli cab of the same composite in a form *E *E ]e > c33 / c13

(3.23)

Figure 3.8. Effective piezoelectric coefficients e3* j (m, ȡ) (in C / m2), d 3* j (m, ȡ) (in pC / N), 8 g3* j (m, ȡ) (in mV m / N), and h3* j (m, ȡ) (in 10 V / m) that have been calculated for the 0–3 (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II) ceramic / araldite composite containing spheroidal * , curve 2 inclusions with ȡ = 0.01 (a), ȡ = 0.10 (b) or ȡ = 1 (c). Curve 1 corresponds to 102 e31 * (graphs a and b) or 102 * (graph c), curve 3 corresponds to * corresponds to e33 (graphs e33 d 31 –1 * (graph c), curve 4 corresponds to 10 * (graphs a and b) or * a and b) or 10 d 31 d 33 d 33 (graph –1 * * c), curve 5 corresponds to 10 g 31 (graphs a and b) or g 31 (graph c), curve 6 corresponds to * (graphs a and b) or 10–1 * (graph c), curve 7 corresponds to * , and curve 8 10–2 g 33 g 33 h31 –1 * * (graph c) (reprinted from Glushanin et al. corresponds to 10 h33 (graphs a and b) or h33

[68], with permission from Elsevier)

Non-monotonic Volume-fraction Dependences of Effective Properties

Figure 3.8 (continued)

69

70

Electromechanical Properties in Composites Based on Ferroelectrics

Both the increase in the volume fraction m of the ceramic inclusions and an increase in their aspect ratio ȡ result in the reduction in the ]e value for small *E *E changes in the c33 / c13 ratio. A violation of condition 3.23 leads to a change in * curve on increasing ȡ (compare curves 3 in graphs a–c of the shape of the d 31 Figure 3.8). A fulfilment of condition 3.23 promotes the non-monotonic volume* fraction behaviour of d 31 at ȡ << 1. The transition from prolate spheroidal inclusions (ȡ << 1) to spherical inclusions (ȡ = 1) leads to a significant redistribution of the internal electric and mechanical fields in the composite sample. As a consequence, a significant decrease in | e3* j |, | d 3* j |, | g 3* j |, and | h3* j | at m = const (compare graphs in Figures

3.8a and c) and ]e is observed in the 0–3 composites studied on increasing the aspect ratio ȡ in the range 0 < ȡ < 1. In contrast, some increase in |]d | is observed in the same range. This trend is the result of a relatively slow decrease in the * (m) dependence at ȡ = 1 (see curve 3 in Figure 3.8c). d 31 The range 0 < ȡ < ȡ* is of practical interest due to the large ]e and | g 3* j | values, where the ȡ* is taken from the aforementioned range [0.11; 0.16]. The studied 0–3 composites also have relatively large hydrostatic piezoelectric coefficient g h* . Its local maximum can be represented as g h*,m = g h* (mh, ȡ) at ȡ = const. The g h*,m values, calculated for the (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II) / polymer composite at 0.01 ” ȡ ” 0.10, changes from 90.3 mV m / N to 52.6 mV m / N (the composite with the araldite matrix) or from 114 mV m / N to 54.7 mV m / N (the composite with the polyurethane matrix). The calculated g h*,m values are comparable with the experimental g h*,exp values [3, 52] known for a variety of the 0–3 PbTiO3-type ceramic / polymer composites ( g h*,exp = (47–100) mV m / N) and significantly exceed an experimental value g h*,exp = 8 mV m / N [3] related to the 0–3 PZT / polyurethane composite. The next interesting example of the high piezoelectric sensitivity is related to the 0–3 (Pb1–xCax)TiO3 ceramic / polyurethane composites. The (Pb1–xCax)TiO3 ceramic in the molar concentration range of 0.10 ” x ” 0.30 exhibits non(1) monotonic dependences of both the piezoelectric anisotropy values e33 / e31(1) and (1) (Table 3.3). The d 33(1) / d 31(1) and an almost constant piezoelectric coefficient g 33

(1) on x (see Table 3.3) and the large d 33(1) / | d 31(1) | values result slight dependence g 33 in a narrow range of values of the hydrostatic piezoelectric coefficient, i.e., 17.5 * mV m / N ” g h(1) (x) ” 18.9 mV m / N. As a consequence, the local maxima g 33 = ,m

* (m, ȡ, x)|ȡ; max g 33 * h

x= const

* = g 33 (m3, ȡ, x) and g h*,m = max g h* (m, ȡ, x)|ȡ;

x= const

=

g (mh, ȡ, x) undergo small changes at 0.10 ” x ” 0.30 and ȡ = const. It is obvious that the presence of prolate ceramic inclusions in the 0–3 composite favours an * increase in the g 33 and g h* values in comparison with the monolithic ceramic

Non-monotonic Volume-fraction Dependences of Effective Properties

71

* component and both the g 33 and g h*,m values remain large for fixed values of the ,m * values molar concentration x. The volume fraction m3 that corresponds to the g 33 ,m

is varied within a narrow range: the m3 values are almost constant for ȡ = const and 0.02 ” m3 ” 0.12 for 0.01 ” ȡ ” 0.20. The volume fraction mh related to the g h*,m values exhibits similar behaviour at the variation of the molar concentration x: mh § const for ȡ = const and 0.02 ” mh ” 0.19 for 0.01 ” ȡ ” 0.20. The relatively low volume fractions m3 and mh of the ceramic inclusions lead to a small effective dielectric permittivity H 33*V and the relatively large effective piezoelectric * . Both these factors, along with the small dielectric anisotropy of the coefficient d 33

PbTiO3-type ceramics (i.e., H 11(1),V / H 33(1),V d 1.3) [77, 80] favour the high piezoelectric sensitivity of the 0–3 composite with the prolate ceramic inclusions. The lesser (1) * / g 33 values are a result of moderate g h*,m / g h(1) values in comparison with the g 33 ,m

]d values. As is known from the data on the (Pb1–xCax)TiO3 ceramic (Table 3.3), max( d 33(1)

/ | d 31(1) |) is attained in the vicinity of x = 0.25. The volume-fraction dependences of * (1) / g 33 of the 0–3 (Pb0.75Ca0.25)TiO3 the normalised piezoelectric coefficient g 33 ceramic / polyurethane composite show that a significant increase in the piezoelectric sensitivity is attained at ȡ << 1 and 0 < m d 0.20 (curves 1–3 in * Figure 3.9). Increasing ȡ and/or m results in a reduction of g 33 values (see, for * value (compare curves example, curve 4 in Figure 3.9) due to a decrease of the d 33 *V 4 in Figures 3.8b and c) or an increase of the H 33 value as m increases. * (1) The calculated g 33 / g 33 ratios (curves 1–4 in Figure 3.9) are comparable with experimental values (sets 5 and 6 in Figure 3.9) for the related 0–3 PbTiO3-based composites [51, 81]. The calculated curves 1–3 in Figure 3.9 and values of local * maximum g 33 demonstrate the promising performance of the 0–3 composite with ,m

highly prolate spheroidal inclusions made of the highly anisotropic ferroelectric * (1) ceramic. In this case large g 33 / g 33 values are attained, even in the presence of * (1) / g 33 values (see sets the non-piezo-active polymers, whereas the experimental g 33 5 and 6 in Figure 3.9) correspond to the 0–3 ceramic / polymer composites with both components being piezo-active. Thus, the large piezoelectric anisotropy ( d 33(1) /| d 31(1) | >> 1) of the ceramic component and the system of the aligned prolate

spheroidal ceramic inclusions favour the high piezoelectric sensitivity of the 0–3 composites studied. Now we consider how the piezoelectric anisotropy d 33(1) / d 31(1) of the ceramic * / d 33(1) in the 0–3 component influences the normalised piezoelectric coefficient d 33 composites studied. Despite the large anisotropy of the piezoelectric coefficients d 3(1j) and positive signs of both the piezoelectric coefficients e3(1j) (see inequalities

* (m) dependence calculated for the 3.22) in the PbTiO3-type ceramics, the d 33

72

Electromechanical Properties in Composites Based on Ferroelectrics

Figure 3.9. Calculated (curves 1 – 4) and experimental (sets 5 and 6) volume-fraction * (1) dependences of the g 33 / g 33 ratio of the 0–3 ceramic / polymer composites. Curves 1, 2, 3, and 4 are plotted for the (Pb0.75Ca0.25)TiO3 ceramic / polyurethane composite with ȡ = 0.01, 0.05, 0.10, and 0.15, respectively, experimental dependences 5 and 6 are related to the PbTiO3 ceramic / 70 / 30 mol.% copolymer of vinylidene fluoride and trifluoroethylene composite [81] and (Pb,Ca)TiO3 ceramic / 75 / 25 mol.% copolymer of vinylidene fluoride and trifluoroethylene composite [51], respectively (reprinted from Glushanin et al. [68], with permission from Elsevier)

related 0–3 composites is analogous to that determined [60] for other 0–3 composites based, for example, on the BaTiO3 or Pb(Zr,Ti)O3 ceramics. The * piezoelectric coefficient d 33 is written for transversely isotropic materials (fmm symmetry) in accordance with Equations 2.12 and 3.12 in the form [68] * * *E *E = e31 [2( s13*E / s33 ) + ] e] d 33 s33

(3.24)

* In the 0–3 PbTiO3-based composites examined here, e31 (m, U) > 0 (see, for *E (m, U) > 0 for 0.01 d example, curves 1 in Figures 3.8a – c), s13*E (m, U) < 0 and s33

m d 0.50 and 0.01 d U d 1. The piezoelectric anisotropy value ]e(m, U)» U = const > 0 *E changes from a few tens (at U = 0.01) to a few units (at U = 1). The s13*E / s33 ratio *E = – c13*E / ( c11*E + c12*E ), i.e., we have involved in Equation 3.24 reduces to s13*E / s33 *E < 1 at various m and U values. The piezoelectric coefficient d 33* (m, U) | s13*E | / s33

Non-monotonic Volume-fraction Dependences of Effective Properties

73

* exhibits a monotonic increase [68] at U = const due to the increase in e31 ( m, U ) > 0 and the large ]e(m, U) values. Some examples of the volume-fraction dependence of the normalised piezoelectric coefficient d 33* / d 33(1) of the 0–3 ceramic / polymer composites are

shown in Figure 3.10. The largest d 33* / d 33(1) values (curves 1 and 2 in Figures 3.10a and c, and curve 5 in Figure 3.10b) are related to the PbTiO3-based composites with highly prolate inclusions. The simple comparison of values of the * / d 33(1) calculated for the PbTiO3- and normalised piezoelectric coefficient d 33 Pb(Zr, Ti)O3-based composites at fixed aspect ratios U << 1 (see curves 1, 2, 5, * and 6 in Figure 3.10b) suggests that the lower d 33 / d 33(1) values are attained in the 0–3 composites based on the Pb(Zr, Ti)O3-type ceramics. Such a response is due to the relatively small anisotropy of the piezoelectric coefficients d 33(1) / d 31(1) and the

* / Figure 3.10. Volume-fraction dependences of the normalised piezoelectric coefficient d 33

d 33(1) of the 0–3 ceramic / polymer composites: (a) curves 1, 2, 3, and 4 are calculated for the (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II) / araldite composite with ȡ = 0.01, 0.05, 0.10, and 0.15, respectively; experimental dependences 5 and 6 are related to the PbTiO3 ceramic / 70/30 mol.% copolymer of vinylidene fluoride and trifluoroethylene composite [81] and (Pb, Ca)TiO3 ceramic / 75/25 mol.% copolymer of vinylidene fluoride and trifluoroethylene composite [51], respectively, (b) curves 1, 2, 3, and 4 are calculated for the (Pb0.75Ca0.25)TiO3 ceramic / araldite composite with ȡ = 0.01, 0.05, 0.10, and 0.15, respectively; curves 5 and 6 are calculated for the PZT ceramic / polymer composite with ȡ = 0.01 and 0.10 [60], respectively, and (c) curves 1, 2, 3, and 4 are calculated for the (Pb0.75Ca0.25)TiO3 ceramic / polyurethane composite with ȡ = 0.01, 0.05, 0.10, and 0.15, respectively (reprinted from Glushanin et al. [68], with permission from Elsevier)

74

Electromechanical Properties in Composites Based on Ferroelectrics

Figure 3.10 (continued)

Non-monotonic Volume-fraction Dependences of Effective Properties

75

(1) (1) equality sgn e33 = –sgn e31 in the Pb(Zr, Ti)O3-type ceramics [10, 40]. In addition,

the large piezoelectric anisotropy d 33(1) / d 31(1) of the PbTiO3-type ceramic inclusions and the relatively soft polymer matrix promote the normalised piezoelectric * coefficient d 33 / d 33(1) | 1 over a wide volume-fraction range (see curve 1 in Figure 3.10c), and this tendency is similar to that in the 1–3 ceramic / polymer composites [32] independently of the signs of the piezoelectric coefficients eij(1) of the ceramic component. In general, the presence of the highly prolate ceramic inclusions with the large piezoelectric anisotropy d 33(1) / d 31(1) and the piezoelectric coefficients eij(1) obeying inequalities 3.22 lead to a simple combination of electromechanical constants that would form the piezoelectric response of the 0–3 composite along the poling axis. It is seen that advantages of the 0–3 PbTiO3-based composites are associated with * * (1) the significantly large g h* , d 33 / d 33(1) and g 33 / g 33 values and the possibility of tailored piezoelectric anisotropy. 3.3.3 0–3 Composites with a Hierarchy of Inclusions In this section we will consider a relationship between the microstructure and electromechanical properties in the 0–3 composite based on the Pb(Zr, Ti)O3-type ceramic. It is important to note that microgeometrical features of a distribution of the ceramic inclusions within a polymer matrix can result in either pure 0–3 or mixed connectivity [82, 83]. The microstructure of the composite with mixed connectivity is often represented by a system of parallelepipeds (or cubes) [2, 50– 52, 54, 82] in the matrix. This composite is often described in terms of 0–3, 1–3 or 3–3 connectivity patterns [38, 59] that influence the effective electromechanical properties in a variety of aspects. Jayasundere et al. [83] proposed a finite-element model suitable for the ceramic / polymer composite with 0–3 / 1–3 mixed connectivity. This form of mixed connectivity is thought to be a representative microstructure where the average size of the inclusion is comparable to the thickness of the composite sample. On increasing the volume fraction of the ceramic component, the 1–3 connectivity elements become the main component responsible for the piezoelectric properties in the composite studied [83]. In this case, predicted properties based on the 0–3 model of the composite with isolated spherical inclusions are lower [56] than the experimental values of the * , dielectric permittivity H 33*V , etc. piezoelectric coefficient d 33 A new approach to the study of the electromechanical properties in the ceramic / polymer composites is concerned with a hierarchy of the inclusions [84, 85] that provide 0–3 connectivity over a wide volume-fraction range. To compare the behaviour of the effective parameters of different composites, one can introduce the normalised dielectric permittivity

EH = H 33*V (mFC) / H 33*V (1) = H 33*V (mFC) / H 33(1),V

(3.25)

76

Electromechanical Properties in Composites Based on Ferroelectrics

and the normalised piezoelectric coefficient * * * Ed = d 33 (mFC) / d 33 (1) = d 33 (mFC) / d 33(1)

(3.26)

where mFC is the volume fraction of the ceramic component. The experimental data on EH and Ed as a function of mFC for 0–3 ceramic / polymer composites based on Pb(Zr, Ti)O3 [55, 56] or Pb(Ca, Ti)O3 [51, 52] ceramics show that changes in the EH(mFC) dependence (Equation 3.25) are different, even in composites with a volume fraction mFC that is varied over a wide range. There are also distinctions between the experimental EH(mFC) dependences of these 0–3 composites [51, 52, 55, 56]. The main reasons for the differences between the EH(mFC) dependences are associated [85] with different H 33*V (0) / H 33*V (1) ratios, where H 33*V (0) and H 33*V (1) characterise dielectric properties of the polymer and ceramic component, respectively. Features of microstructure, poling conditions and differences between the electromechanical properties of the components are important in forming the Ed(mFC) dependence from Equation 3.26. In addition, evaluations made for EH and Ed within the framework of the model of the 0–3 composite [68] with spherical ceramic inclusions demonstrate that good agreement between the calculated and experimental [51, 52, 55, 56] values are attained only when mFC d 0.1. It seems probable that microstructural elements of the composite are having a considerable influence on the electromechanical properties at mFC > 0.1. Our further analysis of this influence is carried out within the framework of the following model [84, 85]. A heterogeneous medium is represented by two types of layers that randomly alternate along the poling axis OX3 (Figure 3.11). The first layer, with the volume fraction 1 – m, comprises of spherical ceramic inclusions and a polymer matrix. These inclusions have a relatively small volume fraction n0 and are uniformly distributed in the aforementioned layer. In the second layer, the inclusions in the layer with the volume fraction m form a structure similar to a cylindrical rod (Figure 3.11, insert). It should be indicated for comparison that, according to scanning electron micrographs [86] of the fracture surfaces of the 0–3 PZT-based composite with mFC = 0.49 and 0.67, the ceramic inclusions are distributed quasiuniformly. This implies that the ceramic component can be represented as a series of chains, rod-like elements or other quasi-continuous structures and remain distributed relatively uniformly in separate regions. In the same composite with mFC = 0.67 [86], the volume fraction of the regions with a quasi-continuous arrangement of the ceramic inclusions becomes dominant. The presence of the rod-like elements oriented along the OX3 axis favours poling of the composite, particularly at high volume fractions mFC. It is assumed [85] that the electromechanical constants of the rod-like elements are approximately equal to the constants of the ceramic. This approximation may be valid at low volume fractions of the polymer component within these elements and with an almost continuous distribution of the ceramic inclusions along the OX3 axis in separate regions of the composite. In the model of the 0–3 composite (Figure 3.11), the rod-like elements are approximated by circular cylinders, and their volume fraction n1 in the above-described layers is varied from 0 to S / 4. The

Non-monotonic Volume-fraction Dependences of Effective Properties

77

Figure 3.11. Schematic diagram illustrating the 0–3 ceramic / polymer composite structure with the hierarchy of inclusions. m is the volume fraction of layers comprising the rod-like elements (see the insert on the left side) that are characterised by the volume fraction n1 in the separate layer. 1 – m is the volume fraction of layers comprising spherical inclusions that are characterised by the volume fraction n0 in the separate layer. (X1X2X3) is the rectangular co-ordinate system. The ceramic component is shown as a shaded region (reprinted from Glushanin and Topolov [85], with permission from IOP Publishing)

upper bound of n1 equals the ratio of the area of a circle, inscribed into a square lying in the (X1OX2) plane, to the area of this square. The volume fraction of the ceramic component in the 0–3 composite shown in Figure 3.11 equals

mFC = n1 m + n0 (1 – m)

(3.27)

The presence of the three parameters m, n0 and n1 in Equation 3.27 suggests that the effective electromechanical properties of the 0–3 composite can be determined in three stages associated with procedures of averaging of these parameters. In the first stage of averaging, the electromechanical properties in the layer containing the spherical ceramic inclusions in the polymer matrix (0–3 connectivity) are determined as a function of n0 by means of the EFM (see Section 3.3.1). In the second stage of averaging, the electromechanical properties in the layer containing the rod-like ceramic inclusions in the polymer matrix (1–3 connectivity) are determined as a function of n1 by means of the EFM [32]. There is a small difference between the effective constants of the 1–3 ceramic / polymer composite with the finite elongated rods [87, 88] and the effective constants evaluated for similar structures with the infinite rods [32]. The difference depends on the aspect ratio Dr, i.e., the height-to-width ratio of the separate ceramic rod. On increasing the Dr value, this difference decreases and becomes less than 10% for Dr t 5, as shown for effective electromechanical coupling factors kt,eff, k31,eff, k33,eff,

78

Electromechanical Properties in Composites Based on Ferroelectrics

piezoelectric coefficients d31,eff, d33,eff and other parameters of the 1–3 PZT / polymer composite [87] with a periodic structure. Elastic moduli cabE ,eff , piezoelectric coefficients eij ,eff and dielectric permittivities H [pp,eff are calculated as functions of the volume fraction of the ceramic rods [88] for the 1–3 PZT-5 / polymer composites with the finite (Dr = 5) or infinite (Dr o f) cylindrical rods. According to Berger et al. [88], the effective constants cabE ,eff (ab = 13 and 33), eij ,eff (ij = 15, 31 and 33) and H [pp,eff (p = 1 and 3) of the 1–3 composite with Dr = 5

show very good agreement with those evaluated for a similar composite with Dr o f: in the range of n1 = 0.1–0.6, differences between the similar parameters do not exceed 5%. For the effective elastic moduli cabE ,eff with ab = 11, 12 and 44, the aforementioned differences attain about 10 – 15% at n1 > 0.5 and less than 10% at n1 < 0.5. The effective constants of the individual 0–3 and 1–3 layers are used in the third stage of averaging [85] on the volume fraction m (2–2 series-connected composite, see Figure 3.1a) according to formulae in ref. [10] that take into account the electromechanical interaction between the piezo-active layers distributed randomly along the poling direction.2 As a result, the effective electromechanical constants of the 0–3 composite studied are calculated on dependence of n0, n1 and m, and finally, the volume-fraction dependences from Equations 3.25 and 3.26 are calculated over the wide range of mFC. For these calculations, the full sets of electromechanical constants of the ceramic and polymer components are required. To the best of our knowledge, the full sets of such constants have not been published for the components of the 0–3 composites studied in refs. [55, 56]. To minimise possible discrepancies, it is possible to use experimental constants of PZ 27 ceramic [89] and 75 / 25 mol.% copolymer of vinylidene fluoride and trifluoroethylene [29] for calculations and comparison with the experimental data [55]. As with PKI 502 ceramic [55], PZ 27 belongs to the Navy II ceramic group [89], so that d 33(1) / d 33( 2 ) , d 33(1) / H 33(1),V and H 33(1),V / H 33( 2 ),V ratios calculated using data on the PZ 27 ceramic and the 75 / 25 mol.% copolymer of vinylidene fluoride and trifluoroethylene are in good agreement with the analogous ratios evaluated for the components of the 0–3 composite from ref. [55]. The calculated volume-fraction dependences and experimental values of EH and Ed from Equations 3.25 and 3.26 are shown in Figure 3.12. The EH(mFC) dependence undergoes small changes after replacing the piezo-active matrix by the piezo-passive one (compare Figure 3.12a and Figure 3.12c). The presence of two piezo-active components poled in the same direction results in a variation of the Ed value in a wider range (Figure 3.12b) mainly due to the different signs of the piezoelectric coefficients ( d 33(1) > 0 and d 33( 2 ) < 0 [29, 55]). In comparison with ____________ 2

Formulae for the full set of the electromechanical constants of the 2–2 series-composite with the random distribution of the piezo-active layers are collected in Appendix.

Non-monotonic Volume-fraction Dependences of Effective Properties

79

Figure 3.12. Volume-fraction dependences 102EH(mFC) (graphs a and b) and 102Ed(mFC) graphs c and d) that have been calculated for the 0–3 PZ 27 / 75/25 mol.% copolymer of vinylidene fluoride and trifluoroethylene composite with both the components poled in the OX3 direction (graphs a and b) or with the piezo-passive polymer component (graphs c and d). In all the graphs, n0 = 0.10. Curves 1, 2, 3, 4, 5, and 6 are related to n1 = 0.20, 0.30, 0.40, 0.50, 0.60, and 0.70, respectively. Dotted lines 7, 8, 9, 10, 11, and 12 are related to m = 0.45, 0.55, 0.65, 0.75, 0.85, and 0.95, respectively. Experimental values [55] of the parameters of the related composite are indicated by triangles (reprinted from Glushanin and Topolov [85], with permission from IOP Publishing)

80

Electromechanical Properties in Composites Based on Ferroelectrics

Figure 3.12 (continued)

Figure 3.12b, the Ed(mFC) values in Figure 3.12d vary within a narrower range. A comparison of the calculated and experimental dependences shown in Figures 3.12a and b demonstrates the emphasis of the role of the rod-like inclusions (Figure 3.11) in forming the dielectric and piezoelectric responses of the

Non-monotonic Volume-fraction Dependences of Effective Properties

81

0–3 composite along the poling axis. Even at relatively low volume fractions of the ceramic (mFC = 0.2–0.3), the volume fraction of these inclusions can be n1 | 0.3– 0.4 (see curves 2 and 3 in Figures 3.12a and b) in the layers with the volume fraction m | 0.5 – 0.7 (see dotted lines 8–10 in Figures 3.12a and b). It is obvious that these inclusions provide the largest contribution to the volume fraction mFC (see Equation 3.27) owing to conditions n1m > n0(1 – m) and n0 = 0.1. Figures 3.12a and b show that small differences in the parameters (n1, m) related to the experimental EH and Ed values are attained at constant mFC values from a wide volume-fraction range. This demonstrates a significant correlation between the EH(mFC) and Ed(mFC) dependences in the composite with the components poled in the same direction. In the case of the composite with a piezo-passive matrix, this correlation becomes less considerable (see, for example, a mutual arrangement of curves 1–6 and experimental points in Figures 3.12c and d). This volume fraction behaviour is concerned [85] with poling features and a reduction in electromechanical coupling in the 0–3 composite. Yamada et al. [90] attempted to explain the volume-fraction behaviour of some of the electromechanical constants of a 0–3 PZT / PVDF composite within the framework of a model with a spheroidal ceramic inclusion in the polymer matrix, but without taking into account the electromechanical interaction among the components. Despite this approximation, it was shown [90] that the ceramic inclusion suitable for the prediction of the effective properties would be a prolate spheroid with a3 / a1 = 2.8 (see Figure 3.7 and Equation 3.18). In our model, geometrical characteristics of the prolate ceramic inclusions in the layers (Figure 3.11) change as a result of variations in the parameters n1 and m from Equation 3.27. Approaching mFC o 0.6 corresponds to m | 0.90 – 0.95 (see Figure 3.12), i.e., the 0–3 structure shown in Figure 3.11 becomes very much like a 1–3 structure shown in Figure 3.2. This is in good agreement with the experimental results [55]. Even at mFC | 0.6, a continuous distribution of the ceramic component was not observed [55, 86] in the thickness direction. The composite based on the Pb(Zr, Ti)O3-type ceramic is described by 0–3 connectivity at 0 < mFC d 0.6. It should be added that the model of the 0–3 composite (Figure 3.11) was also applied [85] to interpret the experimental data [51, 52] on the composites based on the PbTiO3type ceramics. Good agreement between the calculated [84, 85] and experimental [51, 52, 55] parameters over the wide mFC range testifies to the correct consideration of the hierarchy of the inclusions in the studied 0–3 ceramic / polymer composites and, therefore, to the plausibility of the developed model. According to this model, in the Pb(Zr, Ti)O3-based composite, the system of rod-like ceramic inclusions plays an important role in forming the dielectric and piezoelectric properties at 0.20 d mFC d 0.60 independently of the poled state of the polymer matrix. In the (Pb, Ca)TiO3-based composite, the similar rod-like inclusions strongly influence the piezoelectric properties at 0.20 d mFC d 0.65 [85]. It seems likely that a further increase in the mFC value would result in a continuous distribution of the ceramic component on the OX3 direction (1–3 connectivity) and smooth changes in the effective electromechanical properties.

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Electromechanical Properties in Composites Based on Ferroelectrics

3.3.4 0–3 Composites with Planar Microgeometry Examples of a 0–3 connectivity pattern with planar interfaces are shown in Figures 1.2 and 3.13. A system of ceramic inclusions is regularly distributed in a continuous polymer matrix and represents a 0–3 piezo-composite that can be manufactured by means of rapid prototyping from the computer-aided design files [6]. Such a composite is completely described by the cubic Banno unit cell (Figure 3.13) with a single inclusion. This inclusion is in the form of a rectangular parallelepiped so that its length, width and height make up the tth, nth and hth parts of the Banno unit cell edge, respectively (Figure 3.13). The electromechanical constants of the ceramic and polymer components are averaged by the matrix method that has also been applied to 2–2, 0–3, 1–1, and 1–3 connectivity patterns [22, 38, 59]. The averaging procedure for the 0–3 composite implies [64] the determination of the effective electromechanical constants of a laminar ceramic–polymer structure by taking into account the electrical and mechanical boundary conditions at interfaces (i) x1 = const, (ii) x2 = const and (iii) x3 = const separately. The ceramic (p = 1) and polymer (p = 2) components are characterised ( p ), E , piezoelectric coefficients d kl( p ) and dielectric by the elastic compliances s ab *E permittivities H (ffp ),V . The full set of effective elastic compliances s ab , piezoelectric coefficients d kl* and dielectric permittivities H *ffV of the 0–3 composite is calculated as a function of the parameters t, n and h that characterise microgeometry and the volume fraction of the ceramic component m = tnh (Figure 3.13). An interesting example of an influence of the geometric factor on the effective parameters of the 0–3 composite is concerned with the variation of length and width of the ceramic inclusion (i.e., 0 < t < 1 and 0 < n < 1, see Figure 3.13) at its constant height (h0 = const, 0 < h0 < 1). This composite poled along the OX3 axis is described by mm2 symmetry, and its effective electromechanical properties obey *E *E conditions s ab (t, n, h0) = s ab (tc, nc, h0), d kl* (t, n, h0) = d kl* (tc, nc, h0) and H *ffV (t, n, h0)

Figure 3.13. Schematic representation of the Banno unit cell for the 0–3 composite with planar interfaces. (X1X2X3) is the rectangular co-ordinate system

Non-monotonic Volume-fraction Dependences of Effective Properties

83

* * = H *ffV (tc, nc, h0) where tc = n and nc = t. For example, d 33 (0.1, 0.7, 0.9) = d 33 (0.7, 0.1, 0.9). Our further consideration in this section is related to the 0–3 PbTiO3-type ceramic / polymer composite. By variation of n and t it is possible to establish maxima of the effective * 2 * * * parameters gh* = g 33 + g 32 + g 31 , (Q33 ) = d33* g 33* and (Qh* ) 2 = d h* g h* that

characterise piezoelectric sensitivity of the composite with the 0–3 particulate microgeometry. The values of these maxima increase with increasing h0 (i.e., the height of the inclusion shown in Figure 3.13), and in the limiting case of h0 = 1 the * 2 composite has 1–3 connectivity, where the absolute maxima of gh* , (Q33 ) and

(Qh* ) 2 are achieved. As follows from results in ref. [64], a difference between values of the local max gh* (t, n, 0.9) and the absolute max gh* does not exceed 5%, * 2 and any trend is therefore typical of the volume-fraction dependences of (Q33 )

and (Qh* ) 2 . Examples of the dependences of the effective parameters on t and n at h0 = 0.95 are shown in Figure 3.14. The corresponding 0–3 composite displays * 2 relatively high values of gh* | 4 g h(1) and (Q33 ) | 15( Q33(1) )2 in comparison with the * 2 parameters of the ceramic component. The locations of max gh* and max (Q33 ) are

realised on conditions that

t << 1 and n o 1 or n << 1 and t o 1

(3.28)

Conditions 3.28 correspond to the use of plate-like or fibre-like ceramic inclusions having the least edge lying along one of the non-polar (OX1 or OX2) axes (see * 2 Figures 3.13 and 3.14a). In contrast to the gh* and (Q33 ) values, the squared hydrostatic figure of merit (Qh* ) 2 exhibits a monotonic dependence on both t and n from ranges of 0 < (t, n) < 0.99 (see, e.g., Figure 3.14c). However, the largest (Qh* ) 2 value remains two orders of magnitude lower than (Qh(1) ) 2 of the bulk (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 ceramic due to the presence of isolated ceramic inclusions and a discontinuous distribution of the ceramic component in the polymer matrix along the poling axis OX3 (Figure 3.13). At such a discontinuous * distribution, the piezoelectric coefficients d33 and d h* remain relatively small in comparison with the constants of the monolithic ceramic material and do not * 2 provide large (Qh* ) 2 values. A considerable difference between the (Q33 ) and

(Qh* ) 2 values satisfying conditions 3.28 is accounted for by the small volume fraction m of the ceramic component. At low m values the piezoelectric anisotropy d 33(1) / d 31(1) of the ceramic is unable to influence a redistribution of internal electric and mechanical fields and possibly promote an increase in the (Qh* ) 2 value of the composite. Thus, our study on the 0–3 piezo-composites enables us to determine volumefraction dependences of the effective electromechanical properties and related

84

Electromechanical Properties in Composites Based on Ferroelectrics

Figure 3.14. Effective parameters g h* (a), (b) and (Qh* ) 2 (c) that have been calculated for the 0–3 (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II) / elastomer composite with inclusions shown in Figure 3.13, where h = 0.95, 0.01 d t d 0.05 and 0.01 d n d 0.99 (reprinted from Bowen and Topolov [64], with permission from Elsevier)

Non-monotonic Volume-fraction Dependences of Effective Properties

85

Figure 3.14 (continued)

parameters and establish important extreme points thereof. Due to the lack of connectivity of the piezo-active ceramic inclusions, the piezoelectric coefficients * and d h* are fairly low. At the same time, the reduced dielectric permittivity H 33*V d33 * and gh* . Therefore, such 0–3 promotes the enhanced piezoelectric coefficients g33 composites are suitable for sensing applications (e.g., hydrophones or sensor elements), rather than active transducers. It is important to note that high piezoelectric sensitivity depends not only on the microgeometry of the composite (for example, prolate inclusions oriented along the poling axis), but also on the volume fraction of the ceramic. The common feature of the 0–3 ceramic / polymer composites studied in Sections 3.3.2 and 3.3.4 is that the relatively small volume fraction of the ceramic component and the presence of the ceramic inclusions being prolate with respect to the poling axis OX3 can be regarded as a necessary condition for high piezoelectric sensitivity.

