Ceramics And Composite Materials: New Research

CERAMICS AND COMPOSITE MATERIALS: NEW RESEARCH

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CERAMICS AND COMPOSITE MATERIALS: NEW RESEARCH

B.M. CARUTA EDITOR

Nova Science Publishers, Inc. New York

Copyright © 2006 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com

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Published by Nova Science Publishers, Inc.

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CONTENTS Preface Chapter 1

Chapter 2

vii New Approaches for Estimating and Improving the GrainBoundary Conduction in Stabilized Zirconia Jong-Heun Lee Effective Elastic Moduli of Alumina, Zirconia and AluminaZirconia Composite Ceramics Willi Pabst and Eva Gregorová

1

31

Chapter 3

Progress in Bioceramic Materials P. N. De Aza

Chapter 4

Novel Bioactive Hydroxyapatite-Base Ceramics and GelatinImpregnated Composites V.S. Komlev and S.M. Barinov

133

Thermal Stresses in Particle-Matrix System and Related Phenomena. Application to Sic-Si3N4 Ceramics Ladislav Ceniga

147

Novel Bone-Repairing Materials: Bioactive Organic-Inorganic Hybrids Masanobu Kamitakahara

197

Effect of Sample Size and Distribution Parameters in Estimation of Confidence Lower Bounds for Weibull Percentiles Burak Birgoren

215

Chapter 5

Chapter 6

Chapter 7

Index

101

237

PREFACE In stabilized zirconia, which is one of the representative solid electrolytes, the grainboundary resistivity is known to be ~102-104 times higher than the grain-interior one. For the applications such as solid oxide fuel cells and electrochemical gas sensors operated at the intermediate (~600oC) and low temperature (~400oC) regime, the improvement in ionic conduction across the grain boundary becomes significant. For this, a precise estimation of the grain-boundary conduction via impedance spectroscopy is essential. In Chapter 1, New Approaches for Estimating and Improving the Grain-Boundary Conduction in Stabilized Zirconia, Jong-Heun Lee suggests the new methods for improving and estimating the grainboundary conduction in stabilized zirconia. In the first part, the various approaches to improve the grain-boundary conduction are discussed. The addition of Al2O3 is known to scavenge the siliceous grain-boundary phase. However, this might deteriorate the graininterior conduction when the sintering temperature becomes very high (>1600°C). Therefore, new routes for improving the grain-boundary conduction using two-stage sintering process are suggested. The formation of a Si-containing phase in a discrete configuration and the dewetting of the intergranular liquid phase were suggested to be the mechanisms for scavenging via pre- and post-sintering heat treatments, respectively. In the second part, a local impedance technique using a sub-millimeter-scale electrode array was suggested to estimate the spatially uneven distribution of the grain-boundary resistivity, which was named as ‘Millicontact Impedance Spectroscopy’. The fundamentals and validity of the technique were explained, and the analyses of the dynamic rearrangement of an intergranular liquid phase are given as an example. In the second chapter, Effective Elastic Moduli of Alumina, Zirconia and AluminaZirconia Composite Ceramics, Willi Pabst and Eva Gregorová investigate from the theoretical point of view, with a side-glance on experimental results and applications. In the first section alumina, zirconia and alumina-zirconia composites are introduced as structural materials, relations of elastic moduli to other properties are recalled and targets of microstructural design are formulated. In the second section elastic properties are defined from the viewpoint of rational mechanics for anisotropic and isotropic materials in general. The difference between adiabatic and isothermal elastic moduli is explained and estimated for alumina and zirconia. In the third section effective elastic properties are defined and discussed from the viewpoint of micromechanics and composite theory. General formulae are given for the calculation of effective elastic moduli of polycrystalline materials from monocrystal data. Further, the Voigt-Reuss bounds for the effective elastic moduli of multiphase materials are

viii

B. M. Caruta

given, as well the Hashin-Shtrikman bounds for the special case of two-phase materials. For porous materials the dilute approximations are recalled as well as the predictions following from the self-consistent, Mori-Tanaka, differential, Gibson-Ashby and Coble-Kingery approach as well as the functional equation approach recently advocated by the authors. A comprehensive survey of model relations for the porosity dependence of elastic moduli is given, including exponential and power-law relations and a new relation recently proposed by the authors, which seems to be the simplest relation allowing for the occurrence of a percolation threshold (critical porosity). In the fourth section all these micromechanical bounds and relations are applied to the alumina-zirconia system. Using a theoretically sound and experimentally confirmed set of elastic moduli (and Poisson ratios) for dense (i.e. fully sintered) polycrystalline alumina and zirconia the Hashin-Shtrikman bounds of dense alumina-zirconia composites are calculated and compared to experimentally measured values. Several predictions for porous alumina, zirconia and alumina-zirconia composites are compared to the data measured for ceramics with convex interconnected pores prepared by the starch consolidation casting technique. A master fit curve is given for porous ceramics with this type of matrix-inclusion microstructure and explicit numerical expressions are given throughout. The last section gives examples of the mathematical modeling of other effective properties and their dependence on composition and microstructure and an outlook is given to future research aims. In particular, the significance of interfaces is emphasized and ideas on the way from micromechanics to nanoscience − towards a general mixture theory − are outlined. A strong interest in the use of ceramics for biomedical engineering applications developed in the late 1960’s. Used initially as alternatives to metallic materials in order to increase the biocompatibility of implants, bioceramics have become a diverse class of biomaterials presently including three basic types: relatively bioinert ceramics maintain their physical and mechanical properties in the host and form a fibrous tissue of variable thickness; surface reactive bioceramics which form a direct chemical bonds with the host; and bioresorbable ceramics that are dissolved with the time and the surrounding tissue replaces it. In Progress in Bioceramic Materials, P. N. De Aza gives a review of the composition, physicochemical properties and biological behaviour of the principal types of bioceramics, based on the literature and some of our own data. The materials include, in addition to bioceramics, bioglasses and bio-glass-ceramics. Special attention is given to structure as the main physical parameter determining nor only the properties of the materials, but also their reaction with the surrounding tissue. In addition, biomaterials with appropriate three-dimensional geometry (appropriate porous structure) are highly recommended because are able to become osteoinductive (capable of osteogeneis), and can be effective carriers of bone cell seeds. A new way of preparing “in situ” porous bioactive ceramics with interconnected porosity from a dense bioactive ceramic is presented. To this purpose the binary system wollastonite- tricalcium phosphate is selected, taking into account the different bioactivity behavior of both phases. The present status of research and development of bioceramic is characterized as first step in the solution of complex problems at the confluence of materials science and engineering, biology and medicine. In Novel Bioactive Hydroxyapatite-Base Ceramics and Gelatin-Impregnated Composites, V.S. Komlev and S.M. Barinov introduce a study aimed at the development of novel ceramic and composite materials intended for application in bone tissue engineering. A method to

Preface

ix

produce porous spherical hydroxyapatite (HA) granules was proposed. The method is based on liquids immiscibility effect using the HA/gelatin suspension and oil as liquids. The granules of 50 to 2000 µm diameter contain open pores of 10 nm to 10 µm size in amount up to 45 vol.%. A route for the fabrication of porous HA ceramics having two population of open pores was developed. Ceramics contain intragranular, up to 10 µm size pores and intergranular, hundreds micrometer size interconnected pores were prepared. The interconnections in the intergranular pores are the pathway to conduct cells and vessels for the bone ingrowth. The gelatin solution impregnation into the pores was demonstrated to enhance strength significantly. The strength of HA/gelatin composites was increased by 6 to 10 times compared to that of initial ceramics. The ceramics can further be filled with a drug resulting in a drug delivery system of slow drug release rate. In Chapter 5, Thermal Stresses in Particle-Matrix System and Related Phenomena. Application to Sic-Si3N4 Ceramics, Ladislav Ceniga deals with elastic thermal stresses in an isotropic particle-matrix system of homogeneously distributed spherical particles in an infinite matrix, divided into cubic cells containing a central spherical particle embedded in a matrix of a dimension equal to an inter-particle distance. Originating during a cooling process as a consequence of the difference in thermal expansion coefficients between the matrix and the particle, and investigated within the cubic cell, the thermal stresses of the isotropic multiparticle-matrix system are thus functions of the spherical particle volume fraction, v, and are transformed for v = 0 to those of the isotropic one-particle-matrix system. With regard to the particle-matrix system yield stress, the thermal stresses are derived for such temperature range within which the particle-matrix system exhibit elastic deformations. Similarly, the particle-matrix boundary adhesion strength is also considered. In addition to the thermal-stress-induced elastic energy of the cubic cell, the elastic energy gradient within the cubic cell, calculated by two equivalent mathematical techniques and representing a surface integral of the thermal-stress-induced elastic energy density, is presented to derive the thermal-stress strengthening in the spherical particle and the cubic cell matrix. Considering a curve integral of the thermal-stress-induced elastic energy density, the critical particle radii related to crack formations in ideal-brittle particle and matrix, functions describing crack shapes in a plane perpendicular to a direction of the crack formation in the particle and the matrix, and consequently particle and matrix crack dimensions are derived along with the condition concerning a direction of the particle and matrix crack formation. The former parameters for v = 0 are derived using a spherical cell model for the spherical cell radius, Rc → ∞. The derived formulae are applied to the SiC-Si3N4 multi-particle-matrix system, and calculated values of investigated parameters are in an excellent agreement with those from published experimental results. In general, artificial materials implanted into bone defects are encapsulated by a fibrous tissue. Some ceramics, such as Bioglass®, sintered hydroxyapatite and glass-ceramic A-W, however, form a bone-like apatite layer on their surfaces in the living body and bond to living bone through this apatite layer, i.e. they show bioactivity. Although these bioactive ceramics are used clinically as important bone-repairing materials, they are essentially brittle and hence limited in their applications. It is desirable to develop new types of deformable bioactive materials. Organic–inorganic hybrids prepared by a sol-gel method are expected to exhibit

x

B. M. Caruta

bioactivity as well as deformability. Polydimethylsiloxane (PDMS)-CaO-SiO2-based hybrids were prepared. To evaluate their potential of bone-bonding property, their ability of apatite formation was examined in a simulated body fluid (SBF) in Chapter 6, Novel Bone-Repairing Materials: Bioactive Organic-Inorganic Hybrids by Masanobu Kamitakahara. Since the apatite formation on bioactive materials in the body can be reproduced even in SBF, the author can estimate the bioactivity of the material by using SBF. Mechanical properties were also examined. Some of them showed apatite-forming ability and mechanical properties analogous to those of human cancellous bone. These CaO-containing hybrids, however, showed a decrease in mechanical strength in SBF. These hybrids containing no CaO do not form apatite on their surfaces in SBF. The author then prepared CaO-free PDMS-TiO2 and poly(tetramethylene oxide) (PTMO)-TiO2 hybrids in which anatase was precipitated by hotwater treatments. These hybrids also showed apatite-forming ability on their surfaces in SBF and flexibility, and they showed little decrease in mechanical strength in SBF. In conclusion, it was revealed that we can design new bioactive materials for bone-repair by controlling the compositions and structures in organic-inorganic hybrid systems. The Weibull distribution is widely used for modelling fracture strength of composite materials. Estimating lower percentiles of the Weibull distribution has been a major concern, because small sizes of fracture strength experiments leads to unreliable estimates. Therefore, several recent studies have focused on estimating confidence lower bounds on the lower percentiles, namely A-basis and B-basis material properties. The general linear regression methods with different probability indices and weight factors, and the maximum likelihood method have been used for the estimation; the performance of these methods were compared in recent studies and different methods were proposed using different criteria for comparison. Aside from this ambiguity, the criteria and comparison studies do not allow any practical interpretation. In particular, experimenters would like to know, in practical terms, how well a method performs with respect to others and how its performance is affected by the sample size. Burak Birgoren, in Effect of Sample Size and Distribution Parameters in Estimation of Confidence Lower Bounds for Weibull Percentiles, intends to clarify answers to these questions. Percent departures of the lower bounds from the true percentiles convey such a practical meaning. Simulating these departures, probabilistic upper bounds for maximum percent departures are determined; tables are provided for computing the upper bounds for a wide range of sample sizes. Using the upper bounds as a criterion for comparison, it is observed that the maximum likelihood method is the best as compared to several variations of the general linear regression methods. Its superiority is illustrated by comparing the sample sizes necessary for different methods to achieve the same level of performance. For the maximum likelihood method, the upper bound levels are shown to increase at a higher rate for sample sizes less than 30 as compared to larger sizes. Also, the Weibull modulus is observed to have a very serious effect on the upper bounds. Finally, a two-stage experimental setup is proposed for sample size determination; after running a pilot experiment, the setup enables the experimenter to calculate the number of additional experiments necessary for lowering the upper bounds to a certain level. Its implementation is illustrated on a practical example.

In: Ceramics and Composite Materials: New Research ISBN: 1-59454-370-4 Editor: B.M. Caruta, pp. 1-30 © 2006 Nova Science Publishers, Inc.

Chapter 1

NEW APPROACHES FOR ESTIMATING AND IMPROVING THE GRAIN-BOUNDARY CONDUCTION IN STABILIZED ZIRCONIA Jong-Heun Lee* Division of Materials Science and Engineering, Korea University, Seoul 136-701, Korea

Abstract In stabilized zirconia, which is one of the representative solid electrolytes, the grain-boundary resistivity is known to be ~102-104 times higher than the grain-interior one. For the applications such as solid oxide fuel cells and electrochemical gas sensors operated at the intermediate (~600oC) and low temperature (~400oC) regime, the improvement in ionic conduction across the grain boundary becomes significant. For this, a precise estimation of the grain-boundary conduction via impedance spectroscopy is essential. This article suggests the new methods for improving and estimating the grain-boundary conduction in stabilized zirconia. In the first part, the various approaches to improve the grain-boundary conduction are discussed. The addition of Al2O3 is known to scavenge the siliceous grain-boundary phase. However, this might deteriorate the grain-interior conduction when the sintering temperature becomes very high (>1600°C). Therefore, new routes for improving the grain-boundary conduction using two-stage sintering process are suggested. The formation of a Si-containing phase in a discrete configuration and the dewetting of the intergranular liquid phase were suggested to be the mechanisms for scavenging via pre- and post-sintering heat treatments, respectively. In the second part, a local impedance technique using a sub-millimeter-scale electrode array was suggested to estimate the spatially uneven distribution of the grainboundary resistivity, which was named as ‘Millicontact Impedance Spectroscopy’. The fundamentals and validity of the technique were explained, and the analyses of the dynamic rearrangement of an intergranular liquid phase are given as an example.

*

Corresponding author Jong-Heun Lee, Ph. D., Associate Professor, Division of Materials Science and Engineering, Korea University, 1, 5-ka, Anam-dong, Sungbuk-ku Seoul 136-701, South Korea. Tel: 82-2-3290-3282, Fax: 82-2-928-3584. Email: [email protected]

Jong-Heun Lee

2

1

Introduction

When divalent or trivalent oxides such as CaO, MgO, Y2O3, Yb2O3 and Sc2O3 are added to ZrO2, the high-temperature cubic phase becomes stabilized at room temperature. [1] The effective negative charge of the dopant cation in the Zr4+ site is compensated by the formation of a doubly ionized oxygen vacancy (VO••), which leads to the stabilized ZrO2 being a pure oxygen ion (O2-) conductor. [1,2] A high ionic conductivity is essential in order to achieve a high-energy efficiency and a low operation temperature in a solid oxide fuel cell (SOFC) and an electrochemical sensor. [3] The improvement in the ionic conductivity in stabilized zirconia has been studied extensively in the viewpoints of effective dopants and their optimum concentrations. [1,2,4,5] Among the types of stabilized zirconia, Yb2O3-stabilized or Sc2O3-stabilized zirconia (YbSZ or ScSC) showed the highest conductivity, which is explained by the minimum lattice mismatch between the dopant cation (Yb3+ and Sc3+) and the host Zr4+ ion. [4] However, these efforts to increase the grain-interior conductivity are in the mature stage. Siliceous phase [VO••] O O

2-

2-

(a)

(b)

Fig.1 The origins of the grain boundary resistance. (a) space charge layer and (b) siliceous intergranular phase

Another approach is to minimize the grain-boundary resistance. The grain boundary is known to be specifically ~102-104 times more resistive than the grain interior. [6-8] Two reasons for the origins for high grain-boundary resistivity have been suggested. One is the existence of a space charge layer near the grain boundary with the lower oxygen vacancy concentration, which becomes significant in pure materials. (Fig.1(a)) [9-12] The other is an oxygen-ion-blocking siliceous phase or film at the grain boundary. (Fig.1(b)) [8,13] Even 100 ppm of SiO2 is known to significantly deteriorate the grain-boundary conduction. [14] Considering that SiO2 is the most ubiquitous impurity, siliceous contamination is almost inevitable during ceramic processing. Therefore, the studies on improving the grain-boundary conduction have been focused on the addition of a scavenger to clean the resistive grainboundary phase. Thus far, Al2O3 is known to be the most effective scavenger.[14-20] However, when the sintering temperature becomes >1600oC, the improvement in the grainboundary conduction is known to be nullified by the deterioration of grain-interior conduction. [21,22] This is related to the incorporation of Al2O3 into the ZrO2 lattice and the subsequent defect association between AlZr’ and VO••.[23] Therefore, in order to avoid the adverse effect of Al2O3, two new approaches for improving the grain boundary conduction without the addition of Al2O3 were suggested.

New Approaches for Estimating and Improving…

3

These methods are based on the configuration change in the intergranular phase from continuous one to a rather discrete one by the additional heat treatments before or after the sintering process. The mechanism for enhancing the grain-boundary conduction was investigated. The key idea of this method, which is the sensitive correlation between the grain-boundary resistance and intergranular phase, can be also applied to a new research field. In the second part of this contribution, a local impedance method combined with a submillimeter-scale-electrode array, which is called ‘millicontact impedance spectroscopy’, was suggested to investigate the spatially-uneven grain boundary. The dynamic rearrangement of the intergranular liquid during liquid-phase sintering, which had not been observed in scanning electron microscopy, could be successfully observed via the millicontact impedance technique.

2

The Model of Grain-Boundary Resistivity in Stabilized Zirconia Ceramics

The complex impedance spectrum of stabilized zirconia generally consists of three semicircles. (Fig.2) From the low frequency, three semicircles can be attributed to the contributions by electrode polarization, grain boundary and grain interior, respectively. Typically, the grain-boundary capacitance (Cgb) is much larger than the grain-interior capacitance (Cgb) because the grain boundary is very thin compared with grain size. However, the grain-boundary resistance (Rgb) was reported to be comparable with the grain-interior resistance (Rgi). Therefore, the relaxation time constant for the grain boundary (τgb), the product between Rgb and Cgb, becomes markedly larger than that that of the grain interior (τgi). This is the reason why the contributions from the grain interior and grain boundary can be separated effectively in a complex impedance plane.

τgi Cgi Rgi

« τgb « τep « Cgb « Cep ≅ Rgb < Rep

-Z”(Ω)

freq. Z’(Ω)

Fig.2 Schematic complex impedance spectrum of the stabilized zirconia. R, C, and τ mean the resistance, capacitance and relaxation time constant, respectively. From the low frequency, subscripts ‘ep’, ‘gb’, and ‘gi’ mean the electrode polarization, grain boundary and grain interior, respectively.

In a stabilized zirconia specimen with a defined thickness (l) and electrode area(A) (Fig.3(a)), the grain-interior resistivity (ρgi) can be calculated by normalizing the deconvoluted Rgi value using a shape factor.

Jong-Heun Lee

4

Rgi = ρ gi

l A

(1)

l

Porous Pt electrode A O2-

(a) dg

δgb

O2-

(b) Fig.3 (a) Schematic diagram showing the ionic conduction in polycrystalline stabilized zirconia specimen with a resistive grain boundary and (b) its simplification by the brick layer model. (A: electrode area, l: specimen thickness, dg: average grain size, δgb : grain boundary thickness)

However, because l is not the dimension of the grain boundary, normalization of Rgb by shape factor does mean not the specific grain-boundary resistivity (ρgbsp) but the apparent grain-boundary resistivity (ρgbapp). app Rgb = ρ gb

l A

(2)

In order to calculate ρgbsp, the number and thickness of the grain boundary across the electrode needs to be considered. In order to make the analysis simple, the brick layer model is generally being used as an effective approximation. [7,13,24-26] This model assumes that the electrically identical cubic grains (size=dg) are surrounded by the same the grain boundary thickness (δgb). (Fig.3(b)) When the grain boundary is highly conducting compared with the grain interior, ionic conduction will become a competition between the fast ionic conduction along the narrow grain boundary and the slow ionic conduction along the wide grain interior. Therefore, a parallel equivalent circuit is established. (Fig.4(a)) However, in stabilized zirconia, the grain boundary is generally 102-104 times more resistive than the grain interior. In this situation, the conduction along the resistive grain boundary can be neglected. Therefore, a series of two RC lumps can be summarized as an equivalent circuit. (Fig.4(b))

New Approaches for Estimating and Improving…

5

dg δgb Rgb Cgb O2Rgi Cgi

(a) parallel equivalent circuit: ρgi » ρgbsp Rgi

Rgb

Cgi

Cgb

O2-

(b) serial equivalent circuit: ρgi « ρgbsp Fig.4 (a) Schematic diagrams showing the ionic conduction in specimen with highly conductive grain boundary and the corresponding parallel equivalent circuit. (b) Schematic diagrams showing the ionic conduction in the specimen with a highly resistive grain boundary and the corresponding serial equivalent circuit.

Under the assumption of a brick layer model and serial equivalent circuit, ρgbsp can be calculated by the following equation: [8]

ρ gbsp = ρ gbapp

dg

δ gb

(3)

However, in real situations, a precise estimation of δgb is very difficult because the grainboundary segregation and intergranular liquid phase are too thin to observe and the thickness of the space charge layer is difficult to determine. Therefore, the normalization of ρgbapp by the grain-boundary density (d: number of grain boundary per unit length) can be used to compare the grain-boundary conduction in a specific manner. [26] The grain-boundary density can be calculated by the reciprocal of the average grain size (dg).

Rgbs =

ρ gbapp d

app sp = ρ gb d g = ρ gb δ gb

(4)

The Rgbs value means the resistance for the unit grain-boundary area and its physical meaning is the product between ρgbsp and δgb. Therefore, without a precise determination of δgb, whether the grain-boundary conduction has improved or deteriorated can be determined.

6

Jong-Heun Lee

As a measure for the improvement in the grain-boundary conduction, the scavenging factor SF can be defined as follows:

SF =

Rgbs ( AS ) Rgbs ( BS )

(5)

where Rgbs(BS) and Rgbs(AS) are the Rgbs values before and after the scavenging reaction, respectively. SF<1 and SF>1 mean an improvement and deterioration of grain-boundary conduction, respectively and a SF below 1 means the more effective scavenging of the resistive intergranular phase.