3.4 Composites with 3–3 Connectivity, Increasing the Piezoelectric Sensitivity and Applications In the 3–3 composite both the components being continuously distributed in three dimensions (Figure 1.2) form complicated interpenetrated structures. The presence of the piezo-active framework offers possibilities to vary both the microgeometry and the effective electromechanical properties in the composite over wide ranges. In contrast to the 0–3 composite system, the continuous distribution of the piezoactive component between electrodes provides favourable conditions for poling a

86

Electromechanical Properties in Composites Based on Ferroelectrics

3–3 composite sample. The 3–3 ceramic / polymer composites are of interest due to a series of effective parameters that exhibit a non-monotonic volume-fraction behaviour. For example, max d h* is typical of the 3–3 composites based on the * and max kt* were experimentally Pb(Zr, Ti)O3-type ceramics [91], max d33

established in the 3–3 (Pb, Ca)(Ti, Co, W)O3 ceramic / epoxy composite [92], * max g33 and max [ (Qh* ) 2 ] were found in the 3–3 PZT / epoxy composites [93] with a certain arrangement of ceramic elements. In the 3–3 PZT / polymer composite [94] with a three-dimensional lattice of ceramic rods distributed regularly in a * and max [ (Qh* ) 2 ] are also attainable. polymer matrix, max d 33 Recently, a variety of methods and technologies have been applied to fabricate the 3–3 ceramic / polymer composites. Skinner et al. [95] first fabricated these composites by means of the lost-wax method and replamine process. A simpler method developed in refs. [91, 96] is based on the so-called burned-out plastic spheres (BURPS) technique. In this method, a ceramic powder, for example, of the Pb(Zr, Ti)O3 type, and plastic spheres are mixed in an organic binder. The sintering procedure makes it possible to form a ceramic skeleton with a well-developed structure of pores. The skeleton is later back-filled with a polymer, and as a result, 3–3 connectivity is observed. The corresponding ceramic / polymer composites are manufactured to provide the volume fraction of the ceramic component up to 80% [92] and performance can be better than that of the related 0–3 and 1–3 composites. Other methods for manufacturing the 3–3 composites include relic processing [97, 98], distorted reticulated ceramics [99], rapid prototyping [6], and robotic deposition technique (robocasting) [94]. The rapid prototyping and robocasting methods enable one to attain a fine-scale periodic structure with 3–3 connectivity in a wide volume-fraction range. The 3–3 ceramic / polymer composites are traditionally characterised by relatively high piezoelectric sensitivity, low density and relatively low dielectric permittivity [91–94]. A next stage in the development of the 3–3 structure can be concerned with creation of a piezo-composite that would provide extremely high piezoelectric sensitivity at a relatively large volume fraction m of the ceramic component, i.e., a piezo-active material for which conditions [64] * * (m)]2 / [ Q33 (1)]2 >> 1, [ Qh* (m)]2 / [ Qh* (1)]2 >> 1, gh* (m) / gh* (1) >> 1, [ Q33 (3.29) and m a 1 – m

would be fulfilled simultaneously. It is obvious that an increase in piezoelectric sensitivity of the composite is closely connected with increasing the connectivity of its piezo-active component: this tendency is clearly observed on comparing the effective parameters of the 1–3 and 0–3 composites (see Sections 3.2.1 and 3.3.2). In the 3–3 composite system the connectivity of both the components is maximum. This high degree of connectivity favours high piezoelectric activity at relatively low dielectric permittivity, and as a consequence, such a combination of the dielectric and piezoelectric properties can result in a fulfilment of conditions 3.29.

Non-monotonic Volume-fraction Dependences of Effective Properties

87

A typical Banno unit cell for the 3–3 composite is shown in Figure 3.15a, where the interpenetrating components are indicated. It is assumed that the components are distributed regularly and the composite is poled along the OX3 axis of the rectangular co-ordinate system (X1X2X3). As the dielectric permittivity H 33(1),V of the piezo-ceramic is significantly greater than that of the polymer, it will dominate the effective permittivity H 33*V of the composite. As follows from analysis of Figure 3.15b, the piezoelectric column parallel to the poling direction OX3 contributes to H 33*V of the composite (volume 1). The other volumes shown in Figure 3.15b contribute little to H 33*V since they consist of either the low-permittivity polymer (volume 2) or the piezo-active component being in series with a polymer in the OX3 direction (volumes 3 and 4). The effective dielectric permittivity of the studied composite can therefore be approximated [100] by a formula H 33*V = H 33(1),V m

(H )

(3.30)

where m(H) = L2 / (L + l)2 is the volume fraction of the piezo-active component contributing to H 33*V . * , the fraction of a force To calculate the effective piezoelectric coefficient d 33 experienced by the piezo-active component must be determined. If a force F3 is applied on the OX3 direction, the total force is distributed through four volumes indicated in Figure 3.15b. The polymer component will experience less force, being of significantly higher compliance than the ceramic and has led to the

Figure 3.15. Banno unit cell and distribution of forces in the 3–3 ceramic / polymer composite: (a) model of the interpenetrating structures of two components (piezoelectric and polymer phases) with volume fractions depending on the piezoelectric column width L and length l; (b) force F3 in the OX3 direction is distributed through volumes 1 – 4; (c) force F1 in the OX1 direction is distributed through volumes 5 – 8 (reprinted from Bowen and Topolov [64], with permission from Elsevier)

88

Electromechanical Properties in Composites Based on Ferroelectrics

assumption in some models of complete stress transfer into the piezo-ceramic [100]. However, the actual distribution of the force between the four volumes depends on the respective compliances of the volumes and the cross-sectional area of each volume. For volume 1, the elastic compliance is s33(V 1) = s33(1), E and the area of an intersection by the (X1OX2) plane is a(V1) = L2. For volume 2, the compliance is s33(V 2) = s33( 2 ) and the area of the similar intersection is a(V2) = l2. Volumes 3 and 4 are identical in geometry and consist of the polymer and piezo-active components connected in series in the OX3 direction. For both these volumes, the area of the intersection by the (X1OX2) plane is a(V3)= a(V4)= Ll and the elastic compliance connected with the OX3 direction can be calculated using a series model by means of a formula [64] (V 3 ) = s33(V 4) = m(V3) s33(1), E + (1 – m(V3)) s33( 2 ) = (L s33(1), E + l s33( 2 ) ) / (L + l) s33

(3.31)

where m(V3) is the volume fraction of the piezo-active component in volume 3. At high or intermediate volume fractions of the ceramic component, only volume 1 * can be considered to contribute to the effective piezoelectric coefficient d 33 of the 3–3 composite. Volume 2 consists only of the piezo-passive polymer while the piezo-active ceramics in volumes 3 and 4 are in series with the polymer in the OX3 * direction, which results in a negligible contribution to the d33 value. * of the composite shown in Figure The effective piezoelectric coefficient d33 3.15 can therefore be calculated [100, 101] from the following formula: * = d 33(1) u fraction of load experienced by volume 1 d33 = d 33(1) (L / s33(1), E )[(L2 / s33(1), E ) + (l2 / s33( 2) ) + 2 L l (L + l) / (L s33(1), E + l s33( 2) )]–1

2

(3.32)

* By varying the L and l values, the piezoelectric coefficient d33 can be calculated as a function of the volume fraction of the piezo-ceramic and the aspect ratio of the unit cell. The same procedure is undertaken to calculate the * piezoelectric coefficient d 31 of the composite [100]. For this calculation, one can

consider a force F1 acting in the OX1 direction: as seen from Figure 3.15c, this force is distributed through volumes 5–8. The volume of the piezo-ceramic that * contributes to the d 31 value is the cube in volume 6, as indicated in Figure 3.15c (the remainder of volume 6 is neglected since it is in series in the OX3 direction with the polymer component), and the piezo-ceramic in volume 7. Based on these * statements, d 31 can be calculated using a formula [64] * = d 31(1) [L /(L + l)]-1 u fraction of load experienced by volume 6 and 7 d 31 = d 31(1) [L / (L + l)]{[L l (L + l) / (L s11(1), E + l s11( 2 ) ) ] + (L2 / s11(1), E )]{(l2 / s11(1), E ) + [2L l (L

+ l) / (L s11(1), E + l s11( 2 ) )] + (L2 / s11(1), E )}–1

(3.33)

Non-monotonic Volume-fraction Dependences of Effective Properties

89

Thus, the volume-fraction dependences of the effective hydrostatic piezoelectric coefficients d h* and g h* of the 3–3 ceramic / polymer composite can be determined using Equations 3.30–3.33. Calculations were made for 3–3 composites consisting of a piezo-ceramic and a (2) (1), E = 10 s ab ) [64]. polymer being ten times more compliant than the ceramic (i.e., sab The most important results on the dielectric and piezoelectric responses of the studied 3–3 composites are graphically represented in Figure 3.16. As would be expected, there is a gradual reduction in H 33*V as the low-permittivity polymer replaces the ceramic (Figure 3.16a). Figure 3.16b shows that the hydrostatic piezoelectric coefficient d h* reaches a maximum at polymer volume fractions about 0.4–0.5 for the PCR-63, PCR-62 and Pb(Zr1–xTix)O3 ceramics. The highest d h* values are achieved in the composites based on ceramics with the highest d 33(1) values (e.g., PCR-63 among the ceramic compositions chosen). It should be mentioned for comparison that in the 1–3 composites, the maximum in d h* is generally achieved at higher polymer volume fractions of 0.7–0.8 [43]. The mechanism by which d h* increases for the 3–3 composites is similar to that for 1– 3s [29, 32]: in both cases the composite structure leads to decoupling of * * mechanical stress and a large reduction in the | d 31 | value relative to d 33 . It is for this reason that the ceramic components that experience the greatest enhancement of d h* relative to d h(1) are those with a small anisotropy value d 33(1) /| d 31(1) | (i.e., with a relatively high | d 31(1) | value), such as PCR-62 and PCR-63 (see Figure 3.16b and Table 2.1). For the composite based on the modified PbTiO3 ceramic,3 no increase in d h* is observed (Figure 3.16b) as the ceramic component inherently exhibits a high d 33(1) / | d 31(1) | ratio due to its relatively high d 33(1) (51 pC / N), low d 31(1) (–4.4 pC / N) and initial high d h(1) values. Incorporating polymer into this modified PbTiO3 ceramic merely serves to gradually decrease d 33(1) and the associated the

d h(1) value (Figure 3.16b). For a particular ceramic / polymer composite system, * * relative to d 33 (and as a consequence, the resulting increase in the reduction in d 31 * d h ) is greater for 3–3 composites than 1–3 composites. This is due to the relatively high stiffness of the ceramic along the OX1 and OX2 axes of the 3–3 composite (Figure 3.15a), which acts to stiffen the structure in these directions and reduce * | d 31 |. It is for this reason that top and bottom metallic cover plates are used in 1–3 composites, where stiffening the structure in the OX1 and OX2 directions really results in the increase in d h* [43]. The hydrostatic piezoelectric coefficient g h* increases for all the composites ____________ 3

In this section the modified PbTiO3 ceramic means the composition of (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II). The full set of electromechanical constants of this ceramic is given in Table 2.1.

90

Electromechanical Properties in Composites Based on Ferroelectrics

1200

Relative permittivty H33T

PCR-63 PCR-62

1000

x=0.4 x=0.3

800

x=0.2 x=0.1

600

Mod PbTiO3

400 200 0 0

0.2

0.4

0.6

0.8

1

Polymer volume fraction

a

dh (in 10

-12

-1

mV )

100

PCR-63

90

PCR-62

80

x=0.4

70

x=0.3 x=0.2

60

x=0.1

50

Mod PbTiO3

40 30 20 10 0 0

0.2

0.4

0.6

0.8

1

Polymer volume fraction

b *V Figure 3.16. Effective relative dielectric permittivity H 33 / H 0 (a), hydrostatic piezoelectric

coefficient d h* (b), hydrostatic piezoelectric coefficient g h* (c), and squared hydrostatic figure of merit ( Qh* )2 (d) that have been calculated for 3–3 ceramic / polymer composites with elements of microgeometry shown in Figure 3.15a. Ferroelectric ceramics used as piezo-active components include modified PbTiO3 ((Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (II)), PCR-62, PCR-63, and Pb(Zr1–xTix)O3 ceramics where x = 0.1, 0.2, 0.3, and 0.4 (reprinted from Bowen and Topolov [64], with permission from Elsevier)

Non-monotonic Volume-fraction Dependences of Effective Properties

91

350 PCR-63

gh (in 10-3 V m-1 Pa-1)

300

PCR-62 x=0.4

250

x=0.3

200

x=0.2 x=0.1

150

Mod. PbTiO3

100 50 0 0

0.2

0.4

0.6

0.8

Polymer volume fraction c

d Figure 3.16 (continued)

1

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Electromechanical Properties in Composites Based on Ferroelectrics

(Figure 3.16c) as polymer is incorporated into the 3–3-type ceramic skeleton and the ceramic volume fraction decreases. This is generally a consequence of the increase in d h* (Figure 3.16b) and decrease in H 33*V (Figure 3.16a) of the 3–3 composite. The highest hydrostatic piezoelectric coefficient g h* is observed for the composite based on the modified PbTiO3 ceramic, due to its relatively low dielectric permittivity ( H 33(1),V / H 0 = 170) and relatively high piezoelectric coefficient ( d 33(1) = 51 pC / N). The squared hydrostatic figure of merit (Qh* ) 2 reaches a maximum at polymer volume fractions of approximately 0.8 and, unlike 0–3 ceramic / polymer composites, large ratios of (Qh* ) 2 / (Qh(1) ) 2 are attained (Figure 3.16d). As an example, a 3–3 PCR-63 / polymer composite exhibits max[ (Qh* ) 2 ] = 3.4u10–12 Pa–1 (Figure 3.16d), while the ceramic component is characterised by a value of (Qh(1) ) 2 = 0.04u10–12 Pa–1. It should be added that a similar max[ (Qh* ) 2 ] is observed for the 1–3 ceramic / polymer composites, where the maximum is located at slightly higher polymer volume fractions of 0.9 (see Section 3.2.3 and refs. [32, 43, 44]). The 3–3 connectivity leads to relatively large d h* , g h* and (Qh* ) 2 values and represents various examples of fulfilment of conditions 3.29. Such performance of the 3–3 composites studied in this section promotes both signal-generation and signal-sensing applications assuming that the composite has an optimum architecture. The optimum architecture in terms of choice of the ceramic volume fraction and choice of the ceramic composition depends on the application and the appropriate figure of merit that needs to be maximised.

3.5 Connectivity – Non-monotonic Behaviour – Manufacturing In this chapter we have analysed and summarised results on non-monotonic volume-fraction dependences of effective electromechanical properties and related parameters in a variety of D–E ceramic / polymer composites. As follows from a series of review papers [2–6, 20, 31, 47, 51, 102] written in the last decades, these composites are of interest for both research and applications. We have applied different methods to evaluate the effective parameters of these composites and to find extreme points of the effective parameters. Along with extreme points, appropriate methods for manufacturing the D–E composites have been mentioned in Table 3.4. Undoubtedly, the data listed in Table 3.4 cannot be restricted by the connectivity patterns considered in Chapter 3. It is believed that the available list of connectivity indices and examples of the non-monotonic behaviour of the effective parameters will be renewed in the future due to advances in manufacturing, experimental studies and promising applications of the D–E composites.

Non-monotonic Volume-fraction Dependences of Effective Properties

93

Table 3.4. Non-monotonic effective parameters and methods for manufacturing the D–E ceramic / polymer composites

D–E

Examples of extreme points of effective parameters

2–2

*E *V * [9], max e31 [9, 10, 40], max H 11 [9], min c11 * 2 max kt* [7,8,13], max[( Q33 ) ] [11] (series

connection) or max kt* [13], max d h* [12], max g h* [12] (parallel connection) 1–3

* * * [37,39], max d 33 [38], max e33 [35, min d 31 * * 38,40,41], max g 33 [33,42], max h33 [33],

max d h* [29,32,34,40], max g h* [32,34,43], * 2 max kt* [29,40], max[( Q33 ) ] [44],

max[( Qh* )2] [30, 34, 43, 44]

0–3

* * * min d 31 [68], max g 33 [68], min g 31 [68],

* 2 max g h* [64,68], max[( Q33 ) ] [64]

3–3

* * max d 33 [92], max g 33 [93], max d h* [64,91],

max kt* [92], max[( Qh* )2] [64,93]

1–1

* * * * max d31 or min d31 [22], min d 32 or max d 32 * * * [22], max e31 or min e31 [22], min e32 or

Methods for manufacturing the composites Dice and fill [4,5], injection moulding [4,5, 14], tape lamination [4,5, 14], jet machining [4,5, 16], co-extrusion [4,5], rapid prototyping [6] Rod placement (pick and place) [4,5], dice and fill [4,5], injection moulding [4,5,14], co-extrusion [4, 5], tape lamination [4,5, 14], rapid prototyping [6], lost mould [4], dielectrophoresis [4, 46], robocasting [94] Dielectrophoresis [58], pressing mixture of ceramic and polymer [102], colloid processing method [102], rapid prototyping [6], mixing and hot rolling [3,54] Lost-wax method and replamine process [95], BURPS [91,96], relic procession [97,98], distorted reticulated ceramics [99], rapid prototyping [6], robocasting [94] Laminating the layers [103]

* * * max e32 [22], min e33 [22] or max e33 [41], * * * min g 31 [22], min g 32 [22], max g 33 [22],

* 2 * * [22], min k 32 [22], max[( Q33 )] max k 31

[22], max[( Qh* )2] [22]

References [1]

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4 Piezoelectric Response of Porous Ceramic and Composite Materials Based on Pb(Zr, Ti)O3

4.1 Porous Materials, General Characteristic and Manufacturing This chapter will describe the study of porous materials based on ferroelectric ceramics of the Pb(Zr, Ti)O3 type. The ceramic compositions are chosen, as a rule, near the MPB to optimise the electromechanical properties, and many examples of the electromechanical constants of the appropriate Pb(Zr, Ti)O3-type ceramics are provided in Table 2.1. Porous materials, whether they are a composite or a ceramic (Figures 1.6 and 2.3a), or contains closed pores (air inclusions, voids) or open pores (air channels), exhibit electromechanical properties that depend on the volume fraction and microgeometry of pores [1–5]. As shown by Banno [1], Wersing et al. [2, 3], Khoroshun et al. [6], and other authors, the electromechanical properties of porous ferroelectric ceramics can be described on the basis of averaging procedures similar to those suitable for the piezo-active composites. Following this principle, we will consider piezoelectric performance and features of the porous ceramics and composites as two groups of related heterogeneous materials. The piezoelectric and other physical properties of a porous materials depend on the properties of the piezo-active component, porosity mp (i.e., the volume fraction of the pores), the shape and size of the pores and their orientation. Results of the large body of work [1–4, 6–15] on porous Pb(Zr, Ti)O3-type ceramics show that these materials are characterised by reduced values of the piezoelectric coefficients *E *D * * * , h33 and | d 31 |, elastic moduli cab and cab , dielectric permittivities H *ffV and e33 * * * H *ff[ , electromechanical coupling factors k33 , | k31 | and | k p |. Although there is a reduction in the above-mentioned properties, there is also a simultaneous increase * * * in the piezoelectric coefficients g 33 , | g 31 |, g h* , and d h* , figures of merit Q33 and

Qh* in porous ceramics [10–12, 14, 15], as compared to the monolithic ceramic * and d 33 * electromechanical coupling factor kt remain almost constant [10–12] over wide

material. It is also known that the piezoelectric coefficient

102

Electromechanical Properties in Composites Based on Ferroelectrics

mp ranges. It should be mentioned that the Pb(Zr, Ti)O3-type ceramics with the perovskite-type structure represent a vast group of ferroelectric ceramic materials that have the piezoelectric coefficients [16] ( FC ) ( FC ) < 0, e33 > 0 and e15( FC ) > 0 e31

(4.1)

(see Table 2.1).1 Inequalities 4.1 differ from similar conditions 3.22 written for the PbTiO3-type ceramics. As a result, in contrast to the PbTiO3-type ceramics, the Pb(Zr, Ti)O3-type materials do not exhibit the large anisotropy of the piezoelectric ) coefficients d 3( FC [16]. j A variety of methods for manufacturing porous materials based on ferroelectric ceramics have been proposed and developed in the last decades. These methods are similar to those concerned with the 3–3 composites (Section 3.4). For example, to manufacture porous ceramics, Skinner et al. [17] used a coral replamine technique, Creedon et al. [18] used polymer foams that were coated with ceramic particles. In refs. [3, 10, 14, 19–26] a ceramic powder and plastic pellets (spheres) were mixed with an organic binder and compacted. The compacted elements were subjected to heating and a burnout cycle (BURPS technique). A polymeric sponge technique with a subsequent burnout cycle was also applied [14, 22] to manufacture porous ceramics and composites with various microstructures. As noted by Kara et al. [14], the BURPS and polymeric sponge techniques are relatively easy to implement for a large-scale production. After the burnout cycle Hayashi et al. [23] carried out hot-isostatic pressing of the sintered samples without encapsulating. The porous PZT ceramic manufactured in this way was characterised by 3–3 connectivity and a homogeneous microstructure with a bimodal distribution of small and large pores in the ceramic matrix [23]. The authors of refs. [24–26] modified the BURPS method and studied the role of the pore-forming agent. Various patterns of porosity in the PZT-based samples were attained using agents such as polyvinyl chloride, polymethyl methacrylate, polyethylene oxide [24], polyethylene glycol, saccharose, and ammonium oxalate [26]. The performance of porous ceramics was studied [23–26] in connection with their microgeometry, pore size and other factors. The aerogel process [27] and tape-casting methods [28] have also been employed to manufacture porous ceramics. Tape-casting technology combined with a fugitive ink was applied by Kahn et al. [9] to form arrays of ordered platelike air pores in a ceramic matrix. A pattern with a regular arrangement of pores was formed on an ink-deposition screen that was used for subsequent deposition of the ink onto a green ceramic tape. The tapes were subsequently stacked and laminated, and organic components were finally burned out to form the porous medium. A porous ceramic with 3–1 connectivity was manufactured by Bast and Wersing [29] using the multilayer method, i.e., a combination of photolithography with stacking, pressing and firing the ceramic sheets. Lamellar pores oriented ____________ 1

In Chapter 4 we use superscript “(FC)” to denote the electromechanical constants and other parameters of the monolithic ferroelectric ceramic.

Piezoelectric Response of Porous Ceramic and Composite Materials

103

parallel or perpendicular to the poling axis were formed by Lopatin et al. [30] in ceramic samples. It is obvious that each method enables a distinctive microstructure, size and arrangement of pores to be obtained. Their volume fraction mp can be varied in specific ranges. For example, in the samples prepared using the BURPS technique, mp is usually varied between 0.30 and 0.70, the foam or sponge technique enables mp = 0.85–0.90 to be obtained [14, 21, 22]. Various porous ceramics and composites with mp = 0.09–0.92 were manufactured using the BURPS and sponge techniques and studied in refs. [14, 21]. It should be noted that an increase in the mp value results in various fractions of closed and open porosity [21] and results in changes in the connectivity of porous samples. However, experimental works contain restricted data on features of microstructure of the porous ceramics and composites (see, e.g., [14, 18, 29, 31]). These factors make it impossible to experimentally analyse the behaviour of the effective electromechanical properties of the porous ceramics and / or composites in the complete mp range for a fixed pore size and morphology.

4.2 Volume-fraction Dependences of Parameters of Porous Piezo-active Materials Experimental dependences of a series of the effective parameters )* ()* = d 3* j , * * , h33 , H 33*V , k3* j , etc.) of the porous ceramic on its porosity mp in a range 0 g 3* j , e33

< mp d 0.66 were first generalised in ref. [10]. These dependences were measured on porous samples made of the Pb(Zr, Ti)O3-type ceramics such as PCR-1 (with ( FC ) ( FC ) the large g 33 value), ZTSSt-2 (with large d 33 value) and ZTStBS-2 (with the ( FC ) ( FC ) and d 33 values). The porous ceramic materials were g 33 characterised by 3–3 connectivity. Based on numerous experimental data, Lopatin and Lupeiko [10] approximated the )*(mp) dependence by means of a formula

moderate

)*(mp) / ) (FC) = A (mp2 + B)-1 + Cmp + G r H

(4.2)

In Equation 4.2 coefficients A, B, C, G, and H were calculated for different parameters of the porous ceramics by the least-square method and ) (FC) = )*(0). Examples of the )*(mp) / ) (FC) ratios are given below for the cases of d()* / ) (FC)) * * * * , e33 , h33 , and H 33*V ), d()* / ) (FC)) / dmp | 0 ()* = d 33 ) and / dmp < 0 ()* = d 31 * * and g 33 ) [10]: d()* / ) (FC)) / dmp > 0 ()* = g 31 * ( FC ) / d 31 = –10.239 (mp2 + 2.0)–1 – 2.429 mp + 6.128 r 0.045, d 31 * ( FC ) / d 33 = –0.206 (mp2 + 1.0)–1 + 0.047 mp + 1.202 r 0.044, d 33 * ( FC ) / g 31 = –2.319 (mp2 – 1.0)–1 – 1.260 mp – 1.270 r 0.011, g 31 * ( FC ) / g 33 = –5.316 (mp2 – 0.95)–1 + 0.625 mp – 4.531 r 0.010, g 33

104

Electromechanical Properties in Composites Based on Ferroelectrics * ( FC ) / e33 = –15.373 (mp2 + 2.5)–1 – 2.737 mp + 7.129 r 0.020, e33 * ( FC ) / h33 = –0.421 (mp2 + 0.4)–1 – 1.746 mp + 2.055 r 0.039, and h33 2 –1 H 33*V / H 33( FC ),V = –81.195 (mp + 10) – 1.731 mp + 9.108 r 0.023.

In refs. [2–5, 11–13] devoted to porous piezo-ceramics, a variety of approaches were proposed to predict the dependences of the effective electromechanical properties and related parameters on porosity mp. The study by Wersing et al. [3] is based on a comparative analysis of models and formulae for calculating these dependences in the wide mp range. The models involved with the analysis and porous ferroelectrics include Wagner’s model, Marutake’s model and the unit-cell models related to 3–0 or 3–3 connectivity [2, 3]. It is shown that Bruggeman’s formulae (first applied to isotropic heterogeneous media) [32] can be used to describe the isotropic and quasi-isotropic properties of a porous ceramic. Such a description is correct in a case when there are only small anisotropies of the dielectric and elastic properties, and the good agreement between calculated and experimental results [3] demonstrates the correctness of the approach proposed by Bruggeman [32]. Based on the modified cube model, Banno collected formulae [1] for the effective electromechanical constants of a ceramic with closed pores (3–0 connectivity). The calculated data obtained using these formulae are in agreement with various experimental dependences of the effective parameters of porous ceramics on mp in a range of 0 < mp < 0.2 [1]. Simultaneously, the theoretical treatment within the framework of the Banno unit cell with planar interfaces [1, 12] cannot examine the dependence of the electromechanical properties on the size and distribution of pores within the ceramic medium. In contrast to the work of refs. [1, 12], Dunn and Taya [4, 11] and Wu [13] developed an approach to determine the full set of effective electromechanical constants of a transversely isotropic piezo-active material containing spherical pores. The procedure is similar to that described in Section 3.3.1 where the effective properties of the 0–3 ceramic / polymer composites are considered. The methodology is based on the EFM, i.e., the Mori – Tanaka method applied to a heterogeneous material with a piezo-active matrix. As is shown in ref. [11], this method accounts for the electromechanical interaction at small and moderate mp values (0 < mp d 0.5). It should be added for comparison that the touching of identical sized spheres would be at volume fractions mp > S / 6, i.e., the upper value 0.5 is near the lower bound related to sphere touching. Numerical results illustrate an effect of the shape of air pores on the effective electromechanical properties in the porous ceramics. An agreement between the experimental and calculated values of a series of parameters is attained [11] for the porous ceramics of the Pb(Zr, Ti)O3 type with various microstructures. Problems of modelling of the electromechanical properties in the porous ceramics were also discussed in refs. [15, 26, 33–39]. Levassort et al. [26] considered representative unit cells for 3–3 and 3–0 connectivity patterns as well as a 3–3 unit cell with a 3–0 matrix. Based on the unit-cell concepts, Levassort et al. analysed different cases of the open- and closed-

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porosity patterns and calculated a series of the effective parameters for porous PZT ceramics. Comparing the calculated and experimental data, it was shown that elements of both 3–0 and 3–3 connectivities would be important at mp t 0.25. Results on the finite-element modelling of the effective electromechanical properties in the porous PZT ceramic are discussed in refs. [37–39]. Kar-Gupta and Venkatesh [37] put forward a classification of porous ceramics. According to this classification, these ceramics are characterised by 3–0, 3–3 or 3–1 connectivity. The porous ceramics with 3–1 connectivity can be regarded as either “longitudinally” porous or “transversely” porous materials, depending on a mutual arrangement of the porosity axis and the poling direction. A further systematisation of data on the effective properties is discussed in ref. [37]. Features of microgeometry and symmetry of the porous 3–1 material are considered in ref. [38]. It is shown that the full set of electromechanical constants in such a material can comprise as many as 45 independent values, and the number of values depends on the symmetry of the monolithic material, the porosity orientation and distribution therein [38]. Asai et al. [39] presented and discussed results on the finite-element modelling of the properties in the porous PZT ceramics with complex three-dimensional microstructures. An approach developed in ref. [39] is concerned with an iterative solver that is suitable for the prediction of effective electromechanical properties in a wide mp range. Important examples of the influence microgeometry on porous PZT ceramics and related composites are studied in refs. [40–42]. It is of interest to note that the authors of refs. [40–42] used optimal design methods and then constructed and manufactured novel porous materials intended for hydrophone and other piezotechnical applications. As follows from a comparison, good agreement between the predicted and measured parameters [40] is attained. However, a discrepancy between the predicted and measured parameters [42] is accounted for by an insufficient polarisation caused by a field distribution (or non-uniform electric field) during poling. Examples of experimental dependences of the normalised parameters )* / )(PZT-5H) on the volume fraction of the PZT-5H ceramic mPZT = 1 – mp in porous materials are shown in Figures 4.1–4.5. These dependences were determined for both porous ceramics (PZT-5H–air) and polymer-impregnated ceramic foams (PZT-5H–polymer) over wide mPZT ranges. Some technological aspects of manufacturing these materials and features of their microstructure were considered and discussed in refs. [14, 20, 21]. Low-density foams (mPZT < 0.3) were prepared by coating the polymeric foams with a ceramic suspension and by subsequent sintering. The porous materials with mPZT > 0.3 were processed using a variety of volatile additives added to the ceramic powder before uniaxial pressing. The fugitive additives include polymethyl methacrylate (PMMA), polyethylene oxide (PEO), self-raising flour (SR), and corn starch (CS), which all produce porosity within the ceramic medium on volatilisation during heat treatment. The scatter in the data in graphs (Figures 4.1–4.5) is thought to originate from the different microstructures formed by the different inclusion types and sizes. For example, PMMA-based materials exhibit cracking parallel to the pressing (and poling) * direction [14, 21]: this cracking leads to fairly large d 33 / d 33( PZT 5 H ) ratios (Figure