3

Improvement of Grain-Boundary Conduction by the Addition of Al2O3

Many additives such as Fe2O3 [6], Bi2O3 [6], TiO2 [27], and Al2O3 [6,13-20,27-29] have been used to improve the grain-boundary conduction of stabilized zirconia via the scavenging of resistive siliceous phase. Among the additives used, Al2O3 is known to enhance the ionic conduction in the grain boundary. [14-20] However, there are also contrasting reports. [6,26,29] Table 1 summarizes the literature on the positive and negative role of Al2O3. [1318,26,30] The improvement in the grain-boundary conduction is usually explained in terms of the scavenging of the siliceous phase by Al2O3. Butler and Drennan [19] examined the TEM microstructure of Al2O3-added Y2O3/Yb2O3-stabilized zirconia ceramics. They reported that most intragranular Al2O3 particles contain Zr-rich or (Si+Zr)-rich inclusions and that intergranular Al2O3 particle are frequently associated with (Si and Al)-rich amorphous cusp. From the observation, they suggested that Al2O3 scavenged the siliceous second phase during the pinning time that was assisted by fast grain-boundary diffusion. In order to examine the effect of Al2O3 configuration on the scavenging efficiency, Lee et al. [21] added various Al2O3 particles sized 0.3, 4, and 10 µm to 8 mol% of Yttria-Stabilized Zirconia (8YSZ) via ball milling and ultrasonic dispersion. The pure 8YSZ powder without Al2O3 was also ball-milled or ultrasonically dispersed to exclude the effect of background impurities, which might be introduced during the mixing process, with the results shown in Figure 5. The small and large black spots in Fig.5(b) and (c) represent the fine and coarse Al2O3 particles added, respectively. The 0.3 µm Al2O3 particles dispersed well (Fig.5(b)) at the inter- and intra-granular points. Although the same mol% of Al2O3 was added, the average spacing between the 10 µm Al2O3 particles was very long (~20 µm) compared with that between the 0.3 µm Al2O3 particles because the total number of Al2O3 particles was quite small. (Fig.5(c)) Despite these large differences in the concentration and size of Al2O3, all the specimens showed almost the same scavenging factor (Rgbs(AS)/Rgbs(BS) =~0.2). (Fig.5(a)) This strongly supports the suggestion by Butler and Drennan’s [19] that scavenging occurs via the fast grain-boundary diffusion. The majority of research results showed an improvement in grain-boundary conduction as a result of Al2O3 addition.

New Approaches for Estimating and Improving…

7

Table 1. The effect of Al2O3 addition on the grain-boundary conduction of stabilized zirconia

Materials

Al2O3

Tsint(oC)

σgi

1 Rgbs

2.5YSZ

10 wt% (~20 mol%)

1600

↓

↑

8YSZ

1 wt% (~2 mol%)

1500

↓

↑

9.9YSZ

2 mol%

1350

↑

9YSZ 8YSZ

1.5,2 mol% 1 mol%

1600 1600

↓

↑ ↑

8YSZ

0.5 mol%

1700

↓

↓

8YSZ

0.4 mol%

1650

↓

↓

8.5YSZ

0.78 mol% 0-1wt% Al2O3 0-1wt% SiO2

1267

↓

↓

3YSZ

15CSZ

1 mol%

or σ gb

app

Comments

Ref. [15]

Solubility limit of alumina= 0.6~1.2 mol% at 1500oC Solubility limit of alumina= 0.1mol% at 1300oC

[16]

[17] [18] [14]

Solubility limit of alumina= 0.5mol% at 1700oC Explained by the increase of grainboundary space-charge potential

[26]

[29] [6]

Minimum σ gb at app

1500

↓

[13]

SiO2/Al2O3=1 1450

↑

1600

↓

Dissolving of alumina at high Tsint Æ deterioration of grainboundary conduction

[30]

However, the deterioration in grain-boundary conduction via the addition of Al2O3 has also been reported. [6,26,29] This inconsistent tendency might be due to the different siliceous impurity and/or the different interaction between Al2O3 and siliceous impurity. For example, Gödikemier et al. [13] varied the SiO2/Al2O3 molar ratio systematically in 3 mol% Yttria-Stabilized Zirconia (3YSZ) and attained the largest Rgbs value near SiO2/Al2O3=1. Both the SiO2-rich or Al2O3-rich compositions showed smaller Rgbs values than Rgbs at SiO2/Al2O3=1. Lee et al. [30] added Al2O3 sized 4 and 10 µm to 15 mol% Calcia-Stabilized Zirconia (15CSZ) and attained two opposite roles of Al2O3 in grain-boundary conduction (Fig.6). When the sintering temperature is lower than 1525oC, the grain-boundary conductivity improved up to 15 times by the addition of Al2O3. However, as the sintering temperature becomes higher than 1550oC, the grain-boundary conductivity deteriorates significantly. Figure 6(b) and (c) show typical microstructures for the 15CSZ specimens doped with 1 mol% of coarse Al2O3 particles (size=10 µm). The 10 µm sized Al2O3 particle preserved its shape (points A and B in Fig.6(b)) when sintered at 1450oC, while most of the Al2O3 particles disappeared (see empty pores C in Fig.6(c) sized about ~10 µm) and a wetting

Jong-Heun Lee

8

intergranular liquid phase could be observed (see arrows in Fig.6(c)). Therefore, The deterioration in the grain-boundary conduction at >1525oC can be attributed to the formation of wetting intergranular liquid phase as a result of the dissolution of Al2O3 into siliceous intergranular liquid.

BM: Ball Milling

UD: Ultrasonic Dispersion

Rgbs(AS)/Rgbs(BS)

1.0 0.8 0.6 0.4 0.2

ρgb

2.0

1500

Rgbs

1.5

2

2000

1000

1.0

500

0.5

0

8Y

02A03 1A03

8Y

02A4 02A10 1A10

Rgbs[Ω.cm ]

ρgb

app

[Ω.cm]

0.0

0

Samples (a) (b)

(c)

10 µm

10 µm

Fig.5 The grain-boundary resistivity (ρgb), resistance per unit grain-boundary area (Rgbs), and scavenging efficiency (SF) of 8YSZ specimens with various concentrations and sizes of Al2O3 particles. (8Y: 8mol% Yttria-Stabilized Zirconia, 02A03: the addition of 0.2 mol % Al2O3 sized 0.3 µm, 02A4: the addition of 0.2 mol % Al2O3 sized 4 µm, 1A10: the addition of 1.0 mol % Al2O3 sized 10 µm) All the specimens were sintered at 1500oC for 4h and the impedance was measured at 400oC in air, according to [21].

The above results suggest that the role of Al2O3 is not always the same but depends on the degree of siliceous contamination, the sintering temperature and the temperature for forming the intergranular liquid phase. When a high sintering temperature is combined with a large concentration of siliceous impurity, the possibility for the deterioration of grainboundary conduction due to Al2O3 dissolution into the intergranular liquid becomes large. In order to minimize such possibility, a lower level of siliceous impurity is essential. Indeed, when the siliceous impurity is low, the addition of Al2O3 is known to effectively improve the

New Approaches for Estimating and Improving…

9

grain-boundary conduction of the stabilized zirconia. However, direct evidence for the scavenging role of Al2O3 in specimens with a segregation-level impurity is difficult to find. This arises from the difficulties in the observation of impurity segregation or the extremely thin intergranular phase. For example, even a sub-monolayer segregation of the siliceous phase, which is very difficult to estimate, is known to affect the grain-boundary conduction. [8] Even if there were a siliceous film at the grain boundary, the observation of many nanometer-scale intergranular films with statistical confidence is still difficult.

10

Rgbs(AS)/Rgbs(BS)

15CSZ-1A4 Deterioration of 15CSZ-1A10

G.B. conduction

1

(c)

0.1 Improvement of G.B. conduction

1450

1500

1550

1600 o

Sintering temperature( C) (a) Scavenging resistive g.b. phase by 10 µm Al2O3 particles

(b) A

Dissolution of Al2O3 into intergranular liquid

(c) C b) A C

B B

10 µm

C B

C 10 µm

Fig.6 (a) The scavenging efficiency (SF) of the 15CSZ specimens as a function of varying sintering temperatures: 15CSZ1A4: 1-mol%-Al2O3(size=4µm)-doped 15CSZ specimens, 15CSZ1A10: 1-mol%Al2O3(size=10µm)-doped 15CSZ specimens. The microstructures of 15CSZ1A10 specimens sintered (b) at 1450oC and (b) 1600oC for 4h, according to [30].

Lee et al. examined the spatial distribution of 100 ppm-level-siliceous impurity in 8YSZ and 1-mol%-Al2O3-doped 8YSZ specimens using imaging secondary ion mass spectroscopy. [14,21] From the secondary-ion images of 28Si-, and 27Al16O- for two specimens, it was found that the siliceous impurity is segregated at the grain boundary in pure 8YSZ specimen and gathers near the Al2O3 particles in 1-mol%-Al2O3-doped 8YSZ specimen. This provided the direct and visual evidence for the presence of siliceous segregation and its scavenging as a result of Al2O3 addition.

10

Jong-Heun Lee

On the other hand, Guo [29] reported that the addition of Al2O3 changes the structure of the space charge layers as well as the grain-boundary contamination. He formulated the deterioration of the grain-boundary conduction by the increase in the grain-boundary spacecharge potential within the solubility limit of Al2O3. However, he suggested that the addition of Al2O3 above the solubility limit improves the grain-boundary conduction via a scavenging reaction.

4 4.1

Improvement in Grain-Boundary Conduction Via Two-Stage Sintering Approach without Additive Improvement in the Grain-Boundary Conduction by Pre-sintering Heat Treatment

Although the addition of Al2O3 is a convenient approach for improving the grain-boundary conduction, it is also known to decrease the grain-interior conductivity. [16,21,22,26] In Table 1, it can be seen that the addition of Al2O3 always deteriorates the grain-interior conduction. This is explained in terms of the defect association between AlZr’ and VO•• [23] after the incorporation of Al2O3 into ZrO2 lattices. Therefore, it tends to become significant at higher sintering temperatures (typically > 1600oC). [21,22,26] In order to prevent the adverse effect of Al2O3, Lee et al. suggested a new scavenging method without the addition of Al2O3. [22] The schematics for the scavenging mechanisms by Al2O3 additive and pre-sintering HT are compared in Fig.7. In contrast to ‘additive scavenging’, which uses the interactive reaction between Al2O3 and siliceous intergranular phase (Fig.7(a)), the new process employs a two-stage sintering process. (Fig.7(c)) This process was named as ‘precursor scavenging’ because it can be achieved by the precursor itself without any additives. Figure 8 shows the variation in the impedance spectra by additive scavenging and precursor scavenging in 8YSZ. [22] For additive scavenging, 1 mol% of Al2O3 with various sizes (A03: 0.3 µm, Asol: Al2O3 sol, A10: 10 µm) were added. For precursor scavenging, the pure 8YSZ specimens were heat-treated at 1200oC for 40h prior to sintering at 1500oC for 4h. (denoted as 8Y-1200(40)-1500 specimen) In the figure, ‘BM’ and ‘UD’ denote ball milling and ultrasonic dispersion, respectively. The semicircle in the intermediate frequency regime corresponds to the grain-boundary contribution. When various forms of Al2O3 were added, the grain-boundary resistivity decreased to a large extent. (Fig.8(a),(b),(d), and (e)) This can be explained by the scavenging reaction of Al2O3. Approximately the same degree of improvement in the grain-boundary conduction was achieved as a result of pre-sintering heat treatment (HT) at 1200oC. (Fig.8(c) and (f)) Note that ρgi increased in the additive (Al2O3)scavenged specimen (Fig.8(e)) while it remained constant in the precursor-scavenged specimen (Fig.8(f)) when sintered at 1600oC. This clearly demonstrates that precursor scavenging can improve the grain-boundary conduction significantly without affecting the grain-interior one even at high sintering temperatures.

New Approaches for Estimating and Improving…

1600

(a)

1200

Al2O3 g.b.

Additive scavenging

800 400 0

o

Temperature ( C)

11

(b)

1200

Siliceous g.b. phase

800 400 0

(c)

1200 800

Precursor scavenging

1200(40)-1500

400

Discrete Si-containing phase

HT Temp. HT time

g.b.

0 0

20

40

60

Time (h) Fig.7 The temperature profiles for sintering and a schematic diagram of scavenging mechanism. (a) Scavenging by the Al2O3 additive (additive scavenging), (b) before scavenging and (c) precursor scavenging by pre-sintering heat treatment(HT).

(a) γ

8Y-BM-1500 8Y-UD-1500

-|Z|sinθ [kΩ.cm]

(d)

β

δ

1

δ

8Y-BM-1600 8Y-UD-1600

α

β γ

α

0

(b)

(e)

1

8Y1A03-BM-1500

δ

8Y1A03-BM-1600

δ

8Y1Asol-BM-1500

γ

8Y1A10-UD-1500

β

8Y1Asol-UD-1600

α

8Y1A10-UD-1600

α γ

β

ρgi↑

0

(c)

(f)

1

δ

α : 543 Hz β : 5.12 kHz γ : 48.4 kHz δ : 586 kHz

δ γ

8Y-1200(40)-1500

β

α 8Y-1200(40)-1600

γ βα

0 4

5

6

7

8

4

5

6

7

8

|Z|cosθ [kΩ.cm]

Fig.8 Complex impedance spectra at 400oC of the various 8YSZ specimens sintered at (a), (b), (c) 1500 and (d), (e), (f) 1600oC. (8Y: 8YSZ, BM: Ball Milling, UD-Ultrasonic Dispersion, 1A03: the addition of 1 mol% of Al2O3 sized 0.3 µm, 1A10: the addition of 1 mol% of Al2O3 sized 10 µm, 1Asol: the addition of 1 mol% of Al2O3 sol, 1200(40)-1500: heat treatment at 1200oC for 40 h prior to sintering at 1500oC for 4h) The semicircles indicated by arrows mean the grain-boundary contributions, according to [22].

Jong-Heun Lee

12 2.0

2.0

(a)

ρgbapp

1.0

0.5

0.5

2

1.0

Rgbs[Ω.cm ]

1.5

ρgb

app

[kΩ.cm]

Rgbs 1.5

0.0 1050

0.0

1100

1150

1200

1250

1300

1350

HT temp. before sintering at 1500oC (oC)

2.0

2.0

(b) ρgbapp

1.0

1.0

ρgb

ρgb

app

and Rgbs regime of

scavenging by Al2O3 0.5

Rgbs[Ω.cm2]

1.5

Rgbs

app

[kΩ.cm]

1.5

0.5

0.0

0.0

0

10

20

30

40

HT time before sintering (h) Fig.9 The variation of apparent grain-boundary resistivity (ρgbapp) and resistance per unit grainboundary area (Rgbs) at 400oC with varying (a) HT temperature (HT time=10h) and (b) HT time (HT temperature=1200oC) prior to sintering at 1500oC for 4h. Pure 8YSZ specimens with the 100 ppm of SiO2 impurity were used, according to [22].

Figure 9 shows the effect of the heat-treatment temperature (HTT) and holding time of HT on the grain-boundary conduction. The Rgbs and ρgbapp values showed the minimum at HTT=1200oC (Fig.9(a)). At the optimal HTT (1200oC), holding time > 20h was necessary to achieve the saturated scavenging efficiency. Because the reaction between the precursor and impurity occurs at the impurity level, the elucidation of the precursor-scavenging mechanism will require a further detailed study. However, one possible explanation is the gathering of a siliceous intergranular phase into a discrete Si-containing phase such as ZrSiO4 by additional HT. The formation of ZrSiO4 is known to be quite difficult [31,32] due to its small formation energy [33] and large activation energy for the combination reaction between ZrO2 and SiO2.

New Approaches for Estimating and Improving…

13

[34] For example, only a low concentration of ZrSiO4 is reportedly formed even by the calcination of the precursor at 1500-1600oC when the heating rate was 300oC/h. [34] ZrSiO4 is known to form at > ~1200oC when prepared by the sol-gel method [35] and grow via an epitaxial mechanism. [36] Indeed, the seeding of 0.5 – 2 wt% ZrSiO4 is known to accelerate ZrSiO4 formation. [35-37] Therefore, the long duration at 1200oC results in sufficient ZrSiO4 nucleation and growth during the subsequent heating at higher temperatures. If the epitaxial growth mechanism is considered, the increase in the Rgbs value at HTT≥ 1250oC can be attributed to the low nucleus (ZrSiO4) density due to the fast passing through the region of nucleation. The conductivity improvement via HT at T=1200oC despite the subsequent sintering at T=1500-1600oC indicates that the Si-gathered phase formed during HT is stable up to the sintering temperature. Pure ZrSiO4 is known to dissociate at >1700oC [34] although this depends on the sample purity. [38,39] Therefore, the thermal stability of ZrSiO4 up to the sintering temperature (1500-1600oC) coincides well with the above explanation. Table 2. The 8YSZ specimens with the various siliceous impurity concentrations and their scavenging efficiency as a result of the additional HT at 1200oC for 40h prior to sintering at 1500oC for 4h, according to [42].

Spec. 8Y 8Y-PS 8Y100 8Y100-PS 8Y160 8Y160-PS 8Y310 8Y310-PS 8Y1000 8Y1000-PS † ‡

Heat treatment and sintering 1500† 1200(40)-1500‡ 1500 1200(40)-1500 1500 1200(40)-1500 1500 1200(40)-1500 1500 1200(40)-1500

Impurity conc. (ppm by weight) SiO2

Al2O3

Fe2O3

Rgbs ( BS )

Rgbs ( AS )

Rgbs ( AS )

2

2

Rgbs ( BS )

(Ω⋅cm )

(Ω⋅cm )

1.20

As received

0.20

100

29

28

160

28

25

310

29

24

1000

30

26

1.45 0.19 3.76 0.87 1.58 1.49 1.43 1.31

0.167 0.131 0.231 0.943 0.916

sintered at 1500oC for 4h. heat-treated at 1200oC for 40h prior to sintering at 1500oC for 4h

The 8 mol% Ytterbia-Stabilzied Zirconia (8YbSZ) specimens with an 80 and 170 ppmlevel background impurity showed a similar precursor-scavenging behavior except for the slight difference in the optimum HTT (1250oC). [40,41] This indicates that precursor scavenging can be extended to other stabilized zirconia systems. Other supporting evidence for the scavenging siliceous phase as a result of pre-sintering HT is the decrease in grain size. The grain size of 8YbSZ specimen containing 170 ppm of SiO2 was 8.1 µm, which is larger than the 5.2 µm for the 8YbSZ specimen containing 80 ppm of SiO2. This means that the trace siliceous impurity enhances grain growth. As a result of the pre-sintering HT at 1250oC, the grain-boundary conduction not only improved but the grain size decreased from 8.1 to 5.9 µm. This decrease in grain size can be attributed to the scavenging of siliceous impurity. Indeed, all the YSZ and YbSZ specimens showed a decrease in grain size by the pre-sintering

Jong-Heun Lee

14

HT and a higher scavenging efficiency always accompanied the larger decrease in grain size. [22, 40-42] However, precursor scavenging was not always valid for all the stabilized zirconia specimens. Table 2 shows the 8YSZ specimens containing 0-1000 ppm of siliceous impurities and their scavenging efficiencies by pre-sintering HT at 1200oC. [42] As shown in the table, the specimens with a low SiO2 impurity (< 200 ppm) showed an approximately 5-fold increase in grain-boundary conduction, while those with the high SiO2 impurity (> 300 ppm) did not show an improvement. This was explained by the inhomogeneous distribution of the siliceous phase for the heavily-SiO2-doped specimens and its influence on the scavenging reaction. The slow kinetics for forming ZrSiO4 might be one reason for the difficult precursor scavenging of a large siliceous impurity concentration.

4.2

The Improvement in the Grain-Boundary Conduction by Post-sintering Heat Treatment

The ‘precursor scavenging’ with pre-sintering HT is a convenient and effective approach to improving the grain-boundary conduction of stabilized zirconia with the segregation-level siliceous impurity. However, an intergranular liquid is usually formed when the siliceous impurity is abundant. [43,44] In this condition, the scavenging by forming a discrete Sicontaining phase cannot be applied and a new concept is necessary. This study attempted to enhance the grain-boundary conduction via the configurational change in the intergranular liquid. 1600

(a)

1200

(b) 40

800 105 Hz

400

1200 800 400 0

0

−Ζ"[kΩ.cm]

(c)

o

Temp.( C)

0

101 Hz

103 Hz

(d)

40 105 Hz

103 Hz

101 Hz

0

(e)

1200

(f) 40

800 400

105 Hz

0

103 Hz 101 Hz

0 0

10

20

Time(h)

30

0

40

80

Ζ'[kΩ.cm]

120

160

Fig.10 The heating schedules of 15CSZ specimens and their corresponding impedance spectra at 400oC in air: (a),(b) normal sintering at 1550oC for 4h, (c),(d) pre-sintering HT at 1300oC for 10h + sintering 1550oC for 4h. (e),(f) sintering 1550oC for 4h + post-sintering HT at 1300oC for 10h, according to [45].

New Approaches for Estimating and Improving…

15

Figure 10 shows the complex impedance of the 15CSZ specimens with the variation in the HT schedules. [45] In the specimen sintered at 1550oC, the size of the grain-boundary semicircle was almost identical to that of the grain-interior semicircle. (Fig.10 (a) and (b)). Although the pre-sintering HT at 1300oC for 10h decreased the grain-boundary resistivity, the change was not significant. (Fig.10 (c) and (d)) However, post-sintering HT at 1300oC for 10h greatly decreased the size of the grain-boundary semicircle. (Fig.10 (e) and (f)) From the systematic investigation on the HTT and HT time (Fig.11), it was found that the optimum HTT is 1300oC and the saturated scavenging efficiency can be attained by HTT at 1300oC for ≥10h. At the maximum scavenging efficiency, the grain-boundary conductivity improved 6.5 times by post-sintering HT.

(c)

1200

1200

Temp.(oC)

1600

o

Temp.( C)

(a) 1600

800 400

800 400

0

0 0

10

20

30

0

10

20

Time(h) 100

100

(b)

(no HT)=78.1 kΩ.cm

(d) ρgb

app

40

50

[kΩ.cm]

60

40

app

40

ρgb

[kΩ.cm]

60

ρgb

(no HT)=78.1 kΩ.cm

app

80

app

80

ρgb

30

Time(h)

20 0 1000

20 0 1100

1200

1300

HT Temperature(oC)

1400

0

10

20

30

40

HT time at 1300oC(h)

Fig.11 The apparent grain-boundary resistivity (ρgbapp) at 400oC for 15CSZ specimens (a), (b) with the variation in the HT temperatures (HT time=10h) and (c),(d) with the variation in the HT time (HT temperature=1300oC), according to [45].