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Electromechanical Properties in Composites Based on Ferroelectrics

* Figure 4.1. Experimental dependence of the normalised piezoelectric coefficient d 33 /

d 33( PZT 5 H ) on the relative density (i.e., volume fraction of the ceramic component mPZT) for different porous materials based on the PZT-5H ceramic (reprinted from Topolov et al. [15], with permission from Maney Publishing)

* Figure 4.2. Experimental dependence of the normalised piezoelectric coefficient d 31 /

d 31( PZT 5 H ) on the relative density (i.e., volume fraction of the ceramic component mPZT) for different porous materials based on the PZT-5H ceramic (reprinted from Topolov et al. [15], with permission from Maney Publishing)

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Figure 4.3. Experimental dependence of the normalised hydrostatic piezoelectric coefficient d h* / d h( PZT 5 H ) on the relative density (i.e., volume fraction of the ceramic component mPZT) for different porous materials based on the PZT-5H ceramic (reprinted from Topolov et al. [15], with permission from Maney Publishing)

*V Figure 4.4. Experimental dependence of the normalised dielectric permittivity H 33 /

H 33( PZT 5 H ),V on the relative density (i.e., volume fraction of the ceramic component mPZT) for different porous materials based on the PZT-5H ceramic (reprinted from Topolov et al. [15], with permission from Maney Publishing)

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Electromechanical Properties in Composites Based on Ferroelectrics

Figure 4.5. Experimental dependence of the normalised hydrostatic piezoelectric coefficient g h* / g h( PZT 5 H ) on the relative density (i.e., volume fraction of the ceramic component mPZT) for different porous materials based on the PZT-5H ceramic (reprinted from Topolov et al. [15], with permission from Maney Publishing) * 4.1) along with small d 31 / d 31( PZT 5 H ) and H 33*V / H 33( PZT 5 H ),V ratios (Figures 4.2 and 4.4) in a wide mPZT range. * Figure 4.1 shows the dependence of the piezoelectric coefficient d 33 on mPZT

for foams and porous materials manufactured from the different inclusion * materials. The general trend observed is a decrease in the d 33 value with * at mPZT < 0.75 and a decreasing mPZT. This trend shows a distinct change in d 33 slighter change at mPZT > 0.75. It is found that different processing methods result * in different d 33 values for the same relative density or porosity mp, and, moreover, * the condition d 33 | const established in a wide mp range [10–12] is violated. This * implies that the piezoelectric coefficient d 33 also depends on features of microstructure, manufacturing and poling conditions and other factors that are to be taken into consideration in future studies. In other words, many of the models assume that the piezo-active material is fully poled along a common axis without regard to regions surrounding the air inclusions or pores, and features of inhomogeneous poling should be taken into consideration for future modelling activities. Variations among additives are related to different microstructures formed in * the porous materials. The variation of the piezoelectric coefficient d 31 with mPZT is

shown in Figure 4.2. The general trend observed here consists in a decrease in the

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* ( PZT 5 H ) * / d 31 ratio with decreasing mPZT, but at a more rapid rate than d 33 / d 31

d 33( PZT 5 H ) (see Figure 4.1), particularly in the high density range, at mPZT > 0.5. * ( PZT 5 H ) * Such a decrease in d 31 / d 31 or | d 31 | suggests that an electromechanical interaction between some piezo-active elements aligned in the poling direction OX3 becomes weaker as the porosity of the material surrounding these elements * increases. The rapid decrease in | d 31 | and the subsequent increase in the * * piezoelectric anisotropy d 33 / | d 31 | results in high values of the hydrostatic

piezoelectric coefficient d h* (Figure 4.3). A maximum of d h* is present at volume fractions mPZT from a range 0.5 – 0.75. The highest d h* values are attained using PMMA as the inclusion material. The foam-based materials, while having a modest d h* value (100 – 200 pC / N), still show a higher value than that for the monolithic PZT-5H ceramic (around 50 pC / N). These foam materials have many other effective property advantages in their favour, including low density and acoustic impedance, although the relatively low strength of these materials should also be considered. Figure 4.4 shows how the normalised dielectric permittivity H 33*V / H 33( PZT 5 H ),V depends on the volume fraction mPZT. The results for samples made with PEO and SR have similar values over the wide mPZT range. However, the samples *V made with PMMA have lower H 33 values than expected; and is a result of the presence of cracks [14, 21] that are oriented with their length mainly perpendicular to the poling direction OX3. In contrast, the samples made with CS have a higher *V permittivity H 33 , possibly due to the relatively small pore size. For a further comparison, it should be added that the porous PZT ceramic manufactured using *V capsule-free hot-isostatic pressing [23] shows high dielectric permittivity H 33 along with good connectivity on the electric field direction. The hydrostatic piezoelectric coefficient g h* reaches fairly large values (i.e., 50 g h( PZT 5 H ) and more) at volume fractions mPZT < 0.5 (Figure 4.5) due to relatively high hydrostatic piezoelectric activity (Figure 4.3) and low dielectric permittivity (Figure 4.4). The large g h* values are important for hydroacoustic applications. It should be noted that the samples made with PMMA have the large g h* values * arising from their suppressed H 33*V and d 31 values caused by cracking of these samples [14] during manufacture. It can also be seen that the samples made with CS have a lower g h* value than the samples made with PEO and SR, owing to the

large H 33*V of the samples made with CS (see Figures 4.4 and 4.5). The PZT-5H– air-foam-based materials exhibit the large g h* values but are difficult to use due to their relatively poor mechanical strength. However, the PZT-5H–polymer-foambased materials, while having a lower g h* value, are more robust and still well suited to hydrophone applications.

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Electromechanical Properties in Composites Based on Ferroelectrics

In the next section we consider an example of prediction of the effective electromechanical properties in porous ceramics and composites and compare results of modelling and known experimental data. This consideration is concerned with porous materials based on the Pb(Zr, Ti)O3-type ceramics.

4.3 Model of the Modified Layered Composite: Application to Porous Materials The possibility of creating a three-component piezo-composite based on ferroelectric ceramics that exhibit considerable piezoelectric anisotropy was first discussed by Topolov and Turik [43]. Such a model can be adapted to describe a porous material with different arrangements of porosity and second phase. As follows from results [44] on the 2–2 series-connected composite shown in Figure 3.1a, absolute values of the piezoelectric anisotropy ]d can be higher than the d 33( n ) / | d 31( n ) | ratios related to the piezo-active components (n = 1 or 2). An increase in |]d | is possible even in the presence of two piezo-active components with small d 33( n ) / | d 31( n ) | ratios (for example, less than 3) [44]. The influence of the 2–2-type structure on ]d can be enhanced in a modified layered composite [43]. Such a structure contains two types of series-connected layers (see Figure 3.1a). The layer of the first type is made of a polymer matrix that is reinforced by ceramic rods distributed continuously along the poling axis OX3, i.e., 1–3 connectivity takes place in this layer. The layer of the second type comprises of ceramic only. The aforementioned layers alternate in the OX3 direction and promote a redistribution of internal electric and mechanical fields. This redistribution results in large |]d | and |]e | values [43] in specific ranges of volume fractions of the ceramic components. Based on results [43], a similar model containing two types of piezo-active layers (Figure 4.6) is used to describe the electromechanical properties of porous ceramics and composites [15]. It is assumed that the relatively small decrease in * the piezoelectric coefficient d 33 with increasing porosity mp results from a continuous distribution of the piezo-ceramic framework along the poling axis OX3, and it can take place with the presence of elements in the structure with 1–3 connectivity. According to the model from ref. [15], the layer of the first type represents a piezo-passive matrix reinforced by ceramic cylindrical rods being lengthy and parallel to the poling axis OX3. The layer of the second type is assumed to be ceramic and lengthy in the OX1 and OX2 directions. These layers, which randomly alternate along the OX3 direction, are the main elements of the structure of porous ceramic or composite materials (Figure 4.6). Volume fractions of their piezo-active (mPZT) and piezo-passive (mp) components are determined by the formulae mPZT = msmr + 1 – ms and mp = ms(1 – mr). An additional effect of porosity can be taken into account if the polymer matrix in the layers of the first type is considered as a cellular structure with uniformly distributed spherical air inclusions, and the volume fraction of these inclusions in the matrix equals min.

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Figure 4.6. Schematic diagram showing the structure of the porous piezo-active ceramic or porous composite based on the piezo-ceramic. The dashed area is the monolithic ceramic component with the remanent polarisation vector PR nn OX3, the clear area is the piezopassive medium such as air, polymer or porous polymer (reprinted from Topolov et al. [15], with permission from Maney Publishing)

The effective parameters )* = )*(mPZT) of the porous material schematically shown in Figure 4.6 are determined using formulae for piezo-composites in three stages. Averaging on min is carried out on the basis of formulae [4] (0–3 connectivity), averaging on mr is based on formulae [45] (1–3 connectivity) and averaging on ms is performed with formulae [6] (2–2 connectivity, see also Appendix). In calculating the effective electromechanical constants of the 1–3-type layer, the approximations concerned with the aspect ratio Dr of the ceramic rods (see Section 3.3.3) should also be considered. It is assumed that the rods elongated in porous samples in the OX3 direction (Figure 4.6) are characterised by values of the aspect ratio Dr t 5. Examples of the )*(mPZT) dependences calculated for porous materials are shown in Table 4.1 and Figure 4.7. Table 4.1 also shows the dependences of the normalised piezoelectric coefficients d 3l* / d 3(lPZT 5 H ) on the parameters mr and ms. The axes in the graphs of Figure 4.7 are to the same scale as the experimental data in Figures 4.1 – 4.5 for direct comparison. The data from Table 4.1 and Figure 4.7 contain the )*(mPZT) dependences calculated for four different cases. Cases I and II are related to the PZT-5H / polyethylene and PZT-5H / elastomer composites, respectively. Case III is related to the PZT-5H / porous elastomer composite with the regular distribution of spherical pores (min = 0.3) in the polymer matrix. Case IV is related to the PZT-5H ceramic / air composite (i.e., limiting case of the most compliant matrix).

4.4 Comparison of Calculated and Experimental Results The calculated and experimental results can be compared in two ways. First, a comparison of the normalised parameters )* / )(PZT-5H) for cases I and II suggests that the substitution of the polymer component by a more compliant one (e.g., polyethylene o elastomer) results in (i) a slight increase in H 33*V (Figure 4.7b),

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Table 4.1. Calculated normalised piezoelectric coefficients d 3l* / d 3(lPZT 5 H ) of composite materials based on the PZT-5H ceramic (reprinted from Topolov et al. [15], with permission from Maney Publishing) ms 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.10 0.15 0.20 0.25

mr

mp

mPZT

* ( PZT  5 H ) / d 31 d 31

Case I: PZT-5H – polyethylene 0.08 0.092 0.908 0.413 0.10 0.135 0.865 0.388 0.10 0.180 0.820 0.327 0.13 0.218 0.782 0.355 0.12 0.264 0.736 0.299 0.12 0.308 0.692 0.273 0.10 0.360 0.640 0.215 0.10 0.405 0.595 0.202 0.09 0.455 0.545 0.175 0.06 0.517 0.483 0.117 0.05 0.570 0.430 0.0984 0.04 0.624 0.376 0.0821 Case II: PZT-5H – Elastomer 0.08 0.092 0.908 0.445 0.10 0.135 0.865 0.412 0.10 0.180 0.820 0.346 0.13 0.218 0.782 0.368 0.12 0.264 0.736 0.308 0.12 0.308 0.692 0.279 0.10 0.360 0.640 0.218 0.10 0.405 0.595 0.201 0.09 0.455 0.545 0.171 0.06 0.517 0.483 0.112 0.05 0.570 0.430 0.0913 0.04 0.624 0.376 0.0737 Case III: PZT-5H – porous elastomer 0.08 0.092 0.908 0.460 0.10 0.135 0.865 0.422 0.10 0.180 0.820 0.354 0.13 0.218 0.782 0.373 0.12 0.264 0.736 0.311 0.12 0.308 0.692 0.280 0.10 0.360 0.640 0.217 0.10 0.405 0.595 0.199 0.09 0.455 0.545 0.166 0.06 0.517 0.483 0.106 0.05 0.570 0.430 0.0833 0.04 0.624 0.376 0.0638 Case IV: PZT-5H – Air 0.08 0.092 0.908 0.466 0.10 0.135 0.865 0.426 0.10 0.180 0.820 0.358 0.13 0.218 0.782 0.375

* ( PZT  5 H ) / d 33 d 33

0.724 0.745 0.720 0.771 0.738 0.728 0.672 0.667 0.634 0.519 0.468 0.409 0.872 0.885 0.872 0.900 0.883 0.878 0.847 0.843 0.823 0.744 0.703 0.650 0.962 0.967 0.963 0.972 0.966 0.965 0.955 0.954 0.947 0.917 0.900 0.876 1 1 1 1

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Table 4.1 (continued) ms

mr

mp

mPZT

* ( PZT  5 H ) / d 31 d 31

* ( PZT  5 H ) / d 33 d 33

0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

0.12 0.12 0.10 0.10 0.09 0.06 0.05 0.04

0.264 0.308 0.360 0.405 0.455 0.517 0.570 0.624

0.736 0.692 0.640 0.595 0.545 0.483 0.430 0.376

0.313 0.281 0.218 0.199 0.166 0.104 0.0811 0.0607

1 1 1 1 1 1 1 1

(ii) a considerable increase in | d 3l* | (Table 4.1) and d h* (Figure 4.7a) and (iii) a more significant increase in g h* (Figure 4.7c) at 0 < mPZT < 0.5. This change in the matrix material influences the piezoelectric anisotropy value ]d of the composite, primarily due to the change in the elastic properties and Poisson ratio of the polymer component. Second, comparing data from cases II, III and IV, it is possible to conclude that * * the changes in H 33*V and d 31 are relatively small. However, the d 33 , d h* and g h* values increase significantly when an elastomer is substituted by a more compliant porous elastomer or by air. Such behaviour is concerned with the weak electromechanical interaction between piezo-active rods in the OX1 and OX2 directions, owing to the piezo-passive matrix, and with conditions favourable to the “rod–layer” electromechanical interaction in the OX3 direction [15]. It is known [44, 45] that softening the matrix surrounding the piezo-active rods in the 1–3 * * , g 33 , d h* , and g h* even at small volume composites provides higher values of d 33 fractions mr of the rods. This can be understood by considering the application of a force to the composite along the poling axis. As the stiffness of the polymer matrix is reduced, more force is experienced by the stiffer piezoelectric material and * * hence there is an increase in d 33 . An important consequence of the slight d 33 (ms,

mr) dependence at small mr values (see cases II and III in Table 4.1) is the equality * ( PZT 5 H ) * 2 / g 33 = (Q33 g 33 ) / (Q33( PZT 5 H ) ) 2

(4.3)

which holds true with accuracy up to a few per cent [15]. The correlation between the calculated (Figure 4.7 and Table 4.1) and experimental (Figures 4.1–4.5) volume-fraction dependences of the effective parameters exists in a wide porosity range. According to the model concepts [15], changes in mp are associated with changing the parameters ms and mr of the layers of the first type (Figure 4.6). As porosity mp increases, ms also increases, i.e., at mpc > mp the inequality msc > ms holds and mr is varied within a narrow range. As seen from Table 4.1, the relation mpc / mp | msc / ms holds at 0.10 < mp < 0.60. However, as mp approaches 0.60, a comparison of the calculated and experimental values of )* / )(PZT-5H) becomes problematic [15]. Our calculations for mp t 0.60 suggest that the calculated and experimental parameters would be in good agreement at mr <

114

Electromechanical Properties in Composites Based on Ferroelectrics 10 Case I

9

Case II

d* h /d h (PZT-5H)

8

Case III

7

Case IV

6 5 4 3 2 1 0 0

0.25

0.5

0.75

1

Rela tive Density

a

1 Case I

0.9

Case II

T H * 33 / H 33

T(PZT-5H)

0.8

Case III

0.7

Case IV

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.25

0.5

0.75

1

Relative Density

b

120 Case I Case II

100

g* h /g h (PZT-5H )

Case III Case IV

80 60 40 20 0 0

0.25

0.5

0.75

1

Relative Density

c Figure 4.7. Calculated normalised effective parameters )* of porous composites based on V (b) and gh (c) (reprinted from Topolov et al. [15], with the PZT-5H ceramic: ) = dh (a), H 33 permission from Maney Publishing)

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115

0.10. However, the structure of the porous material with small mr values at ms > 2/3 (see Figure 4.6) becomes mechanically unstable and may be destroyed by a small amount of mechanical loading. This highlights the range of 0.10 < mp < 0.60 as the most suitable porosity range wherein the model [15] can be used for interpretation of experimental data on porous piezo-active materials. Another example of the application of the model [15] for interpretation of experimental results is concerned with the work in ref. [35] on porous piezo-active materials based on the ZTS-19 ceramic. This ceramic also belongs to the Pb(Zr, Ti)O3-type materials and has piezoelectric coefficients e3( lZTS 19) satisfying conditions (4.1) (see Table 2.1). A comparison of the calculated and experimental [35] parameters (Table 4.2) demonstrates the applicability of the model [15] in a wide porosity range. It should be added that the effective parameters of the porous composite based on ZTS-19 are subjected to small changes (about 1–2%) as Table 4.2. Calculateda and experimental (in parentheses, from ref. [35]) values of normalised effective parameters )* / )(ZTS-19) of the porous ZTS-19 ceramic ms 0.160 0.160 0.180 0.180 – 0.320 0.320 0.340 0.340 – 0.420 0.420 0.440 0.440 – 0.500 0.500 0.520 0.520 – 0.600 0.600 0.620 0.620 –

mr 0.300 0.230 0.330 0.250 – 0.230 0.270 0.280 0.240 – 0.200 0.210 0.210 0.220 – 0.120 0.130 0.130 0.140 – 0.110 0.120 0.110 0.140 –

mp 0.112 0.123 0.121 0.135 0.130 0.246 0.234 0.245 0.258 0.250 0.336 0.332 0.375 0.343 0.350 0.440 0.435 0.452 0.447 0.450 0.534 0.528 0.552 0.533 0.530

H 33*V

* d 31

* g 33

H 33( ZTS 19 ),V

d 31( ZTS 19)

( ZTS 19 ) g 33

* 2 (Q33 ) (Q33( ZTS 19) ) 2

0.729 0.652 0.733 0.650 (0.736) 0.483 0.537 0.536 0.484 (0.533) 0.374 0.388 0.377 0.394 (0.375) 0.215 0.231 0.224 0.239 (0.234) 0.172 0.186 0.167 0.209 (0.172)

0.730 0.653 0.733 0.651 (0.648) 0.485 0.538 0.534 0.482 (0.488) 0.376 0.390 0.379 0.391 (0.385) 0.218 0.234 0.227 0.242 (0.233) 0.175 0.189 0.171 0.212 (0.211)

1.37 1.53 1.36 1.53 (1.36) 2.06 1.85 1.86 2.06 (1.88) 2.65 2.55 2.63 2.54 (2.67) 4.56 4.26 4.39 4.12 (4.34) 5.71 5.28 5.86 4.72 (5.47)

1.37 1.52 1.36 1.53 (1.36) 2.04 1.85 1.86 1.99 (1.88) 2.63 2.53 2.61 2.52 (2.67) 4.48 4.19 4.32 4.06 (4.42) 5.59 5.18 5.74 4.64 (5.47)

Note to Table 4.2 a. Calculations were made within the framework of the model of ref. [15]. In the layers of the first type, ceramic rods (Figure 4.6) are embedded in a porous elastomer matrix. This matrix contains spherical pores with the volume fraction min = 0.3

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Electromechanical Properties in Composites Based on Ferroelectrics

porosity min in the elastomer matrix is varied or even as this matrix is replaced by air. The normalised piezoelectric coefficient d 33* (ms, mr) / d 33( ZTS 19 ) undergoes slight changes (under 5%) as ms and mr are varied in wide ranges to obey condition mp < 0.60. At the same time, d 31* (ms, mr) / d 31( ZTS 19 ) and H 33*V (ms, mr) / H 33( PZT 5 H ),V considerably decrease on increasing mp (Table 4.2). As a result of the constant effective piezoelectric coefficient d33* of the porous material at 0.10 < mp < 0.60, an ( ZTS 19 ) * 2 * equality g 33 / g 33 = (Q33 ) / (Q33( ZTS 19) ) 2 is valid with an accuracy up to a few per cent (compare with Equation 4.3). In general, the calculated effective parameters from Table 4.2 follow the experimental values due to both the monotonic decrease in mr and the monotonic increase in ms as porosity mp increases. A similar trend was shown earlier for d 3l* / d 3(lPZT 5 H ) (Table 4.1), but on decreasing mr in the fairly narrow range only. Thus, the porous piezo-active composite (ceramic) material can be represented as a combination of structural elements that are described by either 1–3 or 2–2 connectivity (Figure 4.6). These elements play an important role in forming the effective electromechanical properties and the variable piezoelectric anisotropy in the range of porosity 0.10 < mp < 0.60. The combination of high piezoelectric activity with high piezoelectric sensitivity in the studied Pb(Zr, Ti)O3-based porous materials promotes their piezotechnical applications as active elements of sensors, hydrophones and ultrasound transducers. In particular, results [46] demonstrate the possible use of ferroelectric “hard” porous ceramics (for example, based on the PCR-78 composition) in ultrasound transducers with high intensity. An application of novel porous ceramics of the PCR group in the piezoelectric transducers of medical devices for therapy and diagnostics was discussed in ref. [47].

4.5 From 3–3 Connectivity to Porous Materials A simple view of understanding the large g h* and d h* values of porous piezo-active composites with 3–3 connectivity can be developed using the Banno unit cell (Figure 3.12a) that consists of a perfectly interpenetrating piezo-active (poled ferroelectric ceramic) and a piezo-passive component (polymer or air). The piezoelectric ceramic is assumed to be fully poled in the OX3 direction. Figure 4.8 shows a side view of the Banno unit cell from Figure 3.12a. When considering the * variation of the effective piezoelectric coefficient d 33 with the polymer volume fraction it is necessary to consider a force applied parallel to the poling axis (as in Figure 4.8a) and the specific piezoelectric elements that contribute to the generation of charge in response to the applied force. In the case of Figure 4.8a the majority of the force is experienced by piezoelectric areas 1 and 2, since the stiffness of the piezo-ceramic component is greater than that of the polymer component. As a result, the piezoelectric coefficient d 33* is dependent on the sizes (and fraction) of areas 1 and 2. As the polymer volume fraction is increased

Piezoelectric Response of Porous Ceramic and Composite Materials

117

Force (d33)

Polymer

1 2

2 a

1

Polymer

3

Polymer

Polymer

2 b

X3

c

Force (d33)

2

Force (d31)

X1

Force (d31)

3 d

Figure 4.8. Cross-sections of the Banno unit cell of the 3–3 piezo-composite and vectors of external forces applied to the sample. The piezoelectric coefficients determined at the direct piezoelectric effect are given in parentheses

(Figure 4.8b) the fraction of areas 1 and 2 gradually reduce and hence d 33* will reduce. * To interpret the behaviour of the piezoelectric coefficient d 31 , we consider the response of the Banno unit cell with a force applied normal to the poling direction, as shown in Figure 4.8c. In this case, the high stiffness of the ceramic results in the majority of the force being experienced by areas 2 and 3. Since area 3 is in series in the OX3 direction with a piezo-passive polymer it does not contribute to the * piezoelectric coefficient d 31 . Hence, only area 2 contributes to the piezoelectric * coefficient d 31 of the composite. As can be seen from Figure 4.8d, as the polymer volume fraction is increased the fraction of piezoelectric contributing to d 31* (area 2

in Figure 4.8d) falls at a much quicker rate than d 33* (areas 1 and 2 in Figure 4.8d). Therefore, while d 33* decreases slowly, d 31* quickly falls to small values (see * and low d 31* results in the large Figures 4.1 and 4.2). This combination of high d 33 hydrostatic piezoelectric coefficient d h* . The reduction in permittivity also results

in large values of the hydrostatic piezoelectric coefficient g h* (see Figures 4.4 and 4.5). The optimum polymer volume fraction for high d h* is ultimately a balance of * having sufficient polymer component to lead to a reduction in d 31 , while avoiding a significant reduction is d 33* (which can occur at high polymer volume fractions,

see, for example, a range of low relative density in Figure 4.1).

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It should be noted that if the second component is air (i.e., a continuous medium with very large compliance) then all of the force is experienced by areas 1 and 2 (Figure 4.8) irrespective of the polymer volume fraction of the ceramic component, and d 33* remains constant. This is clearly not the case in some systems (see, for example, Figure 4.1) and is possibly due to a lack of interconnectivity at small ceramic volume fraction and/or due to the difficulty in poling highly porous samples. Thus, features of volume-fraction dependences of the piezoelectric coefficients d 3l* , d h* and g h* of the porous materials are closely connected with certain elements of microstructure responsible for such piezoelectric properties. It is clear that there is no unified approach to predict the effective electromechanical properties in porous materials. Therefore, the different connectivity patterns (0–3, 1–3, 2–2, 3–0, 3–1, 3–3, etc.) can be taken into consideration for an analysis of experimental data on the electromechanical properties and related parameters of the piezo-active porous materials.

References [1]

Banno H, (1987) Effects of shape and volume fraction of closed pores on dielectric, elastic, and electromechanical properties of dielectric and piezoelectric ceramics – a theoretical approach. Ceramic Bulletin 66:1332–1337 [2] Wersing W, (1986) Composite piezoelectrics for ultrasonic transducers. In: ISAF’86: Proceedings of the Sixth International Symposium on Applications of Ferroelectrics. Lehigh University, Bethlehem, PA, USA, 8–11 June, 1986. IEEE, Piscataway, pp 212–223 [3] Wersing W, Lubitz K and Mohaupt J, (1986) Dielectric, elastic and piezoelectric properties of porous PZT ceramics. Ferroelectrics 68:77–97 [4] Dunn M, (1995) Effects of grain shape anisotropy, porosity, and microcracks on the elastic and dielectric constants of polycrystalline piezoelectric ceramics. Journal of Applied Physics 78:1533–1541 [5] Gómez Alvarez-Arenas T, Montero de Espinosa Freijo F, (1996) New constitutive relations for piezoelectric composites and porous piezoelectric ceramics. Journal of the Acoustical Society of America 100:3104–3114 [6] Khoroshun LP, Maslov BP, Leshchenko PV, (1989) Prediction of Effective Properties of Piezo-active Composite Materials. Naukova Dumka, Kiev (in Russian) [7] Kahn M, (1985) Acoustic and elastic properties of PZT ceramics with anisotropic pores. Journal of the American Ceramic Society 68:623–628 [8] Ting R, (1986) Evaluation of new piezoelectric composite materials for hydrophone applications. Ferroelectrics 67:143–157 [9] Kahn M, Dalzell A, Kovel B, (1987) PZT ceramic–air composites for hydrostatic sensing. Advanced Ceramic Materials 2:836–840 [10] Lopatin SS, Lupeiko TG, (1991) Properties of the porous piezoelectric ceramic of the lead zirconate-titanate type. Neorganicheskie Materialy 27:1948–1951 (in Russian) [11] Dunn ML, Taya M, (1993) Electromechanical properties of porous piezoelectric ceramics. Journal of the American Ceramic Society 76:1697–1706 [12] Banno H, (1993) Effects of porosity on dielectric, elastic and electromechanical properties of Pb(Zr, Ti)O3 ceramics with open pores: a theoretical approach. Japanese Journal of Applied Physics Pt 1 32:4214–4217

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[13] Wu T-L, (2000) Micromechanics determining of electroelastic properties of piezoelectric materials containing voids. Materials Science and Engineering A 280:320–327 [14] Kara H, Ramesh R, Stevens R, Bowen CR, (2003) Porous PZT ceramics for receiving transducers. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 50:289–296 [15] Topolov VYu, Glushanin SV, Bowen CR, (2005) Piezoelectric response of porous ceramic and composite materials based on Pb(Zr, Ti)O3: experiment and modelling. Advances in Applied Ceramics 104:300–305 [16] Madelung O, Nelson DF (eds.) Landolt-Börnstein, (1993) Numerical Data and Functional Relationships in Science and Technology. New Series. Group III: Solid State Physics. Vol. 29, Subvol. b: Piezoelectric, Pyroelectric and Related Constants. Springer, Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest [17] Skinner DP, Newnham RE, Cross LE, (1978) Flexible composite transducers. Materials Research Bulletin 13:599–607 [18] Creedon MJ, Schulze WA, (1994) Axially distorted 3–3 piezoelectric composites for hydrophone applications. Ferroelectrics 153:333–339 [19] Shrout TR, Schulze WA, Biggers JV, (1979) Simplified fabrication of PZT / polymer composites. Materials Research Bulletin 14:1553–1559 [20] Rittenmyer K, Shrout T, Schulze WA, Newnham RE, (1982) Piezoelectric 3–3 composites. Ferroelectrics 41:189–195 [21] Bowen CR, Perry A, Lewis ACF, Kara H, (2004) Processing and properties of porous piezoelectric materials with high hydrostatic figures of merit. Journal of the European Ceramic Society 24:541–545 [22] Kara H, Perry A, Stevens R, Bowen CR, (2002) Interpenetrating PZT / polymer composites for hydrophones: models and experiments. Ferroelectrics 265:317–332 [23] Hayashi T, Sugihara S, Okazaki K, (1991) Processing of porous 3–3 PZT ceramics using capsule-free O2-HIP. Japanese Journal of Applied Physics 30:2243–2246 [24] Praveen Kumar B, Kumar HH, Kharat DK, (2005) Study on pore-forming agents in processing of porous piezoceramics. Journals of Materials Science: Materials in Electronics 16:681–686 [25] Praveenkumar B, Kumar HH, Kharat DK, (2006) Study on microstructure, piezoelectric and dielectric properties of 3–3 porous PZT composites. Journal of Materials Science: Materials in Electronics 17:515–518 [26] Levassort F, Holc J, Ringgaard E, Bove T, Kosec M, Lethiecq M, (2007) Fabrication, modelling and use of porous ceramics for ultrasonic transducer applications. Journal of Electroceramics 19:127–139 [27] Geis S, Fricke J, Lobmann P, (2002) Electrical properties of PZT aerogels. Journal of the European Ceramic Society 22:1155–1161 [28] Cracuin F, Galassi G, Roncari E, Filippi A, Guidarelli G, (1998) Electro-elastic properties of porous piezoelectric ceramics obtained by tape casting. Ferroelectrics 205:49–57 [29] Bast U, Wersing W, (1989) The influence of internal voids with 3–1 connectivity on the properties of piezoelectric ceramics prepared by a new planar process. Ferroelectrics 94:229–242 [30] Lopatin SS, Lupeiko TG, Nesterov AA, Vikhryanova IV, (1988) Properties of lead zirconate-titanate ceramic with oriented pores. Izvestiya Akademii Nauk SSSR. Neorganicheskie Materialy 24:1229–1230 (in Russian) [31] Hikita K, Yamada K, Nishioka M, Ono M, (1983) Effect of porous structure on piezoelectric properties of PZT ceramics. Japanese Journal of Applied Physics 22 (Suppl. 2):64–66