Table 3 shows the changes in the intergranular phase as a result of the post-sintering HT. If the additional grain growth had occurred during post-sintering HT, it can decrease the ρgbapp value by decreasing the grain-boundary density. However, the average grain size did not change as a result of post-sintering HT. This confirms that the significant improvement in grain-boundary conduction emanated not from the changes in the grain-boundary density but from the change in the physico-chemical properties of the grain boundary. From TEM analysis, it was found the dihedral angles in the triple junction and between the two grains increased significantly as a result of the post-sintering HT. In addition, intergranular phases in the specimen with post-sintering HT were crystalline, while only the amorphous intergranular liquid was observed in the specimen without the HT. (see electron diffraction patterns in Table 3)

Jong-Heun Lee

16

The dewetting phenomenon can be explained in relation to the crystallization of an intergranular liquid. Fig.12 illustrates a schematic diagram. As shown in the equation in Fig.12, the dihedral angle of the intergranular liquid was determined by the ratio between the grain-boundary energy (γSS) and solid(matrix)-liquid interface energy (γSL). When the liquid is crystallized, the γSL term changed into solid(matrix)-solid(crystallized intergranular phase) interface energy (γSS’). Because the γSS’ value is usually larger than the γSL value [46], the dihedral angle can increase via crystallization. When a composition of glass melt is crystallized during cooling, the degree of crystallization is generally explained by TimeTemperature-Transformation curve (T-T-T curve). Usually, the optimum crystallization temperature, which is known as the ‘knee temperature’, is present because the crystallite nucleation is low near the melting temperature and crystallization is slow at the low temperature. [47] For crystallization, the cooling schedule should be designed to pass the crystallization regime as much as possible. Therefore, the optimum HTT (1300oC) is believed to be the knee temperature in the T-T-T diagram and the ρgbapp increase with increasing HT temperature from 1300 to 1350oC can be explained in the same way. Liquid melt

Crystalline

Temp.

γSL < γSS’

Amorphous φ↓

γ SS = 2γ SL cos

φSL

Crystalline φ↑

φSL < φSS’ γ SS = 2γ SS ' cos

2

φSS ' 2

Time Fig.12 Schematic diagram showing the crystallization and dewetting of the intergranular phase during post-sintering HT.

Finally, in order to compare the scavenging efficiency as a result of the addition of Al2O3, the 15CSZ specimens were sintered at 1450-1600oC and then heat-treated at 1300oC for 10h. Figure 13 shows the scavenging factor. When sintered at 1450oC, a ~2-fold increase in the grain-boundary conduction occurred, which is quite small compared with the ~15-fold increase by the addition of 1 mol% Al2O3. (see Fig.6(a)). However, at the high sintering temperature (1500-1600oC), a 5-fold increase in grain-boundary conduction was achieved, while the addition of Al2O3 deteriorates the grain-boundary conduction as a result of Al2O3 dissolution into the siliceous intergranular liquid. This suggests that the present post-sintering HT is a useful approach for improving the grain-boundary conduction of 15CSZ particularly when the sintering temperature is high.

New Approaches for Estimating and Improving…

17

Table 3. The comparison of the intergranular liquid phase between the 15CSZ specimens with and without post-sintering HT at 1300oC for 10h, according to [45]. Sintering at 1550oC for 4h + Postsintering HT at 1300oC for 10h

Sintering at 1550oC for 4h Intergranular phase at the triple junction

0.5 µm

0.5 µm Electron diffraction pattern of intergranular liquid

Average grain size φ(triple junction) φ(between two grains)

15.7 µm 49o ± 36o (# of observation =27) 8 o ± 8o (# of observation =28)

15.6 µm 89o ± 24o (# of observation =52) 89o ± 27o (# of observation =30)

Rgbs(AS)/Rgbs(BS)

10

Post-sintering HT > Al2O3 scavenging 1

0.1 1450

1500

1550

1600 o

Sintering temperature( C) Fig.13 The scavenging factor (SF) of 15CSZ specimens by post-sintering HT at 1300oC for 10h. The ρgbapp value was measured at 400oC in air, according to [45].

18

Jong-Heun Lee

5

Estimation of the Intergranular-Liquid-Phase Distribution Using Local Impedance Technique

Impedance spectroscopy has been a useful tool for separating the grain-interior and grainboundary contributions in many electroceramics. [48,49] In most analyses, the brick layer model, which assumes electrically and dimensionally homogeneous grain and grain boundary, was used. [7,13,24-26] (Fig.3(b)) However, the distribution of the ρgi and ρgb values can become spatially uneven for the specimens under an electric field [50-53] or with a functionally graded composition. [54] Therefore, Rodewald, Fleig and Maier proposed the local impedance technique using an array of the micro-scale electrodes. [50,51,55-57] When the specimen thickness was sufficiently large compared with the size of the microelectrode, the impedance between the microelectrode and large counter electrode mainly represents that of the near-microelectrode hemisphere where most of the potential drops [55,58]. This technique is useful because it not only spatially resolves the electrical properties within a large grain but also can be used to directly investigate the individual grain boundary.

(a) Liquid Phase Sintering

(b)

Impedance Measurement

-Z”

-Z” Z’

-Z” Z’

Z’

Fig.14 The schematic diagram of (a) dynamic liquid rearrangement during liquid phase sintering and (b) estimation of the intergranular-liquid distribution by millicontact impedance spectroscopy.

However, the conduction for a single grain boundary can become heterogeneous due to differences in the wetting of a resistive liquid [13], the liquid distribution [59], the grainboundary type (low angle or high angle) [60], boundary microstructure (straight or undulated) etc. Moreover, during liquid phase sintering, the intergranular liquid phase can flow dynamically to reduce the total liquid-vapor interface energy. Generally, the liquid phase is known to agglomerate at the center of the specimen in the initial stage of sintering and the agglomerated liquid flows outward as a result of further sintering. (Fig.14(a)) [61-63] Indeed,

New Approaches for Estimating and Improving…

19

the uneven distribution of the resistive intergranular liquid has been reported in stabilized zirconia. [59,64,65] In this case, the average impedance for a small volume rather than the impedance for a single grain boundary would be more advantageous in examining the spatial variation in grain-boundary conduction. (Fig.14(b))

5.1

Local Impedance Measurement by Successive Thinning of a Specimen

In order to estimate the spatial distribution of the intergranular liquid in 1-mol%-Al2O3(size = 0.3µm)-doped 15CSZ specimen, a series of complex impedance spectra were attained by the successive thinning of a specimen and subsequent measurement. (Fig.15) [66] The ρgbapp decreased to approximately 60% of its original value by removing 1/20 of the original specimen thickness (t) from both outer regions. This strongly suggests that the outer part of specimen has a more resistive grain boundary than the grain interior. The ρgbapp value decreased further as the specimen was further thinned and then saturated at a specimen thickness < 0.6t.

-|Z|sinθ [kΩ.cm]

10

0.7t

5

0.8t

0.9t

1.0t

543.5Hz 48.4kHz

0 25

57.6Hz

5.129kHz

30

35

40

45

|Z|cosθ [kΩ.cm] Fig.15 The complex impedance spectra of 1-mol%-Al2O3(size=0.3µm)-doped 15CSZ specimens sintered at 1600oC for 4h. 1/20 of original sample thickness (t) was removed successively from both outer regions and the corresponding impedance was measured at 450oC in air, according to [66].

The results attained by successive thinning can be analyzed in terms of the successive attachment of the outer layers from the specimen core. Because the addition of every outer layer results in a data variation in a cumulative manner, even a small increase in the cumulative ρgbapp value means a large increase in the real ρgbapp value. Indeed, from the analysis, it was calculated that the grain boundary at the outermost portion is ~15 times more resistive than that at the inner core. From the uniform grain-size distribution throughout the specimen, this heterogeneous ρgbapp profile was attributed to the outward rearrangement of the intergranular liquid phase.

20

Jong-Heun Lee

5.2

Millicontact Impedance Spectroscopy: The Fundamental Concept

Although the measurement of the local impedance by the successive thinning of a specimen is a convenient tool for analyzing the spatially-uneven grain-boundary characteristics, there are some restrictions. Because the data is in a cumulative form, unless the variation in the ρgbapp value is very large and monotonous, it is not easy to observe the presence of spatial variation. Therefore, instead of an indirect thinning method, a straightforward measurement of the local impedance is desirable when one wishes to characterize the spatial variation of the electrical properties with precision. Therefore, a local impedance technique using 1-dimensional or 2-dimensional array of the sub-millimeter-scale electrodes was suggested to estimate spatially the heterogeneous distribution of the resistive intergranular liquid. [67] This was referred to as ‘millicontact impedance spectroscopy’. As shown in Figure 14(b), the core concept is the determination of the spatial distribution in the grain-boundary resistance by measuring the local impedance for a small volume.

(a)

B

A

1

1’

(b)

B

A

Fig.16 Equi-potential contour maps of (a) single crystal and (b) polycrystalline materials with highly resistive grain boundary (ρgb =100ρgi, δgb=0.02dg). The changes of line and color indicate equipotential lines. Potential drop between two neighboring lines is 1/20 of the applied voltage, according to [67]. sp

In order to check the validity of the present technique, the equi-potential contours of single crystal and polycrystalline materials with a highly resistive grain-boundary phase (ρgbsp =100ρgi, δgb=0.02dg) were calculated using a simulation and the results are given in Fig.16. In a single crystal specimen, the local impedance measured between electrode 1 and 1’ are the approximate contribution from the volume element (A) defined by the electrode area and the specimen thickness. In particular, a circumventing conduction path between the electrode edges (see current line B in Fig.16(a)) should be considered and it becomes important when the specimen is thick and the electrode area is small. However, in a thin specimen with a sufficient electrode area, the overall ρgi profile can be attained unless the spatial variation of ρgi value is not large near the electrode edge.

New Approaches for Estimating and Improving…

21

When highly insulating grain boundaries are present, the equi-potential line near the edge becomes more parallel to the electrode (Fig.16(b)), which significantly narrows the circumventing conduction. The narrowing effect will be further enhanced as the grain size decreases or as the ρgbsp value increases. Therefore, millicontact impedance spectroscopy can be used as an effective and valid tool to estimate the spatial distribution of electrical properties in stabilized zirconia with a micro-meter-sized grain and a highly resistive grain boundary.

5.3

Millicontact Impedance Spectroscopy: The Distribution of Intergranular-Liquid Phase

The presence of a small amount of a liquid phase is known to enhance the sintering rate by providing a rapid material-transport path. According to the literature [61-63], the liquid can rearrange dynamically during liquid-enhanced sintering, which results in a spatially uneven liquid distribution. When the liquid is abundant, such a liquid rearrangement process can be observed using a conventional scanning electron microscope (SEM). However, at the small liquid content, SEM is not sufficient for observations because the liquid usually presents as an intergranular film with a few-nanometer thickness. In this case, the complex impedance, which is very sensitive to the configuration of intergranular phase, can be used as a measure for the liquid distribution. Figure 17(a) shows the electrode array for measuring the local impedance. The specimen was cut along the diameter of a cylindrical specimen with a thickness of 0.5 mm. A 2dimensional array of cube-shape electrode (0.5mm X 0.5mm) was made in both surfaces by screen printing a Pt paste and subsequent heat treatment. The impedance was measured at 400oC between two opposite electrodes. Fig.17 (b) shows the complex impedance spectra as a function of the distance from the surface along the axial direction (X=5.25) for the 1-mol%Al2O3-doped 15CSZ specimen sintered at 1650oC. [67] The ρgbapp value attained from the intermediate-frequency (~102-103 Hz) semicircle changed to a large extent from 56 to 3.6 kΩ⋅cm, while the ρgi value attained from the high-frequency semicircle remained constant as 82 kΩ⋅cm. Impedances for all the electrodes were measured, and the Rgbs values were calculated from the ρgbapp value and the average grain size. (Fig.17(c)) Note that the profile of Rgbs was the same as that of ρgbapp because the grain size was uniform throughout the specimen. The Rgbs value was high at the surface, and decreased abruptly at the second electrode showing a saturated value (25-15 Ω⋅cm2) at the core region (x > 2mm). Consequently, the grain boundary at the surface was ~10-15 times more resistive than that located at the inner part. The profile of Rgbs clearly demonstrates that the spatial distribution of the grain-boundary conductivity can be estimated by the present technique. In order to investigate the reason for the large difference in ρgbapp, the Al2O3 concentration and sintering temperature were varied. For this measurement, 1-dimensional array of electrodes (size: 2 mm X 0.5 mm) in Fig.18(a) was employed. Figure 18(b), (c), (d), (e) show the results for the 15CSZ specimens sintered at 1600oC for 4h with the variation in the Al2O3 concentrations. [68] In pure 15CSZ specimen, the ρgbapp value increased with increasing distance from the surface (x) and became saturated at x ≥ 2mm. However, the

Jong-Heun Lee

22

tendency for increasing ρgbapp disappeared at 0.5-mol%-Al2O3 (Fig.18(c)) and completely reversed at 1.0- and 1.5-mol%-Al2O3. (Fig.18(d), (e)) Surface 105

104

103

102

10 Hz

(a) Pt electrode array 2 Area: 0.5 x 0.5 mm Spacing: 0.5 mm

Ω. cm] -|Z|sinθ [k

50 40 30 20 10 0

40

Center 80

120

0.25 1.25 2.25 3.25 4.25 5.25 6.25

160

|Z|cosθ [kΩ.cm]

d from surface(mm)

(b)

Y Center

13 mm

X

(c)

t=0.5 mm Φ=11 mm

Rgbs [Ω.cm 2]

300

200

Y po

m) n (m s i t io

0 1 2 3 4 5 6

100

10

8

nter 4 ce 6

2

0

X position (mm)

Fig.17. (a) Schematic diagram of a 2-dimensional array of electrodes for measuring the local impedance. (b) The impedance spectra of 1-mol%-Al2O3(size 0.3 µm)-doped 15CSZ specimen measured at 400oC in air as a function of distance from the surface along the cylinder axis. (The specimen was sintered at 1650oC for 4h) (c) The distribution of the resistance per unit grain-boundary area (Rgbs) as a function of specimen location, according to [67].

Figure 19 shows the ρgbapp profiles of 1-mol%-Al2O3-doped 15 CSZ specimens as a function of the sintering temperature. [59,69] When sintered at 1650oC for 4h, the grain boundary at the surface was ~15 times more resistive than that in the center region. According to TEM analysis, there was abundant intergranular liquid at the surface, which penetrated almost all the grain boundary, while that in the center region showed a discontinuous configuration with a frequent lenticular pocket. The average dihedral angles of the liquid pockets were ~20° at the surface and ~100° at the center. In addition, the liquid composition was different from each other, which indicates phase separation of the intergranular liquid.

New Approaches for Estimating and Improving…

surface

Cylinder-shape specimen

x(mm)

11 mm

Impedance Measurement

ρgb

app

x=5.25 0.5 mm

[kΩ.cm]

13 mm

x=0.25

center

100 ρ=5.33,d =15. g 80 60 (b) 40 20 ρ=5.49,dg=31. (c) 0 40 ρ=5.45,dg=35.

(a) Pt electrode 0.5 x 2 mm2 Spacing: 0.5mm

23

(d)

20 0 40

ρ=5.45,dg=37.

(e)

20 0 0 1 2 3 4 5 6 x from surface (mm)

Fig.18 (a) Schematic diagram of a 1-dimensional array of electrodes for measuring the local impedance and the profile of the apparent grain-boundary resistivity (ρgbapp) of 15CSZ the specimens at 400oC with the variation of Al2O3 (size=0.3 µm) amounts: (b) pure 15CSZ, (c) 0.5-mol%-Al2O3-doped 15CSZ, (d) 1.0-mol%-Al2O3-doped 15CSZ and (e) 1.5-mol%-Al2O3-doped 15CSZ. All the specimens were sintered at 1600oC for 4h, according to [68].

However, when 1-mol%-Al2O3-doped 15 CSZ specimen was sintered at 1450oC for 0 h, the profile of ρgbapp became opposite. The ρgbapp value was higher in the center region. (Fig.19(e)) This is analogous to the pure 15CSZ specimen sintered at 1600oC (Fig.18(a)). Note that two specimens (Fig.19(e) and Fig.18(b)) show a significantly low density(5.285.33) compared with the other specimens. (5.41-5.45) During liquid phase sintering, it is generally acknowledged that the liquid agglomerates at the center region in the initial stage of sintering to decrease the total solid-liquid interface energy. [61] Indeed, TEM analysis showed that the liquid pocket size in the central region was ~ 3 times larger than that near the surface in the pure 15CSZ specimen (Fig.18(b)). Therefore, the large ρgbapp values in Fig.19(e) and Fig.18(b) can be attributed to liquid coagulation at the initial stage of sintering. From the different composition of wetting and non-wetting intergranular liquids in the specimen sintered at 1650oC (Fig.19(a)), the phase separation can be deduced. As the sintering temperature was increased from 1450 to 1650oC, grain growth and the densification occurred simultaneously. [Fig.19 (d), (c), (b), (a)] During this process, among the two liquids, a liquid with a composition that improves the wetting ability will flow preferentially to the surface along the capillary. The decrease in the pore size as a result of densification will be advantageous for a capillary rise. [70] Besides, grain growth is known to be necessary for liquid to flow into the pores. However, densification was always accompanied by grain growth as a result of the addition of Al2O3 (Fig.18) and the increase in the sintering temperature (Fig.19), which make the separation of the variables difficult. Therefore, the lowdensity specimen (5.27) with almost the same grain size (29.9 µm) was fabricated by

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24

sintering a low-density green body. (Fig.20(c)) A relatively flat profile was attained, which indicates that not only grain growth but also densification is a prerequisite for the capillary action.

60 ρ=5.44,dg=45.2µm

Wetting Liquid

(a)

40

ρgbapp[kΩ.cm]

20

Non-wetting Liquid

0 40 ρ=5.45,dg=35.1µm

(b) 20

Outward Capillary rise

0 40 ρ=5.47,d =17.9µm g

(c)

20 0 20

ρ=5.41,dg=9.7µm

(d) Liquid

Phase Coagulation Separation

0 40

(e) 20

ρ=5.28

0

0

1

2

3

4

5

6

x from surface (mm) Fig.19 The profile of the apparent grain-boundary resistivity (ρgbapp) of 1.0-mol%-Al2O3(size=0.3 µm)doped 15CSZ the specimens with the variation of the sintering time and temperatures: (a) sintered at 1450oC for 0h, (b) sintered at 1450oC for 4h, (c) sintered at 1500oC for 4h, (d) sintered at 1600oC for 4h and (e) sintered at 1650oC for 4h. The impedance was measured at 400oC in air, according to [59,69].

Figure 20(b) shows the ρgbapp profiles when 1-mol%-Al2O3-doped 15CSZ specimen was cooled rapidly (800oC/min) after sintering 1600oC for 4h. Compared to the specimen cooled at a rate of 200oC/min (Fig.20(a)), the profile moved upward. This means a significant increase in the average ρgb value for the entire specimen. The configuration of the intergranular liquid can be changed dynamically during cooling because the wetting/dewetting characteristics depend on temperature. The increase in the overall profile by fast cooling qualitatively means that the high-temperature configuration of the intergranular liquid is more resistive. In addition, the grain-boundary resistance is known to depend on the thermal history, [71] sintering atmosphere, [72], and superplastic deformation

New Approaches for Estimating and Improving…

25

[73]. These phenomena can be better understood when the local electrical properties are characterized using a spatially resolved impedance technique.

40 ρ=5.45,dg=35.1µm

(a)

20

ρgbapp[kΩ.cm]

0

ρ=5.46,dg=35.0µm

(b)

0 40 ρ=5.27,dg=29.9µm

(c)

60 40 20

20 0 0

1 2 3 4 5 6 x from surface (mm)

Fig.20 The profile of the apparent grain-boundary resistivity (ρgbapp) of 1.0-mol%-Al2O3-doped 15CSZ the specimens with the variation of cooling rate and specimen density: (a) cooling rate: 200oC/h, density: 5.45 g/cm3, dg: 35.1 µm, (b) cooling rate: 800oC/h, density: 5.46 g/cm3, dg: 35.0 µm, and (c) cooling rate: 200oC/h, density: 5.27 g/cm3, dg: 29.9 µm. All the specimens were sintered at 1600oC for 4h, according to [67].

The redistribution of the liquid phase could also be observed in the 8YSZ specimen containing 0.5 mol% of SiO2 and 0.5 mol% Al2O3. Figure 21 shows the ρgbapp profiles for the specimens sintered at different temperatures. [74] When sintered at 1450oC for O h, liquid coagulation at the center region could be noted. (Fig.21(c)) As the sintering temperature increased, the liquid phase was spread outward to make a uniform distribution. (Fig. 21(b) and (a)) The reason why the concentration of the wetting intergranular liquid was not found near the surface region can be explained by there being no phase separation of a liquid phase in 8YSZ system. Therefore, the liquid coagulation at the center and its subsequent redistribution outward could be observed as a typical liquid phase sintering behavior. Figure 21(d) and (e) show the SEM microstructures of the center and surface regions for the specimen sintered 1450oC O h. despite the significant difference in the ρgbapp value, the microstructure as well as the grain size was the same. This demonstrates that the rearrangement of a small amount of intergranular liquid, which cannot be observed by SEM, can be observed using the local impedance technique.

Jong-Heun Lee

26

30 20

ρgbapp[kΩ.cm]

10

(d)

0 40 30

(e)

20 50 40

5 µm

5 µm

30 0

1

2

3

4

5

x from surface (mm) Fig.21 The profile of the apparent grain-boundary resistivity (ρgbapp) of 8YSZ the specimens containing 0.5 mol% Al2O3 and 0.5 mol% SiO2 with the variation of sintering temperature: The ρgbapp profiles for the specimens (a) sintered at 1550oC for 4h, (b) sintered at 1550oC for 0h, and (c) sintered at 1450oC for 0h. The impedance was measured at 400oC in air. The microstructures of a specimen sintered at 1450oC for 0h (d) at the central region and (e) at surface region, according to [74].

5.4

Millicontact Impedance Spectroscopy: The Investigation of Liquid Penetration

Millicontact impedance spectroscopy can be also applied to the characterization of the functionally graded materials or to the determination of material transport near the interface. For example, the interface reaction between the LaMnO3-based cathode and the silicate impurity diffused from YSZ can form a deleterious lanthanum silicate phase at the interface, which affects the long-term stability of the SOFC.[75] In order to understand this phenomenon, the migration of a siliceous impurity across the YSZ specimen should be characterized. Figure 22(a) shows the experimental setup for providing an artificial impurity on the 8YSZ specimen by dropping a dilute solution.[76] The impedance spectra are given in Fig. 22(b). The ρgi values in the first and second electrodes (x=0.25, 1.25 mm) from the surface were ~7-8 kΩ⋅cm, which was slightly larger than the value of the lower part (~6 kΩ⋅cm). (x=2.25-7.25 mm) The ρgbapp showed more dramatic changes. The uppermost ρgbapp value (at x=0.25) was ~ 32 times larger compared with those at x=1.25-7.27 mm. This

New Approaches for Estimating and Improving…

27

indicates that the penetration depth of the impurity is ~ 0.5-1.5 mm. The main advantage of millicontact impedance spectroscopy is that both the spatial variations of the grain-interior and grain-boundary conductivities can be examined separately.

(a)

CaSi2O5 drop

8YSZ green body

Sintered at 1550oC for 4h

x

3

10 Hz

)

(b) 105 Hz

8

mm ace( surf

TOP

0.25 2.25

6

2 0

6.25 10

BASE

20

|Z|cosθ [kΩ.cm]

30

m

4.25

4

x fro

. cm] θ [kΩ -|Z|sin

10

Fig.22 (a) Schematics of the experimental set up for liquid penetration and the electrode configuration. (b) The Complex impedance spectra of the 8YSZ specimen sintered at 1550oC for 4h after dropping the ethanol solution containing TEOS and Ca(NO3)·4H2O. The total amount of doping was 0.0025g in terms of CaSi2O5 and the impedance was measured at 400oC in air as a function of the distance from the surface (x), according to [76].

6

Conclusions

New methods for improving the grain-boundary conduction of stabilized zirconia were suggested. In contrast to the scavenging of the resistive intergranular phase by the addition of Al2O3, the configurational change in the intergranular phase via a two-stage sintering process was used to enhance the grain-boundary conduction, which was called ‘precursor scavenging’. For the 8YSZ and 8YbSZ specimens with a segregation-level siliceous impurity (70-170 ppm), pre-sintering heat treatment (HT) at 1200-1250oC for > 20h resulted in approximately the same scavenging efficiency as that by Al2O3 addition. The formation of a discrete Sicontaining phase such as ZrSiO4 with the assistance of additional HT was suggested to be the scavenging mechanism. In the 15CSZ specimen with the intergranular liquid, the postsintering HT at 1300oC for > 10h showed ~6.5-fold increase in the grain-boundary conductivity, which could be explained by the dewetting of the intergranular phase due to crystallization. In the second part, a new local impedance technique, called ‘Millicontact Impedance Spectroscopy’, was used to estimate the spatially uneven grain-boundary character. The dynamic rearrangement of a small amount of liquid phase, which could not be observed

Jong-Heun Lee

28

by SEM, was successfully determined using the present local impedance technique combined with a sub-millimeter-scale electrode array. In conclusion, millicontact impedance spectroscopy is a useful tool for characterizing functionally graded materials and the spatial distribution of electrical conductivity in many electroceramics.

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

21. 22. 23. 24. 25. 26. 27. 28.

Estell, T. H.; Flengas, S.N., Chem. Rev. 1970, 70, 339-376. Subbarao, E. C.; Maiti, H. S., Solid State Ionics 1984, 11, 317-338. Steele, B.C. H., J. Mater. Sci. 2001, 36, 1053-1068. Kilner, J. A., Solid State Ionics, 1983, 8, 201-207. Kilner, J. A.; Steele B. C. H., In Nostoichiometric Oxides; Sørensen, O. T., Academic Press: New York, NY, 1981, pp.233-269. Verkerk，M. J.; Winnubst, A. J. A.; Burggraaf, A. J, J. Mater. Sci. 1982, 17, 3113-3122. Verkerk, M. J.; Middelhuis, B. J.; Burggraaf, A. J., Solid State Ionics 1982, 6, 159-170. Aoki, M.; Chiang, Y. -M.; Kosacki, I.; Lee, J. -R.; Tuller, H.; Liu, Y. J. Am. Ceram. Soc. 1996, 79, 1169-1180. Guo, X., Solid State Ionics 1995, 81, 235-242. Guo, X.; Sigle, W; Fleig, J.; Maier, J., Solid State Ionics 2002, 154-155, 555-561. Guo, X.; Computational Materials Science 2001, 20, 168-176. Lee, J. –S.; Kim, D. –Y. J. Mater. Res. 2001, 16, 2739-2750. Gödickemier, M.; Michel, B.; Orliukas, A.; Bohac, P.; Sasaki, K.; Gauckler, L.; Henrich, H.; Schwander, P.; Kostorz, G.; Hofmann, H.; Frei, O. J. Mater. Res. 1994, 9, 1228-1240. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Komatsu, M.; Haneda, H. J. Am. Ceram. Soc. 2000, 83, 1273-1275. Rajendran, S.; Drennan, J.; Badwal, S. P. S., J. Mater. Sci. Lett. 1987, 6, 1431-1434. Feighery, A. J.; Irvine, J. T. S., Solid State Ionics 1999, 121, 209-216. Filal, M.; Petot, C.; Mokchah, M.; Chateau, C.; Carpentier, J. L., Solid State Ionics 1995, 80, 27-35. Guo, X.; Tang, C. -Q.; Yuan, R. -Z., J. Euro. Ceram. Soc. 1995, 15, 25-32. Butler, E. P.; Drennan, J. J. Am. Ceram. Soc. 1982, 65, 474-478. Drennan, J.; Badwal, S. P. S. In Science and Technology of Zirconia III, Somiya, S.; Yamamoto, N.; Yanagida, H., American Ceramic Society: Colombus, OH, 1988, pp.807817. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Komatsu, M.; Haneda, H. Electrochemistry 2000, 68, 427-432. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Komatsu, M.; Haneda, H., J. Electrochem. Soc. 2000, 147, 2822-2829. Macdrodt, W. C.; Woodrow, P. M., J. Am. Ceram. Soc. 1986, 69, 277-280. Dijk, T. van; Burggraaf, A. J., Phys. Stat. Solid A (a) 1981, 63, 229-240. Ioffe, M.; Inozemtsev, M. V.; Lipilin, A. S.; Perfilev, M. V.; Karpachov, S. V., Phys. Stat. Solid A 30, 87 (1975). Miyayama, M.; Yanagida, H.; Asada, A., Am. Ceram. Soc. Bull. 1986, 65, 660-664. Radford, K. C.; Bratton, R. J., J. Mater. Sci. 1979, 14, 66-69. Yuzaki, A.; Kishimoto, A. Solid State Ionics 1999, 116, 47-51.

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29. Guo, X., J. Am. Ceram. Soc. 2003, 86, 1867-1873. 30. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Takenouchi, S., Ceramics International 2001, 27, 269-276. 31. Spearing, D. R., J. Am. Ceram. Soc. 1997, 81, 1964-1966. 32. Mori, T.; Yamamura, H.; Kobayashi, K.; Mitamura, T., J. Am. Ceram. Soc. 1992, 75, 2420-2426. 33. Ellison, A. J. G.; Navrotsky, A. J. Am. Ceram. Soc. 1992, 75, 1430-1433. 34. Kanno, Y., J. Mater. Sci. 1989, 24, 2415-2420. 35. Mori, T; Yamamura, H.; Kobayashi, K.; Mitamura, T., J. Mater. Sci. 1993, 28, 49704973. 36. Vilmin, G.; Komarneni, S.; Roy, R., J. Mater. Sci. 1987, 22, 3556-3560. 37. Shi, Y.; Huang, X.; Yan, D., J. Euro. Ceram. Soc. 1994, 13, 113-119. 38. Butterman, W. C.; Foster, W. R., Am. Mineral. 1967, 52, 880-885. 39. Curtis, C. E. Snowman, H. G., J. Am. Ceram. Soc. 1953, 36, 190-198. 40. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Drennan, J.; Kim, D. –Y. J. Am. Ceram. Soc 2000, 84, 2734-2736. 41. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Drennan, J.; Kim, D. –Y., J. Electrochem. Soc. 2002, 149, J35-J40. 42. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Drennan, J.; Kim, D. –Y., J. Mater. Res. 2001, 16, 2377-2383. 43. Mecartney, M. L., J. Am. Ceram. Soc. 1987, 70, 54-58. 44. Beekmans, N. M.; Heyne, L., Electrochimca Acta. 1976, 21, 303-310. 45. Jung, Y. S.; Lee, J. -H.; Lee, J. –H.; Kim, D. –Y., J. Electrochem. Soc. 2003, 149, J49J53. 46. Chiang, Y. –M.; Birnie III, D.; Kingery, W. D., Physical Ceramics, John Wiley, New York, NY, 1997, p.362 and p.427. 47. Chiang, Y. –M.; Birnie III, D.; Kingery, W. D., Physical Ceramics, John Wiley, New York, NY, 1997, p.432. 48. Bonanos, N.; Steele, B. C. H.; Butler, E. P.; Johnson, W. B.; Worrell, W. L.; Macdonald, D. D.; McKubre, M. C. H. in Impedance Spectroscopy; Macdonald, J. R., Wiley: New York, NY, 1987, pp.191-316. 49. Sinclair, D.C.; West, A. R., J. Appl. Phys. 1989, 66, 3850-3856. 50. Rodewald, S.; Fleig, J.; Maier, J. J. Am. Ceram. Soc. 2000, 83, 1969-1976. 51. Rodewald, S.; Fleig, J.; Maier, J. J. Euro. Ceram. Soc. 1999, 19, 797-801. 52. Waser, R.; Baiatu, T.; Härdtl, K. –H. J. Am. Ceram. Soc. 1990, 73, 1645-53. 53. Waser, R.; Baiatu, T.; Härdtl, K. –H. J. Am. Ceram. Soc. 1990, 73, 1654-1662. 54. Sánchez-Herencia, A. J.; Moreno, R.; Jurado J. R., J. Euro. Ceram. Soc. 2000, 20, 16111620. 55. Fleig, J.; Rodewald, S.; Maier, J., Solid State Ionics 2000, 136-137, 905-911. 56. Fleig, J.; Rodewald, S.; Maier, J., J. Am. Ceram. Soc. 2001, 84, 521-530. 57. Fleig, J.; Maier, J., Solid State Ionics 1996, 85, 9-15. 58. Skapin, A. S.; Jamnik, J.; Pejovnik, S. Solid State Ionics 2000, 133, 129-138. 59. Lee, J. –H.; Lee, J. H.; Kim, D. –Y., J. Am. Ceram. Soc. 2002, 85, 1622-1624. 60. Fisher, C. A.; Matsubara, H., J. Euro. Ceram. Soc. 1999, 19, 703-707. 61. Kwon, O. –J.; Yoon, D. N. In Sintering and Related Phenomena; Kuczynski, G. C., Plenum Publishing Company: New York, NY, 1980, pp.233-269.

30 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76.

Jong-Heun Lee Yoo, Y. -S.; Kim, J. –J.; Kim, D. –Y., J. Am. Ceram. Soc. 1987, 70, C322-C324. Kim, Y. S.; Park, J. K.; Yoon, D. N., J. Powder Metall. 1985, 21, 30-37. Chaim, R.; Brandon, D. G.; Heuer, A., Acta. Metall. 1986, 34, 1933-1939. Souza, D. P. E. De; Souza, M. F. De., J. Mater. Sci. 1999, 34, 4023-4030. Lee, J. -H.; Mori, T.; Li, J. –G.; Ikegami, T.; Takenouchi, S. J. Euro. Ceram. Soc. 2001, 21, 13-17. Lee, J. -H.; Jung, Y. –S.; Woo, H. –S.; Chung, Y. –C.; Kim, D. –Y., J. Euro. Ceram. Soc. 2004, 24, 1129-1133. Lee, J. –H.; Lee, J. H.; Jung, Y. –S.; Kim, D. –Y., J. Am. Ceram. Soc. 2003, 86, 15181521. Choi, J. –H.; Lee, J. –H.; Kim, D. -Y.; J. Kor. Ceram. Soc. 2002, 39, 818-821. Leed, J. S. Principles of Ceramic Processing, Wiley, New York, NY, 1995, p.25. Badwal, S. P. S.; Drennan, J., J. Mater. Sci. 1989, 24, 88-96. Badwal, S. P. S.; Hushes, A. E. J. Euro. Ceram. Soc. 1992, 10, 115-12. Drennan, J.; Swain, M. V.; Badwal, S. P. S., J. Am. Ceram. Soc. 1989, 72, 1279-1281. Jung, Y. –S.; Lee, J. -H.; D. –Y., J. Euro. Ceram. Soc. 2003, 23, 499-503. Kuščer, D.; Holc, J.; Hrovat, M.; Bernik, S.; Samardžija, Z.; Kolar, D. Solid State Ionics 1995, 78, 79-85. Jung, Y. -S.; Choi, J. –H.; Lee, J. –H. Mater. Sci. & Eng. B. 2004, 108, 219-222.

In: Ceramics and Composite Materials: New Research ISBN: 1-59454-370-4 Editor: B.M. Caruta, pp. 31-100 © 2006 Nova Science Publishers, Inc.

Chapter 2

EFFECTIVE ELASTIC MODULI OF ALUMINA, ZIRCONIA AND ALUMINA-ZIRCONIA COMPOSITE CERAMICS Willi Pabst and Eva Gregorová Department of Glass and Ceramics Institute of Chemical Technology in Prague (ICT Prague) Technická 5, 166 28 Prague, Czech Republic

Abstract The effective elastic moduli of polycrystalline alumina and zirconia as well as aluminazirconia composite ceramics are investigated from the theoretical point of view, with a sideglance on experimental results and applications. In the first section alumina, zirconia and alumina-zirconia composites are introduced as structural materials, relations of elastic moduli to other properties are recalled and targets of microstructural design are formulated. In the second section elastic properties are defined from the viewpoint of rational mechanics for anisotropic and isotropic materials in general. The difference between adiabatic and isothermal elastic moduli is explained and estimated for alumina and zirconia. In the third section effective elastic properties are defined and discussed from the viewpoint of micromechanics and composite theory. General formulae are given for the calculation of effective elastic moduli of polycrystalline materials from monocrystal data. Further, the Voigt-Reuss bounds for the effective elastic moduli of multiphase materials are given, as well the Hashin-Shtrikman bounds for the special case of two-phase materials. For porous materials the dilute approximations are recalled as well as the predictions following from the self-consistent, Mori-Tanaka, differential, Gibson-Ashby and Coble-Kingery approach as well as the functional equation approach recently advocated by the authors. A comprehensive survey of model relations for the porosity dependence of elastic moduli is given, including exponential and power-law relations and a new relation recently proposed by the authors, which seems to be the simplest relation allowing for the occurrence of a percolation threshold (critical porosity). In the fourth section all these micromechanical bounds and relations are applied to the alumina-zirconia system. Using a theoretically sound and experimentally confirmed set of elastic moduli (and Poisson ratios) for dense (i.e. fully sintered) polycrystalline alumina and zirconia the Hashin-Shtrikman bounds of dense alumina-zirconia composites are calculated and compared to experimentally measured values. Several predictions for porous alumina, zirconia and alumina-zirconia composites are compared to the

Willi Pabst and Eva Gregorová

32

data measured for ceramics with convex interconnected pores prepared by the starch consolidation casting technique. A master fit curve is given for porous ceramics with this type of matrix-inclusion microstructure and explicit numerical expressions are given throughout. The last section gives examples of the mathematical modeling of other effective properties and their dependence on composition and microstructure and an outlook is given to future research aims. In particular, the significance of interfaces is emphasized and ideas on the way from micromechanics to nanoscience − towards a general mixture theory − are outlined.

1

Introduction

Macroscopically (i.e. from a continuum viewpoint), brittle materials (e.g. ceramics, including glasses) exhibit purely elastic response upon loading until fracture occurs when a critical stress level, the strength, is exceeded. Since under usual conditions (i.e. moderate temperatures and pressures) fracture is commonly encountered before the region of nonlinear elasticity, plasticity, creep, viscoelasticity or viscosity can be attained, linear elasticity is the appropriate theoretical framework to treat the stress-strain behavior of most ceramics and glasses. The elastic properties of ceramics, defined as tensorial or scalar coefficients in linear constitutive equations [Torquato 2002, Pabst & Gregorová 2003a], determine their (static and dynamic) mechanical behavior and can have a decisive influence on many other properties, cf. e.g. [Billington & Tate 1981, Cook & Pharr 1994, Davidge 1979, Green 1998, Irwin 1958, Komeya & Matsui 1994, Lawn 1993, Menčík 1992, Munz & Fett 1999, Salmang & Scholze 1982, Wachtman 1996]. According to the early theories of brittle fracture [Griffith 1920, Orowan 1934, 1949] the theoretical strength σ 0 of a ceramic material should be directly proportional to the square root of the elastic modulus E (tensile modulus), i.e.

σ0 ∝ with

Eγ , d

γ being the specific surface energy (surface tension) and d the size of a typical defect

(Griffith) or interatomic spacing (Orowan). That means that the elastic modulus, together with the specific surface energy, determines the maximum strength which a material can principally achieve when it can be prepared in a defect-free state. For example, the smooth, defect-free surface is the inherent reason for the high strength of glass fibers compared to bulk glass. At the same time it is evident, however, that pores (which can be viewed as defects) will reduce the strength by more than what should expected based on the decrease of E alone (large pores more than small ones). Similarly, the critical value of the stress intensity factor K IC , the so-called fracture toughness, which arises within the theory of linear fracture mechanics [Irwin 1958], is directly proportional to the elastic modulus

K IC ∝ Eγ . There have been attempts to relate the elastic constants to other, operationally defined, mechanical “properties”, like hardness or wear performance. It is remarkable, e.g. that using

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

33

Sanditov’s simple formula for the (micro-) hardness H of glasses [Sanditov 1977, Volf 1988]

 1 − 2ν  H = ⋅E ,  6 + 6ν  with ν being the Poisson ratio, one obtains for alumina and zirconia (inserting for E and ν the values recommended below) hardness values of 29 GPa and 10 GPa, respectively, which are not too far from the typical Vickers, Knoop or Berkovich indentation hardness values experimentally measured and reported in the literature, which are approx. 24 GPa and 12 GPa, respectively (orientational values only), cf. also [Conway 1986, Lawn & Howes 1981, Luo & Stevens 1999, Marshall et al. 1982, Zeng et al. 1992]. Moreover, as a result of Hertzian contact mechanics [Hertz 1882, Johnson 1985, Lawn 1993] the fracture toughness is related to the elastic modulus and the indentation hardness via relations of the type

K IC ∝

E  f  ⋅ , H  c3/ 2 

where f is the loading force and c the radius of the cracks originating from the indentation. In contrast to hardness, which is approximately twice as high for alumina than for zirconia, both strength and fracture toughness are usually significantly higher for zirconia ceramics and alumina-zirconia composites than for pure alumina ceramics. For zirconia they can attain peak values of approx. 1500 MPa and 15 MPa⋅m1/2, respectively, compared to approx. 400 MPa and 4 MPa⋅m1/2 for alumina (orientational values only) [Becher & Rose 1994, Cawley & Lee 1994, Hannink et al. 2000, Lee & Rainforth 1994, Munro 1997, 2002, NIST 2002, Wachtman 1996]. Elastic properties, together with the (tensile) strength σ and the thermal expansion coefficient α (and possibly thermal conductivity or thermal diffusivity) also determine a material’s response to thermal shocks. Accordingly, all thermal shock parameters R are of the form [Davidge 1979, Menčík 1992, Salmang & Scholze 1982, Wachtman 1996]

R∝

σ (1 − ν ) . αE

That means, the thermal shock resistance is best for high-strength, low-expansion materials with low elastic constants (low tensile modulus and low Poisson ratio). From an atomistic point of view (discrete approach) the elastic constants of a material are dependent on bond strength (force constant) and bond density (number of bonds per unit volume). That means, all factors controlling bond strength and bond density will have an influence on the elastic constants, e.g. composition (type of ions, including their size and deformability), crystal structure (particle packing, coordination number, molar volume) and bonding type (covalent or ionic) [Born & Huang 1954, Newnham 1975, Pauling 1968, Sahimi 2003a]. Although in principle it may be possible to calculate the elastic constants from the force-distance relationships between atoms (viewed as harmonic oscillators) [Born & Huang

Willi Pabst and Eva Gregorová

34

1954], this has been successful only for the simplest structures and bonding types and is usually not done in practice. With increasing temperature the bond strength and density decrease, due to lattice expansion (i.e. due to anharmonic effects), resulting in a decrease in the elastic moduli [Ashcroft & Mermin 1976, Kittel 1988]. It is common practice to measure elastic constants and their temperature dependence experimentally, either via static (e.g. three- or four-point bending, XRD or neutron scattering in connection with strain gauges) or via dynamic tests (e.g. sound velocity methods or resonant frequency techniques). It is well known from solid state physics, cf. e.g. [Ashcroft & Mermin 1976, Kittel 1988], that the elastic constants determine the velocity of sound waves in solids. For example, the velocity of transversal waves (shear waves) vT is given by

vT =

G

ρ

and the velocity of longitudinal waves (compressional waves) v L is given by the relation

vL =

3K + 4G = 3ρ

(1 − ν ) , ρ (1 + ν ) (1 − 2ν )

E

where G is the shear modulus, K the bulk modulus, E the tensile modulus, ν the Poisson ratio and ρ the density (bulk density of the material). Similar relations hold for surface waves (Rayleigh waves). Therefore sound wave velocity measurements (e.g. by Brillouin scattering, pulse-echo or ultrasonic techniques) can vice versa be used to determine the elastic constants experimentally [Aksel & Riley 2003, Asmani et al. 2001, Hardy & Green 1995, Lima et al. 2003, Martin et al. 1998a, 1998b, Shen & Hing 1997], e.g. from the shear wave velocity vT the tensile modulus can be calculated via the relation

E = 2 G (1 + ν ) = 2 ρ vT2 (1 + ν ) . Materials selection and engineering design is a complex task which must take account of various requirements depending on the intended applications. In many applications proper materials selection and design implies a compromise between high stiffness (elastic modulus), strength and fracture toughness and at the same time minimal weight, i.e. density. When this is the case, certain ratios between the elastic moduli or strength and various powers of density (e.g. E

ρ , E ρ 2 or E ρ 3 , depending on the prevailing loading mode) can be the critical

selection and design parameters, cf. e.g. [Brezny & Green 1994, Gibson & Ashby 1997]. Certain requirements concerning mechanical performance and reliability must be fulfilled in all applications, e.g. thermally or electrically insulating materials (dense alumina for electronic circuit supports, thermally insulating closed-pore alumina or zirconia), catalyst supports, filter membranes and membrane supports (open-pore alumina), ionic conductors, oxygen sensors (dense zirconia) and oxygen-ion-conducting membranes in solid oxide fuel cells, oxygen separators and catalytic membrane reactors (open-pore zirconia) [Atkinson &