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[32] Bruggeman DAG, (1935) Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. I. Dielektrizitätskonstanten und Leitfähigkeiten der Mischkörper aus isotropen Substanzen. Annalen der Physik 24:636–664 [33] Rybjanets AN, (2002) Porous piezo-ceramic materials. Part 1. Structure, technology of processing and mathematical models. In: Proceedings of the International Scientific-Practical Conference “Basic Problems of Piezoelectric Device Designing” – Piezotechnics-2002. 17–21 September 2002. Tver State University, Tver, pp 48–55 (in Russian) [34] Rybjanets AN, (2002) Porous piezo-ceramic materials. Part 2. Electrophysical properties. In: Proceedings of the International Scientific-Practical Conference “Basic Problems of Piezoelectric Device Designing” – Piezotechnics-2002. 17–21 September 2002. Tver State University, Tver, pp 56–64 (in Russian) [35] Vorontsov AA, Glushanin SV, Topolov VYu, Panich AE, (2003) Features of the piezoelectric effect and piezoelectric sensitivity of Pb(Zr, Ti)O3-based porous ceramics and composites. In: Bol’shakov VI (ed.) Aims for Future of Engineering Science. Collection of Articles. Issue 5. Gaudeamus, Dnepropetrovsk, pp 107–114 (in Russian) [36] Rybjanets AN, Nasedkin AV, Turik AV, (2004) New micro structural design concept for polycrystalline composite materials. Integrated Ferroelectrics 63:179–182 [37] Kar-Gupta R, Venkatesh TA, (2006) Electromechanical response of porous piezoelectric materials. Acta Materialia 54:4063–4078 [38] Kar-Gupta R, Venkatesh TA, (2007) Electromechanical response of porous piezoelectric materials: Effects of porosity distribution. Applied Physics Letters 91:062904 – 3 p [39] Asai M, Takano N, Uetsuji Y, Taki K, (2007) An iterative solver applied to strongly coupled piezoelectric problems of porous Pb(Zr, Ti)O3 with nondestructive modelling of microstructure. Modelling and Simulation in Materials Science and Engineering 15:597–617 [40] Crumm AT, Jalloran JW, Silva ECN, Montero de Espinosa F, (2007) Microconfigured piezoelectric artificial materials for hydrophones. Journal of Materials Science 42:3944–3950 [41] Reilly CJ, Halloran JW, Silva ECN, (2007) Piezoelectric hydrophones from optimal design, Part I: fabrication. Journal of Materials Science 42:4824–4831 [42] Reilly CJ, Halloran JW, Silva ECN, Montero de Espinosa F, (2007) Piezoelectric hydrophones from optimal design, II: properties. Journal of Materials Science 42:4810–4814 [43] Topolov VYu, Turik AV,* (1998) Modified layered composite with high anisotropy of * piezoconstants eij and dij . Technical Physics Letters 24:441–443 [44] Topolov VYu, Turik AV, (2000) Non-monotonic concentration dependence of electromechanical properties of piezoactive 2–2 composites. Journal of Physics D: Applied Physics 33:725–737 [45] Grekov AA, Kramarov SO, Kuprienko AA, (1989) Effective properties of a transversely isotropic piezoelectric composite with cylindrical inclusions. Mechanics of Composite Materials 25:54–61 [46] Rybjantes AN, (2002) Physical properties and applications of ferroelectrically hard porous ZTS-based piezo-ceramics. In: Proceedings of the International ScientificPractical Conference “Basic Problems of Piezoelectric Device Designing” – Piezotechnics-2002. 17–21 September 2002. Tver State University, Tver, pp 65–72 (in Russian)

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[47] Sitalo EI, Smotrakov VG, Eremkin VV, Ivanov PV, Tsikhotsky ES, (2002) Technology of processing and properties of the porous piezo-ceramic. In: Proceedings of the International Scientific-Practical Conference “Basic Problems of Piezoelectric Device Designing” – Piezotechnics-2002. 17–21 September 2002. Tver State University, Tver, pp 305–308 (in Russian)

5 Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectric Single Crystals

5.1 Relaxor-ferroelectric Solid Solutions and Engineering In the last decades, crystallographic and compositional engineering has become an important tool [1] in the development of advanced piezo-active materials. An important group of such materials comprises perovskite-type solid solutions with a general formula (1 – x)Pb(B1, B2)O3 – xPbTiO3. The complex perovskites Pb(B1, B2)O3 are disordered compounds containing ions of metals from the following groups: B1 = Mg, Zn, Ni, Fe, Sc, Yb, and In (low valence) and B2 = Nb, Ta and W (high valence) [1–3]. The combination of metal ions with low and high valences results in a series of physical properties that distinguish Pb(B1, B2)O3 from “normal” (ordered) perovskite-type ferroelectrics such as PbTiO3. The complex perovskites Pb(B1, B2)O3 exhibit a broad and frequency-dispersive dielectric maxima and contain polar nanoregions (with ferro- or antiferroelectric ordering) in a non-polar phase over a wide temperature range [2, 3]. They are characterised by the relaxation dielectric polarisation [3] and are called relaxors [2] or ferroelectric relaxors. The solid solutions of (1 – x)Pb(B1, B2)O3 – xPbTiO3 combine physical properties of the relaxor-type and “normal” ferroelectric components, and excellent properties are often observed near the MPB [2–4]. To achieve a high piezoelectric activity, the relaxor-ferroelectric solid solutions are engineered by compositional adjustment with a corresponding decrease in Curie temperature [4]. Of particular interest for piezotechnical applications are domain-engineered / heterophase single crystals of (1 – x)Pb(Mg1/3Nb2/3)O3 – xPbTiO3 (PMN–xPT) and (1 – y)Pb(Zn1/3Nb2/3)O3 – yPbTiO3 (PZN–yPT) with molar concentrations x and y, respectively, taken in the vicinity of the MPB. In these single crystals engineered non-180q domain structures [5, 6], intermediate ferroelectric phases [7–10] and domain switching and rotation [11–13] are observed. These single crystals poled along certain crystallographic directions, for example, along [001] or [011] of the perovskite unit cell [14–16], exhibit very high piezoelectric activity and significant electromechanical coupling. In [001]-poled polydomain PMN–0.33PT single

124

Electromechanical Properties in Composites Based on Ferroelectrics

crystals at room temperature, the piezoelectric coefficient d33 and electromechanical coupling factor k33 can exceed 2800. pC / N and 90%, respectively [14]. According to data in ref. [17], the piezoelectric coefficient d33 in [001] poled heterophase PZN–0.08PT single crystals can attain values up to 12 000 pC / N at the electric-field-induced phase transition. In [001]-oriented PZN–0.08PT single crystals [13] d33 becomes as large as 16 350 pC / N under the electric field E | Ec and d33 = 2500 pC / N at E > Ec, where Ec is the coercive field (a threshold field for a non-180q domain-wall motion). High strains (over 0.5%) [1, 4] are achieved in the [001]-poled polydomain single crystals with compositions of the 3m point group under an electric field E || [001]; although the spontaneous polarisation vectors in domains are parallel to the [111], [ 1 11], [1 1 1], and [ 1 1 1] directions [5, 18]. Some interconnections between the structure and properties in the PMN–xPT and PZN–yPT solid solutions have been reviewed in ref. [19]. The high piezoelectric activity and large electromechanical coupling in these solid solutions are closely connected with the intermediate monoclinic phases [11, 12, 19] that occur close to the MPB. The high piezoelectric activity in these solid solutions can be driven by a polarisation rotation induced by the electric field E [11, 12], and this rotation between the single-domain states in the tetragonal (4mm) and rhombohedral (3m) phases can be implemented in different ways. The corresponding rotation paths found in ref. [11] are the minimum free-energy paths and are associated with different intermediate ferroelectric phases that would be observed in the studied solid solutions in certain ranges of molar concentration x or y, electric field E and temperature T. The aforementioned d33 and k33 values are considerably more than those related to the conventional ferroelectric ceramics based on Pb(Zr1–zTiz)O3 [20–22], and the outstanding performance of the PMN–xPT and PZN–yPT single crystals could lead to a revolution in electromechanical transducer technology [23, 24] and piezoelectric materials science. It should be added that the PMN–xPT and PZN– yPT single crystals with domain-engineered structures exhibit relatively stable electromechanical properties at room temperature. Full sets of the electromechanical constants of these single crystals were measured on samples (crystal cuts) poled along the [001] (Table 2.2) or [011] direction [16] of the perovskite unit cell. The outstanding electromechanical properties in these single crystals near the MPB suggest that these materials would be good candidates for modern piezoelectric sensors, large-strain actuators, highly sensitive medical ultrasonic transducers, hydrophones, ultrasonic imaging devices, and other piezotechnical devices [4, 24]. In the sections below we analyse the possibilities and advantages of using the relaxor-ferroelectric single crystals as components of the piezo-composites. We demonstrate how these single crystals improve performance of the novel piezocomposites, as compared with the performance of the conventional ferroelectric ceramic / polymer composites, and promote the formation of a series of nonmonotonic volume-fraction dependences of the effective parameters.

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125

5.2 0–3 Single Crystal / Ceramic Composites In this section we consider single crystal / ceramic composites with 0–3 connectivity, where the PMN–xPT or PZN–yPT single crystals are used as isolated inclusions with high piezoelectric activity. The motivation to study these composites is due to a number of factors: (i) The single crystal / ceramic composites based on PbTiO3 have been previously examined [25], and features of their effective electromechanical properties were studied [26, 27] taking into account the composition, domain structure and microstructure. (ii) The 0–3 Pb(Ni1/3Nb2/3)O3 – PbTiO3 single crystal / Pb(Ni1/3Nb2/3)O3 – Pb(Zr, Ti)O3 ceramic composites containing the relaxor-ferroelectric components were manufactured by advanced hybrid sintering that combines microwave heating and hot pressing [28] and show good characteristics. These composites have a good interface state between the single-crystal inclusions and the ceramic matrix so that neither cracks nor re-solidification of melted material would be observed at the boundaries separating the components [28]. (iii) The modified PbTiO3 ceramics are good candidates to be used as matrix components in the 0–3 composites. PbTiO3 ceramics are characterised [29, 30] by moderate piezoelectric activity (e.g., the piezoelectric coefficient d33 can be about 50 times less than that of the PMN–xPT or PZN–yPT single crystals) with a large anisotropy of the piezoelectric coefficients (i.e., d33 / | d31| t 10) or electromechanical coupling factors (i.e., k33 /| k31| t 10 or kt / | kp | t 10). The piezoelectric coefficients e3j of these ceramics and single crystals are generally of equal order of magnitude, but the piezoelectric coefficients e31 differ in sign (see Tables 2.1 and 2.2). (iv) An alternative to the relaxor-ferroelectric single crystals could be textured ceramics [31] wherein a high degree of crystallographic texture improves the piezoelectric properties. According to data [31], textured PMN–xPT ceramics (0.25 < x < 0.35) are processed by the homo-template grain growth, and a favourable orientation of templates is achieved during the tape-casting process. An important feature of such ceramics is the small influence of connectivity and spatial distribution of the phases on the effective electromechanical properties. Our further consideration is carried out within the framework of the model of a 0–3 composite with spheroidal inclusions (Section 3.3). Now, the single-crystal inclusions are uniformly aligned in the ceramic matrix and form a periodic structure. The main crystallographic axes X, Y and Z of the single crystals are parallel to the co-ordinate axes OX1, OX2 and OX3, respectively. The semi-axes of the inclusions are a1, a2 = a1 and a3 (Figure 3.7). The single-crystal inclusions are poled along Z || [001] || OX3, and the ceramic matrix is characterised by the remanent polarisation vector being also parallel to the OX3 axis. To avoid confusion between the volume fractions of the inclusions in different composite groups, hereafter we denote the volume fraction of the single-crystal inclusions with mcr. The volume fraction of the ceramic matrix surrounding these inclusions equals 1 – mcr.

126

Electromechanical Properties in Composites Based on Ferroelectrics

The effective properties of the 0–3 single crystal / ceramic composite [32] depend on the volume fraction mcr and the aspect ratio U = a1 / a3 of the inclusions and are determined by means of the EFM (see Equations 3.19 and 3.20). Below we show how the electromechanical constants of the piezo-active components promote a non-monotonic volume-fraction dependence of the effective piezoelectric * * coefficients e3* j and d 3* j , changes in sgn e31 and sgn d 31 , along with an increase in the hydrostatic piezoelectric coefficients * * * * + 2 e31 and d h* = d 33 + 2 d 31 eh* = e33

(5.1)

To find a composite with optimum parameters from Equations 5.1, different combinations of the PMN–xPT or PZN–yPT single crystals (see Table 2.2) and PbTiO3-type ceramics (see Table 2.1) were analysed on varying the volume fraction mcr from 0 to 0.5 and the aspect ratio U from 0.1 to 10. An example of nonmonotonic behaviour of the effective parameters of the 0–3 composite (Figure 5.1) 5

4

e*31

3

1

1: U = 0.1 ... 5: U = 0.5 ... 10: U = 1 11: U = 5 12: U =10

2 3

2

4

1

0

12

-1 0,1

0,2

0,3

a

0,4

5 6 7 8 10 9 11 0,5

mcr

Figure 5.1. Volume-fraction dependences of effective piezoelectric coefficients e3* j (in C / m2) and d 3* j (in pC / N) and normalised hydrostatic piezoelectric coefficients eh* / eh(FC ) = * * ( 2) ( 2) * * ( 2) ( 2) ( e33 + 2 e31 ) / ( e33 + 2 e31 ) and d h* / d h(FC ) = ( d 33 + 2 d 31 ) / ( d 33 + 2 d 31 ), which have been calculated for the PZN–0.08PT single crystal / (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic composite with 0–3 connectivity. In graphs (a), (b), (e), and (g), curves 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 are related to the aspect ratios U = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1, 5, and 10, respectively. In graph (c), curves 1, 2, and 3 are related to U = 1, 5, and 10, respectively. In graphs (d) and (i), curves 1, 2, 3, 4, 5, and 6 are related to U = 0.1, 0.2, 0.3, 0.4, 0.5, and 1, respectively. In graphs (f) and (h), curves 1, 2, 3, and 4 are related to U = 0.9, 1, 5, and 10, respectively (reprinted from Topolov and Kamlah [32], with permission from IOP Publishing)

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

9,0 8,5

e*33

8,0 7,5

127

6 5 4 3

1: U = 0.1 2: U = 0.2 3: U = 0.3 4: U = 0.4 5: U = 0.5 6: U = 0.6 7: U = 0.7 8: U = 0.8

2

8 7

1

7,0 6,5 6,0 5,5 0,1

0,2

0,3

0,4

0,5

mcr

b 1 9,0

2

e*33

8,5

8,0

3

1: U = 1 2: U = 5 3: U = 10

7,5

7,0

6,5 0,1

0,2

0,3

c

0,4

0,5

mcr

Figure 5.1 (continued)

is considered [32] when the PZN–0.08PT single-crystal inclusions are surrounded by poled (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic. This ceramic is characterised ( 2) ( 2) [29] by a large anisotropy of the piezoelectric coefficients e3( 2j ) (i.e., e33 / e31 | 14.2, ( 2) ( 2) see Table 2.1) and d 3( 2j ) (i.e., d 33 / d 31 | –11.6).1 It turns out that the large

____________ 1

Following notations used for the 0–3 composite in Section 3.3, we denote the electromechanical constants of the inclusion and the matrix with superscripts “(1)” and “(2)”, respectively.

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Electromechanical Properties in Composites Based on Ferroelectrics

2,4 2,2

e*h / e

(FC) h

2,0 1,8

1

1: U = 0.1 2: U = 0.2 3: U = 0.3 4: U = 0.4 5: U = 0.5 6: U = 1

2 3

1,6

4 5

1,4 1,2

6

1,0 0,8 0,1

0,2

0,3

0,4

0,5

mcr

d 20

1 10 0

d*31

2 -10 -20 -30 -40 -50

1: U = 0.1 2: U = 0.2 3: U = 0.3 4: U = 0.4 5: U = 0.5 6: U = 0.6 7: U = 0.7 8: U = 0.8

3 4 5 6 7 8 0,1

0,2

0,3

e

0,4

0,5

mcr

Figure 5.1 (continued)

anisotropy of the piezoelectric properties predetermines a non-monotonic volumefraction behaviour of both groups of effective piezoelectric coefficients, e3* j (Figures 5.1a and b) and d 3* j (Figures 5.1e and g). The piezoelectric coefficient * (mcr) increases considerably at U d 0.2 (Figure 5.1a), whereas the piezoelectric e31 * coefficient e33 (mcr) undergoes a small increase in a restricted range of mcr (Figure

5.1b). It is seen that the more distinct changes in the e3* j and d 3* j values take place at 0.1 d U < 0.5 (curves 1–5 in Figures 5.1a and b), as the shape of the inclusion is transformed from a needle (U << 1) to a sphere (U = 1). In this case, the electric

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

129

0

-10

d*31

-20

-30

-40

1: U = 0.9 2: U = 1 3: U = 5 4: U = 10

1 4 2 3

-50 0,1

0,2

0,3

f 130 120 110 100

d*33

90 80

0,4

0,5

mcr 8 7 6 5 4

1: U = 0.1 2: U = 0.2 3: U = 0.3 4: U = 0.4 5: U = 0.5 6: U = 0.6 7: U = 0.7 8: U = 0.8

3 2

70 60 50

1

40 30 0,1

0,2

0,3

0,4

0,5

mcr

g Figure 5.1 (continued)

depolarisation field of the inclusion becomes more intensive and, simultaneously, a certain relief of the internal mechanical stress occurs. Increasing the aspect ratio U at mcr = const results in moderate changes in the e3* j values (Figures 5.1a–c). The studied 0–3 composite exhibits a unique e3* j (U) dependence at mcr = const: * * for this material the e31 changes monotonically, whereas e33 changes nonmonotonically (compare graphs in Figures 5.1a and c). This behaviour of the piezoelectric coefficients of 0–3 composites was initially reported in ref. [32]. The * * unusual increase in e31 and decrease in e33 at small mcr values and 0 < U d 0.2 (see

curves 1 and 2 in Figures 5.1a and b) is explained by an electromechanical

130

Electromechanical Properties in Composites Based on Ferroelectrics 2 1 130

3

120

4

110

d*33

100 90 80

1: U = 0.9 2: U = 1 3: U = 5 4: U = 10

70 60 50 0,1

0,2

0,3

h 1,8 1,7

d*h / d

(FC) h

1,6 1,5

0,4

0,5

mcr 1

1: U = 0.1 2: U = 0.2 3: U = 0.3 4: U = 0.4 5: U = 0.5 6: U = 1

2

1,4

3

1,3

4

1,2

5

1,1 1,0

6

0,9 0,1

0,2

0,3

i

0,4

0,5

mcr

Figure 5.1 (continued)

interaction between the highly piezo-active and highly anisotropic components. Based on condition 2.13, we can represent the piezoelectric coefficients e3* j of the studied composite as follows: * * * * * * *E *E = d 31 ( c11*E + c12*E ) + d 33 and e33 = 2 d 31 e31 c13 c13*E + d 33 c33

(5.2)

* * In Equations 5.2 the first terms are small due to condition d 33 >> _ d 31 _, which is (2) (2) directly associated with the inequality d 33 >> | d 31 | being correct for the modified

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

131

* PbTiO3 ceramic [29]. The term d 33 c13*E rises with increasing mcr, since a condition * c13(1),E > c13( 2),E holds, and a slight decrease in d 33 (see curves 1 and 2 in Figure 5.1g)

is compensated due to the ratio c13(1),E / c13( 2),E > 4 (see Tables 2.1 and 2.2). In contrast * *E to these relations, the term d 33 from Equations 5.2 decreases on increasing mcr c33 * due to a decrease in d 33 and an inequality c33(1),E < c13( 2),E . A further increase of the aspect ratio U and the volume fraction mcr of inclusions changes the balance of the * above-mentioned terms in Equations 5.2: as a consequence, a change in sgn e31 (see

curves 8–12 in Figure 5.1a) is observed and the shape of the e3* j curves changes (Figures 5.1a and b). As with the piezoelectric coefficients d 3* j (mcr) (Figures 5.1e and g), the volume-fraction dependences e3* j (mcr) are also non-monotonic and a * change in sgn e31 takes place at 0.1 d U d 0.2 (Figures 5.1a and b). It is known that

prolate-shaped inclusions cause a slight electric depolarisation field and an intensive mechanical stress field, and this combination of the internal fields promotes the non-monotonic volume-fraction dependences of various piezoelectric coefficients. It is of practical interest to study how the electromechanical interaction of the highly piezo-active and highly anisotropic components influences the hydrostatic response of the 0–3 composite. The next example is concerned with the effective hydrostatic piezoelectric coefficients from Equations 5.1. For a better comparison with the effective properties, the normalised hydrostatic piezoelectric coefficients eh* / eh(FC ) and d h* / d h(FC ) (Figures 5.1d and i) are calculated, where hydrostatic ( 2) ( 2) parameters of the ceramic component are eh(FC ) = e33 + 2 e31 = eh* (0) and d h(FC ) =

d 33( 2) + 2 d 31( 2) = d h* (0). As follows from data shown in Figures 5.1a, b, e, and g, the prolate inclusions with 0.1 d U d 0.2 lead to a considerable increase in both * * * * (mcr) and d 31 (mcr) as well as a small decrease in both e33 (mcr) and d 33 (mcr) at e31 mcr < 0.2. This volume-fraction dependence promotes a distinct enhancement of the effective hydrostatic piezoelectric properties (see curves 1 and 2 in Figures 5.1d and i) as compared to those of the ceramic component. In the vicinity of max d h* (mcr) at U = 0.1, the normalised hydrostatic piezoelectric coefficients are characterised by a relation eh* / eh(FC ) | d h* / d h(FC ) | 2. This result can be compared with data on the hydrostatic piezoelectric coefficient eh* calculated for the 1–3 PZT-4 ceramic / epoxy composite [33]: the maximum value of eh* / eh(FC ) for this 1–3 composite does not exceed 2.7 and is attained at volume fractions of the piezoelectric ceramic rods in a range of 0.8 < m < 0.9. For purely geometrical reasons, however, values of m > S / 4 for the volume fractions of cylindrical rods in a square arrangement are impossible. As the volume fraction m approaches a realistic value of 0.5, eh* / eh(FC ) decreases and becomes

132

Electromechanical Properties in Composites Based on Ferroelectrics

smaller than that calculated for the studied 0–3 PZN–0.08PT-based composite with 0.1 d U d 0.2 (see curves 1 and 2 in Figure 5.1d). Thus, despite the fairly small value of the hydrostatic piezoelectric coefficient (1) (1) (1) of the PZN–0.08PT single crystal ( e33 + 2 e31 | | e31 |, see data from Table 2.2), this component can play an important role in forming the remarkable hydrostatic piezoelectric properties in the 0–3 single crystal / ceramic composite. In the next section we analyse the role of this material in forming the piezoelectric response in the 1–3-type composite.

5.3 0–1–3 Single Crystal / Ceramic / Polymer Composites In this section we consider 1–3-type composites that consist of piezo-active 0–3 single crystal / ceramic rods and a piezo-passive polymer matrix (Figure 5.2). The cylindrical extended rods are uniformly distributed in the matrix so that a square arrangement in the (X1OX2) plane takes place. The volume fraction of the rods and the surrounding matrix are m and 1 – m, respectively, whereas the single-crystal inclusions in each rod are characterised by a volume fraction mcr. It is assumed that the largest semi-axis a3 of each inclusion is considerably less than the radius of the rod and these inclusions are regularly distributed in the rod, i.e., centres of symmetry of the inclusions are located in the sites of a simple tetragonal lattice. The main crystallographic axes x, y and z in each single-crystal inclusion are parallel to the co-ordinate axes OX1, OX2 and OX3, respectively. The composite as a whole has a remanent polarisation vector Pr || OX3 and is described by 0–1–3 connectivity. The electromechanical properties of the rod are determined by means of the EFM, as described for the 0–3 composite (see Sections 3.3 and 5.2), and

Figure 5.2. Diagram of the 0–1–3 composite containing single-crystal (n = 1), ceramic (n = 2) and polymer (n = 3) components. (X1X2X3) is the rectangular co-ordinate system, m and mcr denote the volume fractions of rods and spheroidal inclusions in them, respectively. Directions of the spontaneous polarisation vector Ps || OX3 in the single-crystal inclusion and of the remanent polarisation vector PR || OX3 in the surrounding ceramic medium are indicated by arrows

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

133

*E characterised by the elastic moduli cab , piezoelectric coefficients eij* and dielectric

permittivities H kp*[ . The effective electromechanical properties )** in the 0–1–3 composite2 are determined using the same EFM and the Eshelby tensor [34] for the **E cylindrical inclusion. The elastic moduli cab (m, mcr, U), piezoelectric coefficients eij** (m, mcr, U) and dielectric permittivities H kp**[ (m, mcr, U) of this composite are

regarded as functions of the volume fractions m and mcr and the aspect ratio U. The 0–1–3 PZN–0.08PT / (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic / polymer composite is an interesting example of a piezo-active material that is characterised by a non-monotonic behaviour of the piezoelectric coefficients and changes in their signs (Figure 5.3). Based on the results [32], two kinds of non-monotonic behaviour can be described. First, both e3**j and d 3**j non-monotonically depend on the volume fraction mcr (see Figures 5.3a, d, g, and j) at m = const and U = 0.1. Such a behaviour is observed due to the non-monotonic dependences of the piezoelectric coefficients e3* j (mcr) and d 3* j (mcr) of the related 0–3 single crystal / ceramic composite (Figures 5.1a, b, e, and g). Second, the piezoelectric coefficient ** of the 0–1–3 composite passes a minimum (Figures 5.3g, h and i) when d 31 varying the volume fraction m of the rods at constant parameters mcr and U of the single-crystal inclusions. A volume-fraction dependence similar to d 31** (m, mcr, U) at mcr = const and U = const cannot be found among those determined for the more conventional 1–3 Pb(Zr, Ti)O3-type ceramic / polymer composites [35–37]. In addition, the piezoelectric coefficient d 33** (m, mcr, U) of the studied 0–1–3 composite at mcr = const and U = const changes monotonically and attains saturation at m | 0.2 (Figures 5.3j–l). Such a configuration of the d 33** curves is typical of the 1–3 ceramic / polymer composites [34–36]. The unusual behaviour of d 31** (m, mcr, U) at mcr = const and U = const (Figures 5.3g–i) may be explained [32] by taking into account the influence of the elastic and piezoelectric anisotropy of the two piezo-active components on the effective piezoelectric activity of the composite. It is important to underline, for example, the significant differences in the c11( n),E / c13( n ),E or c33( n ),E / c13( n ),E ratios where superscripts n = 1 and 2 are related to the single-crystal and ceramic components, respectively. (1) ( 2) = –sgn e31 holds for the chosen single-crystal and Furthermore, an equality sgn e31 ceramic components. The shape of curves 1–3 in Figures 5.3g–i depends on the aspect ratio U of the inclusions in the rod. At U = 0.1, the piezoelectric coefficient ** has a minimum and passes zero (Figure 5.3g). As mcr and U increase, a d 31 saturation trend of d 31** (m, mcr, U) < 0 is observed (Figures 5.3g and h). It follows from the results in ref. [32] that the relaxor-ferroelectric component ** has a significant influence on e31 (m, 0.4, 1) (see curves 3–5 in Figures 5.3b and c). ____________ 2

In this section we actually consider a “composite in a composite“ and use two asterisks to denote the effective electromechanical properties of the studied 0–1–3 composite.

134

Electromechanical Properties in Composites Based on Ferroelectrics

5 3

5

2 4

e**31

3

2

1

1

U = 0.1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

4

0 0,2

0,4

0,6

0,8

1,0

m a 0,4

1

0,3

2

0,2 0,1

3

e**31

0,0 -0,1 -0,2 -0,3 -0,4

U=1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

4

-0,5

5 0,2

0,4

0,6

0,8

1,0

m b Figure 5.3. Volume-fraction dependences of the effective piezoelectric coefficients e3**j (in C / m2) and d 3**j (in pC / N) that have been calculated for the 0–1–3 PZN–0.08PT / (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic / araldite composite. In all graphs, curves 1, 2, 3, 4, and 5 are related to volume fractions of the single-crystal inclusions mcr = 0.1, 0.2, 0.3, 0.4, and 0.5, respectively. The aspect ratios of these inclusions are given as follows: U = 0.1 (graphs (a), (d), (g), and (j)), U = 1 (graphs (b), (e), (h), and (k)), and U = 10 (graphs (c), (f), (i), and (l)) (reprinted from Topolov and Kamlah [32], with permission from IOP Publishing)

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

1

0,2

2

0,0

e**31

-0,2

-0,4

135

U = 10 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

3

4 -0,6

-0,8

5 0,2

0,4

0,6

0,8

1,0

m c

5 7

4

6

3

e**33

5 4 3 2

U = 0.1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

1

2

1 0 0,2

0,4

0,6

m d Figure 5.3 (continued)

0,8

1,0

136

Electromechanical Properties in Composites Based on Ferroelectrics

5 4 3 2 1

9 8 7

e**33

6 5 4 3

U=1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

2 1 0 0,2

0,4

0,6

0,8

1,0

m e

5 4 3 2 1

8 7 6

e**33

5 4 3

U = 10 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

2 1 0 0,2

0,4

0,6

m f Figure 5.3 (continued)

0,8

1,0

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

3 4 2

20

5

15

d**31

10

5

137

U = 0.1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

1

0

-5

-10 0,2

0,4

0,6

0,8

1,0

m g 0 -5 -10

U=1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

1 2

-15

d**31

-20

3

-25 -30

4

-35 -40 -45

5

-50 0,2

0,4

0,6

m h Figure 5.3 (continued)

0,8

1,0

138

Electromechanical Properties in Composites Based on Ferroelectrics 0

-10

U = 10 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

1 2

d**31

-20

3 -30

4 -40

5

-50 0,2

0,4

0,6

0,8

1,0

m i

5 40

1 4 2 3

d**33

30

20

U = 0.1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

10

0 0,2

0,4

0,6

m j Figure 5.3 (continued)

0,8

1,0

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

139

5 130 120 110

4

100 90

3

d**33

80 70

2

60

1

50 40 30 20 10

U=1 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

0 0,2

0,4

0,6

0,8

1,0

m k

5 110 100

4

90 80

3

d**33

70

2 1

60 50 40 30 20 10

U = 10 1: mcr = 0.1 2: mcr = 0.2 3: mcr = 0.3 4: mcr = 0.4 5: mcr = 0.5

0 0,2

0,4

0,6

m l Figure 5.3 (continued)

0,8

1,0

140

Electromechanical Properties in Composites Based on Ferroelectrics

** At mcr t 0.4 the e31 curves assume negative values at large volume fractions m. The above-given examples of the volume-fraction dependence of the piezoelectric coefficients e3**j (m, mcr, U) and d 3**j (m, mcr, U) at mcr = const and U = const suggest

that the 0–1–3 composite exhibits a piezoelectric anisotropy that can be varied in a wide range. To describe specific features of piezoelectric anisotropy, one can ** ** , d 33** / d 31** and k 33** / k 31** and we will consider introduce the following ratios: e33 / e31 the PZN–0.08PT / (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic / polymer composite with prolate single-crystal inclusions as rods (U = 0.1). The criteria for a large piezoelectric anisotropy of the composite are written in the following form [32]: ** ** | e33 / e31 | t 10

(5.3)

| d 33** / d 31** | t 10

(5.4)

| k 33** / k 31** | t 10

(5.5)

Table 5.1 contains data on the volume fractions mcr and m for which conditions 5.3–5.5 hold true. The volume fraction mcr of the single-crystal inclusions in the rods is varied from 0.1 to 0.5. An intersection m1,opt d m d m2,opt is related to the optimal volume-fraction ranges, where conditions 5.3–5.5 hold simultaneously. The width of this intersection is defined as 'm = m2,opt – m1,opt. It is found that 'm strongly depends on the elastic properties of the polymer component and on the volume fraction mcr of the single-crystal component (Table 5.1). The softer polymer component, elastomer, leads to a relatively wide range of optimal volume fractions mi,opt: at 0.1 d mcr d 0.5 values 'm | 0.2 – 0.3 are attained. The relatively wide ranges of m obeying condition 5.4 (Table 5.1) illustrate the significant influence of the “rods–matrix” composite structure (Figure 5.2, right) on the volume-fraction behaviour of the piezoelectric coefficients e3**j : this behaviour is similar to that known for the 1–3 ferroelectric ceramic / polymer composites [35–37]. The wider ranges of m obeying condition 5.5 in comparison with those related to condition 5.3 (see Table 5.1) are a result of the prolate shape of the inclusions (Figure 5.2, left). Such inclusion shapes, along with differences in the elastic moduli of the single-crystal and ceramic components, strongly influence the elastic properties of the 0–3 composite (see Section 5.2). An important consequence is that the piezoelectric anisotropy of the 0–1–3 composite depends *E *E on the s11*E / s33 ratio, where sab are the elastic compliances of the 0–3 single crystal / ceramic composite. Finally, the results given in Table 5.1 represent another example of the non-monotonic volume-fraction behaviour of the effective parameters. As is shown for the elastomer-containing composite, the calculated dependence of the range width 'm on the volume fraction mcr first decreases with increasing mcr and then increases again. This behaviour is a result of the unusual

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

141

Table 5.1. Ranges of the volume fraction m of rods (U = 0.1, Figure 5.2) in the 0–1–3 PZN– 0.08PT / (Pb0.9625La0.025)(Ti0.99Mn0.01)O3 (I) ceramic / polymer composite (reprinted from Topolov and Kamlah [32], with permission from IOP Publishing) Polymer Araldite

Volume fraction mcr 0.1 0.2 0.3 0.4 0.5

Polyurethane

0.1 0.2 0.3 0.4 0.5

Elastomer

0.1 0.2 0.3 0.4 0.5 Polymer

Araldite

Polyurethane

Elastomer

Ranges of the volume fraction m obeying conditions 5.3 | 5.4 | 5.5 0.336 d m d 0.669 0.198 d m d 0.809 0 < m d 0.413 --0.196 d m d 0.398 0.131 d m d 0.519 --0.160 d m d 0.335 0.113 d m d 0.433 --0.185 d m d 0.387 0.129 d m d 0.494 --0.281 d m d 0.584 0.187 d m d 0.716 0.357 d m d 0.681 0 < m d 0.676 0.162 d m d 0.861 0.221 d m d 0.419 0 < m d 0.333 0.121 d m d 0.598 0.187 d m d 0.358 0 < m d 0.127 0.110 d m d 0.510 0.214 d m d 0.410 0 < m d 0.154 0.125 d m d 0.573 0.312 d m d 0.604 0 < m d 0.346 0.169 d m d 0.781 0.461 d m d 0.744 0 < m d 0.918 0.058 d m d 0.968 0.313 d m d 0.501 0 < m d 0.832 0.057 d m d 0.858 0.276 d m d 0.441 0 < m d 0.780 0.058 d m d 0.800 0.310 d m d 0.496 0 < m d 0.787 0.064 d m d 0.840 0.422 d m d 0.680 0 < m d 0.836 0.075 d m d 0.939 Range of the Volume 'm = m2,opt – m1,opt optimal volume fraction fraction mcr m1,opt d m d m2,opt 0.1 0.077 0.336 d m d 0.413 0.2 ----0.3 ----0.4 ----0.5 ----0.1 0.319 0.357 d m d 0.676 0.2 0.112 0.221 d m d 0.333 0.3 ----0.4 ----0.5 0.034 0.312 d m d 0.346 0.1 0.283 0.461 d m d 0.744 0.2 0.188 0.313 d m d 0.501 0.3 0.167 0.274 d m d 0.441 0.4 0.186 0.310 d m d 0.496 0.5 0.268 0.422 d m d 0.680

volume-fraction dependences of the piezoelectric coefficients e3* j (mcr) and d 3* j (mcr) of the related 0–3 composite (Figures 5.1a, b, e, and g) and by the following relations between the elastic moduli of the single-crystal (n = 1) and polymer (n = 3) components in the 1–0–3 composite: c11(1),E / c12(1),E | c11(1),E / c13(1),E | c11(3) / c12(3) and (1),E ( 3) | 1 at c11(3) = c33 . c11(1),E / c33

142

Electromechanical Properties in Composites Based on Ferroelectrics

Thus, as follows from the results analysed in this section, the piezoelectric properties in the 0–1–3 composites containing highly piezo-active (relaxorferroelectric) and highly anisotropic (ferroelectric ceramic) PbTiO3-based components have no analogues among the properties in any existing 1–3-type piezo-composites. The important role of the relaxor-ferroelectric single-crystal component as well as the anisotropy of both the elastic and piezoelectric properties in the single-crystal and ceramic components has been emphasised. Studying the effect of the piezo-passive polymer component on the piezoelectric anisotropy factors from formulae 5.2–5.4 enables us to conclude that the softer polymer is preferred for creating a highly anisotropic 0–1–3 composite. Other representatives of the 1–3-type composites based on the [001]-poled relaxor-ferroelectric single crystals and the effective electromechanical properties will be considered in Sections 5.4 and 5.5.

5.4 1–3 Single Crystal / Polymer Composites Examples of the application of the relaxor-ferroelectric single-crystal components of the advanced 1–3 composites were considered in experimental papers [38, 39]. In both studies the dice-and-fill technique [40, 41] was used to manufacture the 1–3 single crystal / polymer composites with various volume fractions of the PZN– 0.08PT [38] or PMN–0.33PT [39] single crystals. Growth of high-quality PMN– xPT [42] and PZN–yPT [43] single crystals and the formation of engineereddomain structures [5, 6] opens up new possibilities for creating piezo-composites based on these single crystals. With regards to 1–3 composites, it is of interest to analyse a variety of possible single crystal / polymer composites. In this section we consider several volume-fraction dependences of the effective parameters of the 1– 3 PMN–xPT / araldite and PZN–yPT / araldite composites with single-crystal compositions chosen near the MPB. It is assumed that the 1–3 composite sample has been poled under the electric field E || OX3 and contains a system of single-crystal rods in the form of rectangular parallelepipeds (Figure 5.4). These parallelepipeds are continuous in the OX3 direction and their lateral faces are parallel to the (X1OX3) and (X2OX3) planes. The main crystallographic axes X, Y and Z of the single crystals are oriented as follows: X || OX1, Y || OX2 and Z || OX3. The single-crystal rods are regularly distributed in the polymer matrix. A full set of effective electromechanical constants of the 1–3 composite shown in Figure 5.4 is determined using the matrix method [44–46] and the 9 u 9 matrices from Equation 3.1. These matrices are now related to the single-crystal (n = 1) and polymer (n = 2) components. Figure 5.4 shows the 1–3 structure with planar microgeometry used for the determination of the effective properties based on averaging the aforementioned electromechanical constants of the components and taking into account boundary conditions at xi = const (i = 1 and 2). These boundary conditions imply [32, 45, 46] a continuity of normal components of the mechanical stress Viv, tangential components of the mechanical strain [rt, normal components of the electric displacement Di, and tangential components of the electric field Ef. An averaging

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

143

Figure 5.4. Banno unit cell of the 1–3 single crystal / polymer composite. The spontaneous polarisation vector of the single crystal (see the shaded parallelepiped) is denoted by an arrow. (X1X2X3) is the rectangular co-ordinate system. m and 1 – m are volume fractions of the single-crystal and polymer components, respectively

procedure is implemented in a long-wave approximation that is valid when a wavelength of an external acoustic field is much longer than the thickness of the separate rod in the composite. The matrix || C* || of the effective electromechanical properties in the 1–3 composite is expressed by Equation 3.2, where m is the volume fraction of the single-crystal rods and || M || is the matrix related to the boundary conditions at xi = const (i = 1 and 2). According to the results in refs. [44, 45], || M || depends on elements of || C(n) ||, and the structure of the || C* || matrix is similar to that shown in Equation 3.1. The matrix elements of || C* || represent volume-fraction dependences of the effective elastic compliances *E (m), piezoelectric coefficients dij* (m) and dielectric permittivities H *ppV (m) of sab the 1–3 composite. Based on these effective constants and Equations 2.12 – 2.15, we evaluate the piezoelectric coefficients eij* (m), g ij* (m) and hij* (m) as well as other effective parameters in a range of 0 < m < 1. We will now consider the maxima of different effective parameters of the 1–3 composites [47] based on the PMN–xPT or PZN–yPT single crystals where x = 0.30, 0.33, 0.42 and y = 0.045, 0.07, 0.08. Some results on maximum values are shown in Table 5.2 and Figures 5.5 and 5.6 (curve 1 in all graphs). Among the aforementioned single-crystal components, of particular interest are PMN–0.42PT and PMN–0.33PT. In the 1–3 PMN–0.42PT / araldite composite the largest * maximum values are attained for the following parameters [47]: max g 33 (m) = (1) 19.3 g 33 , max g h* (m) = 23.3 g h(1)

* (1) , where the and max h33 (m) = 1.08 h33

(1) (1) , etc. are listed in Table 5.3. The piezoelectric coefficients g 33 , g h(1) , h33 piezoelectric response of the 1–3 PMN–0.42PT / araldite composite is closely

144

Electromechanical Properties in Composites Based on Ferroelectrics

* * Table 5.2. Maximum values of effective parameters g 33 (in mV m / N), h33 (in 108 V / m),

–12 –1 –12 –1 * 2 * 2 d h* (in pC / N), g h* (in mV m / N), k t* , (Q33 ) (in 10 Pa ), and (Qh ) (in 10 Pa ) that have been calculated for the 1–3 relaxor-ferroelectric single crystal / araldite composites (reprinted from Bezus et al. [47], with permission from IOP Publishing)

Single-crystal component PMN–0.30PT

* max g 33

* max h33

26.1 382 (0.414 – 0.547) (0.013)a PMN–0.33PT 496 37.8 (0.017) (0.501 – 0.532) PMN–0.42PT 519 56.9 (0.024) (0.653 – 0.709) PZN–0.045PT 317 19.5 (0.021) (0.311 – 0.389) PZN–0.07PT 359 23.1 (0.023) (0.419 – 0.480) PZN–0.08PT 352 22.3 (0.019) (0.334 – 0.390) * 2 Single-crystal max k t* max[ (Q33 ) ] component PMN–0.30PT 0.819 106 (0.552) (0.160) PMN–0.33PT 0.861 144 (0.559) (0.250) PMN–0.42PT 0.729 46.3 (0.640) (0.078) PZN–0.045PT 0.720 108 (0.532) (0.338) PZN–0.07PT 0.750 133 (0.537) (0.460) PZN–0.08PT 0.765 136 (0.422) (0.422)

max d h*

max g h*

244 (0.434) 274 (0.509) b

103 (0.013) 130 (0.016) 140 (0.024) 180 81.0 (0.438) (0.019) 194 93.9 (0.455) (0.021) 206 90.6 (0.409) (0.018) max[ (Qh* ) 2 ] 6.16 (0.099) 7.45 (0.115) 3.46 (0.082) 4.62 (0.141) 5.43 (0.146) 5.38 (0.131)

Notes to Table 5.2 a. The optimal volume fractions mopt of the single-crystal component that correspond to the * maximum values (i.e., max)* = )*(mopt)) are written in parentheses. In the h33 column * – * (m) ” 107 V / m ranges of the volume fractions obey a condition max h33 h33 b. For this composite, the monotonic dependence of the hydrostatic piezoelectric coefficient d h* (m) has been established

(1),V associated with the relatively low dielectric permittivity H 33 of the PMN–0.42PT (1),V single crystal: according to experimental data [48], H 33 / H 0 = 660, i.e., one order (1),V of magnitude smaller than H 33 / H 0 of other PMN–xPT single-crystals (Table 5.3) with compositions across the MPB. Despite the fairly low volume fractions 0.01 < * (m) and max g h* (m) (see Table 5.2 and curves mopt < 0.02 corresponding to max g 33

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* Figure 5.5. Volume-fraction dependences of the piezoelectric coefficients g 33 (a, in mV m * (c, in 108 V / m) that have been calculated for the PMN– / N), g h* (b, in mV m / N) and h33 0.42PT single crystal / araldite (curve 1) and PMN–0.42PT single crystal / porous araldite (mp = 0.3) (curve 2) composites. The porous matrix contains spherical pores (reprinted from Bezus et al. [47], with permission from IOP Publishing)

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Electromechanical Properties in Composites Based on Ferroelectrics

Figure 5.5 (continued)

Figure 5.6. Volume-fraction dependences of the hydrostatic piezoelectric coefficient d h* (a, * in pC / N), electromechanical coupling factor k t* (b), squared figure of merit ( Q33 )2 (c, in

*

10–15 Pa–1), and squared hydrostatic figure of merit ( Qh )2 (d, in 10–15 Pa–1) that have been calculated for the PMN–0.33PT single crystal / araldite (curve 1) and PMN–0.33PT single crystal / porous araldite (mp = 0.3, spherical pores) (curve 2) composites (reprinted from Bezus et al. [47], with permission from IOP Publishing)

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

Figure 5.6 (continued)

147

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Electromechanical Properties in Composites Based on Ferroelectrics

Figure 5.6 (continued) Table 5.3. Experimental values of the room-temperature electromechanical constants of the relaxor-ferroelectric single crystals (reprinted from Bezus et al. [47], with permission from IOP Publishing) Single crystal

d 33(1) ,

pC / N PMN–0.30PT [48] 1981 PMN–0.33PT [14] 2820 PMN–0.42PT [48] 260 PZN–0.045PT [49] 2000 PZN–0.07PT [15] 2455 PZN–0.08PT [38] 2890 Single crystal PMN–0.30PT [48] PMN–0.33PT [14] PMN–0.42PT [48] PZN–0.045PT [49] PZN–0.07PT [15] PZN–0.08PT [38]

(1) , g 33

h33(1) ,

mV m / N 28.7 38.8 44.5 44.0 49.3 42.4 (1),V H 33 / İ0 kt(1) 7800 8200 660 5200 5622 7700

0.62 0.64 0.62 0.50 0.47 0.45

d h(1) ,

g h(1) ,

108 V / m 24.6 33.7 53.2 17.0 20.7 17.7 (1) 2 ( Q33 ),

pC / N mV m / N 139 2.1 160 2.0 78 13.4 60 2.0 47 0.9 –20 –0.2 (1) 2 ( Qh ) ,

10–12 Pa–1 56.9 109 11.6 88.0 121 123

10–12 Pa–1 0.292 0.320 1.05 2.64 0.423 0.004

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

149

*V (1),V 1 in Figures 5.5a and b) as well as an inequality H 33 (mopt) << H 33 , the PMN– xPT-based composites (x = 0.33 or 0.42) are of practical interest at volume * (1) fractions m = 0.05–0.10. In this range the ratios g 33 (m) / g 33 | 8–13 and g h* (m) / * * (m), the condition h33 (m) | const holds g h(1) | 10–15 are attained. In contrast to g 33 for 0.1 < m < 0.9 (curve 1 in Figure 5.5c). This constancy is associated with the boundary conditions for the electric-field and mechanical-stress components and, * (1) *[ (1),[ as a consequence, a relation e33 (m) / e33 | H 33 (m) / H 33 holds true for all the 1–3 PMN–xPT / polymer or PZN–yPT / polymer composites studied in ref. [47]. The 1–3 PZN–0.08PT / araldite composite is characterised by the large * max[ Q33 (m)]2 value (Table 5.2) that is primarily the result of the large d 33(1) and

H 33(1),V values of the single-crystal component (see Table 5.3). The optimal volume * fraction mopt corresponding to max[ Q33 (m)]2 is nearly equal to that found for

max d h* (m), but this closeness is not typical of the other 1–3 composites listed in Table 5.2. In addition, the PZN–0.08PT / araldite composite does not possess large * * values of max g 33 (m), max d h* (m) and max g h* (m), but the ratio max[ Q33 (m)]2 / max[ Qh* (m)]2 | 25 is the highest value for the composites listed in Table 5.2. Such features of the composite are associated with the elastic and piezoelectric properties of the PZN–0.08PT single crystal. Maximum values of the effective parameters of the PZN–yPT-based composites are located in relatively narrow ranges (see Table 5.2). Such behaviour is a result of the monotonic dependences [47] of the electromechanical properties of the PZN–yPT single crystals [15, 38, 49] at 0.045 d y d 0.08. As is seen from Table 5.2, the high values of max d h* (m) = 1.71 d h(1) , max kt* (m) = 1.25 kt(1) , * max[ Q33 (m)]2 = 1.31( Q33(1) )2, and max[ Qh* (m)]2 = 21.1( Qh(1) )2 are attained in the

PMN–0.33PT-based 1–3 composite. The PMN–0.33PT single crystal is characterised [14] by the largest values of the elastic compliances | sa(13),E |, (1),V piezoelectric coefficients | d 3(1j) | and dielectric permittivity H 33 in the range of

molar concentrations 0.30 ” x ” 0.42. These outstanding properties strongly influence the effective electromechanical properties and their maxima in the related composite. A further comparison of the calculated [47] and experimental [39] parameters of the PMN–0.33PT-based composite (Table 5.4) suggests that the good agreement between these parameters is attained in a wide m range. Some differences between the calculated and measured values in Table 5.4 can be associated with the finite sizes of the single-crystal rods [50] and their piezoelectric and dielectric properties. (1) (1),V values of the PMN– As is known from data in ref. [39], the measured d 33 and H 33 0.33PT single crystal are less than those determined in experimental work [14] and used in calculations [47]. Thus, the results represented in this section show how the PMN–xPT and PZN– yPT single crystals actively influence the effective electromechanical properties in

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Electromechanical Properties in Composites Based on Ferroelectrics

Table 5.4. Calculated and experimental (in parentheses, from ref. [39]) values of normalised parametersa of the 1–3 PMN–0.33PT single crystal / epoxy composite (reprinted from Bezus et al. [47], with permission from IOP Publishing)

kt* / kt(1) *D (1), D / c33 c33

m = 0.37 1.32 (1.30)

m = 0.48 1.34 (1.36)

m = 0.62 1.35 (1.38)

m = 0.77 1.31 (1.34)

0.295 (0.300)

0.371 (0.332)

0.474 (0.431)

0.605 (0.598)

Note to Table 5.4 *D (1), D ratios have been calculated using the room-temperature a. The kt* / kt(1) and c33 / c33 electromechanical constants of the PMN–0.33PT single crystal [14] and epoxy [39]

1–3 single crystal / polymer composites. A further increase in specific effective parameters can be attained by use of a porous structure of the matrix surrounding the single-crystal rods. Different examples of high-performance 1–3-type composites comprising ferroelectric ceramic rods and the porous polymer matrix were considered in refs. [51–53]. In the next section we consider how the porous polymer matrix would improve the piezoelectric sensitivity of the 1–0–3 composites based on the relaxor-ferroelectric single crystals.

5.5 1–0–3 Single Crystal / Porous Polymer Composites In this section we study a 1–0–3 composite that contains single-crystal rods in the form of rectangular parallelepipeds (Figure 5.7). These rods are regularly distributed in a lengthy porous polymer matrix, wherein the shape of closed air pores is described by an equation (x1 / a1)2 + (x2 / a1)2 + (x3 / a3)2 = 1 relative to the axes of the rectangular co-ordinate system (X1X2X3), where either an equality a1= a3 or an inequality a1 > a3 holds for the spherical or oblate spheroidal pores, respectively. The centres of these pores are assumed to occupy the sites of a simple cubic (at a1 = a3) or tetragonal (at a1 > a3) lattice. The semi-axis a1 is considerably less than the length of the side of the square being the intersection of the rod by the (X1OX2) plane (Figure 5.7). In this connection one can regard the porous matrix as a medium with effective properties [52] that depend on porosity mp and the aspect ratio U = a1 / a3 of the pores. The single-crystal component is characterised by the volume fraction m, and the volume fraction of the porous matrix is 1 – m. The composite sample is poled by an electric field E || OX3. Based on results in ref. [54], the poling procedure can be carried out after inserting the aligned rods into the matrix and covering the composite sample with electrodes parallel to the (X1OX2) plane. Below, we consider two examples of volume-fraction dependences of the effective electromechanical properties in the 1–0–3 composite. The first example is concerned with a composite containing spherical pores distributed regularly in the matrix. The effective parameters )* = )*(m, mp) of such a composite based on PMN–xPT (x = 0.33 or 0.42) are represented in Figures 5.5 and 5.6 (see curves 2 in all the graphs), and the benefits of using a porous matrix are clearly demonstrated. Using a porous matrix (0 < mp d 0.3) rather than a

Effective Properties in Novel Piezo-composites Based on Relaxor-ferroelectrics

151

Figure 5.7. Schematic diagram of the 1–3-type composite. The spontaneous polarisation vector of the single crystal is denoted by an arrow. (X1X2X3) is the rectangular co-ordinate system. A spheroidal air pore is shown in the inset (reprinted from Bezus et al. [47], with permission from IOP Publishing) * monolithic one (mp = 0) results in only a modest change in h33 (m, mp) (compare curves 1 and 2 in Figure 5.5c). This independence on mp is closely connected with * (1) *[ (1),[ *[ (1),[ interrelations e33 (m, mp) / e33 | H 33 (m, mp) / H 33 and H 33 << H 33 that remain

true in a wide m range. The 1–0–3 PMN–0.33PT / porous araldite (mp = 0.3) composite is of interest because of the following maximum values [47] of the effective parameters: max kt* (m, 0.3) = 1.32 kt(1) , max d h* (m, 0.3) = 2.66 d h(1) , * max[ Q33 (m, 0.3)]2 = 2.19( Q33(1) )2, and max[ Qh* (m, 0.3)]2 = 72.6( Qh(1) )2. These

effective parameters produce new possibilities for creating advanced piezo-active composites containing the relaxor-ferroelectric components. The maximum values of the above-mentioned parameters are higher than those measured on 1–3 piezocomposites containing ceramic or single-crystal rods. For example, the experimental values of max kt* are 0.83, 0.80 and 0.70 for the PZN–0.08PT single crystal / polymer [38], PMN–0.33PT single crystal / epoxy [39] and PZT-7A ceramic / polymer [40] composites, respectively. The experimental max d h* value in the PZT-7A ceramic / Spurr epoxy composite [40] does not exceed 120 pC / N. * An increase in polymer porosity mp leads to an increase in g 33 , g h* , d h* , * 2 ( Q33 ) , and ( Qh* )2 in a wide m range as compared with the case of mp = 0 (see

curves 1 and 2 in Figures 5.5a, b, and 5.6a, c and d). Increasing d h* (m, mp) in comparison with d h* (m, 0) (see Figure 5.6a and Table 5.5) correlates with changes in the c11,m / c12,m ratio of the elastic moduli of the porous matrix. The piezoelectric

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Electromechanical Properties in Composites Based on Ferroelectrics

* * coefficient d 33 (m, mp) of the studied 1–0–3 composite increases faster than | d 31 (m, mp) | on increasing the volume fraction m. This behaviour stems from the fact that a piezoelectric interaction between the rods in the (X1OX2) plane is reduced to a greater degree by a porous matrix than by a monolithic one. The balance of d 3* j in

this composite favours an increase in max d h* (m, mp) at mp = const, as follows from * * (m, mp) is due to Table 5.5. A change in the balance between d 33 (m, mp) and d 31 changes in c11,m / c12,m and the elastic stiffness of the matrix. In particular, as is seen from Table 5.5, increasing c11,m results in the highest volume fractions mopt corresponding to max d h* , and such a shift takes place for both the 1–3 and 1–0–3

composites. As follows from data on the 1–0–3 composites (Table 5.5), the higher mopt value corresponds to a lower c11,m / c12,m ratio. The correlation between max d h* and c11,m / c12,m [47] allows prediction of the hydrostatic piezoelectric activity of the 1–0–3 composites based on relaxor-ferroelectric single crystals. The second example is concerned with a similar PMN–0.33PT-based composite containing spheroidal pores distributed regularly in the polymer matrix. The volume fraction of the single-crystal rods m, porosity mp and the aspect ratio U of the pores are now regarded as three independent arguments in a function )*(m, mp, U) of the effective parameters of the 1–0–3 composite. The 1–0–3 PMN– 0.33PT / porous araldite composite at mp = 0.3 is characterised by the following features of the piezoelectric response [47]. * Even at relatively small pore aspect ratios (2 d U d 5), points of max g 33 (m, 0.3,

U) and max g h* (m, 0.3, U) appear at 0.005 < m < 0.01. The microgeometry of such a * porous matrix displaces the maximum points of [ Q33 (m, 0.3, U)]2 and [ Qh* (m, 0.3, * 2 U)]2 to lower m values: e.g., for ( Q33 ) from 0.088 at U = 2 to 0.060 at U = 5. The

Table 5.5. Values of max d h* (in pC / N), mopt a (at mp = const) b and c11,m / c12,m that have been calculated for the 1–3-type composites based on PMN–0.33PT single crystals (reprinted from Bezus et al. [47], with permission from IOP Publishing) Components Polyethylene (mp = 0) Elastomer (mp = 0) Spurr epoxy (mp = 0) Araldite (mp = 0) Porous araldite (mp = 0.10) Porous araldite (mp = 0.15) Porous araldite (mp = 0.20) Porous araldite (mp = 0.25) Porous araldite (mp = 0.30)

max d h*

mopt

c11, m / c12, m

174 225 266 274 323 348 373 399 426

0.497 0.256 0.519 0.509 0.456 0.434 0.414 0.395 0.376

1.17 1.22 1.75 1.77 1.92 1.99 2.06 2.12 2.18

Notes to Table 5.5 a. From equation d h* (mopt, mp) = max d h* b. mp is the volume fraction of the spherical pores distributed regularly in the polymer matrix

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153

* Table 5.6. Effective piezoelectric coefficients d h* (in pC / N), g 33 , g h* (in mV m / N), * 2 ) , ( Qh* )2 (in 10–12 Pa–1), and electromechanical coupling squared figures of merit ( Q33 * factors k33 and kt* that have been calculated for the 1–0–3 PMN–0.33PT single crystal / porous araldite composite at mp = 0.30 (reprinted from Bezus et al. [47], with permission from IOP Publishing)

ȡa

md,hyd b

d h*

* g 33

g h*

10 100 200 300 400 500 ȡa

0.252 0.120 0.096 0.083 0.077 0.073 * 2 ( Q33 )

998 1780 1920 1980 2010 2040 ( Qh* )2

149 317 399 459 495 522

74.1 252 335 396 432 459

* k33

kt*

10 100 200 300 400 500

300 710 910 1050 1140 1210

73.9 449 644 783 869 934

0.941 0.946 0.946 0.946 0.947 0.947

0.920 0.940 0.942 0.942 0.943 0.943

Notes to Table 5.6 a. The aspect ratio of the spheroidal pores in the polymer matrix b The volume fraction md,hyd of the single-crystal rods obeys condition max d h* (m, 0.3, ȡ) =

d h* (md,hyd , 0.3, ȡ) at the fixed ȡ value

maximum points of d h* (m, 0.3, U) also shift to lower m values, but this displacement to lower m becomes appreciable in the wide range of U. These results suggest that for piezotechnical applications one can propose the 1–0–3 composites that possess the maximum hydrostatic piezoelectric activity in addition to considerable piezoelectric sensitivity and electromechanical coupling. Taking the aspect ratio U from a wide range, a series of effective parameters that are expected at max d h* (m, 0.3, U) (Table 5.6) have been calculated. It can be seen that an * * 2 , g h* , d h* , ( Q33 ) , and ( Qh* )2 at increase in the U value results in an increase in g 33 * almost constant and large values of electromechanical coupling factors k33 and kt* . The effect of the aspect ratio on the above-given parameters is mainly associated with a change in the c11,m / c33,m ratio of the elastic moduli of the porous matrix. Increasing the ȡ value corresponds to softening of the polymer matrix in the OX3 direction, and this softening enables the control of the piezoelectric response of the 1–0–3 composite at mp = const. It should be added that the values of max d h* (m,

0.3, ȡ) from Table 5.6 are larger than those evaluated recently [55] for the 0–0–2–2 composites containing one or two relaxor-ferroelectric components (for example, PMN–0.33PT and PZN–0.045PT) and two monolithic polymer components. This

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Electromechanical Properties in Composites Based on Ferroelectrics

advantage, especially at 5 < ȡ < 10, is associated with the important role of the porous matrix with the variable c11,m / c33,m ratio. It is of interest to consider the possible mechanical properties of the porous polymers examined in the work. The porosity of the polymer matrix was varied [47] in the range 0 < mp < 0.3. With regard to the typical mechanical properties of the porous polymer with mp = 0.2–0.3, it is observed that the typical decrease in stiffness and strength of such materials is about one order of magnitude [56, 57], relative to the pore-free material. For a sensor material consisting of a porous polymer with a high stiffness piezo-active material (e.g., the single-crystal rod in the studied 1–3-type composites, see Figures 5.4 and 5.7), most of any applied mechanical load will be experienced by the stiffer piezo-active component. As a result, the polymer strength is unlikely to be a major limitation. Similarly, for an actuator material consisting of a porous polymer and piezoelectric ceramic, the relatively small piezoelectric strain is unlikely to exceed the failure strain of polymer in the matrix.

5.6 High Performance in Wide Ranges In this chapter we have considered the effective electromechanical properties and related parameters of novel piezo-composites based on single crystals of relaxorferroelectric solid solutions, such as PMN–xPT or PZN–yPT, with compositions chosen near the MPB. A combination of the single-crystal and ceramic or singlecrystal and polymer components in these composites with different microgeometry and connectivity provides a series of examples of non-monotonic volume-fraction behaviour of all four types of the piezoelectric coefficients (see Figures 5.1, 5.3 and 5.5). Moreover, the piezoelectric properties, electromechanical coupling factors and figures of merit of these composites clearly indicate their advantages over conventional 0–3 or 1–3 ceramic-based composites. In general, the results discussed in this chapter can aid in the design and selection of single-crystal and ceramic components as well as a search for proper microgeometry of the porous matrix. It is believed that the development of high-quality and high-performance composites based on the relaxor-ferroelectric single crystals opens up new possibilities of controlling the effective parameters in wide volume-fraction ranges (see, for example, data from Tables 5.1, 5.2, 5.4 – 5.6). Particularly high values of * the effective piezoelectric coefficients g 33 , g h* and d h* , electromechanical coupling * 2 factor kt* and squared figures of merit ( Q33 ) and ( Qh* )2 in certain m ranges as well

as relatively large piezoelectric coefficients | d 3* j | in the wide m range suggest that the related composites are suitable for hydrophone, sensor, actuator, and transducer applications. An electromechanical interaction between the relaxor-ferroelectric single crystals and modified PbTiO3 ceramics in the 0–3 composites also provides an opportunity to vary the piezoelectric anisotropy over wide ranges.

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[19] Noheda B, (2002) Structure and high-piezoelectricity in lead oxide solid solutions. Current Opinion in Solid State and Materials Science 6:27–34 [20] Xu Y, (1991) Ferroelectric Materials and Their Applications. North-Holland, Amsterdam London New York Toronto [21] Jaffe B, Cook WR, Jaffe H, (1971) Piezoelectric Ceramics. Academic Press, London New York [22] Gorish AV, Dudkevich VP, Kupriyanov MF, Panich AE, Turik AV, (1999) Piezoelectric Device-making. Vol. 1: Physics of Ferroelectric Ceramics. Radiotekhnika, Moscow (in Russian) [23] Peng J, Luo H, He T, Xu H, Lin D, (2005) Elastic, dielectric, and piezoelectric characterization of 0.70Pb(Mg1/3Nb2/3)O3 – 0.30PbTiO3 single crystal. Materials Letters 59:640–643 [24] Oakley CG, Zipparo MJ, (2000) Single crystal piezoelectrics: A revolutionary development for transducers. In: IEEE Ultrasonic Symposium: Proceedings. October 22–25, 2000. Caribe Hilton San Juan, Puerto Rico. IEEE, Piscataway, pp 1157–1167 [25] Smotrakov VG, Eremkin VV, Alyoshin VA, Tsikhotsky ES, (2000) Preparation and study of single-crystal–ceramic composite. Izvestiya Akademii Nauk. Seriya Fizicheskaya 64:1220–1223 (in Russian) [26] Topolov VYu, Glushanin SV, (2002) Effective electromechanical properties of ferroelectric piezoactive composites of the crystal–ceramic type based on (Pb1–xCax)TiO3. Technical Physics Letters 28:279–282 [27] Glushanin SV, Topolov VYu, Turik AV, (2003) Prediction of piezoelectric properties of crystal-ceramic composites. Crystallography Reports 48:491–499 [28] Takahashi H, Tukamoto S, Qiu J, Tani J, Sukigara T, (2003) Property of composite ceramics composed of single crystals and ceramic matrix using hybrid sintering. Japanese Journal of Applied Physics. Pt 1 42:6055–6058 [29] Ikegami S, Ueda I, Nagata T, (1971) Electromechanical properties of PbTiO3 ceramics containing La and Mn. Journal of the Acoustical Society of America 50:1060–1066 [30] Nagatsuma K, Ito Y, Jyomura S, Takeuchi H, Ashida S, (1985) Elastic properties of modified PbTiO3 ceramics with zero temperature coefficients. In: Taylor GW (ed.) Ferroelectricity and Related Phenomena. Vol. 4: Piezoelectricity. Gordon and Breach Science Publishers, New York London Paris Montreux Tokyo, pp 167–176 [31] Levassort F, Thi MP, Hemery H, Marechal P, Tran-Huu-Hue L-P, Lethiecq M, (2006) Piezoelectric textured ceramics: Effective properties and application to ultrasonic transducers. Ultrasonics 44:e621–e626 [32] Topolov VYu, Kamlah M, (2004) Piezoelectric properties of PbTiO3-based 0–3 and 0–1–3 composites. Journal of Physics D: Applied Physics 37:1576–1585 [33] Dunn ML, Taya M, (1993) An analysis of piezoelectric composite materials containing ellipsoidal inhomogeneities. Proceedings of the Royal Society (London), Pt A 443:265–287 [34] Huang JH, Yu S, (1994) Electroelastic Eshelby tensors for an ellipsoidal piezoelectric inclusion. Composites Engineering 4:1169–1182 [35] Grekov AA, Kramarov SO, Kuprienko AA, (1989) Effective properties of a transversely isotropic piezoelectric composite with cylindrical inclusions. Mechanics of Composite Materials, 25:54–61 [36] Taunaumang H, Guy IL, Chan HLW, (1994) Electromechanical properties of 1–3 piezoelectric ceramic / piezoelectric polymer composites. Journal of Applied Physics 76:484– 4 89 [37] Topolov VYu, Turik AV, (1999) Nonmonotonic concentration dependence of piezoelectric coefficients of 1–3 composites. Journal of Applied Physics 85:372–379

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6 Comparison of Results on Two-component Piezo-composites

6.1 The Pros and Cons of Different Methods This chapter presents data on the effective electromechanical properties determined by different methods or averaging procedures. It is well known that determining the effective properties of composites offers challenges and difficulties, even in the case of relatively simple two-component piezo-composites with well-defined microgeometry. These difficulties are closely connected with functions that would describe structural inhomogeneities and their spatial distributions [1, 2]. Calculations of the effective electromechanical constants of piezo-composites are based on methods that are approximate and require the internal fields affected by numerous inclusions in a continuous matrix to differ from the electric and elastic fields caused by a single inclusion in the same matrix. Moreover, any deviation from a regular distribution of the inclusions in the matrix influences the effective properties [2]. Methods based on direct averaging of the properties over a macroscopic volume (analytical methods and FEM), methods based on regularisation of structure, stochastic differential equations or virial expansion, as well as a self-consistent method [1–6] are often applied to determine the effective properties of the matrix of piezo-composites. In these composites the inclusions can be approximated by ellipsoids or spheroids and electromechanical interactions such as “inclusion – inclusion” and “inclusion – matrix” are taken into account in the averaging procedures [1, 4–6]. The spheroidal shape of the inclusion enables one to consider transitions from an oblate spheroid to a layer or from a prolate spheroid to a circular cylindrical rod. The matrix method based on coupled electroelastic Eshelby tensors [7] is applicable to the 0–3 composite with spheroidal inclusions (Section 3.3.1). This method is associated with different schemes of averaging [4–6], for example, the dilute approach, EFM (Mori – Tanaka method generalised for piezoelectric media), effective medium method, differential scheme, etc. It is obvious that the electromechanical interaction between components, along with the piezoelectric activity of at least one of the composite components, influences its effective properties and coupled effects. A

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comparison of the effective properties evaluated using different schemes or methods would be of value when studying the advanced piezo-composites and their performance. In the following sections we consider a series of examples whereby a comparison is made related to ceramic / polymer composites with 0–3, 1–3 or 2–2 connectivity. As in Chapters 2–5, we assume that the effective electromechanical properties are determined in the long-wave approximation, i.e., when wavelengths of external acoustic or electric fields are much longer than the sizes of inclusions in a composite sample.