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

35

Selçuk 2000, Fleig et al. 2003, Koga 2003, Tsuru 2003, Yamada 2003] and other functional applications using electrical, magnetic or optical properties. Structural applications proper, however, are those where mechanical performance is of primary concern. For alumina and zirconia they range from engine parts, ball bearings, thread guides and cutting tools [Komeya & Matsui 1994, Schwartz 1992, Takebayashi 2003, Fukuhara 2003] to bioinert implants for bone and joint replacement [Boutin et al. 1988, Christel et al. 1989, Hulbert 1993, Kohn & Ducheyne 1992, Kokubo et al. 2003, Park & Lakes 1992, Soltesz & Richter 1984]. The latter may certainly be considered as one of the most demanding applications. The reasons are manifold. A typical origin for post-operative or long-term implant failure is aseptic loosening, caused by the large difference in elastic moduli between bone and implant. The order-ofmagnitude of the tensile moduli is between approx. 0.2 GPa for cancellous (trabecular) bone and approx. 20 GPa for dense (cortical) bone, while titanium alloy implants (the group of commonly used metal implants with the lowest elastic moduli) have tensile moduli of approx. 100 GPa, which is clearly considered to be “elasto-incompatible” [Kohn & Ducheyne 1992, Ontañón et al. 2000, Park & Lakes 1992, Sumner et al. 1998, Weinans et al. 2000]. Naturally, “elastocompatible” solutions are generally not sought among ceramic materials but among polymer-based composites, since for ceramic implant materials like alumina and zirconia the elastic mismatch between bone and implant is even greater than for metals. Alternative solutions to circumvent this principal problem, e.g. by allowing for ingrowth of viable bone tissue into a porous implant and thus forming of a continuous and strong bone-implant interface, is largely complicated by the severe and very specific microstructural demands: the porosity must be open and connected, while the pore size (channel size) should be at least 150 µm for bony ingrowth to be effective [Park & Lakes 1992]. It is clear that such complex demands are very difficult to fulfil, especially with brittle materials, without the danger of a critical deterioration of other mechanical properties (strength) and wear behavior. Apart from the basic requirements on strength and fracture toughness, less well defined mechanical “properties” like hardness, wear (abrasion and erosion), fatigue and ageing (both intimately connected with corrosion) play a major role in all of these applications. This leads to the very intricate questions of lifetime and reliability, which mostly have to be solved empirically by proof testing [Davidge 1979, Menčík 1992, Salmang & Scholze 1982, Wachtman 1996]. In spite of the frequently cited inertness of alumina [Cawley & Lee 1994, Lee & Rainforth 1994] it has to be kept in mind that, similar to other oxide ceramics, also alumina is liable to stress-corrosion cracking and subcritical crack growth in the presence of water vapor, which can seriously limit the lifetime of loaded alumina parts in quite common environments. Zirconia can additionally undergo hydrothermal ageing (i.e. a transformation from tetragonal to monoclinic at moderately elevated temperatures in the presence of water vapor), a phenomenon which led to the general prohibition of steam sterilization for zirconia implants and seems to affect also the long time-behavior of zirconia parts in physiological solutions and in vivo at body temperature [Chevalier et al. 1997, 1999, 2004 Gremillard et al. 2004], although in the early 1990es it was believed to be completely inert with respect to ageing in physiological solutions [Christel et al. 1989, Kohn & Ducheyne 1992]. Although − as is evident from these considerations − it should not be concealed that some microstructural targets probably cannot be achieved with ceramic materials and some of the earlier expectations have turned out to be too optimistic, for many of the above applications alumina, zirconia and alumina-zirconia composite ceramics are quite suitable. Without doubt, during the second half of the 20th century alumina has developed into the most frequently

36

Willi Pabst and Eva Gregorová

used advanced engineering ceramic, while from the 1970es onwards zirconia has gained its remarkable reputation as “ceramic steel” for its unusally high fracture toughness values (still one order of magnitude lower than typical fracture toughness values of metals, however), which is a consequence of the two toughening mechanisms occurring in zirconia and zirconia-containing composite ceramics, viz. transformation toughening and microcrack toughening, cf. [Cawley & Lee 1994, Cook & Pharr 1994, Fantozzi & Olagnon 1993, Green et al. 1989, Hannink et al. 2000, Komeya & Matsui 1994, Lee & Rainforth 1994, Wachtman 1996]. In the present chapter we are focusing on the elastic behavior of alumina, zirconia and alumina-zirconia composite ceramics as expressed by their effective elastic properties, which are mechanical key properties in the sense exposed above. However, since the viewpoint adopted in this chapter is primarily a theoretical one, with only a side-glance on practical issues and experimental experience, the exposition will be general enough for many of the relations to be applicable to other materials as well (at least in the field of ceramics) and may serve as a guideline of analogous research in other ceramic systems. The second section (following this introductory section) defines elastic properties from the viewpoint of rational mechanics for anisotropic and isotropic materials in general. In the third section effective elastic properties are discussed from the viewpoint of micromechanics and composite theory. General formulae are given for the calculation of effective elastic moduli of polycrystalline materials from monocrystal data. Further, the Voigt-Reuss bounds for the effective elastic moduli of multiphase materials are given, as well the Hashin-Shtrikman bounds for the special case of two-phase materials. A rather comprehensive survey of model relations for the porosity dependence of elastic moduli is given, including exponential and power-law relations and a new relation recently proposed by the authors. In the fourth section all these micromechanical bounds and relations are applied to the alumina-zirconia system. The last section gives examples of the mathematical modeling of other effective properties and their dependence on composition and microstructure and an outlook is given to future research aims. In particular, the significance of interfaces is emphasized and some ideas on the way from micromechanics to nanoscience − towards a general mixture theory − are outlined.

2

Elastic Moduli from the Viewpoint of Rational Mechanics

The elastic behavior of brittle materials can be described within the framework of linear elasticity theory. The reason is that, as a response to loadings above a critical value, the stress-strain curve is cut off and brittle materials react by (brittle) fracture, i.e. the materials rupture and loose integrity at stress or deformation levels lower than those that would be necessary for the material to enter the nonlinearly elastic regime. For more details the reader can refer e.g. to [Atanakovic & Guran 2000, Billington & Tate 1981, Gurtin 1972, Haupt 2000, Landau & Lifshitz 1986, Love 1927, Nye 1957, Šilhavý 1997, Slaughter 2002, Sneddon & Berry 1958, Truesdell & Noll 2003, Truesdell & Toupin 1960] and the references cited therein. For the purposes of the present chapter a rather special theory can be used, which is linear with respect to the material model (physical linearization of the constitutive equations) and to the kinematic measures (geometrical linearization of the strain measures for small deformations). We note in advance that, although this theory must be considered as rather special from the rational mechanics point of view, it is the usual setting in which elastic

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

37

properties of ceramics and most other materials are treated in the literature. With respect to the applications intended in this chapter we first present the linear elastic properties of anisotropic materials and then those of isotropic materials, cf. also [Pabst & Gregorová 2003a].

2.1

Elastic Properties of Anisotropic Materials

In the case of small deformations (i.e. invoking geometrical linearization) Hooke's law for anisotropic elastic solids (i.e. the physically linearized constitutive equation) can be written in direct tensor notation as

T = C E,

(1)

where T is the Cauchy stress tensor (a symmetric second-order tensor), C the stiffness tensor (a fully symmetric fourth-order tensor, also called elasticity tensor or tensor of elastic constants) and

E≡

(

1 grad u + grad u T 2

)

(2)

the so-called small strain tensor (a symmetric second-order tensor), with u being the displacement vector and superscript T denoting tensor transposition. Using the more explicit tensor index notation, Hooke's law can also be written in the form

Tij = C ijkl E kl .

(3)

Alternatively, the constitutive equation of linearly elastic materials can be written in the form

E = S T = C -1 T

(4)

(inverse Hooke’s law), where S = C is the compliance tensor (again a fully symmetric fourth-order tensor, also called tensor of elastic coefficients), with the superscript –1 denoting tensor inversion. In tensorial index notation -1

Eij = S ijkl Tkl .

(5)

Due to the symmetry of T and E , the number of components of the stiffness and compliance tensors is reduced from a total of 81 to 36 independent ones. Thus, it is possible to represent these fourth-order tensors alternatively in the form of (6 x 6) matrices (which, of course, do not have the transformation properties of a tensor), and to express Hooke's law and inverse Hooke’s law in direct matrix notation (engineering notation) as

Willi Pabst and Eva Gregorová

38

(σ ) = (C ) (ε )

,

(ε ) = (S ) (σ )

(6) (7)

or, alternatively, in matrix index notation as

respectively. In this notation

σ i = C ij ε j ,

(8)

ε i = S ij σ j ,

(9)

(σ ) and (ε ) are

6-dimensional column vectors referring to

stress and strain, respectively, the components of which are defined as follows

σ 1 ≡ T11 , σ 2 ≡ T22 , σ 3 ≡ T33 , σ 4 ≡ T23 , σ 5 ≡ T31 , σ 6 ≡ T12

(10 a-f)

ε 1 ≡ E11 , ε 2 ≡ E 22 , ε 3 ≡ E33 , ε 4 ≡ 2 E 23 , ε 5 ≡ 2 E31 , ε 6 ≡ 2 E12

(11 a-f)

and (C ) and (S ) are the stiffness and compliance matrices, respectively, the elements of which are given by the substitution of index pairs as follows:

11 → 1 , 22 → 2 , 33 → 3 , 23 → 4 , 31 → 5 , 12 → 6 . When it is assumed, as is usually done,1 that the stiffness and compliance tensors are additionally symmetric with respect to their diagonals, the total number of independent components is reduced from 36 to 21 (so-called Green elasticity or hyperelasticity, in contrast to the so-called Cauchy elasticity, where this is not the case). Thus in the most general case of well-defined anisotropy (triclinic monocrystals) the (6 x 6) stiffness or compliance matrices or, alternatively, the fourth-order stiffness or compliance tensors, have 36 elastic constants or coefficients, respectively, 21 of which can be assumed to be independent, cf. [Pabst & Gregorová 2003a]. For reasons of convenience we confine ourselves to the stiffness matrices in the sequel. It is understood, however, that completely analogous relations and symmetry considerations are valid in the case of the compliance matrices.

1

This assumption is based on a more fundamental assumption concerning the elastic energy (stored energy function). If the elastic energy, which is a potential function for the stress tensor, vanishes in the unstrained state and can be expressed by a symmetric quadratic form, then the stiffness matrix is symmetric, i.e. the elasticity tensor is fully symmetric.

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

39

The stiffness matrix of triclinic monocrystals can be written as:

 C11 C12   C 21 C 22 C C (Ctriclinic ) =  31 32  C 41 C 42   C 51 C 52 C  61 C 62

C13 C 23

C14 C 24

C15 C 25

C 33

C 34

C 35

C 43

C 44

C 45

C 53 C 63

C 54 C 64

C 55 C 65

C16   C 26  C 36   C 46   C 56  C 66 

(12)

For the off-diagonal elements in this stiffness matrix and the following ones we automatically assume validity of the symmetry condition for Green elasticity (hyperelasticity), i.e. C ij = C ji (13) For materials of higher symmetry (monocrystals or polycrystalline bodies) the number of independent elastic constants is further reduced as follows: •

Monoclinic monocrystals (13 independent elastic constants):

 C11   C 21 C (C monoclinic ) =  31  0   C 51  0  •

C12 C 22

C13 C 23

0 0

C15 C 25

C32

C33

0

C35

0

0

C 44

0

C 52 0

C 53 0

0 C 64

C 55 0

0   0  0   C 46   0  C 66 

(14)

Orthorhomic monocrystals and orthotropic polycrystalline bodies (9 independent elastic constants):

 C11   C 21 C (C orthorhombic ) =  31  0   0  0 

C12 C 22

C13 C 23

0 0

0 0

C 32

C 33

0

0

0

0

C 44

0

0 0

0 0

0 0

C 55 0

0   0  0   0   0  C 66 

(15)

Willi Pabst and Eva Gregorová

40 •

Trigonal monocrystals (6 independent elastic constants):

 C11 C12   C 21 C 22  (Ctrigonal ) =  CC31 CC32  41 42  0 0   0 0 

C13

C14

C 23 C 33 0 0 0

C 24 0 C 44 0 0

0 0 0 0 C 55 C 65

0   0  0   0  C 56  C 66 

(16)

with the additional conditions

C11 = C 22 , C13 = C 23 , C 44 = C 55

(17)

and

C14 = −C 24 = C 56 , C 66 =

1 (C11 − C12 ) 2

(18)

Note that most textbooks claim that in the case of trigonal and tetragonal monocrystals it is necessary to distinguish two cases of elastic symmetry, depending on the crystal class

4 ) and m one with 6 independent elastic constants (point groups 32 , 3m , 3 m , 422 , 4mm , 4 2m , 4 2 2 ). As shown in a recent paper by Forte and Vianello [Forte & Vianello 1996], this mmm

(point group): one with 7 independent elastic constants (point groups 3 , 3 4 , 4 ,

claim is wrong. For tetragonal monocrystals (6 independent elastic constants) the general form of the orthorhombic (orthotropic) stiffness matrix, Equation (15), together with conditions (17) applies. The same holds true for materials with higher than tetragonal symmetry. In these cases, however, the following additional conditions hold for the non-zero elements: •

Hexagonal monocrystals and transversely isotropic polycrystalline bodies (5 independent elastic constants):

C 66 = •

1 (C11 − C12 ) 2

(19)

Cubic monocrystals (3 independent elastic constants):

C11 = C 22 = C 33 , C12 = C 23 = C 31 , C 44 = C 55 = C 66

(20)

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ... •

Isotropic materials (2 independent elastic constants):

C11 = C 22 = C 33 , C12 = C 23 = C 31 , C 44 = C 55 = C 66 =

2.2

41

1 (C11 − C12 ) . 2

(21)

Elastic Properties of Isotropic Materials

In the case of isotropic materials the stiffness matrix can be written as follows (using the definitions C12 ≡ λ and C 44 ≡ µ ):

λ λ  λ + 2µ  λ + 2µ λ  λ  λ λ + 2µ (Cisotropic ) =  λ0 0 0   0 0 0   0 0 0  The elastic constants (elastic moduli)

0 0 0

µ 0 0

0 0 0 0

µ 0

0  0 0  0 0  µ 

(22)

λ and µ are called Lamé constants (or Lamé

moduli, units [GPa]). Switching over from matrix notation to tensor notation the elasticity tensor is

Cijkl = λ δ ij δ kl + µ (δ ik δ jl + δ il δ jk )

(23)

where the δ 's are Kronecker deltas. Inserting the stiffness tensor given by Equation (23) into Hooke's law, Equation (3), yields the Cauchy-Hooke law for isotropic materials,

[

]

Tij = λ δ ij δ kl + µ (δ ik δ jl + δ il δ jk ) E kl = λ δ ij δ kl E kl + 2µ Eij = λ δ ij E kk + 2µ Eij (24) Switching over from tensor index notation to direct tensor notation this corresponds to

T = λ ⋅ (tr E ) ⋅ 1 + 2 µ E ,

(25)

where tr denotes the trace of a tensor and 1 is the second-order unit tensor. The inverse Cauchy-Hooke law for isotropic materials is

E=

λ 1 T− ⋅ (tr T ) ⋅ 1 . 2µ 2 µ (3λ + 2µ )

(26)

42

Willi Pabst and Eva Gregorová

The two Lamé constants occurring in Equations (22) through (26) are one possible choice of elastic constants which can be used in the case of isotropic materials. Depending on the application in question, other elastic constants can be more advantageous, e.g. the tensile modulus (Young’s modulus) E (units [GPa]), the shear modulus G (units [GPa]), the bulk modulus K (units [GPa]) and the Poisson ratio ν (dimensionless). Some of these constants are preferable from the practical point of view, since they can be relatively easily determined by standard test procedures ( E and G ), while others are preferable from the theoretical point of view, e.g. for micromechanical calculations ( G and K ). Note, however, that even in the case of isotropic materials always two of these elastic constants are needed to determine the elastic behavior completely. In terms of E and ν the Cauchy-Hooke law for isotropic materials can be written as

T=

 E  ν ⋅ (tr E ) ⋅ 1 E+  (1 + ν )  (1 − 2ν ) 

(27)

and the inverse Cauchy-Hooke law as

E=

(1 + ν ) T − ν ⋅ (tr T) ⋅ 1 .  (1 + ν ) E  

(28)

From Equation (27) it is evident that for the Poisson ratio ν the values 0.5 and − 1 are not allowed. Actually, as a consequence of the second law of thermodynamics the following inequality must hold for isotropic materials [Gurtin 1972, Torquato 2002]:

− 1 < ν < 0 .5 .

(29)

Although it has been known very early that anisotropic materials (e.g. monocrystals) can exhibit negative Poisson ratios in certain directions [Love 1927], until very recently most textbooks on the mechanics of materials suggested that according to experience with real materials the Poisson ratio should be positive for all isotropic materials (i.e. 0 < ν < 0.5 ), cf. e.g. [Slaughter 2002]. This misleading suggestion was obviously supported by physical intuition. Guided by a correct interpretation of the physical meaning of the Poisson ratio (viz. that in the limit of incompressible materials ν → 0.5 , while the case ν → 0 corresponds to a compressible material that does not contract in uniaxial extension perpendicular to the extension direction), it was for a long time considered as evident that isotropic materials are not allowed to extend perpendicular to the extension direction during uniaxial extension. Research in materials science for more than two decades has clearly shown, however, that this apparently obvious conclusion is wrong and that isotropic materials with negative Poisson ratio, so-called “auxetic materials”, do exist and can be designed and produced [Almgren 1985, Bathurst & Rothenburg 1988, Caddock & Evans, Choi & Lakes 1992, Evans 1989, Evans & Caddock 1989, Evans et al. 1992, Friis et al. 1988, Gibson et al. 1982, Lakes 1987, 1991, 1992, Milton 1992, 2002, Rothenburg et al. 1991, Sigmund 1994, 1995, Torquato 2002]. These materials show the unexpected behavior, that when extended in one direction,

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

43

they extend in all perpendicular directions. Of course, almost all ceramic materials, including glasses, exhibit Poisson ratios in the range 0.1 − 0.4 and in the absence of more precise information one usually estimates the value of the Poisson ratio for a ceramic materials between 0.2 (or 0.17) and 0.3 (or 0.33). Nevertheless, newer research in ceramic science does take the possibility of negative Poisson ratios into account [Roberts & Garboczi 2000]. In a similar way as the Young modulus E and the Poisson ratio ν are connected to the uniaxial extension test, the shear modulus G and the bulk modulus K are connected to simple shear and isotropic deformation (i.e. dilatation or compression). Note that, accidentally, it turns out that the shear modulus G equals the second Lamé constant µ . Since for isotropic materials only two of the elastic constants are independent, the knowledge of any pair of them is sufficient to calculate the other constants and thus to describe the elastic behavior of isotropic materials completely. For easy reference in this chapter we list the most important interrelations between the elastic constants:

E=

µ (3λ + 2µ ) 9 KG = 2G (1 + ν ) = 3K (1 − 2ν ) = 3K + G λ+µ G=µ=

K=

E 3K (1 − 2ν ) 3EK = = 2(1 + ν ) 2(1 + ν ) 9K − E

(30)

(31)

2G (1 + ν ) E EG 3λ + 2 µ = = = 3 3 (1 − 2ν ) 3 (1 − 2ν ) 3 (3G − E )

(32)

λ E − 2G 3K − E 3K − 2G = = = 2 (λ + µ ) 2 (3K + G ) 2G 6K

(33)

ν=

In the present chapter we will refer to these relations as the “elasticity standard relations”. All elastic constants (and of course also its tensorial counterparts, the stiffness tensors) are temperature-dependent, e.g.

E = E (T ) ,

(34)

T being the absolute temperature. From the theoretical point of view such a temperature dependence arises naturally within the framework of linear thermoelasticity [Carlsson 1972, Šilhavý 1997]. Note, however, that an alternative and equally valid theory of linear thermoelasticity can be built by replacing the temperature with its Legendre-transformed counterpart, the specific entropy s [Šilhavý 1997]. This results in entropy-dependent elastic constants, e.g. E~ = E~ (s ) .

(35)

44

Willi Pabst and Eva Gregorová

Concluding this section we would like to add some practical remarks. It seems that no complete theory is available to predict the temperature (or entropy) dependence of the elastic constants and thus usually they have to be determined from experimental measurements. Elastic constants determined by static measurements, e.g. from load-deflection curves in three-point bending via the formula [Menčík 1992]

E=

F L3 48 I Y

(36)

are called isothermal, because the measurement is slow enough to permit arbitrary heat exchange, so that during measurement the specimen is in thermal equilibrium with its environment. In Equation (36) F is the loading force, L the distance between the supports (effective specimen length), Y the deflection of the specimen below the load and I the moment of inertia of the cross-section, e.g. for circular and rectangular cross sections

I=

π D4 64

and

I=

bh 3 , 12

(37 a, b)

respectively (with D being the specimen diameter, b the specimen width and h the height of the specimen in the direction of deflection). On the other hand, elastic constants determined via dynamic measurements, e.g. by resonant frequency techniques via the formula [Aksay & Riley 2003] 2 E~ = ρ ⋅ (2vL )

(with

(38)

ρ being the bulk density, L the specimen length and v the frequency of the

longitudinal mode of vibration, i.e. the ratio of velocity of the compressional wave and its wavelength) are called adiabatic or isentropic, because the measurement is so fast that heat exchange will not occur to any sensible degree during signal registration, so that the specimen is de facto thermally isolated until measurement is completed. A rough estimate of the difference between isothermal and adiabatic elastic constants (e.g. E and E~ , respectively) can be made as follows: Since it is known that the isothermal and adiabatic shear moduli are equal [Šilhavý 1997], i.e.

~=G G

(39)

and on the other hand that the isothermal and adiabatic bulk moduli are related by the ratio of specific heats at constant pressure (stress) and at constant volume (deformation), c P and cV , respectively [Ashcroft & Mermin 1976, Callen 1960, Šilhavý 1997], i.e.

K~ c P = , K cV

(40)

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

45

~

the following formula can be given for the adiabatic tensile modulus E :

9

E~ =

cP KG cV

c 3 P K +G cV

.

(41)

~

~

In order to obtain an estimate for the difference between K and K or E and E , respectively, we invoke the well-known relation for the difference between the specific or molar heats at constant pressure and volume c P and cV , respectively, for isotropic condensed phases,

c P − cV = TVK (3α ) , 2

(42)

known from equilibrium thermodynamics [Guggenheim 1957, Kittel 1988, Meyer 1977], where T is the Kelvin temperature, V the specific or molar volume, respectively and K the isothermal bulk modulus (i.e. the inverse compressibility) and α the coefficient of linear thermal expansion for the temperature in question. Further, we invoke the approximate relation

cV =

3 α VK , Γ

(43)

known from solid state physics, where the so-called Grüneisen constant Γ is assumed to be 2, a typical value for crystalline solids [Ashcroft & Mermin 1976, Grüneisen 1926, Meyer 1977, Salmang & Scholze 1982]. For example, inserting for alumina and zirconia the molar volumes 25.5 cm3/mol and 20.2 cm3/mol, respectively (assuming densities of 4.0 g/cm3 and 6.1 g/cm3, respectively) or alternatively the specific volumes 0.250 cm3/g and 0.164 cm3/g, respectively, the approximate linear thermal expansion coefficients 8 ⋅ 10

−6

K-1 and

10 ⋅ 10 −6 K-1, respectively [Menčík 1992, Munro 1997, NIST 2002] and the bulk moduli 247 GPa and 184 GPa, respectively (see below), we obtain at room temperature (25 °C or 298 K) differences of the molar or specific heats c P − cV of 1.081 J⋅mol-1⋅K-1 ( = 0.0106 J⋅g-1⋅K-1) for alumina and 0.997 J⋅mol-1⋅K-1 ( = 0.0081 J⋅g-1⋅K-1) for zirconia. For the absolute values of cV the rough estimates obtained via Equation (43) using Γ = 2 are 75.6 J⋅mol-1⋅K-1 ( = 0.741 J⋅g-1⋅K-1) for alumina and 55.8 J⋅mol-1⋅K-1 ( = 0.453 J⋅g-1⋅K-1) for zirconia. These values are in satisfactory agreement with literature values [Munro 1997, NIST 2002, Salmang & Scholze 1982]. With this input information at hand, Equations (40) and (41) (the latter in connection with approximate values for the shear and bulk moduli, cf. Table 8 below for the definite values) can now be used to obtain estimates for the differences that have to be expected at room temperature between adiabatic elastic constants (measured via dynamic techniques) and isothermal elastic constants (measured via static techniques). For alumina and zirconia

Willi Pabst and Eva Gregorová

46

cP = 1.015 cV

⇒

K~ = 1.015 K

and

E~ = 1.003 E ,

(44)

cP = 1.018 cV

⇒

K~ = 1.018 K

and

E~ = 1.002 E ,

(45)

respectively. Note that the adiabatic elastic constants are always slightly higher than the isothermal ones, since always c P > cV . It is evident, however, that for alumina and zirconia near room temperature the difference is small enough to be negligible for most practical purposes (1-2 % for the bulk modulus and 0.2-0.3 % for the tensile modulus).