6.2 0–3 Versus 2–2 or 1–3 The model of a 0–3 composite (Figure 3.7) with spheroidal inclusions that are regularly distributed in a matrix is important for further comparison with 2–2 and 1–3 structures. By varying the aspect ratio U or semi-axes ai of the inclusion described by Equation 3.18, it is possible to consider the volume-fraction behaviour of the effective properties in limiting cases, at U >> 1 or U << 1. The 0–3 composite with highly oblate inclusions (U >> 1) resembles a 2–2 series-connected composite (Figure 3.1a) whereas the 0–3 composite with highly prolate inclusions (U << 1) resembles a 1–3 composite with continuous rods parallel to the poling axis OX3. Examples of volume-fraction behaviour of the properties in the related 0–3 (U >> 1) and 2–2 composites with one or two piezoelectric components are given in Tables 6.1 and 6.2. Calculations were carried out within the framework of the EFM (0–3 connectivity, Equation 3.20) or the matrix method (2–2 connectivity, Equation 3.2). The electromechanical constants of the components were taken from Tables 2.1 and 2.2 and experimental data [8]. It is known that the EFM is applicable to the 0–3 composite (Figure 3.7) provided that its matrix is either transversely isotropic or isotropic [4–7]. In the matrix method [9–11], which is applied to the piezo-composites with planar interfaces, no restrictions regarding the symmetry of the components are introduced. The volume-fraction dependences calculated for the 2–2 ceramic / polymer and single crystal / ceramic composites (Tables 6.1 and 6.2) suggest that their effective properties satisfy an inequality

|)ij*» U = 102 – )ij*» U = 103 | < |)ij*» U = 103 – )ij*» 2–2 |

(6.1)

where )* = e*, d* and H*V. Condition 6.1 holds true in different cases, for instance, when absolute values of the piezoelectric coefficients | e3* j (m)| and | d 3* j (m)| increase monotonously (Table 6.1) or when | e3* j (m)| and | d 3* j (m)| show a monotonic decrease and increase, respectively (Table 6.2). Inequality 6.1 is due to slight changes in the aspect ratio U of the oblate inclusions and by corresponding monotonic changes in the Eshelby tensors [7]. Increasing U implies that interfaces separating the components in the composite approach the plane x3 = const (Figure 3.1a) and both the components are distributed continuously in the OX1 and OX2

Comparison of Results on Two-component Piezo-composites

161

Table 6.1. Volume-fraction dependences of the piezoelectric coefficients e3* j (in C / m2) and *V d 3* j (in pC / N) and relative dielectric permittivity H 33 / H0, which have been calculated for

the 0–3 and 2–2 PZT-5 ceramic / araldite composites by means of the EFM ma

* 102 e31

* 103 e33

0.05 0.1 0.2 0.3 0.4 0.5 ma

(U = 10 ) –0.238 –0.508 –1.16 –2.03 –3.20 –4.87 * 102 e31

(U = 10 ) 0.207 0.463 1.18 2.31 4.19 7.51 * 103 e33

0.05 0.1 0.2 0.3 0.4 0.5 ma

(U = 10 ) –0.292 –0.617 –1.38 –2.37 –3.66 –5.45 * 102 e31 (2–2

0.05 0.1 0.2 0.3 0.4 0.5

composite) –0.300 –0.632 –1.42 –2.41 –3.72 –5.53

2

3

H 33*V / H0 (U = 102)

* d 33 (U = 102) 0.233 0.362 0.554 0.744 0.973 1.28

(U = 10 ) 0.229 0.509 1.28 2.49 4.46 7.88 * 103 e33 (2–2

* d 31 (U = 102) –0.190 –0.283 –0.401 –0.502 –0.614 –0.759 * d 31 (U = 103) –0.211 –0.302 –0.414 –0.509 –0.618 –0.759 * (2–2 d 31

* d 33 (U = 103) 0.257 0.384 0.568 0.751 0.974 1.28 * (2–2 d 33

(U = 103) 4.21 4.44 5.00 5.71 6.66 7.99 H 33*V / H0 (2–2

composite) 0.232 0.515 1.29 2.51 4.49 7.93

composite) –0.214 –0.305 –0.415 –0.510 –0.618 –0.759

composite) 0.261 0.387 0.569 0.752 0.975 1.27

composite) 4.21 4.44 5.00 5.71 6.66 7.98

2

3

4.21 4.45 5.01 5.74 6.70 8.04 H 33*V / H0

Note to Table 6.1 a. Volume fraction of the ceramic component Table 6.2. Volume-fraction dependences of the piezoelectric coefficients e3* j (in C / m2) and *V d 3* j (in pC / N) and relative dielectric permittivity H 33 / H0, which have been calculated for

the 0–3 and 2–2 PMN–0.33PT single crystal / PCR-7M ceramic composites by means of the EFM ma 0.05 0.1 0.2 0.3 0.4 0.5

* e31

* e33

(U = 102) –9.00 –8.54 –7.70 –6.97 –6.33 –5.78

(U = 102) 30.1 29.2 27.6 26.2 25.0 23.9

* d 31 (U = 102) –345 –344 –349 –361 –383 –417

* d 33 (U = 102) 750 750 760 787 833 904

H 33*V / H0 (U = 102) 4980 4790 4480 4260 4140 4120

162

Electromechanical Properties in Composites Based on Ferroelectrics Table 6.2 (continued)

ma

* e31

* e33

* d 31

* d 33

H 33*V / H0

0.05 0.1 0.2 0.3 0.4 0.5 ma

(U = 103) –8.99 –8.53 –7.69 –6.96 –6.33 –5.77 * (2–2 e31

(U = 103) 30.1 29.2 27.6 26.2 25.0 23.9

(U = 103) 750 748 757 783 828 899

(U = 103) 4980 4780 4460 4250 4120 4090

* (2–2 e33

(U = 103) –345 –344 –348 –360 –381 –414 * (2–2 d 31

* (2–2 d 33

H 33*V / H0 (2–2

0.05 0.1 0.2 0.3 0.4 0.5

composite) –9.00 –8.53 –7.69 –6.96 –6.33 –5.77

composite) 30.1 29.2 27.6 26.2 25.0 23.9

composite) –345 –344 –348 –359 –380 –414

composite) 749 748 757 783 828 898

composite) 4980 4780 4460 4240 4120 4090

Note to Table 6.2 a. Volume fraction of the single-crystal component

directions. Such an arrangement leads to considerable mechanical stress relief in comparison with that in a 0–3 composite containing, for example, spherical inclusions (U = 1). A transition from the 0–3 composite with highly prolate inclusions (U << 1) to the 1–3 composite also results in relatively small changes in the electromechanical properties (Table 6.3). These changes are expressed by a formula

|)ij*»U = 10–2 – )ij*»U = 10–3 | < |)ij*»U = 10–3 – )ij*»1–3 |

(6.2)

It is worth noting that the similarity of conditions 6.1 and 6.2 occurs at two different transformations of the 0–3 structure. A decrease of the piezoelectric activity of the 0–3 composite on increasing the aspect ratio U (Table 6.3) is a result of an increase in the depolarisation field in the ferroelectric ceramic inclusions (see Figure 3.7). It is clear that the 1–3 composite with the structure shown in Figure 3.2 exhibits some advantages over the 0–3 composite at U o 0 due to the continuous distribution of the piezo-active component along the poling axis OX3. Another interesting case of the transition from 0–3 to 1–3 connectivity is concerned with the (Pb0.75Ca0.25)TiO3 ceramic / araldite composite. Its ceramic component is characterised by the piezoelectric coefficients e3(1j) > 0 and large anisotropy of the piezoelectric coefficients d 3(1j) at sgn d 31(1) = –sgn d 33(1) < 0 (see Table * 3.3). The presence of this component causes a small increase of e31 and a decrease * of | d31 | (Table 6.4) as the aspect ratio U increases at m = const. On increasing U the

Comparison of Results on Two-component Piezo-composites

163

Table 6.3. Volume-fraction dependences of the piezoelectric coefficients e3* j (in C / m2) *V and d 3* j (in pC / N) and relative dielectric permittivity H 33 / H0, which have been calculated

for the 0–3 and 1–3 PCR-7M ceramic / araldite composites by means of the EFM ma 0.05 0.1 0.2 0.3 0.4 0.5 ma 0.05 0.1 0.2 0.3 0.4 0.5 ma 0.05 0.1 0.2 0.3 0.4 0.5 ma

0.05 0.1 0.2 0.3 0.4 0.5

* e31

* e33

* d 31

* d 33

H 33*V / H0

(U = 10–2) –0.0293 –0.0621 –0.141 –0.243 –0.378 –0.565

(U = 10–2) 1.58 3.19 6.50 9.91 13.4 17.1

(U = 10–2) –74.8 –112 –154 –182 –205 –227

(U = 10–2) 203 298 398 458 504 545

(U = 10–2) 121 276 631 1030 1470 1950

* e31

* e33

* d 31

* d 33

H 33*V / H0

(U = 5u10–3) –0.0331 –0.0697 –0.156 –0.266 –0.409 –0.604

(U = 5u10–3) 1.81 3.62 7.27 10.9 14.6 18.3

(U = 5u10–3) –90.5 –138 –190 –222 –247 –267

(U = 5u10–3) 246 368 493 561 606 640

(U = 5u10–3) 145 338 779 1260 1760 2290

* e31

* e33

* d 31

* d 33

H 33*V / H0

(U = 10–3) –0.0348 –0.0731 –0.163 –0.276 –0.423 –0.620 * (1–3 e31

(U = 10–3) 1.91 3.81 7.61 11.4 15.2 18.9

(U = 10–3) 268 405 544 614 657 686

(U = 10–3) 157 370 857 1380 1910 2450

* (1–3 e33

(U = 10–3) –98.5 –151 –210 –244 –267 –286 * (1–3 d 31

* (1–3 d 33

H 33*V / H0

composite)

composite)

composite)

composite)

–0.0349 –0.0733 –0.163 –0.277 –0.424 –0.621

1.91 3.82 7.63 11.4 15.2 18.9

–99.0 –152 –211 –245 –269 –287

269 407 547 618 660 689

(1–3 composite) 158 372 862 1380 1920 2460

Note to Table 6.3 a. Volume fraction of the ceramic component

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Electromechanical Properties in Composites Based on Ferroelectrics

Table 6.4. Volume-fraction dependences of the piezoelectric coefficients e3* j (in C / m2), 9 d 3* j (in pC / N), g 3* j (in mV m / N), and h3* j (in 10 V / m), which have been calculated for

the 0–3 (U << 1) and 1–3 (Pb0.75Ca0.25)TiO3 ceramic / araldite composites by means of the EFMa and FEMb Method and U EFM, U = 0.05

EFM, U = 0.01

EFM, U=0

FEM, U=0

mc

* 102 e31

* e33

* d31

* d33

* g31

* g33

* 102 h31

* h33

0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5

0.999 2.24 3.80 5.83 8.56 0.972 2.16 3.66 5.61 8.23 0.960 2.14 3.63 5.56 8.17 0.960 2.14 3.63 5.59 8.31

0.263 0.556 0.884 1.25 1.67 0.440 0.885 1.34 1.79 2.26 0.459 0.919 1.38 1.84 2.31 0.459 0.919 1.38 1.84 2.31

–4.85 –5.14 –4.82 –4.31 –3.72 –6.58 –6.54 –5.90 –5.12 –4.30 –6.73 –6.65 –5.99 –5.19 –4.35 –6.73 –6.65 –5.99 –5.18 –4.30

16.2 19.7 21.5 22.7 23.8 21.5 24.6 25.8 26.5 26.9 22.0 25.0 26.1 26.8 27.2 22.0 25.0 26.1 26.8 27.2

–32.4 –19.1 –12.3 –8.24 –5.64 –38.0 –21.0 –13.1 –8.66 –5.88 –38.4 –21.1 –13.1 –8.69 –5.89 –38.4 –21.1 –13.1 –8.67 –5.83

108 73.3 54.7 43.5 36.1 124 78.7 57.1 44.8 36.8 126 79.3 57.3 44.9 36.9 126 79.1 57.3 44.8 36.8

6.86 8.67 10.1 11.8 13.8 5.94 7.45 8.78 10.3 12.2 5.82 7.31 8.63 10.1 12.1 5.82 7.31 8.63 10.2 12.3

1.80 2.15 2.36 2.53 2.69 2.69 3.05 3.20 3.29 3.36 2.78 3.14 3.28 3.36 3.42 2.78 3.14 3.28 3.36 3.42

Notes to Table 6.4 a. Calculations performed using Equation 3.20 and the Eshelby tensor from ref. [7] b. Calculations performed using the COMSOL package [12] c. Volume fraction of the ceramic component * * larger increase in the piezoelectric coefficient h31 in comparison with e31 (Table

*[ 6.4) is a result of the effective dielectric permittivity H 33 being about one order of

*[ magnitude smaller than H 33 of the above-considered PCR-7M ceramic / araldite composites. With reference to 1–3 connectivity, Table 6.4 shows that values of all the piezoelectric coefficients calculated by means of the EFM are in good agreement with the values obtained using the FEM. Some difference between the values of the piezoelectric coefficients )31* () = e, d, g, or h) is observed at m t 0.4 and may be associated with features of the electromechanical interaction between the piezoelectric rods in the (X1OX2) plane. At large volume fractions (m) the interaction is more intensive than that predicted on the basis of the EFM. This is similar to discussions by Berger et al. [13] in work considering the prediction of the effective electromechanical properties in 1–3 PZT-5 ceramic / polymer composites. In Section 6.3 we will consider a series of examples of the volumefraction dependences determined for the 1–3 ceramic / polymer composites by means of different methods.

Comparison of Results on Two-component Piezo-composites

165

6.3 Data on 1–3 Composites The shape, size and arrangement of the ceramic rods in the 1–3 piezo-composites influence their effective electromechanical properties [13–16]. The volumefraction behaviour of these properties is often considered for composites that contain ferroelectric ceramic rods in the form of either circular cylinders or rectangular parallelepipeds and are characterised by a regular distribution of these rods. 6.3.1 Composites with Cylindrical Ceramic Rods *E The effective electromechanical constants cab , eij* and

H *pp[ of the 1–3 ceramic /

polymer composite shown in Figure 3.2 were calculated [13] by means of the asymptotic homogenisation method (analytical solution) and the FEM. The FEM was applied to the 1–3 structure with a specific arrangement of the rods, and examples of the volume-fraction dependence of the effective electromechanical properties are shown in Figure 6.1. It should be noted that Berger et al. [13] used superscript “eff“ to denote the effective constants (this superscript is equivalent to an asterisk in our notation) and fibre volume fraction in Figure 6.1 denotes the *E * volume fraction m of the ceramic component (rods) in our notations. The c13 , e31 ,

e15* , and H 11*[ curves show a good correlation in the wide volume-fraction range *E * (Figures 6.1c, f–h). In addition, according to results from ref. [13], the c33 , e33 *[ and H 33 calculated by the FEM are also in good agreement with the similar

a Figure 6.1. Volume-fraction dependence of elastic moduli graphs a – e), piezoelectric coefficients *[

e

eff kl

(i.e.,

e

* kl

eff c ab

(i.e.,

*E cab

in our notations,

in our notations, graphs f and g) and

dielectric permittivity H 11 (i.e., H 11 in our notations, graph h) which have been calculated for the 1–3 PZT-5 ceramic / polymer composite by either the FEM or the asymptotic homogenisation method (AHM). In the case of FEM calculations the square (Squ) and hexagonal (Hex) arrangements of the ceramic rods in the polymer matrix have been considered (reprinted from Berger et al. [13], with permission from Elsevier) eff

166

Electromechanical Properties in Composites Based on Ferroelectrics

b

c

d

e Figure 6.1 (continued)

Comparison of Results on Two-component Piezo-composites

167

f

g

h Figure 6.1 (continued)

parameters calculated by the asymptotic homogenisation method in the same volume-fraction range. A complex study on the effective electromechanical properties of 1–3 composites has recently been carried out [17]. Kar-Gupta and Venkatesh developed an analytical model suitable for prediction of the effective properties in the general case, i.e., when the 1–3 composite consists of two piezo-active and elastically anisotropic components. A scheme for determining the effective electromechanical constants (up to 45 constants) has been proposed to characterise the properties in the wide range of volume fractions of the ceramic component and to take into account symmetry features of the components. This scheme has been applied to the following classes of the 1–3 composites: “longitudinal” composites

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Electromechanical Properties in Composites Based on Ferroelectrics

with components poled on the longitudinal direction (OX3) and “transverse” composites wherein the matrix is poled on the transverse direction (OX1) and the rods are poled on the longitudinal direction (OX3). These two classes provide 20 material combinations with different symmetry and properties. Kar-Gupta and Venkatesh compared the effective electromechanical properties determined within the framework of the analytical model with those predicted using the FEM. Greater agreement between FEM and analytical results [17] is achieved for 1–3 composites where the dielectric properties of the piezo-active rods are dominant. Less agreement is achieved where the dominant dielectric properties are related to the matrix of the 1–3 composites. 6.3.2 Composites with Parallelepiped-shaped Ceramic Rods In this section it is assumed that the 1–3 ceramic / polymer composite contains rods in the form of a rectangular parallelepiped aligned parallel to the poling axis OX3 with square bases in the (X1OX2) plane [14]. The rods are regularly distributed in a continuous matrix so that a square pattern of rods is apparent in the (X1OX2) plane. The effective electromechanical properties of the 1–3 composite with parallelepiped-shaped rods that are continuous on the OX3 direction can be determined using the matrix method [10, 11, 14] applied to the 1–E (E = 1, 2 and 3) and 2–2 composites (see Section 3.1.1). The averaging procedure involves full sets ( n ), E of the elastic compliances sab , piezoelectric coefficients d kl(n ) and dielectric permittivities

),V H (n pp of the components (n = 1 and 2). Averaging is implemented on

the volume fraction m of the ceramic component at 0 < m < 1 due to the planar microgeometry of this 1–3 composite. Two examples of the volume-fraction behaviour of the effective piezoelectric coefficients and hydrostatic parameters of the 1–3 composites examined are shown in Table 6.5.1 The ceramics used as composite components are distinguished (Table 2.1) by the signs of the (1) (1),[ piezoelectric coefficients e31 , by values of dielectric permittivity H 33 and by the (1), E anisotropy of the elastic moduli cab . The effective parameters of the 1–3 composites with the same components have been calculated by means of the FEM [15] and within the framework of the differential medium theory [16] (see the two types of curves in Figure 6.2). The 1–3 composite considered in ref. [15] contains ceramic rods in the form of a rectangular parallelepiped (Figure 6.3) with a finite height h. A maximum difference between the similar effective parameters calculated using the matrix method (Table 6.5, h o f) and the FEM (Figure 6.2) for a mesh shown in Figure 6.3 does not exceed 19% at volume fractions ____________ 1

Hydrostatic electromechanical coupling factor of the 1–3 composite studied is defined [10, *V –1/2 * * * 15] as k h* = d h* ( sh*E H 33 ) , where d h* = d 31 + d 32 + d 33 is the hydrostatic piezoelectric

*E *E *E *E coefficient, sh*E = s11 + s12 + s13 + … + s33 is the hydrostatic elastic compliance and

H 33*V is the dielectric permittivity.

Comparison of Results on Two-component Piezo-composites

169

* * Table 6.5. Volume-fraction dependences of the piezoelectric coefficients d 31 , d 33 and d h*

(in pC / N), squared hydrostatic figure of merit (Qh* ) 2 (in 10–15 Pa–1), and hydrostatic electromechanical coupling factor k h* , which have been calculated for the 1–3 ceramic / Stycast polymer composites with parallelepiped-shaped rods by means of the matrix method Ceramic component PZT-5A

(Pb, Ca)TiO3

m

* d 31

* d 33

d h*

(Qh* ) 2

k h*

0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7

–26.3 –45.7 –73.6 –94.1 –111 –124 –136 –147 –8.42 –11.7 –13.4 –12.8 –11.5 –9.93 –8.29 –6.68

87.1 146 221 266 297 319 336 349 28.0 40.9 53.2 59.1 62.5 64.8 66.5 67.7

29.6 47.6 65.1 70.2 69.5 65.6 60.0 53.4 13.6 20.9 30.1 36.6 42.0 46.7 51.0 54.9

1750 2080 1740 1260 888 614 418 279 1700 2280 2460 2430 2400 2370 2360 2340

0.121 0.138 0.136 0.125 0.114 0.104 0.094 0.087 0.120 0.145 0.161 0.173 0.185 0.200 0.218 0.242

a Figure 6.2. Effective parameters calculated for the 1–3 ceramic / polymer composites within either the framework of the FEM (bold line) or the differential effective medium theory * * (dotted line): piezoelectric coefficients d 31 , d 33 and d h* (in pC / N) of the PZT-5A ceramic / Stycast composite (a) and (Pb, Ca)TiO3 ceramic / Stycast composite (b), squared hydrostatic figure of merit (Qh* ) 2 (c, in 10–15 Pa–1) and hydrostatic electromechanical coupling factor k h* (d, in per cent) of both the composites (reprinted from Nelli Silva et al. [15], with permission from Springer)

170

Electromechanical Properties in Composites Based on Ferroelectrics

b

c

d Figure 6.2 (continued)

Comparison of Results on Two-component Piezo-composites

171

Figure 6.3. 1–3 composite unit cell discretised by the 3 u 3 u 3 finite element mesh (reprinted from Nelli Silva et al. [15], with permission from Springer) * 0.05 d m d 0.7. The h value mainly affects the piezoelectric coefficient d 31 and, to a lesser degree, the hydrostatic parameters listed in Table 6.5. It should be added that for calculations of the effective parameters (Figure 6.2), Nelli Silva et al. [15] ( n ),E ),[ used the electromechanical constants cab , ekl(n ) and H (n of the components, pp ( n ),E whereas our averaging procedure based on formulae [10, 11] involves sab , d kl(n )

and

),V H (n pp . On comparing data shown in Table 6.5 and Figure 6.2, the difference

between the effective parameters is less pronounced in the PZT-5A-based composite at large m values and in the (Pb, Ca)TiO3-based composite at small m values. Such volume-fraction behaviour can be considered to be due to the differing anisotropy of the ceramic components in the 1–3 composites studied. 6.3.3 Variational Bounds for Effective Properties in 1–3 Composites with Cylindrical Ceramic Rods Effective parameters of a 1–3 ceramic / polymer composite with cylindrical rods are calculated within the framework of the EFM (matrix formalism) [3–6], the FEM [12, 13] or different micromechanical models (analytical formulae are given, for example, in refs. [1, 2, 8, 17–19, 23, 24]). The effective electromechanical constants calculated using these methods are in good agreement over a wide volume-fraction range. The 1–3 ceramic / polymer composite is also of interest due to rigorous variational bounds [20, 21] determined for the effective electromechanical properties. These bounds were derived on the basis of variational principles proposed by Hashin and Shtrikman [22] and generalised by Bisegna and Luciano [20, 21] for the case of piezo-active heterogeneous media. Bisegna and Luciano first applied the generalised principles [20] to 1–3 piezo-composites with a regular arrangement (square or circular) of ceramic rods in a polymer matrix. The generalised principles enabled the prediction of tighter bounds (in comparison with

172

Electromechanical Properties in Composites Based on Ferroelectrics

the Hashin – Shtrikman and other bounds) and was initially demonstrated for the electromechanical constants of a PZT-type ceramic / polymer composite [20, 21]. The lower and upper bounds can be determined [20, 21] for the following sets of electromechanical constants of the 1–3 composite: *D (i) elastic moduli cab , piezoelectric coefficients hkl* and dielectric *[ impermeabilities E pp (first set) and

*E (ii) elastic compliances sab , piezoelectric coefficients d kl* and dielectric

impermeabilities H *ppV (second set). The rigorous lower and upper bounds [12] are compared with the effective electromechanical constants listed in Table 6.6. For this comparison a composite based on the PbTiO3-type ceramic was chosen due to the large anisotropy of its * * piezoelectric coefficients d3l* . It is seen that the effective constants | d31 |, d33 and

H 33*V of the 1–3 composite studied attain the upper bound at various m values * Table 6.6. Elastic moduli ca*3D (in 109 Pa), piezoelectric coefficients h33 (in 109 V / m) and

9 *V / H0, d 3l* (in pC / N), dielectric impermeability E33*[ (in 10 m / F), dielectric permittivity H 33

and rigorous bounds on these properties in the 1–3 (Pb0.75Ca0.25)TiO3 ceramic / araldite composite with cylindrical rodsa Evaluations

md

c13*D

*D c33

* h33

* | d 31 |

* d 33

E33*[

H 33*V / H0

Using EFMb Lower boundc Upper boundc Using EFM Lower bound Upper bound Using EFM Lower bound Upper bound Using EFM Lower bound Upper bound Using EFM Lower bound Upper bound Using EFM Lower bound Upper bound

0.1 0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6

4.76 4.76 6.52 5.21 5.21 8.97 5.78 5.78 11.8 6.52 6.52 15.1 7.52 7.51 18.9 8.94 8.94 23.5

25.0 25.0 25.8 42.4 42.4 44.1 59.9 59.9 62.6 77.5 77.5 81.3 95.2 95.2 100 114 113 120

2.78 2.78 2.91 3.14 3.14 3.29 3.28 3.28 3.45 3.36 3.36 3.54 3.42 3.41 3.61 3.46 3.46 3.67

6.73 3.63 6.73 6.65 2.74 6.65 5.99 2.06 5.99 5.19 1.59 5.19 4.35 1.24 4.35 3.51 0.977 3.51

22.0 21.5 22.0 25.0 24.6 25.0 26.1 25.9 26.1 26.8 26.6 26.8 27.2 27.0 27.2 27.4 27.3 27.4

6.06 6.06 6.08 3.42 3.41 3.43 2.38 2.38 2.39 1.82 1.82 1.83 1.48 1.48 1.49 1.24 1.24 1.25

19.8 19.7 19.8 35.6 35.6 35.7 51.5 51.5 51.6 67.5 67.4 67.5 83.4 83.3 83.4 99.3 99.3 99.3

Notes to Table 6.6 a. Square arrangement of the ceramic rods is assumed to be in the (X1OX2) plane b. Electromechanical constants determined by means of the EFM (see Equation 3.20) and constitutive piezoelectric equations (see Equations 2.12–2.19) c. Electromechanical constants determined on the basis of bounding [12, 20, 21] d. Volume fraction of the ceramic component

Comparison of Results on Two-component Piezo-composites

173

*D *D * *[ (Table 6.6). The effective constants c13 , c33 , h33 , and E 33 of the same composite attain the lower bound. This feature of the volume-fraction dependence of the effective electromechanical properties is closely connected with the structure of the 1–3 composite (Figure 3.2) that contains ceramic rods aligned and poled on the OX3 direction. The structure promotes, for example, a fulfilment of conditions * (1) * (1) (m) = d 33 (see curve 2 in Figure 3.4a) and d 31 (m) = d 31 [8, 23, 24] with an d 33

accuracy to a few per cent at m > 0.1. The rigorous bounds shown in Table 6.6 exhibit relatively narrow ranges of the electromechanical properties of the *D * * *[ *V composite (e.g., c33 , h33 , d 33 , E 33 , and H 33 ). As follows from the data shown in Tables 6.4 and 6.6, the EFM is undoubtedly a reliable approach to predict the electromechanical properties in the 1–3 ceramic / polymer composite in the wide volume-fraction range. 6.3.4 Variational Bounds for Effective Properties in 1–3 Composites with Parallelepiped-shaped Ceramic Rods Another case of developing bounds of the effective electromechanical properties is related to the 1–3 composite with ceramic rods in the form of a rectangular parallelepiped. It is assumed that these rods are continuous in the OX3 direction and regularly distributed in the polymer matrix so that a square arrangement takes place in the (X1OX2) plane. The composite as a whole is poled along the OX3 direction, and their effective properties can be determined by different methods (see Section 6.3.2). Next we consider the effective electromechanical constants calculated using the matrix method [10, 11, 14] and the rigorous lower and upper bounds found [25] for the piezoelectric and dielectric constants (Table 6.7) of the 1–3 composite based on the PbTiO3-type ceramic. It is seen that values of the piezoelectric coefficients h3* j are slightly more than the lower bounds (Table 6.7) found in the wide m range. As with the data in Table 6.6, the rigorous bounds on dielectric *[ *[ impermeability E 33 form narrow ranges, and the E 33 values calculated by the matrix method are also within these bounds (see Table 6.7). Similar trends are observed in 1–3 composites with different ceramic (Pb1–xCax)TiO3 compositions (Table 3.3) with a variety of piezoelectric anisotropies. The results of the rigorous lower and upper bounds determined for the 1–3 ceramic / polymer composites can lead to a further comparison of the effective electromechanical properties of piezo-composites with different microgeometries. Such a comparison, for example, would be important when predicting the effective electromechanical properties of composites with one or two piezo-active components (e.g., ceramic and polymer or single crystal and ceramic) with inclusions of variable aspect ratio.

174

Electromechanical Properties in Composites Based on Ferroelectrics

* * * Table 6.7. Piezoelectric coefficients d 3l* (in pC / N), 102 e31 and e33 (in C / m2), g 3l (in

* * *[ mV m / N), 102 h31 and h33 (in 109 V / m), dielectric impermeability E 33 (in 109 m / F), *V dielectric permittivity H 33 (in 10–10 F / m),a and rigorous bounds [25],b which have been

calculated for the 1–3 (Pb1–xCax)TiO3 ceramic / araldite composite with rods in the form of the rectangular parallelepiped x

mc

* d 31

* d 33

* 102 e31

* e33

* g 31

0.20

–5.91 –6.02 –5.59 –5.01 –4.39 –3.75 –3.12

19.3 21.9 23.0 23.5 23.9 24.1 24.3 * 102 h31

0.809 1.59 2.55 3.79 5.47 7.87 11.6

0.402 0.803 1.21 1.61 2.01 2.42 2.83

–39.3 –22.6 –14.6 –10.1 –7.15 –5.14 –3.69

x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 mc

* h33

E33*[

H 33*V

0.20

0.1

128 82.5

0.3

60.2

0.4

47.3

0.5

39.0

0.6

33.1

0.7

28.8

mc

7.00 (6.99, 7.01) 4.02 (4.02, 4.03) 2.82 (2.82, 2.83) 2.18 (2.17, 2.18) 1.77 (1.77, 1.77) 1.49 (1.49, 1.49) 1.29 (1.29, 1.29)

x

* d 31

* d 33

2.81 (2.80, 2.89) 3.23 (3.23, 3.33) 3.40 (3.40, 3.52) 3.50 (3.49, 3.62) 3.56 (3.56, 3.70) 3.61 (3.60, 3.75) 3.65 (3.65, 3.80) * 102 e31

1.51

0.2

5.66 (4.41, 25.5) 6.40 (5.65, 31.4) 7.20 (6.71, 35.4) 8.24 (7.92, 39.2) 9.67 (9.46, 43.2) 11.7 (11.6, 47.7) 14.9 (14.8, 53.0)

* e33

* g 31

0.25

–6.55 –6.55 –5.94 –5.16 –4.34 –3.51 –2.69

22.0 25.0 26.1 26.8 27.2 27.5 27.7 * 102 h31

1.23 2.43 3.89 5.78 8.35 12.0 17.7

0.460 0.920 1.38 1.84 2.31 2.78 3.26

–37.5 –20.8 –13.0 –8.65 –5.88 –3.99 –2.63

x

0.1 0.2 0.3 0.4 0.5 0.6 0.7 mc

* h33

E33*[

H 33*V

0.25

0.1

126 79.1

0.3

57.3

0.4

44.9

0.5

36.8

2.79 (2.78, 2.91) 3.14 (3.14, 3.29) 3.29 (3.28, 3.45) 3.37 (3.36, 3.54) 3.42 (3.41, 3.61)

6.06 (6.06, 6.08) 3.42 (3.41, 3.43) 2.38 (2.38, 2.39) 1.82 (1.82, 1.83) 1.48 (1.48, 1.49)

1.75

0.2

7.48 (5.81, 34.2) 8.29 (7.30, 41.2) 9.25 (8.62, 46.2) 10.6 (10.1, 51.0) 12.4 (12.1, 56.2)

* g 33

* g 33

2.66 3.82 4.97 6.13 7.29 8.45

3.15 4.56 5.97 7.38

Comparison of Results on Two-component Piezo-composites

175

Table 6.7 (continued) x

mc

* g 33

* 102 h31

* h33

E33*[

H 33*V

0.25

0.6

31.2 27.1

3.46 (3.46, 3.67) 3.60 (3.50, 3.72)

1.25 (1.24, 1.25) 1.08 (1.07, 1.08)

8.79

0.7

15.0 (14.8, 62.1) 19.0 (18.9, 69.1)

10.2

Notes to Table 6.7 a. Electromechanical constants determined by means of the matrix method [10, 11, 14] * *[ b. The rigorous lower and upper bounds given for h33 and E 33 in parentheses are published as a private communication, with permission of Prof. Dr. P. Bisegna (University of Rome “Tor Vergata”, Italy) c. Volume fraction of the ceramic component

6.4 Data on 0–3 Composites The effective electromechanical properties of the 0–3 ceramic / polymer composite (Figure 3.7) with a square arrangement of spheroidal inclusions in the (X1OX2) plane can be determined through different averaging procedures involving the Eshelby tensor S. These procedures are often concerned with the dilute approach, the EFM (generalised Mori – Tanaka approach) or the effective medium (selfconsistent) method [4, 6, 17]. The matrix of the effective electromechanical properties is written in the general form [4, 6, 26] as || C* || = || C(2) || + m (|| C(1) || – || C(2) ||) || A ||

(6.3)

where || A || is the mechanical strain – electric field concentration matrix that is expressed in terms of the matrices of the electromechanical properties of the components || C(n) || (n = 1 and 2) and of the Eshelby tensor || S || [7] and m is the volume fraction of the ceramic component (inclusions). The || C(n) || matrices from Equation 6.3 have a form similar to that in Equation 3.19, and n = 1 relates to the ceramic inclusion and n = 2 relates to the polymer matrix. The dilute approach neglects the electromechanical interaction between the components. As a consequence, the average electroelastic field in the composite is approximated by a field that appears in an isolated inclusion surrounded by a continuous matrix [6, 26, 27]. In this case Equation 6.3 is represented as || C* || = || C(2) || + m (|| C(1) || – || C(2) ||) [|| I || + || S || || C(2) ||-1 (|| C(1) || – || C(2) ||)]–1 (6.4) where || I || is the identity matrix. The EFM is based on an assumption [4, 6, 26] that || A || from Equation 6.3 is determined from a solution of a micromechanical problem for a single piezo-active inclusion in a continuous matrix subjected to an applied electroelastic field that is equal to an average field in the matrix of the 0–3 composite. Its effective electromechanical properties are determined using Equation 3.20. The effective medium method [4, 6, 26, 27] enables one to consider a single inclusion in a homogenised or effective medium (i.e., a matrix with similar

176

Electromechanical Properties in Composites Based on Ferroelectrics

inclusions), and the effective properties of this system are determined by taking into account the electromechanical interaction between the inclusion and the surrounding medium. In this case the || A || matrix from Equation 6.3 is written in the form || A || = [|| I || + || S || || C* ||-1 (|| C(1) || – || C* ||)]–1

(6.5)

where || C* || characterises the effective properties of the composite. The averaging procedure becomes self-consistent and needs iterations to solve Equation 6.3, wherein the || A || and || S || matrices (see Equation 6.5) depend on the elements of the || C* || matrix. In the section below, numerical results predicted using the aforementioned averaging procedures are compared. The volume-fraction dependences of the effective parameters from Table 6.82 agree with those shown in Figure 6.4. It should be mentioned that refs. [6, 7] devoted to piezo-composites comprise different sets of formulae for the Eshelby tensor S calculated for the spheroidal inclusion. Nevertheless, the difference between the calculated values from Table 6.8 and Figure 6.4 remains less than 5%. A good correlation is observed in wide m and c / a ranges. The effective piezoelectric coefficients of the studied 0–3 composite obey conditions | d 3* j |DIL < | d 3* j |MT < | d 3* j |SC and | e3* j |DIL < | e3* j |MT < | e3* j |SC (Figure 6.4) where subscripts “DIL”, “MT” and “SC” denote the dilute approach, the EFM (Mori – Tanaka approach) and the effective medium (or selfconsistent) method, respectively. A transition from the non-interacting components to interacting components results in higher piezoelectric activity in the 0–3 composite. We see that the electromechanical interaction between the components is more appreciable at c / a o 1 (Table 6.8 and Figure 6.4) and leads to small differences between the constants evaluated by different methods at c / a >> 1. This distinction is closely connected with the internal electric and mechanical fields that are highly dependent on the aspect ratio of the ceramic inclusions. The case of c / a = 1 corresponds to the composite with the spherical inclusions wherein the depolarisation and stress fields cannot be neglected. In the case of c / a = 103 the presence of the needle-shaped inclusions poled along the OX3 axis leads to a removal of the depolarisation field, however, the stress field remains essential and plays an important role in forming the piezoelectric properties in the composite. Apparently, such a stress field would not affect equalities | d 3* j |DIL | | d 3* j |MT | | d 3* j |SC and | e3* j |DIL | | e3* j |MT | | e3* j |SC that hold at c / a = 103 (see data from Table 6.8, Figure 6.4 and ref. [6]). Agreement between the piezoelectric properties determined through different procedures is also observed when comparing the aspect-ratio dependences shown * * * in Table 6.9 and Figure 6.5. The piezoelectric coefficients e33 and eh* = e33 + 2 e31 ____________

The aspect ratio c / a introduced by Fakri et al. [6] equals U -1 = a3 / a1, where a3 = c and a1 = a are semi-axes of the spheroidal inclusion shown in Figure 3.7. 2

Comparison of Results on Two-component Piezo-composites

177

* Table 6.8. Volume-fraction dependences of the piezoelectric coefficients d 31 (in pC / N) * * (1) / e33 and normalised hydrostatic and e31 (in C / m2), normalised piezoelectric coefficient e33

piezoelectric coefficient eh* / eh(1) , which have been calculated for the 0–3 PZT-type ceramic / epoxy composite with spheroidal inclusions Ceramic component PZT-5

Ceramic component PZT-5

Ceramic component PZT-4

Ceramic component PZT-4

m

* d 31

* d 31

a

0.05 0.1 0.2 0.3 0.4 0.5 m

(c / a = 1) –0.132c –0.239 –0.405 –0.526 –0.619 –0.692

(c / a = 10) –5.36d –8.12 –11.0 –12.6 –13.6 –14.3

(c / a = 103)a –50.2e –76.3 –104 –119 –129 –136

* d 31

* d 31

* d 31

b

0.05 0.1 0.2 0.3 0.4 0.5 m 0.05 0.1 0.2 0.3 0.4 0.5 m 0.05 0.1 0.2 0.3 0.4 0.5

* d 31

a

b

(c / a = 10) –5.70g –9.14 –14.0 –18.1 –22.5 –27.9 * (1) / e33 e33

(c / a = 103)b –50.3h –76.6 –105 –122 –133 –142 eh* / eh(1)

* e31

(c / a = 103)a 0.0608i 0.122 0.243 0.365 0.487 0.608 * (1) e33 / e33

(c / a = 103)a 0.188j 0.375 0.750 1.12 1.50 1.88 * eh / eh(1)

(c / a = 103)b –0.0198 –0.0416 –0.0926 –0.157 –0.240 –0.352

(c / a = 103)b 0.0608k 0.122 0.242 0.363 0.483 0.601

(c / a = 103)b 0.187l 0.373 0.740 1.10 1.45 1.7

(c / a = 1) –0.144f –0.286 –0.576 –0.893 –1.27 –1.74 * e31

(c / a = 103)a –0.0189 –0.0377 –0.0754 –0.113 –0.151 –0.189

Notes to Table 6.8 a. Calculated within the framework of the dilute approach (see Equation 6.4) b. Calculated by means of the EFM (see Equation 3.20) * c. The calculated d 31 (m) dependence is to be compared with that shown in Figure 6.4a (DIL curve) * d. The calculated d 31 (m) dependence is to be compared with that shown in Figure 6.4b (DIL curve) * e. The calculated d 31 (m) dependence is to be compared with that shown in Figure 6.4c (DIL curve)

178

Electromechanical Properties in Composites Based on Ferroelectrics

* f. The calculated d 31 (m) dependence is to be compared with that shown in Figure 6.4a (MT curve) * g. The calculated d 31 (m) dependence is to be compared with that shown in Figure 6.4b (MT

curve) * h. The calculated d 31 (m) dependence is to be compared with that shown in Figure 6.4c (MT curve) * (1) i. The calculated e33 (m) / e33 dependence is to be compared with that shown in Figure 6.4d (lower DIL curve) j. The calculated eh* (m) / eh(1) dependence is to be compared with that shown in Figure 6.4d (upper DIL curve) * (1) k. The calculated e33 (m) / e33 dependence is to be compared with that shown in Figure 6.4d (lower MT curve) l. The calculated eh* (m) / eh(1) dependence is to be compared with that shown in Figure 6.4d (upper MT curve)

a Figure 6.4. Effective piezoelectric coefficients calculated for the 0–3 PZT-5 ceramic / epoxy (a – c) and PZT-4 ceramic / epoxy (d) composites within the framework of the dilute approach (DIL curves), the EFM (MT curves) and the effective medium method (SC * * (1) curves): d 31 (in pC / N) at c / a = 1 (a), 10 (b) and 103 (c), e33 (m) / e33 and eh* (m) / eh(1) at c / a = 103 (d). In graph (a) bold dots correspond to experimental data by Furukawa et al. [28] (reprinted from Fakri et al. [6], with permission from Elsevier)

Comparison of Results on Two-component Piezo-composites

b

c

d Figure 6.4 (continued)

179

180

Electromechanical Properties in Composites Based on Ferroelectrics

* (1) Table 6.9. Normalised piezoelectric coefficients e33 / e33 and eh* / eh(1) that have been calculated for the 0–3 PZT-4 ceramic / epoxy composite with spheroidal inclusions

c/a

* (1) / e33 e33

eh* / eh(1) a

* (1) / e33 e33

eh* / eh(1)

(m = 0.2) 1.26u10-4 d 2.15u10-3 1.94u10-2 4.18u10-2 8.90u10-2 0.281 0.582 0.649 0.694 0.733 0.747 0.749 0.750 * eh / eh(1)

(m = 0.5) 7.32u10-4 e 2.42u10-3 1.67u10-2 3.51u10-2 7.36u10-2 0.229 0.473 0.527 0.563 0.595 0.607 0.608 0.608 * (1) e33 / e33

(m = 0.5)a 3.16u10-4 f 5.38u10-3 4.85u10-2 0.104 0.223 0.702 1.46 1.62 1.73 1.83 1.87 1.87 1.87

c/a

(m = 0.2) 2.93u10-4 c 9.66u10-4 6.70u10-3 1.40u10-2 2.95u10-2 9.16u10-2 0.189 0.211 0.225 0.238 0.243 0.243 0.243 * (1) e33 / e33

1 2 5 7 10 20 50 70 100 200 500 700 1000

(m = 0.2)b 4.48u10-4 g 1.46u10-3 9.61u10-3 1.95u10-2 3.88u10-2 0.107 0.198 0.216 0.228 0.238 0.242 0.242 0.243

(m = 0.2)b 2.01u10-4 h 3.21u10-3 2.75u10-2 5.73u10-2 0.116 0.326 0.603 0.659 0.695 0.727 0.738 0.740 0.741

(m = 0.5)b 2.70u10-3 i 8.38u10-3 4.85u10-2 9.00u10-2 0.160 0.348 0.528 0.559 0.578 0.595 0.600 0.601 0.602

(m = 0.5)b 1.33u10-3 j 1.79u10-2 0.135 0.257 0.466 1.03 1.56 1.66 1.71 1.76 1.78 1.78 1.78

1 2 5 7 10 20 50 70 100 200 500 700 1000

a

a

eh* / eh(1)

Notes to Table 6.9 a. Calculated within the framework of the dilute approach (see Equation 6.4) b. Calculated by means of the EFM (see Equation 3.20) * (1) c. The calculated e33 (c / a) / e33 dependence is to be compared with that shown in Figure 6.5a (lower DIL curve) d. The calculated eh* (c / a) / eh(1) dependence is to be compared with that shown in Figure 6.5a (upper DIL curve) * (1) e. The calculated e33 (c / a) / e33 dependence is to be compared with that shown in Figure 6.5b (lower DIL curve) f. The calculated eh* (c / a) / eh(1) dependence is to be compared with that shown in Figure 6.5b (upper DIL curve) * (1) g. The calculated e33 (c / a) / e33 dependence is to be compared with that shown in Figure 6.5a (lower MT curve)

Comparison of Results on Two-component Piezo-composites

181

h. The calculated eh* (c / a) / eh(1) dependence is to be compared with that shown in Figure 6.5a (upper MT curve) * (1) i. The calculated e33 (c / a) / e33 dependence is to be compared with that shown in Figure 6.5b (lower MT curve) j. The calculated eh* (c / a) / eh(1) dependence is to be compared with that shown in Figure 6.5b (upper MT curve)

* (1) Figure 6.5. Normalised piezoelectric coefficients e33 / e33 and eh* / eh(1) of the 0–3 PZT-4 ceramic / epoxy composite at ceramic volume fractions m = 0.2 (a) and 0.5 (b). Calculations were made within the framework of the dilute approach (DIL curves), the EFM (MT curves) and the effective medium method (SC curves) (reprinted from Fakri et al. [6], with permission from Elsevier)

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of the 0–3 ceramic / polymer composite are now studied when the aspect ratio U is varied from 1 (spherical inclusions with c / a = 1) to 10–3 (needle-shaped inclusions with c / a = 103). The comparison of the dependences (Table 6.9 and Figure 6.5) shows that a difference between the parameters calculated using different formulae for the Eshelby tensor S does not exceed 5% at 1 d c / a d 103, i.e., 10–3 d U d 1 in our notations. This conclusion is valid for the parameters calculated on the basis of the dilute approach (Equation 6.4) and the EFM (Equation 3.20) at m = const. It is assumed that the ceramic volume fraction m is taken from a range of 0 < m d 0.5, i.e., low and moderate volume fractions can be considered. Thus, our results on the 0–3 ceramic / polymer composites with spheroidal inclusions suggest that the EFM can be reliably used to predict the electromechanical properties in the wide m and U ranges. Moreover, the transformation of the 0–3 structure shown in Figure 3.7 leads to the 2–2 or 1–3 composite (see Section 6.2). The effective properties in the 2–2 or 1–3 composites are determined in limiting cases, for the aspect ratios U o 0 or U o f, respectively. It should be noted that no jumps are observed in the aspect-ratio dependences of the piezoelectric and dielectric properties (Tables 6.1–6.4) at m = const when approaching U o 0 or U o f. In general, based on data on asymptotic behaviour of the effective properties in the 0–3 ceramic / polymer composites, one can conclude that the EFM is a reliable approach for determining the volumefraction and aspect-ratio dependences of the effective electromechanical properties in these materials. A further comparison of data on the effective electromechanical constants of 0–3 piezo-composites can be made using the FEM.

References [1] [2] [3] [4] [5] [6] [7] [8]

Khoroshun LP, Maslov BP, Leshchenko PV, (1989) Prediction of Effective Properties of Piezo-active Composite Materials. Naukova Dumka, Kiev (in Russian) Sokolkin YuV, Pan’kov AA, (2003) Electroelasticity of Piezo-composites with Irregular Structures. Fizmatlit, Moscow (in Russian) Luchaninov AG, (2002) Piezoelectric Effect in Non-polar Heterogeneous Ferroelectric Materials. Volgograd State Academy of Architecture and Construction, Volgograd (in Russian) Dunn ML, Taya M, (1993) Micromechanics predictions of the effective electroelastic moduli of piezoelectric composites. International Journal of Solids and Structures 30:161–175 Levin VM, Rakovskaja MI, Kreher WS, (1999) The effective thermoelectroelastic properties of microinhomogeneous materials. International Journal of Solids and Structures 36:2683–2705 Fakri N, Azrar L, El Bakkali L, (2003) Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements. International Journal of Solids and Structures 40:361–384 Huang JH, Yu S, (1994) Electroelastic Eshelby tensors for an ellipsoidal piezoelectric inclusion. Composites Engineering 4:1169–1182 Chan HLW, Unsworth J, (1989) Simple model for piezoelectric ceramic / polymer 1– 3 composites used in ultrasonic transducer applications. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 36:434–441

Comparison of Results on Two-component Piezo-composites [9] [10] [11] [12] [13]

[14] [15] [16]

[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27]

183

Akcakaya E, Farnell GW, (1988) Effective elastic and piezoelectric constants of superlattices. Journal of Applied Physics 64:4469– 4473 Glushanin SV, Topolov VYu, (2001) Features of electromechanical properties of piezoelectric composites with elements of connectivity 1–1. Journal of Physics D: Applied Physics 34:2518–2529 Topolov VYu, Glushanin SV, (2002) Evolution of connectivity patterns and links between interfaces and piezoelectric properties of two-component composites. Journal of Physics D: Applied Physics 35:2008–2014 Topolov VYu, Bisegna P, Krivoruchko AV, (2008) Features of electromechanical properties of 1–3 composites based on PbTiO3-type ceramics. Journal of Physics D: Applied Physics 41:035406 – 8 p Berger H, Kari S, Gabbert U, Rodríguez-Ramos R, Bravo-Castillero J, GuinovartDíaz R, (2005) A comprehensive numerical homogenization technique for calculating effective coefficients of uniaxial piezoelectric fibre composites. Materials Science and Engineering A 412:53–60 Levassort F, Topolov VYu, Lethiecq M, (2000) A comparative study of different methods of evaluating effective electromechanical properties of 0–3 and 1–3 ceramic / polymer composites. Journal of Physics D: Applied Physics 33:2064–2068 Nelli Silva EC, Ono Fonseca JS, Kikuchi N, (1997) Optimal design of piezoelectric microstructures. Computational Mechanics 19:397–410 Avellaneda M, Swart PJ, (1994) The role of matrix porosity and Poisson’s ratio in the design of high-sensitivity piezocomposite transducers. In: Garcia E, Cudney H, Das Gupta A (eds.) Adaptive Structures and Composite Materials: Analysis and Applications. American Society of Mechanical Engineers, New York, AD 45:59–66 Kar-Gupta R, Venkatesh T A, (2007) Electromechanical response of 1–3 piezoelectric composites: An analytical model. Acta Materialia 55:1093–1108 Smith WA, (1993) Modeling 1–3 composite piezoelectrics: hydrostatic response. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 40:41–49 Taunaumang H, Guy IL, Chan HLW, (1994) Electromechanical properties of 1–3 piezoelectric ceramic / piezoelectric polymer composites. Journal of Applied Physics 76:484– 4 89 Bisegna P, Luciano R, (1996) Variational bounds for the overall properties of piezoelectric composites. Journal of the Mechanics and Physics of Solids 44:583–602 Bisegna P, Luciano R, (1997) On methods for bounding the overall properties of periodic piezoelectric fibrous composites. Journal of the Mechanics and Physics of Solids 45:1329–1356 Hashin Z, Shtrikman S, (1962) On some variational principles in anisotropic and nonhomogeneous elasticity. Journal of the Mechanics and Physics of Solids 10:335– 342 Jensen H, (1991) Determination of macroscopic electro-mechanical characteristics of 1–3 piezoceramic / polymer composites by a concentric tube model. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control 38:591–594 Grekov AA, Kramarov SO, Kuprienko AA, (1989) Effective properties of a transversely isotropic piezoelectric composite with cylindrical inclusions. Mechanics of Composite Materials, 25:54–61 Bisegna P, (2007) Private communication Dunn M, Micromechanics of coupled electroelastic composites: Effective thermal expansion and pyroelectric coefficients. Journal of Applied Physics 73:5131–5140 Jiang B, Fang D-N, Hwang K-C, (1999) A unified model for piezocomposites with non-piezoelectric matrix and piezoelectric ellipsoidal inclusions. International Journal of Solids and Structures 36:2707–2733

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[28] Furukawa T, Fujino K, Fukada E, (1976) Electromechanical properties in the composites of epoxy resin and PZT ceramics. Japanese Journal of Applied Physics 15:2119–2129

7 Conclusions Eigentlich weiE man nur, wenn man wenig weiE, mit dem Wissen wächst der Zweifel. J.W. von Goethe From a practical point of view, the physical world only matters in so far as it affects us, and the intrinsic nature of what goes on in our absence is irrelevant, provided we can predict the effects upon ourselves. This we can do, just as a person can use a telephone without understanding electricity. Only the most abstract knowledge is required for practical manipulation of matter. But there is a grave danger when this habit of manipulation based upon mathematical laws is carried over into our dealings with human beings, since they, unlike the telephone wire, are capable of happiness and misery, desire and aversion. B. Russell

7.1 From Components to Effective Electromechanical Properties This book has demonstrated to the reader that the analysis of the effective electromechanical properties must take into account a number of factors (components, connectivity, microgeometry, etc.) and the variety of averaging procedures that are available. These procedures become complicated due to electromechanical coupling in the presence of piezo-active components. The role of the ferroelectric components has been considered in detail, demonstrating that their remarkable piezoelectric properties are fundamental in forming the piezoelectric response of the composites with various connectivity patterns and / or different combinations of the components. Our study enables us to conclude that both the ferroelectric ceramic and single-crystal components within piezoelectric composites lead to considerable piezoelectric activity, sensitivity, anisotropy, and hydrostatic piezoelectric response in specific volume-fraction ranges. These properties have been determined for specific composite architectures, connectivity and components. In a number of cases, conventional ferroelectric ceramic / polymer composites with 1–3, 2–2 or 0–3 connectivity have electromechanical properties that can be further improved by using the latest relaxor-ferroelectric single crystals with engineered-domain structures and very high piezoelectric activity. Additional possibilities of improving the performance of the abovementioned ceramic / polymer composites are concerned with porous structures [1], auxetic polymer components [2, 3] or the addition of a third component, e.g., ferroelectric polymer, ceramic or single crystal with specific electromechanical properties. Furthermore, advanced methods of manufacturing (rapid prototyping [4], selective laser sintering, etc.) also often lead to better performance of the

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piezo-composites and enable us to control their microgeometry and sizes of several microstructural elements more effectively. The main steps of our analysis are schematically shown in Figure 7.1.

Figure 7.1. Outline of the study on the piezo-composites

The diagram will be reinforced by new important items associated with future research results and applications of advanced piezo-active and other composites. In this connection we would like to finish by quoting a review paper [5]: “As niche applications become more prevalent in the future, composites and displacementamplifying techniques and materials will proliferate in a continuing effort to widen the force–displacement envelope of performance. These devices, too, will become smarter and smarter as the applications demand”.

References [1] [2] [3] [4] [5]

Topolov VYu, Turik AV, (2001) Porous piezoelectric composites with extremely high reception parameters. Technical Physics 46:1093–1100 Evans KE, Alderson A, (2000) Auxetic materials: functional materials and structures from lateral thinking. Advanced Materials 12:617–628 Topolov VYu, Bowen CR, (2008) Characteristics of 1–3-type ferroelectric ceramic / auxetic polymer composites. Modelling and Simulation in Materials Science and Engineering 16:015007 – 12 p Safari A, Akdogan EK, (2006) Rapid prototyping of novel piezoelectric composites. Ferroelectrics 331:153–179 Haertling GH, (1999) Ferroelectric ceramics: history and technology. Journal of the American Ceramic Society 82:797–818

Appendix Formulae for Effective Electromechanical Constants of 2–2 and 1–3 Piezo-composites

2–2 series-connected composites with two piezo-active transversely isotropic components (Figure 3.1a) distributed stochastically on the OX3 direction are characterised [1] by the effective elastic moduli cij*E , piezoelectric coefficients ekl* and dielectric susceptibilities F *pp[ as follows: ( 2 ),[ (1),[ m1 + F33 m2) c11*E = c11(1), E m1 + c11( 2),E m2 – (m1 m2 /')[( c13(1),E – c13( 2),E )2 ( F33 (1), E ( 2 ),E (1) ( 2) ( 2) (1) (1) ( 2) 2 ( 2 ),E + 2( c13 – c13 )( e31 – e31 )( e33 m1 + e33 m2 ) – ( e31 – e31 ) ( c33 m1 + c33(1),E m2)], ( 2 ),[ (1),[ m1 + F33 m2) c12*E = c12(1),E m1 + c12( 2),E m2 – (m1 m2 /')[( c13(1),E – c13( 2),E )2 ( F33 (1), E ( 2 ),E (1) ( 2) ( 2) (1) (1) ( 2) 2 ( 2 ),E + 2( c13 – c13 )( e31 – e31 )( e33 m1 + e33 m2 ) – ( e31 – e31 ) ( c33 m1 + c33(1),E m2)],

c13*E = c13(1),E m1 + c13( 2),E m2 – (m1 m2 /'){( c13(1),E – c13( 2),E )[( c33(1),E – c33( 2),E )( F33( 2),[ m1 (1),[ + F33 m2) + ( e33(1) – e33( 2) )( e33( 2) m1 + e33(1) m2)] – ( e31(1) – e31( 2) )[( e33(1) – e33( 2) )( c33( 2),E m1 + c33(1),E m2) – ( c33(1),E – c33( 2),E )( e33( 2) m1 + e33(1) m2)]}, *E ( 2 ),[ (1),[ = c33(1),E m1 + c33( 2),E m2 – (m1 m2 /')[( c33(1),E – c33( 2),E )2 ( F33 m1 + F33 m2) c33 (1), E ( 2 ),E (1) ( 2) ( 2) (1) (1) ( 2) 2 ( 2 ),E (1), E + 2( c33 – c33 )( e33 – e33 )( e33 m1 + e33 m2) – ( e33 – e33 ) ( c33 m1 + c33 m2)], *E (1),E ( 2 ),E (1),E ( 2 ),E 2 ( 2 ),E (1),E = c44 m1 + c44 m2 – m1 m2[( c44 – c44 ) / ( c44 m1 + c44 m2)], c44 * ( 2 ),[ (1) ( 2) (1) ( 2) (1), E ( 2 ),E e31 = e31 m1 + e31 m2 – (m1 m2 /'){( e33 – e33 )[( c13 – c13 )( F33 m1

(1),[ ( 2 ),[ (1),[ (1),[ + F33 mB2) + F33 m1 + F33 m2) + ( e31(1) – e31( 2) )( e33( 2) m1 + e33(1) m2) + ( F33 ( 2 ),[ – F33 )[( e31(1) – e31( 2) )( c33( 2),E m1 + c33(1),E m2) – ( c13(1),E – c13( 2),E )( e33( 2) m1 + e33(1) m2)]}, * ( 2 ),[ = e33(1) m1 + e33( 2) m2 – (m1 m2 /'){( e33(1) – e33( 2) )[( c33(1),E – c33( 2),E )( F33 m1 e33 (1),[ (1),[ ( 2 ),[ + F33 m2) + ( e33(1) – e33( 2) )( e33( 2) m1 + e33(1) m2)] + ( F33 – F33 )[( e33(1) – e33( 2) )( c33( 2),E m1 + c33(1),E m2) – ( c33(1),E – c33( 2),E )( e33( 2) m1 + e33(1) m2)]},

188

Appendix ( 2 ),E ( 2 ),E (1),E ) / ( c44 m1 + c44 m2)], e15* = e15(1) m1 + e15( 2) m2 – m1 m2[( e15(1) – e15( 2) )( c44(1),E – c44 *[ (1),[ ( 2 ),[ ( 2 ),E (1),E (1) ( 2) 2 F11 = F11 m1 + F11 m2 + m1 m2[( e15 – e15 ) / ( c44 m1 + c44 m2)], and

F33*[ = F33(1),[ m1 + F33( 2),[ m2 – (m1 m2 /')[( F33(1),[ – F33( 2),[ )2 ( c33( 2),E m1 + c33(1),E m2) (1),[ ( 2 ),[ ( 2 ),[ (1),[ m1+ F33 m2)], + 2( F33 – F33 )( e33(1) – e33( 2) )( e33( 2) m1 + e33(1) m2) – ( e33(1) – e33( 2) )2( F33 (n ),[ where mn, cij( n), E , ekl(n ) , and F pp are the volume fraction, elastic moduli,

piezoelectric coefficients, and dielectric susceptibilities of the nth component, ( 2 ),[ respectively, n = 1 and 2, m2 = 1 – m1, and ' = ( c33( 2),E m1 + c33(1),E m2)( F33 m1 + (n ),[ and dielectric F33(1),[ m2) + ( e33( 2) m1 + e33(1) m2)2. Dielectric susceptibility F pp ),[ permittivity H (n of the strained nth component are interrelated as follows: pp (n ),[ ),[ = H (n – 1. The like relation, F *pp[ = H *pp[ – 1, also holds for the composite. F pp pp

1–3 composites with two piezo-active transversely isotropic components (i.e., long cylindrical rods distributed regularly in a continuous matrix, Figure 3.2) are characterised [2] by the effective elastic moduli cij*E , piezoelectric coefficients dkl* and ekl* and dielectric permittivities H *ppV as follows:

c11*E + c12*E = {( c11( 2),E – c12( 2),E ) < c11E + c12E > + ( c11(1),E + c12(1),E )( c11( 2),E + c12( 2),E )}c–1, –1 c13*E = {2m c13(1),E c11( 2),E + (1 – m) c13( 2),E ( c11(1),E + c12(1),E + c11( 2),E – c12( 2),E )}c , *E = < c33E > – 2m(1 – m)( c13(1),E – c13( 2),E )2 c–1, c33 *E V *E = c55 = c55( 2),E {( c55(1),E + < c55E >)( H11( 2),V + [ H 11 ]) – (1 – m)2 c55(1),E c55( 2),E ( d15(1) c44 V ]) – (1 – m)2 c55(1),E c55( 2),E ( d15(1) – d15( 2) )2}–1, – d15( 2) )2}{ c55( 2),E + [ c55E ])( H11( 2),V + [ H 11 * = {(2 d 31(1) s11( 2),E m + d 31( 2) ( s11(1),E + s12(1),E + s11( 2),E – s12( 2),E )(1 – m))( s33( 2),E m d31 + s33(1),E (1 – m)) – 2( s13( 2),E m + s13(1),E (1 – m))( d 31(1) s13( 2),E m + d 31( 2) s13(1),E (1 – m)) + ( d 33(1) – d 33( 2) )(( s11(1),E + s12(1),E + s11( 2),E – s12( 2),E ) s13( 2),E – 2 s11( 2),E s13(1),E )m(1 – m)}A–1, * = {(( s11( 2),E + s12( 2),E )m + ( s11(1),E + s12(1),E )(1 – m) + s11( 2),E – s12( 2),E )( d 33(1) s33( 2),E m d33 + d 33( 2) s33(1),E (1 – m)) – 2( s13( 2),E m + s13(1),E (1 – m))( d 33(1) s13( 2),E m + d 33( 2) s13(1),E (1 – m)) + 2( d 31(1) – d 31( 2) )( s33(1),E s13( 2),E – s33( 2),E s13(1),E )m(1 – m)}A–1,

d15* = d15( 2) + 4m c55(1),E H11( 2),V ( d15(1) – d15( 2) ){( c55(1),E + < c55E >)( H11( 2),V + [ H 11V ]) – (1 – m)2 c55(1),E c55( 2),E ( d15(1) – d15( 2) )2}–1, * = {2 e31(1) c11( 2),E m + e31( 2) ( c11(1),E + c12(1),E + c11( 2),E – c12( 2),E )(1 – m)}c–1, e31 * = e33(1) m + e33( 2) (1 – m) – 2m(1 – m)( e31(1) – e31( 2) )( c13(1),E – c13( 2),E )c–1, e33

H11*V = H11( 2),V { H11(1),V + < H11V >)( c55(1),E + < c55E >) – (1 – m2) c55(1),E c55( 2),E ( d15(1) V ])( c55(1),E + < c55E >) – (1 – m2) c55(1),E c55( 2),E ( d15(1) – d15( 2) )2}–1, and – d15( 2) )2}{ H11(1),V + [ H11

Appendix

189

H 33*V = H 33(1),V m + H 33( 2),V (1 – m) – m(1 – m){( d 33(1) – d33( 2) )sc+ 2( d 31(1) – d31( 2) )P2[ s33E ] – 4( d 31(1) – d 31( 2) )( d 33(1) – d 33( 2) )[ s13E ]A–1 (n ),V where cij( n ), E , sij( n ), E , ekl(n ) , d kl(n ) , and H pp are the elastic moduli, elastic

compliances, piezoelectric stress coefficients, piezoelectric charge coefficients, and dielectric permittivities of the nth component, respectively, m is the volume fraction of the rods, [X] = mX(2) + (1 – m)X(1), = mY(1) + (1 – m)Y(2), A = ([ s11E + 2 s12E ] + s11( 2),E – s12( 2),E )[ s33E ] – 2[ s13E ] , c = c11( 2),E – c12( 2),E + m( c11( 2),E + c12( 2),E ) + (1 – m)( c11(1),E + c12(1),E ), n = 1 corresponds to the rods, and n = 2 corresponds to the matrix.