3

Effective Elastic Moduli from the Viewpoint of Micromechanics and Composite Theory

It is a long-standing aim of mixture theory to predict the mixture properties (and in general all mixture quantities) from those of the constituents by calculation only, without direct measurement. There are a few fortunate cases where this can be done with an accuracy sufficient for practical purposes. Such is the case for (non-reacting) multiphase mixtures or, more generally, materials with microstructure, particularly in cases where the contribution of the phase boundaries (interfaces) is negligible. The latter can be said for many properties of polycrystalline materials with crystal grains (crystallites) of a size in the tenths-of-micrometer range and above (at least > 0.1 µm), i.e. classical microcomposites, including the special case of monophase polycrystalline materials. In contrast to nanocomposites (and nanocrystalline materials in general) with typical dimensions < 0.1 µm microcomposites (and microcrystalline materials in general) can usually be modeled by considering volume fractions and other microstructural characteristics of the constituent bulk phases only, while the phase boundaries are considered to be sharp, i.e. without assignable volume fraction (in contrast to nanomaterials, where it can be useful to treat the phase boundaries as diffuse, i.e. with a finite thickness). Solid materials of this type are the main subject of the present chapter. With regard to the purpose of this chapter (application to the alumina-zirconia system) we confine ourselves to linearly elastic materials throughout, with special emphasis on one-phase polycrystalline materials and two-phase materials (composites and porous materials). Polycrystalline materials are one-phase materials with microstructure or “morphology” (in contrast e.g. to glasses, which do not exhibit microstructure in this sense and are therefore called “amorphous”) or multiphase materials (composites). They can be uniform (in the absence of macroscopic gradients) but are always heterogeneous (at the microscopic scale). Polycrystalline materials consist of small crystal grains (crystallites), isometric or anisometric (elongated / prolate or flaky / oblate), held together by strong interaction forces (chemical bonds) and exhibiting random orientation or (a certain degree of) preferential orientation. Examples are densely sintered ceramic bodies. Polycrystalline materials with preferential orientation of anisometric crystallites are in general anisotropic, but an adequate structural

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

47

description of “partially anisotropic” materials probably requires the elaboration of new, “fuzzy” symmetry concepts in the future. Short-fiber composites e.g. with only a slight tendency to fiber orientation, can be almost isotropic, although from the viewpoint of classical composite theory they are usually treated as transversely isotropic. In the first case two elastic constants would be needed for a complete description of elastic behavior, in the second five. With respect to the purpose of this chapter (application to the alumina-zirconia system) we confine ourselves in the sequel to polycrystalline materials and particulate composites with isometric particles only. Thus, for evident reasons we consider macroscopically isotropic materials with microstructures exhibiting random orientation and disorder (i.e. non-periodicity). For similar reasons the theoretical treatment is focused on spherical pores, although in principle deviations from sphericity can be accounted for within the newly introduced concept of intrinsic elastic moduli (see below). The effective stiffness tensor C e is defined via the linear constitutive equation

T = Ce E

(46)

(i.e. averaged Hooke’s law for the heterogeneous material from the macroscopic point of view), while its inverse, the effective compliance tensor S e = C e

−1

is defined via the inverse

equation

E = Se T .

(47)

In these relations angular brackets denote volume averages of the Cauchy stress tensor and the small strain tensor, respectively. For the correct use of volume and ensemble averages in connection with random and periodic microstructures the reader should consult [Torquato 2002]. In his classical contribution [Voigt 1889] Voigt has shown, based on the assumption of uniform strain (isostrain assumption), that in the case of isotropic polycrystalline materials with a uniform and random microstructure the scalar elastic moduli can be calculated from the components of the elasticity tensor (stiffness matrix), cf. [Voigt 1910]. In another classical paper [Reuss 1929] Reuss has proposed another result for such materials, based on the assumption of uniform stress (isostress assumption) and using the components of the inverse elasticity tensor (compliance matrix), cf. [Nye 1957, Hearmon 1961, Wachtman 1996, Green 1998]. In both approaches averaging is performed over all possible orientations of the crystallites. Hill [Hill 1952] was apparently the first to recognize that the Voigt and Reuss values were in fact the upper and lower bounds, respectively, to the property in question. Moreover, it turned out that for most practical purposes the Voigt and Reuss bound calculated for polycrystalline materials from monocrystal data are sufficiently close together for their arithmetic average (the so-called Voigt-Reuss-Hill average, abbreviated VRH average in the sequel) to be a satisfactory estimate of the property value [Hill 1952]. Comparison with experimental data has confirmed the validity of the Voigt and Reuss bounds and the usefulness of the VRH average for property prediction purposes [Green 1998]. The procedure to calculate the Voigt and Reuss bounds for elastic moduli of polycrystalline materials from

Willi Pabst and Eva Gregorová

48

monocrystal data is recalled in this chapter, since this procedure can generally serve to establish reliable values of elastic moduli of dense polycrystalline one-phase materials, which are the necessary input information for the prediction of effective elastic moduli of composites or can significantly facilitate data fitting of effective elastic moduli of porous materials (because one fit parameter can be considered as fixed). The effective properties (sometimes also called gross, overall or macroscopic properties) of the multiphase materials with microstructure defined above can in principle be predicted exactly when the properties of the constituent phases and all details of the microstructure are known. Of course the problem with this statement is the fact, that the microstructural details must be known quantitatively in order to properly formalize the abstract statement Effective property = Function (phase properties, microstructure). The theoretical framework to attack this task is called micromechanics (sometimes also mechanics or theory of composites or heterogeneous materials). The interested reader can consult one of the excellent monographs in this field [Beran 1968, Christensen 1979, Markov 2000, Milton 2002, Nemat-Nasser & Hori 1999, Sahimi 2003a, Torquato 2002] for additional information, but it has been attempted to make the present chapter to a large degree selfcontained. Micromechanics provides theoretical concepts for quantifying microstructural information to an arbitrary degree of precision and including it into the description of a material in the form of so-called correlation functions [Beran 1968, Jeulin 2001, 2002, Markov 2000, Sobcyk & Kirkner 2001, Torquato 2002]. The lowest-order microstructural information (one-point correlation function) concerns only the volume fractions of the phases. Thus it is only a measure of composition, replacing the mass fractions or mole fractions from more general mixture theories. Two-, three- and multi-point correlation functions involve information on the size, shape, orientation and mutual arrangement of the phases. Such higher-order correlation functions, however, are extremely difficult to determine. Evidently, for real materials higher-order microstructural information is accessible only via tomographic techniques (direct 3D information) or, approximately, by image analysis of planar sections and the application of stereological approaches (3D information indirectly inferred from 2D information). Of course, computer simulations may be a useful tool to generate model materials and analyze their (virtual) microstructure, cf. e.g. [Roberts & Garboczi 2000] and the extensive work of Torquato and his coworkers cited in [Torquato 2002]. For the remaining part of the present chapter we confine ourselves to isotropic materials, i.e. the scalar effective elastic moduli M (where M stands for the effective tensile modulus E , shear modulus G or bulk modulus K , respectively and – in contrast to Equations (46) and (47) above – the subscript “e ” denoting “effective” is omitted for convenience) as functions of the phase moduli M i and microstructural information of the lowest order, i.e. the volume fractions

φ i (only compositional information). In this sense we introduce the

following basic assumption:

M = f (M i ,φi ) ,

(48)

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ... where M i ( i = 0, 1, 2 … n ) are the phase properties of all n constituent phases and

49

φ i the

volume fractions of these n phases. Note that the volume fractions of all n phases sum up to unity, i.e. n

∑φ i =0

i

= 1,

(49)

so that only n − 1 phase volume fractions are independent. Note that although Equation (48) is a frequently used starting point for modeling material properties and although at first sight it might seem sufficiently general to be of almost universal validity, closer inspection makes clear that this is by no means the case [Torquato 2002]. A simple counter-example demonstrates this: an alumina suspension with 50 vol.% alumina can easily be prepared, but a porous alumina ceramic body containing 50 vol.% of (open, water-saturated) pores can also be prepared. In both cases the alumina volume fraction (or, alternatively, the water volume fraction or porosity) is 0.5, but the first system is a viscous suspension ( E , G , K all zero / undefined, viscosity non-zero / finite), the second an elastic porous solid ( E , G , K all nonzero / finite, viscosity infinite / undefined). Thus, Equation (48) must be considered as a convenient working hypothesis only. Nevertheless, in the lack of more detailed microstructural information it is often the only feasible starting point for micromechanical modeling. It is the purpose of this section to provide, on the basis of Equation (48), combined with a minimum of additional qualitative information (e.g. concerning pore shape and connectivity), an easy-to-grasp micromechanical framework to model porous alumina and zirconia, and (dense and porous) alumina-zirconia composites. With respect to the intended applications the emphasis is on two-phase materials. Of course, the relations given in this section can be used for all brittle materials (ceramics) to mathematically describe the dependence of effective elastic properties on the volume fractions of the constituent phases.

3.1

Effective Elastic Moduli of Polycrystalline Materials

As detailed in the preceding section, the elastic behavior of anisotropic materials (monocrystals and anisotropic composites) is characterized by a fourth-order elasticity tensor, which can be represented, as a consequence of symmetry, as a (6 x 6) matrix. This matrix is again symmetric with respect to its diagonal (due to energetic reasons), so that the maximum number of independent elements is 21 in the least symmetric case (triclinic monocrystal). For higher symmetries this number of independent elements is smaller, cf. the discussion above and [Gurtin 1972, Hearmon 1961, Nye 1957, Pabst & Gregorová 2003a]. When Hooke’s law is written in engineering matrix notation as

σ i = Cij ε j ,

(8)

Willi Pabst and Eva Gregorová

50 where

σ i is the 6-dimensional stress vector and ε j is the 6-dimensional small strain vector,

C ij is called stiffness matrix. On the other hand, using the so-called compliance matrix S ij = C ij−1 (the inverse of the stiffness matrix), Hooke’s law can be written in the inverse form

ε i = S ij σ j .

(9)

Note that the stiffness matrix and the compliance matrix have the same number of independent components (called elastic constants and elastic coefficients, respectively) and zero elements at the same positions, cf. [Nye 1957, Hearmon 1961]. The compliance matrix can be calculated from the stiffness matrix via matrix inversion as follows

S ij =

subdet (C ij ) det (C ij )

,

(50)

where subdet (C ij ) is the subdeterminant obtained by omitting the i th row and the j th column from C ij and det (C ij ) is the determinant of C ij . An analogous relation holds with the roles of S ij and C ij interchanged. While for triclinic and monoclinic crystals the matrix inversion is rather lengthy, for materials of higher symmetry it simplifies considerably. E.g. for orthorhombic monocrystals and orthotropic polycrystalline materials or composites the number of independent components is 9 and one obtains [Hearmon 1961]

S11 =

C 22

C 23

C 23

C 33

C11 C12 C13

C12 C 22 C 23

S 44 =

1 C 44

etc.

C13 C 23 C 33 etc.

(51)

(52)

For materials of even higher symmetry the inversion relations are as follows, cf. e.g. [Hearmon 1961, Green 1998, Pabst & Gregorová 2004a]: •

Trigonal monocrystals (6 independent components):

S11 + S12 =

C 33 C C , S11 − S12 = 44 , S13 = − 13 , K K′ K

(53 a,b,c)

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

S14 = − •

C14 C + C12 C − C12 , S 33 = 11 , S 44 = 11 . K′ K K′

S 33 =

C 33 C 1 , S11 − S12 = , S13 = − 13 , K K C11 − C12

(54 a,b,c)

C11 + C12 1 1 , S 44 = , S 66 = . K C 44 C 66

(54 d,e,f)

Hexagonal monocrystals and transversely isotropic polycrystalline materials or composites (5 independent components):

S11 + S12 =

C 33 C 1 , S11 − S12 = , S13 = − 13 , K C11 − C12 K S 33 =

•

(53 d,e,f)

Tetragonal monocrystals (6 independent components):

S11 + S12 =

•

51

C11 + C12 1 , S 44 = . K C 44

(55 a,b,c)

(55 d,e)

Cubic monocrystals (3 independent components): ` S11 =

C11 + C12 C 1 , S12 = − 12 , S 44 = . K ′′ K ′′ C 44

(56 a,b,c)

In these relations K , K ′ and K ′′ have the following meaning:

K = C 33 (C11 + C12 ) − 2C132 ,

(57 a)

K ′ = C 44 (C11 + C12 ) − 2C142 .

(57 b)

K ′′ = (C11 − C12 ) (C11 + 2C12 ) .

(57 c)

Completely analogous relations are valid with the stiffnesses C ij interchanged by the compliances S ij . Note, however, that for trigonal monocrystals, hexagonal monocrystals (including transversely isotropic polycrystalline materials or composites) and isotropic materials the elastic constant (stiffness) C 66 is given by

Willi Pabst and Eva Gregorová

52

C 66 = 12 (C11 − C12 ) ,

(58)

while the elastic coefficient (compliance) S 66 is given by [Nye 1957]

S 66 = 2 ( S11 − S12 ) .

(59)

This is the only case in which the complete analogy between stiffnesses and compliances is disturbed. Of course, for isotropic materials C 66 and S 66 are identical to C 44 = C 55 and

S 44 = S 55 , respectively, cf. [Pabst & Gregorová 2003a], and therefore Equation (56 c) is redundant, since S 44 = •

1 2

( S11 + S12 ) :

Isotropic materials (2 independent components):

S11 =

C11 + C12 C , S12 = − 12 . K ′′ K ′′

(60 a,b)

Voigt [Voigt 1889, 1910] has shown that, under the assumption that the strain inside the material is uniform (isostrain assumption), the effective elastic moduli of a dense (i.e. porefree) polycrystalline material, e.g. a densely sintered ceramic, composed of crystallites of arbitrary symmetry can be calculated from the 9 elastic constants (stiffnesses) C11 , C 22 , C 33 ,

C 44 , C 55 , C 66 , C12 , C 23 , C 31 , when the material as a whole is statistically or macroscopically isotropic (“quasi-isotropic”). Of course, this requires random orientation of the anisotropic (and possibly anisometric) crystallites. On the other hand, Reuss [Reuss 1929] has shown that, under the assumption that the stress inside the material is uniform (isostress assumption), the effective elastic moduli of a polycrystalline material composed of crystallites of arbitrary symmetry can be calculated from the 9 elastic coefficients (compliances) S11 , S 22 , S 33 , S 44 , S 55 , S 66 , S12 , S 23 , S 31 , when the material is macroscopically isotropic. It should be noted that, according to Voigt and Reuss, even for polycrystals composed of crystallites (monocrystals) with a symmetry lower than orthorhombic / orthotropic only nine elastic constants or elastic coefficients completely determine the elastic response of the polycrystalline aggregate [Green 1998, Hearmon 1961, Hill 1952, Pabst & Gregorová 2004a, Wachtman 1996]. According to Voigt the effective tensile modulus of a macroscopically isotropic polycrystalline material is

EV = the effective shear modulus

( A − B + 3C )( A + 2 B) , 2 A + 3B + C

(61)

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

GV =

53

A − B + 3C , 5

(62)

A + 2B . 3

(63)

and the effective bulk modulus

KV =

In these expressions A , B , C are given by

A=

C11 + C 22 + C 33 , 3

(64)

B=

C12 + C 23 + C 31 , 3

(65)

C=

C 44 + C 55 + C 66 . 3

(66)

According to Reuss the effective tensile modulus of a macroscopically isotropic polycrystalline material is

ER =

5 , 3 X + 2Y + Z

(67)

GR =

5 , 4 X − 4Y + 3Z

(68)

1 . 3( X + 2Y )

(69)

the effective shear modulus

and the effective bulk modulus

KR =

In these expressions X , Y , Z are given by

X =

S11 + S 22 + S 33 , 3

(70)

54

Willi Pabst and Eva Gregorová

Y=

S12 + S 23 + S 31 , 3

(71)

Z=

S 44 + S 55 + S 66 . 3

(72)

Hill [Hill 1952] has shown that the Voigt and Reuss values are the upper bound and the lower bound, respectively, of the effective elastic moduli M of statistically (macroscopically) isotropic polycrystalline materials (Hill’s theorem), i.e.

M R ≤ M ≤ MV .

(73)

Experience with the effective elastic properties of polycrystalline materials has shown that the Voigt bound and the Reuss bound are for practical purposes sufficiently close [Green 1998, Hill 1952]. Therefore, following Hill’s recommendation [Hill 1952], it is common practice to use the arithmetic mean of the Voigt and Reuss values,

M VRH =

MV + M R , 2

(74)

as an estimate (approximate prediction) of the respective effective elastic modulus of dense polycrystalline materials (“VRH average”). There is of course no rational reason to prefer the arithmetic mean over other averages, e.g. the harmonic or geometric mean. But as long as the Voigt and Reuss values are sufficiently close, the various means yield practically identical results.

3.2

Voigt and Reuss Bounds for the Elastic Moduli of Multiphase Materials

When in a multiphase material the properties of the individual phases (constituents) are known, it can be expected that certain bounds on the effective property of the multiphase material (mixture) can be given without further microstructural information apart from compositional information (phase volume fractions). For the elastic moduli (and many other properties that are well defined via linear constitutive equations) this is indeed the case, cf. [Beran 1968, Markov 2000, Milton 2002, Nemat-Nasser & Hori 1999, Sahimi 2003a, Torquato 2002]. It seems plausible e.g. that in a multiphase material the value of an effective property whose carrier are the bulk phases (and not the phase boundaries) should neither be lower nor higher than the value of that property for any of the individual phases. Therefore it can be concluded that such an effective property of the multiphase material, e.g. the effective elastic modulus M , must be some kind of average value of all the phase (constituent) properties, e.g. the elastic moduli M i . The most general average value is the general power mean (weighted by volume fractions) [Pabst & Gregorová 2004a]:

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

M X = (∑ φ i M iN ) N .

55

1

(75)

In this expression the summation extends over all n phases, the exponent N determines the type of mean value and the subscript X denotes this respective type. Setting e.g. N = −1 results in the harmonic mean M H , N → 0 yields the geometric mean M G ,

N = 1 the arithmetic mean M A , N = 2 the quadratic mean M Q , and N = 3 the cubic mean M C etc. For these types of averages the so-called majority relation holds:

… M H ≤ MG ≤ M A ≤ MQ ≤ MC ….

(76)

This is a mathematical law (law of logic), i.e. a proposition that can be rigorously proved without further assumptions. Interestingly, however, it is found that the effective shear and bulk moduli of multiphase materials always lie between the arithmetic and the harmonic mean. This is a physical law (law of nature), i.e. a finding which can be (and has been in micromechanics) rigorously proved for model materials with well defined microstructures. Of course, since its proof is based on model assumptions, its applicability to real materials is and remains, strictly speaking, a question of experience. The arithmetic mean (upper bound) corresponds to the Voigt bound of the shear and bulk modulus, GV and K V , respectively,

GV = G A = ∑ φ i Gi ,

(77 a)

KV = K A = ∑φi K i ,

(77 b)

and the harmonic mean (lower bound) corresponds to the Reuss bounds of the elastic moduli

MR

MR = MH

( φ1

 φ =  ∑ i  Mi

  

−1

⇒

φ 1 1 = =∑ i . MR MH Mi

(78)

In the case of two-phase materials one of the two volume fractions is redundant ≡ 1 − φ and φ 2 ≡ φ , because of Equation (49)) and the Voigt and Reuss bounds reduce

to

GV = (1 − φ ) G1 + φ G2

(79 a)

K V = (1 − φ ) K 1 + φ K 2

(79 b)

Willi Pabst and Eva Gregorová

56 and

1 1−φ φ = + MR M1 M 2 respectively, where

⇒

MR =

M 1M 2 , (1 − φ ) M 2 + φ M 1

(80)

φ is the volume fraction of one of the phases. When, additionally, one of

the phases is the void phase (with zero elastic moduli M 2 = 0 ), the elastic moduli of the solid matrix phase are denoted as M 1 ≡ M 0 as usual, the Voigt bounds reduce to

M V = (1 − φ ) M 0 and the Reuss bounds, Equation (80), degenerate to zero identically. In this case

(81)

φ denotes

the volume fraction of pores and is called porosity. In the micomechanical literature the Voigt-Reuss bounds for multiphase materials are called one-point bounds [Torquato 2002] as a consequence of the fact that only the one-point correlation functions (i.e. the volume fractions) are required as input information. It appears that the first application of the VoigtReuss bounds (one-point bounds) to composites is due to Paul [Paul 1960]. Note that for composites we have written the Voigt bounds only for the shear modulus G and the bulk modulus K , although in the literature it is sometimes tacitly assumed that they hold also for the tensile modulus E [Eduljee & McCullough 1993]. In fact, strictly speaking, the upper bound (Voigt bound) of the tensile modulus EV corresponds to the arithmetic mean E A only if the Poisson ratios of all phases are equal [Sahimi 2003a, Torquato 2002], a case barely encountered in practice. For this reason the Voigt bound of the tensile modulus EV should be calculated from the arithmetic means (Voigt values) of the shear and bulk moduli, GV and K V , respectively, via the standard relation

EV =

9 K V GV . 3K V + GV

(82)

This is possible because the elasticity standard relations must hold for all isotropic continua, whether heterogeneous on the microscale or not [Torquato 2002]. It will be shown below, that in the case of the dense alumina-zirconia composite ceramics (Poisson ratios 0.23 and 0.31 for alumina and zirconia, respectively) the deviation of the Voigt bound EV calculated via Equation (82) from the arithmetic mean turns out to be very small (< 0.6 %), so that in practice the arithmetic mean will be a sufficiently precise approximation of the Voigt bound. In the case of porous materials (degenerate case of two-phase composites where one phase is the void phase) the Voigt bound of the tensile modulus EV corresponds to the arithmetic mean E A exactly, cf. Equation (81).

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

57

We would like to close this subsection with a few words on the effective Poisson ratio. From elasticity theory it is well known that the Poisson ratio has several exceptional properties (e.g. in contrast to the elastic moduli E , G and K it can adopt negative values in principle, as mentioned above). For the effective Poisson ratio of a polycrystalline or multiphase, possibly porous, material the situation seems to be still rather unclear and would need further investigation. There is considerable confusion in the literature concerning this point. However, the following can be said: In general the effective Poisson ratio does not obey the Voigt and Reuss bounds. It is not even true that the effective Poisson ratio must lie between that of the constituent phases [Zimmerman 1994]. Nevertheless, in lack of a better alternative it is often recommended to calculate the effective Poisson ratio of composites via the mixture rule (i.e. as an arithmetic mean weighted by volume fractions). Unfortunately this simple remedy fails in the case of porous materials (degenerate case of composites where one phase is the void phase exhibiting zero elastic moduli). Here the mixture rule for the effective Poisson ratio evidently breaks down, since the effective Poisson ratio is known to approach finite values (mostly assumed to be in the range 0.2-0.33) for porosities close to 100 % (e.g. in foams [Gibson & Ashby 1997, Roberts & Garbozci 2000]), while the close-to-zero value corresponding to 100 % porosity (strictly speaking undefined for pores), and the value corresponding to the inclusion phase in general, is attained only due to a singularity (“boundary layer”) in the ν - φ -diagram [Zimmerman 1991a, 1994]. This is clear evidence of the fact that the effective Poisson ratio can violate the Voigt bound. Markov [Markov 2000] and Zimmerman [Zimmerman 1991a, 1994] discuss these points in great detail and derive the self-consistent, differential and Mori-Tanaka predictions (see below), respectively, of the asymptotic value ν towards which the effective Poisson ratio tends when approaching 100 % porosity. According to the self-consistent and differential approach (see below) this asymptotic value is invariably *

ν * = 0.2 ,

(83)

while according to the Mori-Tanaka approach [Mori & Tanaka 1973] the predicted asymptotic value lies somewhere between the matrix Poisson ratio ν 0 and the value 0.2, and is for porous materials (with spherical voids) given by the formula

ν* =

1 + 5ν 0 . 9 + 5ν 0

(84)

In the case of dense (pore-free) composites where the contrast between the moduli is not too large and therefore the VRH average or, better, the arithmetic average of the HashinShtrikman bounds (HS average, see below) can be expected to yield a satisfactorily sharp estimate of the effective moduli ( M VRH or M HS ) we recommend calculating the effective Poisson ratio via one of the the standard relations (cf. the discussion above)

Willi Pabst and Eva Gregorová

58

ν=

E − 2G 3K − 2G 3K − E = = , 2G 2(3K + G ) 6K

(85)

where the subscripts VRH or HS have been omitted for convenience. It will be shown below that in the alumina-zirconia system the difference between the effective Poisson ratios thus calculated and those calculated by the simple mixture rule is sufficiently small to be neglected (< 1.4 %, corresponding to a maximum absolute error of 0.004, i.e. in the third decimal).

3.3

Hashin-Shtrikman Bounds for the Elastic Moduli of Two-Phase Materials

In contrast to the case of polycrystalline monophase materials (see above) the Voigt-Reuss bounds of two-phase materials are often too far apart to be useful for prediction purposes. Hashin and Shtrikman [Hashin & Shtrikman 1963] derived the best possible bounds on the effective elastic moduli of macroscopically isotropic two-phase composites given just volume-fraction information. Actually the Hashin-Shtrikman bounds are two-point bounds, but accidentally the key integral involving this two-point information has a form that reduces to one-point information, i.e. volume-fraction information, in the case of macroscopically isotropic composites [Torquato 2002]. When K 1 > K 2 and G1 > G 2 (the usual, so-called “well-ordered” case, in contrast to the “badly-ordered” case ( K 1 − K 2 ) ⋅ (G1 − G 2 ) ≤ 0 , considered by Walpole [Walpole 1966a]) the Hashin-Shtrikman upper bounds (in the sequel “HS upper bounds”) for the shear modulus G and the bulk modulus K of two-phase materials take the form

 1 6 (K 1 + 2 G1 )  = G1 +  + φ1 ⋅  ⋅φ2 5 G1 (3 K 1 + 4 G1 )   G2 − G1 −1

G

+ HS

(86)

and −1

K

+ HS

 1  3 = K1 +  + φ1 ⋅  ⋅φ2 , 3 K 1 + 4 G1   K 2 − K1

(87)

−

−

respectively. The Hashin-Shtrikman lower bounds (“HS lower bounds”) G HS and K HS are obtained by interchanging the subscripts 1 and 2, cf. [Beran 1968, Markov 2000, Milton 2002, Torquato 2002]. In the special case where one of the phases is the void phase (with zero elastic moduli G 2 = 0 , K 2 = 0 ), where φ 2 ≡ φ is the porosity and the elastic moduli of the solid matrix phase are denoted as G1 ≡ G0 , K1 ≡ K 0 as usual, the HS upper bounds reduce to

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

 1 6 (K 0 + 2 G0 )  = G0 + − + (1 − φ ) ⋅  ⋅φ , 5 G0 (3 K 0 + 4 G0 )   G0

59

−1

G

+ HS

(88 a)

−1

K

+ HS

 1  3 = K 0 + − + (1 − φ ) ⋅  ⋅φ . K 3 K 4 G + 0 0 0  

(89 a)

and the HS lower bounds degenerate to zero identically. An alternative formulation of the HS upper bounds is +   G HS 15 K 0 + 20 G0 = 1−   ⋅φ , G0  9 K 0 + 8 G0 + (6 K 0 + 12 G0 ) ⋅ φ 

(88 b)

+  3 K 0 + 4 G0  K HS = 1−   ⋅φ . K0  3 K 0 ⋅ φ + 4 G0 

(89 b)

The HS bounds have been theoretically derived for the shear modulus G and the bulk modulus K . The corresponding HS bounds for the tensile modulus E can be obtained via the standard relation [Pabst & Gregorová 2004a]

E HS =

9 K HS G HS . 3K HS + G HS

(90)

In the case of porous materials with a matrix or skeleton Poisson ratio of 0.2 (ν 0 = 0.2 , corresponding to 3K 0 = 4G0 ) it can easily be shown that the HS upper bounds reduce to + E HS G+ K+ 1−φ . = HS = HS = E0 G0 K0 1+ φ

(91)

Note that for this case the HS upper bounds are identical to the Mori-Tanaka predictions for random materials of matrix-inclusion type with spherical inclusions [Mori & Tanaka 1973, Weng 1984, Zimmerman 1994] and that Equation (91) corresponds to a relation known in geophysical context under the name Kuster-Toksöz relation [Kuster & Toksöz 1974, Zimmerman 1991b]. It is shown below that in the alumina-zirconia system the Kuster-Toksöz relation, Equation (91), is an excellent approximation to the HS upper bound for the tensile modulus (error < 0.1 %) and for the shear modulus (error < 2.6 %) but not for the bulk modulus (error < 14.3 %). For the purpose of later reference we note that Equation (91) can be approximated by the following second-order polynomial:

60

Willi Pabst and Eva Gregorová + E HS G+ K+ = HS = HS ≈ 1 − 1.71 φ + 0.71 φ 2 . E0 G0 K0

(92)

Note that, in contrast to the relatively wide Voigt and Reuss bounds, which are security bounds (worst case bounds, which certainly cannot be exceeded), the much tighter HashinShtrikman bounds are optimal or realizable in the sense that microstructures can be devised for which these bounds can be attained. Without microstructural information of higher order (for real materials) or more special assumptions (for model materials) they cannot be improved, cf. [Beran 1968, Markov 2000, Milton 2002, Nemat-Nasser & Hori 1999, Torquato 2002]. Generally the microstructure corresponding to the HS upper bound is the socalled Hashin assemblage [Hashin 1962], consisting of polydisperse composite spheres containing concentric spherical inclusions. In the case of macroscopically isotropic porous materials the Hashin assemblage can be imagined as a porous material consisting of hollow spheres with an infinitely wide size distribution that enables space filling with a fractal microstructure.

3.4

Dilute Approximations for the Effective Elastic Moduli of Porous Materials

In the case of porous materials it is convenient to define a relative elastic modulus as

Mr ≡

M , M0

(93)

where M is the effective elastic modulus (as above) and M 0 the elastic modulus of the matrix phase (in the case of porous materials of the matrix-inclusion type, i.e. porous materials with essentially closed pores) or else the elastic modulus of the solid skeleton phase (in the case of bicontinuous porous materials, i.e. open-pore cellular solids). For convenience we will refer to elastic moduli (and the Poisson ratio) with subscribed index 0 as matrix elastic moduli (and matrix Poisson ratio) in the sequel, irrespective of the underlying concept of pore topology. The Voigt bounds of the relative elastic moduli M rV of porous materials are linearly decreasing with a slope of − 1 ,

M rV =

MV = 1−φ M0

(94)

while the HS upper bounds are nonlinearly decreasing (e.g. with an initial tangent slope of − 1.71 for a material with a matrix Poisson ratio of ν 0 = 0.2 ), cf. Equations (88 b), (89 b) and (92). For very low porosities ( φ → 0 ), where mutual interactions between the pores can be neglected (so-called dilute approximation) it is justified to assume a linear dependence of the relative elastic moduli on the porosity,

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

M r = 1 − [M ] φ ,

61 (95)

where [M ] is the intrinsic elastic modulus defined as

[ M ] ≡ − lim φ →0

M r −1

φ

.

(96)

Note the new sign convention in this definition, which is in contrast to our earlier definition [Pabst & Gregorová 2003c, 2004a, 2004b, 2004c] and to common practice in suspension rheology [Krieger 1972] but contributes much to clarity and simplicity, cf. also [Pabst 2005, Pabst et al. 2006]. Based on the exact classical solution of the spherical-cavity problem (single spherical void embedded in an infinite elastic medium [Goodier 1933, Timoshenko & Goodier 1951]), dilute-limit approximations to the problem of non-interacting spherical cavities (i.e. single pores or a dilute system of pores) in an elastic matrix have been obtained by Dewey [Dewey 1947] and Mackenzie [Mackenzie 1950], cf. also [Christensen 1979, Christensen 2000, Nemat-Nasser & Hori 1999, Torquato 2002]. For the relative shear modulus G r , the relative bulk modulus K r , the relative tensile modulus E r and the relative Poisson ratio ν r we have

Gr = 1 −

15 (1 − ν 0 ) ⋅φ , 7 − 5ν 0

(97)

Kr = 1−

3 (1 − ν 0 ) ⋅φ , 2 (1 − 2ν 0 )

(98)

Er = 1 −

3 (1 − ν 0 ) (9 + 5ν 0 ) ⋅φ , 2 (7 − 5ν 0 )

(99)

vr = 1 +

3 (1 − ν 02 ) (1 − 5ν 0 ) ⋅φ . 2 ν 0 (7 − 5ν 0 )

(100)

An analogous dilute-limit approximation is available for the relative Lamé constant λ . The generalization to triaxial ellipsoids is due to Eshelby [Eshelby 1957]. Explicit formulae for the special limiting cases of needle-like and disk-like inclusions have been given by Torquato [Torquato 2002]. Nemat-Nasser and Hori have shown that these relations have been derived under the assumption of prescribed macrostrain [Nemat-Nasser & Hori 1999]. When macrostress is prescribed the results are of the form [Zimmerman 1991b]

Mr =

1 , 1 + [M ] φ

(101)

Willi Pabst and Eva Gregorová

62

which can be developed into a series expansion in (in

φ and truncated after the first-order term

φ ) to give again the dilute-limit expressions corresponding to Equations (97) through

(100), cf. Equation (95). It is evident that the first-order coefficients (intrinsic elastic moduli) are all functions of the matrix Poisson ratio ν 0 . Note that, according to the dilute approximation, in the special case ν 0 = 0.2 the relative Poisson ratio of a porous material with spherical pores is equal to unity, i.e. the effective Poisson ratio remains unchanged with increasing porosity. Note also that the so-called self-consistent approach [Bruggeman 1937, Budiansky 1965, Hill 1965], which in a certain sense takes interactions into account, results in very similar relations, except for the fact that the intrinsic elastic moduli are functions of the effective Poisson ratio ν instead of the matrix Poisson ratio ν 0 . Of course, in the special case of porous materials with spherical pores and ν 0 = 0.2 both the dilute approximation (in the dilute limit

φ → 0 ) and the self-consistent approach (principally intended for finite φ )

lead to the identical result

M r = 1− 2 φ ,

(102)

i.e. the intrinsic elastic modulus equals 2 for the case of spherical pores in a ν 0 = 0.2 matrix material. Obviously, deviations from this value can be attributed either to deviations of the pore shape from sphericity (including a topological transition from isolated to connected) or to deviations of the matrix Poisson ratio from the “magic” value 0.2. Values of the intrinsic elastic moduli [G ] , [ K ] and [ E ] are listed in Table 1 in dependence of the matrix Poisson ratio ν 0 . Note that for the “normal“ matrix Poisson ratios ν 0 between 0 and 0.5 the intrinsic tensile modulus [ E ] remains very close to 2, increasing from 1.929 (for

ν 0 = 0) to a

maximum value of 2.006 (for ν 0 = 0.268667), followed by a decrease to 1.917 (for ν 0 = 0.5). Due to this “anomalous“ behavior of [E ] the value 2 is attained exactly for two different values of ν 0 (0.2 and 1/3). The limiting values of the bulk and tensile moduli [K ] and [E ] for (extremely auxetic) materials with a (negative) matrix Poisson ratio (ν 0 = − 1 ) are 1, corresponding to the Voigt bound (values < 1 cannot occur). Curiously, the limiting value of the intrinsic shear modulus [G ] for ν 0 = − 1 materials is 2.5, obviously the counterpart of the value 2.5 [Einstein 1906] for the intrinsic shear viscosity occuring in suspension rheology, cf. [Pabst 2004]. Note also that for the “typical“ matrix Poisson ratios ν 0 in the range between 0.17 and 0.33 the values of [G ] are still relatively close to 2 (range 1.875-2.024), while for [K ] this is not the case. For a matrix Poisson ratio of ν 0 = 0.5 (corresponding to a totally incompressible matrix) the intrinsic bulk modulus diverges, i.e. a very small amount of pores would be extremely efficient (detrimental) in such a case. In other words, there would be a singularity in the K - φ -diagram at φ = 0 , where the effective bulk modulus steeply

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

63

falls down from the value of the matrix bulk modulus K 0 to zero. In this connection we note that, due to the condition [ M ] ≥ 1 , all linear relations of the type (95), whether based directly on the dilute approximation or derived from the self-consistent approach, predict a critical porosity

φ C = [ M ] −1 ≤ 1 , for which the effective elastic moduli become zero, i.e. the

material looses integrity. In the context of percolation theory

φ C can be interpreted as a

percolation threshold [Kovačík 1999, Markov 2000, Sahimi 1994, Sahimi 2003a, Stauffer & Aharony 1985, Zallen 1983]. Table 1. Intrinsic elastic moduli [G ] , [K ] , [E ] for different matrix Poisson ratios ν 0 .

3.5

ν0

[G ]

[K ]

[E]

− 1.0

2.5

1

1

− 0.5

2.368

1.125

1.539

− 0.2

2.250

1.286

1.800

− 0.1 0 0.1 0.17 0.2 0.23 0.25 0.26 0.268667 0.27 0.3 0.31 0.333333 0.4 0.5

2.200

1.375

1.870

2.143 2.077 2.024 2 1.974 1.957 1.947 1.939 1.938 1.909 1.899 1.875 1.800 1.667

1.5 1.688 1.886 2 2.139 2.25 2.313 2.371 2.380 2.625 2.724 3 4.5

1.929 1.973 1.994 2 2.004 2.005 2.006 2.006 2.006 2.005 2.004 2 1.980 1.917

∞

Nonlinear Relations for the Effective Elastic Moduli of Porous Materials

Experience shows that usually the porosity dependence of the effective elastic moduli is not linear. Thus, since it is clear that the linear relation (95) claims validity and practical significance only in the case of very small porosities (dilute approximation), there have been numerous attempts to extend it to higher porosities by allowing for a nonlinear dependence. The simplest way to do so is the Coble-Kingery approach [Coble & Kingery 1956], which is

Willi Pabst and Eva Gregorová

64

as follows [Pabst & Gregorová 2003b, 2004a]: Take the linear relation, Equation (95), for the matrix Poisson ratio in question and add a quadratric term in φ (second-order polynomial),

M r = 1 − [M ] φ + α φ 2 .

(103)

Then determine the value of the coefficient α via the condition that M r = 0 at least for

φ = 1 (which is necessary in order not to violate the Voigt bound). In general one obtains [Pabst & Gregorová 2004a, 2004c]

M r = 1 − [ M ] φ + ([ M ] − 1) φ 2 .

(104)

In the special case of porous materials with spherical pores and ν 0 = 0.2 this reduces to

M r = (1 − φ )

2

(105)

[Pabst & Gregorová 2003b, 2003c, 2004a, 2004c]. This relation is identical with the prediction of the differential approach for porous materials with ν 0 = 0.2 [McLaughlin 1977, Norris 1985, Norris et al. 1985, Zimmerman 1991a, 1991b]. An alternative derivation of this relation, using a functional equation approach, has been given recently [Pabst & Gregorová 2003c]. Interestingly, the same result was found for the tensile modulus by fitting a large amount of experimental data on real materials with the semi-empirical Gibson-Ashby model for open-pore cellular solids [Gibson & Ashby 1982, 1997]. Also the Gibson-Ashby relation for the shear modulus has been found to match the general form

Gr = γ (1 − φ )

2

(106)

γ is an adjustable parameter to be determined e.g. by fitting. It has been shown recently, however, that the value γ = 3 (1 + ν 0 ) / 4 proposed by Gibson and Ashby where

[Gibson & Ashby 1982, 1997] is probably too low and should be replaced by γ = 5 (1 + ν 0 ) / 6 in order to ensure that Equation (106) reduces to Equation (105) in the case of porous materials with ν 0 = 0.2 , cf. [Pabst et al. 2006]. Irrespective of the matrix Poisson ratio the binomial relation (105) may be expected to be reasonable for the tensile modulus E and, depending on the matrix Poisson ratio, also for the shear modulus G , but certainly not for the bulk modulus K . This is connected with a principal problem of CobleKingery relations, cf. [Pabst et al. 2006]: if [ M ] > 2 , the second-order polynomials exhibit a minimum with negative M r values, which is clearly nonsense from the physical point of view. Obviously, this problem is harder in the case of the bulk modulus than with the other two moduli, cf. Table 1. Therefore an entirely different relation has been proposed for the

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

65

bulk modulus of low-density cellular solids (i.e. for very high porosities) [Christensen 1986, Warren & Kraynik 1988]:

Kr =

1 − 2ν 0 (1 − φ ) . 3

(107)

Evidently the linear form of this relation prohibits its use in the range of low porosities. In particular, in the case of porous materials with ν 0 = 0.2 it predicts K r = 0.2 (1 − φ ) , which is clearly wrong. A possibility to overcome this problem, in lack of a better alternative, is to combine the Warren-Kraynik-Christensen relation (which can be a realistic prediction in the high-porosity region) with the Coble-Kingery relation (which can be realistic in the lowporosity region) [Pabst et al. 2006]. To ensure in any case positivity of the K r values it has been recommended to use the porosity

φ X at the crossover of the two curves (calculated by

simply equating the Coble-Kingery relation (104) and the Warren-Kraynik-Christensen prediction (107) for the matrix Poisson ratio in question) as the point of continuation, cf. [Pabst et al. 2006]. In the remaining part of this subsection let us concentrate on the tensile modulus. For evident reasons (discussed above) it is in practice unnecessary to emphasize the condition ν 0 = 0.2 in this case, because for pores of spherical shape the intrinsic tensile modulus [ E ] is always very close to the benchmark value of 2 (1.97-2.01), even when the matrix Poisson ratio varies in the wide range between ν 0 = 0.1 and ν 0 = 0.4 (which covers practically the whole range of interest for ceramic materials), cf. Table 1. Notwithstanding the problems with the bulk modulus the Coble-Kingery relations can yield a prediction of the porosity dependence of the relative tensile modulus that fulfils all basic criteria of physical plausibility: it ensures that M r = 0 for φ = 1 and it does not violate the HS upper bound. For example in the case of porous materials with ν 0 = 0.2 the initial tangent slope is − 2 and the curvature is 1, whereas the HS upper bound is characterized approximately by an initial tangent slope of − 1.71 and a curvature of 0.71 in this case, cf. Equation (92). Through the years several other relations have been proposed to predict or describe the porosity dependence of the relative tensile modulus, and recently the most important ones have been collected and rationally classified by the authors [Pabst & Gregorová 2003b, 2003c, 2004a, 2004b, 2004c, Pabst et al. 2006]. In the sequel we give a brief overview of these, in unified notation, using the critical porosity φ C and the newly introduced concept of intrinsic elastic moduli [ M ] wherever feasible. It is understood, that in practice these parameters will usually be treated as fit parameters to be determined from experimentally measured data a posteriori. This is particularly true for the critical volume fraction φ C , where reliable and sufficiently precise a priori estimates will hardly be available [Sahimi 2003a]. Note that for porous materials there exists no reliable benchmark value for φ C comparable to the value of approx. 0.64 (i.e. 64 %) for the packing density of monodisperse rigid spheres in random close packed (rcp) arrangement [Aste & Weaire 2000, Bernal 1959, Bernal & Mason 1960,

66

Willi Pabst and Eva Gregorová

Rintoul & Torquato 1996, Scott 1960]. Nevertheless, in exceptional cases there exist promising results of computer simulations of porous model materials [Roberts & Garbozci 2000]. When these model materials are considered to reflect the microstructure of any real material to a sufficient degree, then of course the following relations may be used for comparing experimental data with model predictions or even for property prediction purposes. The same holds when a sufficient number of data has been measured for materials having certain typical microstructural features in common (e.g. a certain pore space topology). Also in this case (in fact a real experiment in contrast to the virtual computer experiment) the following relations can be used to predict the behavior of similar materials. Spriggs [Spriggs 1961] suggested the use of a simple exponential relation of the form

E r = exp (−[ E ] φ ) ,

(108)

where the intrinsic tensile modulus is principally a parameter to be determined by fitting experimentally measured data. For small porosities ( φ → 0 ) Equation (108) approximates Equation (95), because of the truncated series expansion

exp (−[ E ] φ ) ≈ 1 − [ E ] φ + ... .

(109)

A derivation of the Spriggs relation via the so-called functional equation approach has been given recently by the authors [Pabst & Gregorová 2003c]. Zimmerman was apparently the first to use a differential scheme approach to derive a Spriggs-type relation [Zimmerman 1984, 1991b], which for the special case of porous materials with spherical pores reduces to the simple prediction

E r = exp (−2 φ ) ,

(110)

Of course, the Spriggs relation (including its special case Equation (110)), suffers from the serious principal drawback that E r is not zero for φ = 1 , i.e. the Spriggs relation necessarily violates the HS upper bound and even the Voigt bound. For this reason Hasselman [Hasselman 1962], based on previous work by Hashin [Hashin 1962], suggested a relation which can be written as

Er =

1−φ , 1 + CHφ

(111)

where, again, C H has to be determined principally by fitting experimentally measured data. This relation is clearly nonlinear and monotonically decreasing, and E = 0 is guaranteed for

φ = 1 . Unfortunately, however, the inverse of − C H cannot be interpreted as a critical porosity since in the limit

φ → − 1 C H relation (111) diverges ( E → ∞ ), i.e. − 1 C H

must always lie outside of the interval 0 < φ < 1 . Thus the interpretation of − 1 / C H in

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

67

terms of a critical porosity is principally inadmissible and there is no physical meaning left in the Hasselman relation (111), cf. [Pabst & Gregorová 2004a, 2004c, Pabst et al. 2006]. This finding is also corroborated by the fact that fitting of experimentally measured values invariably leads to positive values of C H [Pabst et al. 2004b, 2005], which implies that the tentative critical porosity − 1 / C H would always be a negative quantity (which is nonsense from the physical point of view). Note, however, that a very special case of the Hasselman relation can in principle be useful for predictive purposes, viz. the case C H = 1 . In this case the Hasselman relation reduces to the Kuster-Toksöz relation [Kuster-Toksöz 1974]

Er =

1−φ , 1+ φ

(112)

which is identical to the Mori-Tanaka prediction and the HS upper bound for porous materials with spherical pores and ν 0 = 0.2 , as discussed above, cf. Equation (91). Recently, it has been recognized [Pabst & Gregorová 2004a, 2004b] that the modified exponential relation

 − [E] φ   E r = exp   1−φ  also circumvents the aforementioned compatibility problem at

(113)

φ = 1 . Most surprisingly, it

seems that this simple fact has not been realized before in the literature. In [Pabst & Gregorová 2004b] a derivation of this relation has been given via a functional equation approach. Trivially, in the case of porous materials with spherical pores the modified exponential leads to the prediction

−2φ   . E r = exp   1−φ 

(114)

Note that this exponential model results in a zero relative tensile modulus only in the limiting case of 100 % porosity. Without doubt, porosities close to 100 % can in principle be achieved, e.g. in some aerogels [Gibson & Ashby 1997]. However, the usual case encountered in practice will be a complete structural breakdown (loss of integrity) at significantly lower porosity levels. In order to allow for the possibility of E r = 0 at porosities lower than 100 % (i.e.