References [1] [2]

Khoroshun LP, Maslov BP, Leshchenko PV, (1989) Prediction of Effective Properties of Piezo-active Composite Materials. Kiev: Naukova Dumka (in Russian) Grekov AA, Kramarov SO, Kuprienko AA, (1989) Effective properties of a transversely isotropic piezoelectric composite with cylindrical inclusions. Mechanics of Composite Materials, 25:54–61

List of Abbreviations

AHM, asymptotic homogenisation method APC, perovskite-type piezoelectric ceramic BURPS, burned-out plastic spheres (technique) CS, corn starch DIL, dilute approach EFM, effective field method FC, ferroelectric ceramic FEM, finite-element method Hex, hexagonal arrangement of rods MPB, morphotropic phase boundary MT, effective field method (Mori – Tanaka method applied to piezo-active media) PCR, piezoelectric ceramic from Rostov-on-Don (Russia) PEO, polyethylene oxide (PEO) PKI, perovskite-type piezoelectric ceramic PMMA, polymethyl methacrylate PMN–xPT, relaxor-ferroelectric solid solutions of (1 – x)Pb(Mg1/3Nb2/3)O3 – xPbTiO3 PVDF, polyvinylidene fluoride PZ, perovskite-type piezoelectric ceramic PZN–xPT, relaxor-ferroelectric solid solutions of (1 – x)Pb(Zn1/3Nb2/3)O3 – xPbTiO3 PZT, piezoelectric ceramic of the Pb(Zr, Ti)O3 type SC, effective medium (self-consistent) method Squ, square arrangement of rods SR, self-raising flour ZTS, ZTSSt, ZTStBS, piezoelectric ceramics of the Pb(Zr, Ti)O3 type

About the Authors

Vitaly Yuryevich Topolov was born in Rostov-on-Don, Russia (former USSR) on November 8th, 1961. He received the qualification “Physicist. Educator” (honours degree) in 1984 along with the degrees “Candidate of Sciences (Physics and Mathematics)” and “Doctor of Sciences (Physics and Mathematics)” in 1987 and 2000, respectively, all from the Rostov State University, Russia. From 1987 to 1991, he worked as a research scientist of the Institute of Physics at the Rostov State University. From 1991 to 2000, he worked as a Senior Lecturer (1991–1992) and an Associate Professor (1992–2000) of the Department of Physics at the Rostov State University. Since 2000, he has been Professor of the same Department. Since December 2006, after reforming the Rostov State University, he is Professor of the Department of Physics at the Southern Federal University (Rostov-on-Don, Russia). Dr. V. Yu. Topolov was also a visiting scientist at the Moscow State University, Russia (former USSR, 1989), University of Saarland, Germany (1994– 1995), Aachen University of Technology – RWTH Aachen, Germany (1998), Karlsruhe Research Center, Germany (2002 and 2003–2004), University of Bath, UK (2006 and 2007), and University of Rome “Tor Vergata”, Italy (2008). His research interests include heterogeneous ferroelectrics, smart materials, domain and heterophase structures, as well as electromechanical effects in ferroelectrics and related materials. He earned the special award from the International Science Foundation (1993) and the Soros Associate Professor title and awards from the International Soros Science-Educational Program and the Open Society Institute (1997, 1998, 2000, and 2001). He presented the best poster at the International Symposium on Ferroelectric Domains (China, 2000) and the best training aids in the Department of Physics, Southern Federal University (Russia, 2006). He is the author of about 280 scientific papers, reviews, conference proceedings, and abstracts. Dr. V. Yu. Topolov has been included in the list of Active Russian Scientists whose papers have been cited over 100 times in periodicals from 2000 to 2006. The biographical sketch by Dr. V. Yu. Topolov was first published in the 26th edition of “Dictionary of International Biography” (IBC, Cambridge, 1998), p. 465.

194

About the Authors

Christopher Rhys Bowen was born on January 18th, 1968 and grew up in Beddau, South Wales (UK). He earned a BSc (First Class) in Materials Science at the School of Materials, University of Bath, UK in 1990 and worked on his DPhil thesis in ceramics under the supervision of Prof. Brian Derby in the Department of Materials, University of Oxford, UK in 1990–1993 (PhD awarded in 1994). In 1993 – 1994, he worked as a Researcher in the Advanced Ceramics Group at the Technical University of Hamburg-Harburg (TUHH), Germany under the supervision of Prof. Nils Claussen. From 1994 to 1996, he was research fellow at the School of Materials, University of Leeds, UK. From 1996 to 1998, he was a Senior Scientist at DERA, Functional Materials Group, UK. He joined the University of Bath in August 1998 and is now a Reader at the same University. The research interests of Dr. C. R. Bowen are concerned with functional ceramics, including ferroelectric ceramics and composites for modern sensors and actuators, as well as with manufacturing and characterisation of these materials. He earned the Thornton and Hazelwood prizes for academic work (1986–1990), Institute of Materials National Lecture Competition award (1993), SET award (2002), and John Willis award for excellence in research and teaching (2003). Dr. C. R. Bowen has published approximately 80 scientific papers, conference proceedings, and abstracts.

Index

anisotropy, 3, 28, 45, 58, 61, 132, 134, 141, 143, 174 dielectric, 71 elastic, 50, 61, 143, 170, 171 piezoelectric, 3, 16, 17, 18, 48, 49, 50, 52, 54, 58, 61, 62, 66, 67, 70, 71, 73, 75, 83, 89, 90, 102, 109, 110, 113, 116, 125, 131, 139, 140, 142, 154 large, 3, 17, 18, 49, 50, 54, 62, 66, 67, 71, 73, 74, 76, 102, 125, 127, 131, 141, 162, 172 applications, 1, 14, 25, 26, 34, 43, 46, 52, 54, 93, 124, 143, 153, 154, 186 acoustic antenna, 29 actuator, 1, 14, 29, 46, 53, 124, 154 hydroacoustic, 46, 62, 109 hydrophone, 8, 29, 46, 53, 63, 85, 105, 109, 116, 124, 154 mechanical engineering, 29 medical device, 46, 53, 62, 116, 124 non-destructive testing, 53 piezotechnical, 17, 18, 29, 34, 35, 62, 105, 116, 123, 124, 153 sensor, 1, 14, 29, 46, 62, 63, 85, 116, 124, 156

signal generation, 92 transducer, 1, 3, 29, 46, 53, 116, 124, 154 araldite, 48, 49, 50, 55, 56, 66, 68, 70, 73, 134, 141, 142, 143, 144, 145, 146, 149, 151, 152, 153, 161, 162, 163, 164, 172, 174, 175, 177 porous, 145, 146, 151, 152, 153 arrangement, 4, 5, 16, 26, 29, 30, 31, 48, 53, 54, 86, 102, 105, 162, 165 irregular, 63 rods, 53, 54, 142, 165 layers, 77 regular, 30, 51, 102, 171 hexagonal, 165 square, 54, 131, 132, 172, 173, 175 quasi-continuous, 76 aspect ratio, 64, 65, 66, 70, 73, 77, 78, 88, 111, 126, 129, 133, 150, 152, 153, 160, 162, 173, 176, 182 averaging procedure, 17, 22, 25, 27, 28, 29, 34, 63, 75, 77, 78, 82, 101, 111, 142, 159, 168, 171, 175, 176, 185 Banno unit cell, 26, 27, 30, 63, 82, 87, 104, 116, 117, 143 boundary conditions, 3, 25, 27, 29, 46, 47, 60, 82, 142, 143, 149

196

Index

ceramic, 1, 2, 3, 5, 17, 29, 43, 45, 46, 47, 49, 53, 58, 61, 63, 64, 65, 67, 71, 73, 76, 77, 79, 82, 83, 84, 86, 87, 89, 93, 94, 101, 102, 114, 116, 123, 131, 165, 174, 185 BaTiO3, 3, 16, 17, 18, 19, 24, 72 ferroelectric, 1, 2, 3, 5, 6, 7, 8, 14, 15, 16, 17, 18, 19, 20, 22, 25, 26, 27, 28, 29, 30, 34, 35, 43, 45, 48, 52, 62, 72, 101, 102, 115 “hard” PZT, 15 magnetostrictive, 24 modified PbTiO3, 19, 89, 92, 125, 131 (Pb, A)(Ti, B)O3-type, 66 Pb(Ca, Ti)O3, 19, 63, 67, 71, 72, 73, 76, 162, 164, 169 (Pb1–xCax)TiO3, see Pb(Ca, Ti)O3 (Pb1–xMex)TiO3, 66 PbTiO3-type, 17, 18, 48, 51, 54, 65, 70, 71, 75, 81, 83, 102, 126, 172, 173 Pb(Zr, Ti)O3-type, see PZT-type Pb(Zr1–xTix)O3, 3, 15, 18, 20, 43, 45, 46, 48, 53, 54, 61, 62, 72, 76, 89, 90 piezoelectric, 1, 15, 16, 20, 24, 26, 86, 88, 89, 90, 110, 111 porous, 8, 26, 27, 101, 102, 103, 104, 105, 109, 110, 111 classification, 105 pyroelectric, 1 PZT-type, 15, 22, 43, 45, 46, 48, 61, 65, 75, 81, 86, 101, 102, 103, 105, 106, 107, 108, 109, 110, 115, 133, 172, 177 “soft” PZT, 15 textured, 125 combination properties, 22, 23, 24 component, 1, 2, 3, 4, 5, 6, 9, 13, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 43, 44, 45, 46, 47, 52, 53, 54, 57, 58, 60, 61, 62, 63, 64, 65, 71, 75, 76, 77, 78, 79, 81, 82, 83, 85, 86, 87, 88, 89, 90, 92, 101, 106, 107, 108, 110, 111, 113, 116,

117, 118, 123, 124, 125, 126, 130, 131, 132, 133, 134, 141, 142, 143, 144, 145, 152, 153, 154, 161, 162, 164, 165, 167, 168, 169, 170, 171, 173, 175, 176, 177, 185, 186 highly anisotropic, 66, 71, 130, 131, 142 piezo-active, 17, 24, 25, 29, 30, 35, 43, 47, 54, 58, 61, 62, 63, 73, 76, 79, 87, 88, 89, 91, 101, 110, 116, 117, 131, 132, 134, 142, 155, 161, 164, 170, 172, 176, 185 piezo-passive, 5, 22, 24, 26, 28, 43, 47, 60, 61, 63, 65, 66, 78, 79, 81, 88, 110, 113, 116, 117, 132, 142 tensor, 25, 26, 65 vector, 25, 26, 60 composite, 1, 2 architecture, 92, 185 cable, 63 ceramic / ceramic, 29 ceramic / polymer, 7, 24, 27, 29, 30, 34, 35, 43, 45, 46, 48, 52, 53, 54, 61, 62, 63, 64, 65, 66, 70, 71, 72, 73, 75, 76, 77, 81, 83, 85, 86, 87, 89, 90, 92, 93, 104, 124, 132, 132, 133, 140, 141, 151, 160, 164, 165, 168, 169, 171, 172, 173, 175, 182, 185 classification, 4, 5 film, 63 high-performance, 154 interpenetrated, 4, 85, 87, 116 matrix, 4, 6 micrographs, 7, 8 modified layered, 110 piezo-active, see piezo-composite piezoelectric–piezomagnetic, 29 porous, 102, 103, 111, 114, 115 single crystal / ceramic, 5, 29, 125, 126, 132, 140, 160 single crystal / polymer, 35, 142, 143, 150, 151 structure, 4, 5, 26, 29, 30, 31, 32,

Index

45, 46, 53, 54, 60, 63, 76, 77, 78, 81, 82, 85, 86, 87, 89, 110, 111, 115, 160, 162, 165, 173, 182 chain-like, 54, 63 complicated, 4, 32, 85 fibrous, 4 granular, 4 irregular, 5 laminar, 4 periodic, see regular regular, 5, 26, 28, 30, 34, 51, 54, 63, 64, 82, 87, 88, 111, 132, 142, 150, 152, 159, 160, 165, 168, 171, 173 stochastic, 5, 26 three-component, 6, 110, 133, 151 two-component, 5, 30, 35, 43, 61, 159 classification, 5, 6 0–3, 5, 6, 7, 22, 26, 27, 28, 29, 30, 62, 63, 64, 65, 66, 68, 70, 71, 72, 73, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 92, 93, 125, 126, 129, 131, 132, 133, 140, 154, 160, 161, 163, 164, 175, 176, 178, 180, 181, 185, 187 1–1, 30, 31, 32, 34, 93 1–3, 6, 7, 22, 24, 27, 29, 30, 31, 34, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 65, 75, 76, 77, 78, 81, 82, 83, 86, 89, 92, 93, 110, 111, 113, 116, 118, 131, 133, 134, 140, 142, 145, 149, 150, 151, 152, 160, 162, 163, 164, 165, 167, 168, 169, 171, 172, 173, 182, 185 formulae, 188 longitudinal, 167 transverse, 168 2–2, 8, 26, 27, 30, 31, 32, 34, 43, 44, 45, 46, 47, 48, 50, 52, 53, 57, 65, 78, 110,

197

110, 111, 116, 160, 161, 162, 168, 182, 185 formulae, 187 parallel-connected, 44, 45, 46, 48, 49, 50, 51, 57 series-connected, 43, 44, 45, 48, 51, 52, 65, 78, 110, 160 composition – structure – properties, 1, 2 conductivity, 24, 34, 51, 52, 64 connectivity, 2, 4, 5, 6, 22, 26, 27, 30, 31, 32, 35 changes, 30, 32 mixed, 29, 75 pattern, 5, 6, 26, 30, 31, 32, 33, 34, 35, 75, 82, 92, 102, 104, 118, 185 evolution, 30, 31, 32, 35 junction, 30, 31 D–E, 5, 6, 30 D–E–J, 6 composite in composite, 133 constituent, 25 copolymer, 72, 73, 78, 79, 80 coupled effect, 29, 159 cracking, 105, 109 Curie group, 2 cymbal, 46 dependence of properties, 22, 24, 43, 49, 52, 61, 73, 79, 101, 104, 106, 107, 108, 126, 143 monotonic, 22, 24, 165 non-monotonic, 24, 43, 45, 49, 52, 57, 61, 67, 70, 71, 87, 94, 124, 126, 131, 132, 134, 153 dielectric permittivity, 12, 15, 18, 19, 21, 22, 24, 25, 28, 34, 43, 46, 48, 49, 52, 54, 57, 67, 76, 78, 83, 86, 87, 88, 91, 93, 101, 107, 109, 133, 134, 143, 144, 149, 161, 163, 164, 165, 168, 172, 174 normalised, 75, 107, 109 differential scheme, 29, 159 dilute concentration limit, 29, 159, 175, 176, 182

198

Index

dilute concentration solution, see dilute concentration limit domain, 12, 15, 16, 17, 31, 67 reorientations, 15, 16 structure, 3, 16, 17, 18, 30, 34, 123, 142 engineered, 3, 18, 124, 142, 185 180q, 16, 17 non-180q, 16, 17, 123 domain-wall displacements, 12, 16, 17 effective medium, 16, 17, 159, 169, 175, 176, 178, 181 eigenstrain, 28 elastic compliance, 13, 15, 16, 28, 34, 46, 51, 67, 82, 83, 87, 88, 89, 118, 140, 143, 149, 168, 172 hydrostatic, 168 elastic modulus, 12, 13, 19, 21, 22, 25, 43, 54, 57, 66, 67, 68, 70, 78, 101, 133, 140, 141, 151, 153, 165, 168, 172 elastomer, 62, 84, 111, 112, 113, 115, 116, 140, 152 electric conductivity, see conductivity electric displacement, 3, 11, 13, 43, 47, 142 electromechanical constants, 14, 15, 16, 17, 18, 20, 22, 25, 27, 28, 29, 34, 46, 47, 51, 57, 61, 65, 67, 75, 76, 78, 81, 82, 101, 104, 105, 111, 124, 126, 142, 148, 150, 159, 160, 165, 167, 171, 172, 173, 175, 182 approximation for porous piezoceramics, 103 formulae for 1–3 composites, 188, 189 formulae for 2–2 composites, 187, 188 scheme for determining, 167 electromechanical coupling factor, 15, 24, 43, 45, 52, 66, 77, 101, 124, 125, 146, 153, 154, 155, 169 hydrostatic, 168 thickness, 43, 45, 52

electromechanical interaction, 3, 6, 7, 16, 27, 29, 34, 63, 78, 81, 104, 109, 113, 154, 159, 164, 175, 176 electrostriction, 11, 24 epoxy, 7, 10, 86, 131, 150, 151, 152, 177, 180 Spurr, 8, 151, 152 Eshelby tensor, 28, 29, 30, 57, 65, 133, 159, 160, 175, 176, 178, 182 experimental data, 3, 12, 16, 18, 19, 21, 22, 48, 60, 67, 70, 71, 72, 73, 74, 76, 78, 79, 80, 81, 86, 103, 104, 105, 106, 107, 108, 111, 113, 115, 148, 150, 160, 178 ferroelectric, 1, 2, 3, 5, 6, 12, 14, 15, 16, 17, 18, 22, 24, 25, 29, 31, 34, 35 relaxor, see relaxor-ferroelectric fibre, 4, 7, 52, 165 field, 12, 13, 15, 16, 24, 27, 28, 29, 34, 47, 48, 52, 64, 105, 132, 161, 162 coercive, 12, 15, 124 effective, 28, 54, 63 elastic, see mechanical field electric, 2, 3, 11, 12, 13, 14, 15, 16, 18, 24, 28, 34, 43, 47, 48, 52, 53, 61, 63, 105, 109, 124, 142, 150, 175 electroelastic, 63, 159, 175 mechanical, 3, 6, 13, 15, 18, 25, 27, 28, 29, 46, 47, 48, 52, 61, 67, 71, 86, 110 figure of merit, 45, 52, 61, 62, 83, 90, 92, 101, 146, 154, 169 hydrostatic, 52, 83, 90, 92, 146, 169 fixture, 53 heterophase, 3, 5, 18, 26, 123, 124 hierarchy-of-properties chain, 3, 17 homogenisation, 26, 165, 167 Hooke’s law, 12 inclusion, 4, 5, 6, 26, 27, 28, 29, 30, 52, 63, 64, 65, 66, 67, 68, 70, 71,

Index

73, 75, 76, 77, 80, 81, 82, 83, 84, 101, 105, 108, 109, 110, 125, 126, 127, 128, 129, 131, 132, 133, 134, 140, 159, 160, 173, 175, 176, 177, 182 aligned, 27, 28, 53, 54, 63, 64, 71, 109, 125, 150, 168 cylindrical, 133 distribution, 27, 28, 30, 64, 77 ellipsoidal, 28, 30, 64 fibre-like, 83 hierarchy, 75, 77, 81 hybrid, 63 needle-shaped, 128, 176, 182 oblate, 159, 160 parallelepiped-shaped, 27, 63, 75, 82, 142, 150, 165, 168, 169, 173, 174 penny-shaped, 28, 30, 65 plate-like, 4, 83 problem, 28 prolate, 63, 67, 70, 71, 73, 75, 81, 85, 140, 159, 162 rod-shaped, 52, 65 shape, 26, 27, 28, 30, 52, 53, 64, 128, 131 spherical, 63, 67, 70, 75, 76, 77, 104, 162, 176, 182 spheroidal, 27, 28, 29, 63, 64, 66, 70, 71, 81, 125, 159, 160, 175, 176, 177, 180, 182 interface, 1, 5, 18, 28, 30, 31, 32, 35, 46, 47, 48, 52, 61, 64, 82, 104, 125, 160 isotropic, 28, 47, 65, 104, 160 layer, 4, 8, 26, 30, 31, 33, 34, 43, 45, 46, 47, 48, 49, 51, 52, 76, 77, 78, 81, 93, 110, 111, 159 parallel-connected, 44, 45, 46, 49, 50, 51, 57 series-connected, 43, 44, 45, 48, 51, 52, 65, 78, 110, 160 0–3-type, 76, 77, 78 1–3-type, 77, 78, 110, 111 long-wave approximation, 47, 143, 160

199

manufacturing, 3, 18, 27, 34, 43, 45, 52, 53, 62, 63, 82, 86, 92, 93, 102, 103, 105, 108, 109, 125, 142, 185 method, see method for manufacturing porous material, 102, 103, 105, 108 0–3 composite, 63, 93 1–1 composite, 93 1–3 composite, 53, 93 2–2 composite, 45, 46, 93 3–3 composite, 86, 93 material, 1, 2, 4, 5 electrostrictive, 1 foam-based, 109 heterogeneous, 1, 24, 25, 26, 28, 32, 101, 104 intelligent functional, 1 magnetostrictive, 1 microinhomogeneous, 1 porous, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 118 smart, 1 matrix, 4, 5, 26, 28, 30, 53, 54, 57, 58, 59, 61, 62, 63, 64, 65, 70, 75, 76, 77, 78, 81, 82, 83, 86, 102, 104, 110, 111, 113, 115, 116, 125, 132, 142, 145, 150, 151, 152, 153, 154, 159, 160, 165, 168, 173, 175 piezo-active, 28, 57, 78, 104 piezo-passive, 28, 61, 65, 66, 78, 81, 110, 113, 132 matrix approach, see matrix method for evaluations measurement conditions, 13 mechanical strain, 3, 11, 12, 13, 14, 24, 25, 28, 43, 46, 47, 124, 142, 154, 175 mechanical stress, 3, 11, 12, 13, 14, 15, 18, 28, 47, 61, 88, 89, 129, 131, 142, 149, 162, 176 method for evaluations, 16, 18, 25, 26, 27, 28, 29, 30 analytical, 22, 51, 54, 159, 171 asymptotic homogenisation, 165, 170

200

Index

conditional moments, 29 differential medium theory, 168 dilute approach, see dilute concentration limit effective field, 28, 29, 57, 63, 64, 77, 104, 126, 133, 138, 159, 160, 161, 163, 164, 171, 172, 176, 178, 180, 182 effective medium, 16, 17, 28, 67, 159, 175, 176, 178, 181 equivalent inclusion, 28, 63 finite element, 26, 29, 34, 159, 164, 165, 168, 171 matrix, 27, 30, 34, 46, 82, 142, 159, 160, 168, 169, 173 Mori – Tanaka, 28, 63, 104, 159, 175, 176 regularisation of structure, 26, 159 self-consistent, 28, 67, 159, 175, 176 method for manufacturing, 93 aerogel process, 102 burned-out plastic spheres, 86, 93, 102, 103 co-extrusion, 45, 53, 93 coral replamine, 102 dice and fill, 45, 53, 93, 142 dielectrophoresis, 53, 63, 93 distorted reticulated ceramics, 86, 93 hot-isostatic pressing, 102, 109 hot rolling, 63, 93 injection moulding, 45, 53, 93 jet machining, 45, 93 lost mould, 53, 93 lost wax, 86, 93 pick and place, 53, 93 polymeric sponge, 102 rapid prototyping, 45, 53, 82, 86, 93, 185 relic processing, 86, 93 replamine, 102 robocasting, 86, 93 tape casting, 102, 125 tape lamination, 45, 53, 93 uniaxial pressing, 105 microgeometry, 3, 5, 25, 30, 43, 45,

53, 63, 75, 82, 83, 85, 90 planar, 82, 142, 168 pore, 101, 102, 105 microstructure, 2, 3, 5, 7, 17, 18, 22, 63, 75, 76, 102, 103, 104, 105, 108, 118, 125 model, 17, 29, 63, 88, 104, 108, 110, 115, 167, 168, 171 Banno cube, 26, 104 finite element, 75, 105, 168 Kar-Gupta and Venkatesh, 167 Levassort et al., 104 Marutake, 104 modified layered composite, 110 Wagner, 104 0–3 composite, 64, 76, 82, 125, 160 moony, 46 morphotropic phase boundary, 3, 5, 18, 101, 123, 124, 142, 144, 154 non-monotonic behaviour, see nonmonotonic dependence of properties pearl chains, 7 perovskite, 3, 15, 16, 18, 62, 66, 102, 123 unit cell, 124 piezo-composite, 1, 2, 3, 5, 6, 11, 22, 24, 25, 26, 27, 28, 29, 30, 32, 34, 35 piezoelectric, 1, 2, 3, 5, 6, 11, 12, 13, 14, 15 activity, 3, 43, 52, 60, 86, 109, 116, 123, 124, 125, 133, 152, 153, 159, 162, 176, 185 hydrostatic, 109, 152, 153 charge coefficient, 3, 13, 189 coefficient, 3, 12, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 28, 34, 43, 44, 46, 47, 48, 49, 50, 51, 52, 54, 57, 60, 61, 62, 65, 66, 67, 68, 70, 71, 72, 73, 75, 78, 82, 83, 85, 87, 88, 92, 101, 102, 108, 110, 115, 116, 117, 118, 124, 125, 126, 127, 128,

Index

129, 130, 131, 133, 134, 140, 141, 143, 149, 151, 154, 160, 162, 164, 168, 171, 172, 172, 173, 176 hydrostatic, 45, 52, 61, 70, 89, 92, 109, 117, 126, 132, 147, 171 normalised, 58, 59, 71, 73, 75, 76, 106, 111, 112, 114, 115, 116, 126, 131, 177, 180, 181 equations, 12, 13 effect, 2, 11, 12, 16, 29 direct, 11, 13, 14, 117 converse, 11, 13, 14 response, 6, 11, 48, 66, 67, 75, 80, 89, 101, 132, 152, 153, 185 sensitivity, 52, 57, 61, 70, 71, 83, 85, 86, 116, 150, 153 strain coefficient, 13 stress coefficient, 13 texture, 2, 3 voltage coefficient, 13 piezoelectricity, 3 piezorubber, 63 point group, 2, 124 polarisation, 11, 12, 13, 64, 105, 123 remanent, 16, 64, 111, 125, 132 spontaneous, 2, 15, 16, 17, 124, 132, 143 poling direction, 32, 46, 48, 78, 87, 105, 109, 117 polyethylene, 66, 111, 152 polymer, 3, 5, 7, 22, 27, 43, 45, 46, 47, 48, 52, 53, 54, 62, 63, 64, 65, 66, 67, 71, 76, 77, 78, 79, 82, 83, 86, 87, 88, 89, 92, 93, 105, 110, 111, 116, 117, 132, 140, 141, 142, 143, 154, 173, 185 porous, 62, 111, 150, 154 polyurethane, 53, 70, 71, 72, 73 pore-forming agent, 102 porosity, 101, 102, 103, 104, 105, 108, 109, 110, 113, 115, 116, 150, 152, 154 pattern, 104 properties, 1, 2, 3

201

combination, see combination properties diagram of changes, 57, 58, 59 dielectric, see dielectric permittivity effective, 3, 7, 17, 22, 23, 24, 26, 28, 29, 30, 32, 34, 35, 47, 51, 52, 57 variational bounds, 171, 173, 174, 176 Bisegna and Luciano, 171 Hashin and Shtrikman, 171, 172 1–3 composite with cylindrical rods, 172, 173 1–3 composites with parallelepiped-shaped rods, 173, 174, 175 elastic, see elastic compliance and elastic modulus electromechanical, 14, 15, 16, 17, 18, 25, 26, 27, 28, 29, 32, 34 ferroelectric, 2, 3 modelling, see model physical, 1, 2, 22 piezoelectric, see piezoelectric coefficient product, 22, 24 sum, 22, 23 pyroelectric coefficient, 12 redistribution of fields, 3, 61, 70, 83, 110 relaxor-ferroelectric, 1, 3, 18, 123, 124, 125, 133, 142, 150, 151, 152, 153, 154, 185 representative volume element, 26, 27, 29, 30 response – actions, 12 rod, 28, 30, 52, 53, 54, 57, 61, 76, 77, 78, 86, 93, 110, 111, 113, 131, 132, 133, 140, 141, 142, 143, 149, 150, 151, 152, 154, 159, 160, 164, 165, 168, 171, 172, 173 finite, 77, 78 infinite, 77, 78 rod-like element, 77, 78

202

Index

single crystal, 2, 5, 11, 12, 14, 15, 16, 17, 18, 22, 25 acentric, 2, 11, 12 ferroelectric, 2, 3 (1 – x)Pb(Mg1/3Nb2/3)O3 – xPbTiO3, 123, 124, 125, 126, 142, 143, 144, 148, 149, 150, 151, 152, 153, 154 (1 – y)Pb(Zn1/3Nb2/3)O3 – yPbTiO3, 123, 124, 125, 126, 142, 143, 144, 148, 149, 154 solid solution, 3, 18, 62, 66, 67, 123, 124, 154 spatial distribution, 2, 29, 125, 159 structure cellular, 110 chain-like, 53, 63

hierarchy of inclusions, 75, 77, 81 irregular, 5, 63 negative, 53 regular, 5, 16, 26 structure – processing – property, 34 synthesis, 2, 32 thermodynamic function, 11, 12 threshold value, 15 transversely isotropic, 28, 47, 54, 61, 65, 72, 104, 160, 187, 188 vinylidene fluoride and trifluoroethylene, 72, 73, 78, 79 volume fraction, 2 dependence, see dependence of properties gradient, 8, 46