φ < 1 ), it is necessary to include a critical volume fraction

φ C in the modulus-porosity relation, which is able to take the possible occurrence of a percolation threshold into account. This results in a Mooney-type exponential relation [Mooney 1951, Pabst & Gregorová 2004a, 2004b]

Willi Pabst and Eva Gregorová

68

   

 − [E] φ E r = exp  1− φ φ C Again, for small porosities ( φ → 0 , which implies

(115)

φ << φ C ) accordance with Equation

(95) is guaranteed. In contrast to the Hasselman relation (111) the physical interpretation of φ C in relation (115) as a critical porosity is in principle admissible. As before, under the assumption of spherical pores we can set [ E ] = 2 , which reduces the number of adjustable fit parameters from two to one. Power-law relations represent another class of fit models, principally different from the exponential relations just presented. The simplest relation of this kind is Archie’s relation [Archie 1942, Markov 2000]

E r = (1 − φ ) [ E ] ,

(116)

a derivation of which via the functional equation approach has been given recently [Pabst & Gregorová 2003c]. Of course, in the case of the tensile modulus the Archie relation can be considered as a generalization of the Coble-Kingery relation

E r = (1 − φ ) 2 . Neither of these relations exhibits a compatibility problem at

(117)

φ = 1 and, again, in the

dilute limit ( φ → 0 ) they reduce to Equation (95), because of the truncated series expansion

(1 − φ ) [ E ] ≈ 1 − [ E ] φ + ... .

(118)

Moreover, as before with the exponential relations, in order to allow for the possibility of the occurence of a percolation threshold, i.e. E r = 0 for porosities φ < 1 , an additional parameter

φ C can be introduced. This results in a Krieger-type power-law relation [Krieger

1972, Pabst 2004], in elasticity context often called Phani-Niyogi relation [Phani & Niyogi 1987a, 1987b]:

 φ E r = 1 −  φC

  

[ E ] φC

(119)

As before, under the assumption of spherical pores we can set [ E ] = 2 , which again reduces the number of adjustable fit parameters from two to one. Evidently, from a principal point of view, there is nothing to be said against the interpretation of φ C in relation (119) as a critical porosity. We would like to emphasize again that, although sometimes considered to be purely empirical, all the exponential and power-law relations mentioned above can be derived

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

69

via a functional equation approach, cf. [Pabst & Gregorová 2003c, 2004b, Pabst 2004]. What makes them semi-empirical fit models in the end is only the fact that, due to variations in pore shape, the first-order coefficient (intrinsic tensile modulus) may not be reliably known and that, due to the difficulty to assess and quantify pore size distribution and connectivity, a priori estimates of the critical porosity are usually not available. Recently, a new relation has been proposed by the authors [Pabst & Gregorová 2004a, 2004c]:

(

) (1(−1φ− φφ) ) .

E r = 1 − [ E ] φ + ([ E ] − 1) ⋅ φ 2 ⋅

C

(120)

This relation has been found heuristically in analogy to the Robinson relation in suspension rheology [Robinson 1949, Mooney 1951, Pabst 2004] and reduces to the linear approximations (95) or (99) in the dilute limit ( φ → 0 , implying φ << φ C ). Furthermore it ensures that E r = 0 when

φ = φ C , as required. Setting [ E ] = 2 our relation adopts the

extremely simple form

 φ E r = (1 − φ ) ⋅ 1 −  φC

  , 

(121)

which seems to be the simplest thinkable relation allowing for a percolation threshold (via the critical porosity φ C ). In the absence of a percolation threshold (i.e. φ C = 1 ) it reduces to the Coble-Kingery relation (117), as required. We recall that [ E ] = 2 is always a reasonable approximation for porous materials with spherical or isometric pores, independently of the precise value of the matrix Poisson ratio ν 0 , cf. Table 1 and the discussion above. Of course, unless the critical porosity

φ C is known a priori (a rather exceptional case) and an application

of Equation (121) for predictive purposes is intended, this discussion is of more or less philosophical character. As long as Equation (121) is used as a fit model its application to cases where the assumption [ E ] = 2 is (approximately) justified is semi-empirical (because its form is a special case of Equation (120)), while its application to cases where the assumption [ E ] = 2 is not justified must be considered as purely empirical. In any case, contrary to the situation with the parameter − 1 / C H occurring in the Hasselman relation (111), cf. the discussion above, there are no principal objections against the interpretation of φ C in terms of a critical volume fraction. In other words, even if taken as a purely empirical fit equation, our relation (121) remains principally meaningful from the physical point of view. In principle, it can be attempted to interpret deviations of the intrinsic tensile modulus determined by fitting with any of the relations above from the benchmark value 2 in terms of an influence of pore shape. This is possible because usually the deviations in [E ] caused by a variation of the matrix Poisson ratio are in practice negligible. Apart from the obvious advice

70

Willi Pabst and Eva Gregorová

that such an interpretation should be based on a reliable determination of [E ] (e.g. by comparing the coincidence of fitting results using several of the relations above) it should be kept in mind, that significant deviations in [E ] require a considerable degree of pore anisometry (elongation or flattening). Non-spherical pores which are more or less isometric (e.g. of polyhedral shape or pores with concave faces) cannot be expected to be responsible for measurable deviations in the intrinsic elastic moduli. Their intrinsic tensile modulus will still be close to 2 ( [ E ] ≈ 2 ). Moreover, this discussion indicates that, due to the principal non-uniqueness of the solution, such an interpretation will always remain on the level of a tentative hypothesis. It can, of course, be useful for the relative comparison of sample series, e.g. when only one processing parameter is changed at one time. In concluding this section we would like to emphasize that of course all semi-empirical relations presented for the relative tensile modulus can be used for fitting experimentally measured data for any kind of elastic modulus (and many other properties as well). Also it may be attempted to interpret the values obtained for the intrinsic properties by fitting in terms of a pore shape influence. However, the intrinsic value of 2 in the case of porous materials with spherical pores and ν 0 = 0.2 is specific to the tensile modulus (where

[ E ] = 2 exactly for ν 0 = 0.2 and ν 0 = 1 / 3 and [ E ] ≈ 2 for all matrix Poisson ratios 0 < ν 0 < 0.5 ) and can be a good approximation for the shear modulus for the matrix Poisson ratios commonly encountered in practice, but certainly not for the bulk modulus or other properties. That means, only for the tensile modulus (and approximately for the shear modulus) the parameter-free special relations (110), (114) and (117) given above (specialized by setting [ E ] = 2 in their more general counterpart relations), including the Kuster-Toksöz relation (112), can be expected to provide useful predictions. Relation (110), of course, is principally disqualified for a completely different reason (viz. because it violates the upper HS bound and even the Voigt bound, cf. the discussion above), while the Kuster-Toksöz relation represents no improvement over the HS upper bound (because it is identical to it). Similarly, we recall that our new relation adopts its elegant and simple form (121) only in the case [ E ] = 2 , otherwise its general form would be (120). This relation, however, is not per se meant to be a predictive model (since the critical porosity

φ C is usually not known a

priori), but a fit equation. As such, of course, it can be used quite universally. The overview presented in this section is rather exhaustive, at least for isometric pores. Of course, apart from the modulus-porosity relations mentioned here, more sophisticated model relations have been proposed, e.g. for anisometric pores [Ondracek 1987, Boccaccini et al. 1993] or remarkable models containing a size ratio information [Boccaccini & Fan 1997], but these are beyond the scope of this chapter. Some well-known relations, e.g. in [Ramakrishnan & Arunachalam 1993], are formally similar to the Hasselman relation (111) and suffer from the same drawbacks, while others, e.g. in [Ishai & Cohen 1967], violate even the HS upper bounds and should therefore be strictly avoided. It seems that all reasonable non-exponential models intending to describe the porosity dependence of the tensile modulus should contain the Coble-Kingery relation (117) as a special case (for the tensile modulus at least this can be considered as a benchmark relation). Models for which this is not the case should be considered with great scepticism. The interested reader can refer e.g. to [Bert 1985,

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

71

Herakovich & Baxter 1999, Kwan et al. 2000, Luo & Stevens 1999, Nielsen 1982, Rice 1998, Wagh et al. 1991] and the references cited therein for some of the (less rational) modulusproperty relations not mentioned here.

4

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia Composite Ceramics

Modern alumina ceramics consist of substantially pure α-Al2O3 (corundum), which is the thermodynamically stable modification at room temperature and atmospheric pressure. Corundum single crystals (monocrystals) are trigonal (rhombohedric), structurally characterized by the point group 3 m (space group R3 m ), cf. e.g. [Lee & Rainforth 1994]. Zirconia ceramics are much more complicated. Apart from a less important orthorhombic phase, there are at least three basic phases which can play a major role in zirconia ceramics: cubic (c-ZrO2), tetragonal (t-ZrO2) and monoclinic (m-ZrO2). In the absence of stabilizing agents the monoclinic phase is the stable one at room temperature and atmospheric pressure. However, as a consequence of the martensitic t-m transformation which occurs during cooling at approx. 950 °C and is connected with a shear strain of approx. 0.16 and with a volumetric expansion of approx. 4 %, sintered parts of pure zirconia cannot be obtained, since cracks are formed and the ceramic part looses integrity. In order to obtain zirconia ceramics that can be used in structural applications other oxides, so-called stabilizing agents, must be added (usually already during powder synthesis), e.g. CaO, MgO, CeO2, Y2O3 and several other oxides, mainly of the rare earth elements. Based on the three fundamental phases, the stabilizing agents and their contents and the typical microstructures one distinguishes at least three types of pure zirconia ceramics: fully stabilized zirconia (purely cubic phase, used mainly in electrochemical applications), partially stabilized zirconia (PSZ, consisting of a matrix of c-ZrO2 with dispersed inclusions or precipitates of t-ZrO2) and tetragonal zirconia polycrystals (TZP). For detailed information the reader can consult one of the many excellent review articles in this field, e.g. [Hannink et al. 2000] and the references cited therein. For a compact overview cf. e.g. [Lee & Rainforth 1994]. If not explicitly stated otherwise, by “zirconia“ we mean the tetragonal phase of zirconia (t-ZrO2) with point group 4 / m m m (space group P 4 2 / n m c ), or else tetragonal zirconia polycrystals (TZP), in this chapter.

4.1

Effective Elastic Moduli of Polycrystalline Alumina and Zirconia

The elastic behavior of trigonal and tetragonal monocrystals is fully described by 6 independent elastic constants (stiffnesses) or, equivalently, elastic coefficients (compliances). The most reliable values currently available for alumina (α-Al2O3) and zirconia (t-ZrO2, more precisely t-ZrO2 with 12 mol% CeO2), which are due to [Wachtman et al. 1960] and [Kisi & Howard 1998], respectively, are listed in Table 2. The alumina data have been measured by X-ray diffraction directly for monocrystals, while the zirconia data have been inferred from neutron diffraction measurements on polycrystalline samples and are in this sense biased, cf. [Kisi & Howard 1998] and the discussion in [Pabst et al. 2004a]. Note that, due to the unavailability of t-ZrO2 monocrystals of sufficient size, unbiased monocrystal data are

Willi Pabst and Eva Gregorová

72

currently not available. The values listed in Table 2 are mutually consistent in the sense that they fulfill the matrix inversion relations, Equations (9) and (10), and several other plausibility criteria, cf. the critical comparison presented in [Pabst et al. 2004a]. Table 2. Components of the stiffness matrix C ij and the compliance matrix S ij for alumina and tetragonal zirconia (t-ZrO2), respectively [Wachtman et al. 1960, Kisi & Howard 1998].

C ij [GPa]

Alumina

t-ZrO2

S ij [GPa-1]

Alumina

t-ZrO2

C11 C33

496.8 ± 1.8

327

0.002353

0.00346

498.1 ± 1.4

264

S11 S33

0.002170

0.00406

C44 C12 C13

147.4 ± 0.2

59

0.006940

0.0170

163.6 ± 1.8

100

− 0.000716

− 0.00096

110.9 ± 2.2

62

S 44 S12 S13

− 0.000364

− 0.00059

C14

− 23.5 ± 0.3

−

S14

0.000489

−

C66 (calc.)

166.6

64

S 66 (calc.)

0.003274

0.0154

Table 3. Components of the stiffness matrix C ij and the compliance matrix S ij for monoclinic zirconia (m-ZrO2 ) according to [Nevitt et al. 1988] and [Chan et al. 1991], respectively.

C ij [GPa]

Nevitt et al.

Chan et al.

S ij [GPa-1]

Nevitt et al.

Chan et al.

C11 C 22 C33

358

361

0.00345

0.00341

426

408

0.00335

0.00503

240

258

S11 S 22 S 33

0.00537

0.00678

C44 C 55

99.1

99.9

0.0114

0.0104

78.7

81.2

S 44 S 55

0.0153

0.0147

C 66

130

126

S 66

0.00873

0.00828

C12 C13

144

142

− 0.00128

− 0.00160

67

55

S12 S13

0.000116

0.000608

C15

− 25.9

− 21.3

S15

0.00173

0.00165

C 23

127

196

S 23

− 0.00178

− 0.00371

C 25

38.3

31.2

S 25

− 0.00268

− 0.00319

C 35

− 23.3

− 18.2

S 35

0.00242

0.00310

C 46

− 38.8

− 22.7

S 46

0.00342

0.00188

In Tables 3, 4, 5 and 6 we list, for reasons of comparison and reference purposes, the components of the stiffness and compliance matrix for monoclinic [Chan et al. 1991, Nevitt et

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

73

al. 1988] and cubic zirconia [Kandil et al. 1984, Kisi & Howard 1998], including the values cited by [Kandil et al. 1984] for cubic zirconia (c-ZrO2) monocrystals in dependence of the yttria content, cf. also [Green 1998]. Table 4. Components of the stiffness matrix C ij and the compliance matrix S ij for c-ZrO2 with 8.1 mol.% Y2O3, according to [Kandil et al. 1984] and [Kisi & Howard 1998].

S ij [GPa-1]

Kandil et al.

S11

0.00274

Kisi & Howard 0.0034

C11

402

Kisi & Howard 390

C44

56

60

S 44

0.0179

0.0168

C12

95

162

S12

− 0.00052

− 0.001

C ij [GPa] Kandil et al.

Table 5. Components of the stiffness matrix C ij for cubic zirconia (c-ZrO2) in dependence of the yttria content (Y2O3 in mol.%), according to [Kandil et al. 1984, Green 1998]. Yttria content [mol.%] 11.1 12.1 15.5

17.9

C ij [GPa]

8.1

C11

401.8

403.5

405.1

397.6

390.4

C44

55.8

59.9

61.8

65.8

69.1

C12

95.2

102.4

105.3

108.6

110.8

Table 6. Components of the compliance matrix S ij for cubic zirconia (c-ZrO2) in dependence of the yttria content (Y2O3 in mol.%), calculated from [Kandil et al. 1984, Green 1998].

-1

Yttria content [mol.%] 11.1 12.1 15.5

S ij [GPa ]

8.1

17.9

S11

0.002737

0.002762

0.002765

0.002849

0.002929

S 44

0.017921

0.016694

0.016181

0.015198

0.014472

S12

− 0.00052

− 0.00056

− 0.00057

− 0.00061

− 0.00065

Table 7 lists the Voigt bounds (subscript V), Reuss bounds (subscript R) and VRH averages of the effective elastic moduli of polycrystalline alumina and tetragonal zirconia (tZrO2), calculated from the components of the respective stiffness and compliance matrices due to [Wachtman et al. 1960, Kisi & Howard 1998], listed in Table 2. For the effective Poisson ratio, for which the Voigt-Reuss bounds do not hold, only a range of values is given, cf. the discussion in [Pabst et al. 2004a]. In order to compare these theoretically calculated values with experimentally measured or extrapolated values for the respective polycrystalline ceramics one has to take account of the fact that real polycrystalline materials always contain a certain amount of porosity.

Willi Pabst and Eva Gregorová

74

Moreover, experimentally determined values depend on the temperature during measurement and the measurement technique (static or dynamic). While the latter seems to play only an insignificant role in the alumina-zirconia system (i.e. the difference between isothermal and adiabatic elastic moduli is negligibly small compared to other, statistical and systematic, errors of measurement), special care has been taken in [Pabst et al. 2004a] to restrict the comparison to experimental values determined at room temperature (25 ± 3 °C) for samples with a total porosity < 3%. For alumina (with a theoretical density of 4.0 g/cm3) and tetragonal zirconia (with a theoretical density of 6.1 g/cm3) the latter requirement corresponds to bulk densities values of at least 3.88 g/cm3 and 5.92 g/cm3, respectively. A detailed comparison, essentially based on [Munro 1997] and the NIST (National Institute of Standards and Technology, Gaithersburg, Maryland, USA) databases [Munro 2002, NIST 2002] has shown that for both alumina and zirconia (t-ZrO2 with 12 mol% CeO2) the agreement between theoretically calculated and experimentally determined values is satisfactory [Pabst et al. 2004a]. Note, however, that in the case of t-ZrO2 with 3 mol% Y2O3 the stiffness and compliance matrix has not been determined so far and thus the effective elastic moduli for this zirconia type must rely solely on experimentally measured values and the proximity to the values of t-ZrO2 with 12 mol% CeO2. Table 8 summarizes the effective elastic moduli and Poisson ratios that can be recommended as reference values for pore-free (i.e. densely sintered), macroscopically isotropic polycrystalline alumina and tetragonal zirconia (t-ZrO2 with 3 mol% Y2O3 or 12 mol% CeO2, respectively). These values have been obtained after careful consideration of theoretical calculated and experimentally determined values available [Pabst et al. 2004a]. Note that, in contrast to many other sets of values occurring in the literature, for each material presented here the ultimate value sets listed in Table 8 naturally obey the standard elasticity relations, Equations (30) through (33). Table 7. Effective elastic properties (elastic moduli and Poisson ratios) of polycrystalline alumina and tetragonal zirconia (t-ZrO2), calculated from the components of the respective stiffness and compliance matrices [Wachtman et al. 1960, Kisi & Howard 1998]; Voigt bound (subscript V), Reuss bound (subscript R) and Voigt-Reuss-Hill average (subscript VRH). Effective elastic property

EV

Alumina 408.2

t-ZrO2 209.9

ER

397.1

192.2

EVRH

402.7

201.1

GV

166.0

82.7

GR

160.7

74.7

GVRH

163.4

78.8

KV

251.4

151.8

KR

250.4

149.3

K VRH

250.9

150.6

0.229−0.236

0.270−0.285

ν (range)

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

75

Table 8. Recommended reference values for the effective elastic properties (elastic moduli and Poisson ratios) of pore-free, macroscopically isotropic polycrystalline alumina and tetragonal zirconia, according to [Pabst et al. 2004a].

400

t-ZrO2 with 3 mol% Y2O3 210

t-ZrO2 with 12 mol% CeO2 200

G0

163

80

76

K0

247

184

175

ν0

0.23

0.31

0.32

Effective elastic property

Alumina

E0

Fig. 1 shows the dependence of the effective tensile modulus of polycystalline zirconia on the yttria content (VRH averages calculated from the monocrystal data for c-ZrO2 listed in Tables 5 and 6 [Kandil et al. 1984, Green 1998], the value for 3 mol% Y2O3 being the value recommended for t-ZrO2 in Table 8 [Pabst et al. 2004a]). 300

Tensile modulus [GPa]

250

200

150

100

50

0 0

2

4

6

8

10

12

14

16

18

20

Yttria content [mole %] Figure 1. Dependence of the effective tensile modulus of zirconia (full squares: c-ZrO2, empty squares t-ZrO2 ) on the yttria content.

According to these data the dependence of the effective tensile modulus E 0 [GPa] of pore-free polycrystalline zirconia on the yttria content (mole fraction) cY can be fitted with the linear expression

E 0 = 205 + 2.05 cY ,

(122)

Willi Pabst and Eva Gregorová

76

(cf. the straight line in Fig. 1). Fig. 2 shows the temperature dependence of the effective tensile modulus E 0 [GPa] of pore-free polycrystalline alumina and zirconia (m-ZrO2) according to the measurements of Goto and Anderson [Goto & Anderson 1989] and the values cited by Munro [Munro 2002], respectively. 450

Tensile modulus [GPa]

400 350 300 250 200 150 100 50 0 0

200

400

600

800

1000

1200

1400

1600

1800

Temperature [°C] Figure 2. Temperature dependence of the effective tensile modulus of alumina (empty squares, upper curve) and zirconia (full squares, lower curve) [Goto & Anderson 1989, Munro 2002].

The solid curves have been obtained by fitting with the Wachtman-Anderson relation [Anderson 1966, 1995, Wachtman et al. 1961, Wachtman 1996]

 T  ∗ E 0 (T ) = E 0 − ϑ ⋅ T ⋅ exp  − 0  ,  T 

(123)

∗

in which E 0 is the effective tensile modulus at absolute zero, T is the Kelvin temperature and

ϑ and T0 are empirical constants. As the temperature decreases to absolute zero the

tangent slope of the E 0 (T ) curve approaches zero, as required by the third law of thermodynamics [Wachtman 1996]. For alumina the Wachtman-Anderson fit is

 297.4  E 0 (T ) = 410 − 0.0583 ⋅ T ⋅ exp  − , T   while for zirconia the Wachtman-Anderson fit is

(124 a)

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

 973.7  E 0 (T ) = 200 − 0.116 ⋅ T ⋅ exp  − . T  

77

(124 b)

In concluding this subsection we remind the reader that from the viewpoint of solid state physics the elastic moduli of crystalline solids are a consequence of bond strength and bond density [Green 1998, Newnham 1975], cf. the introductory section of this chapter. Since the stabilizing agents in zirconia are built into the crystal lattice they modify naturally the elastic modulus (this may also be imagined as being a consequence of the different ionic radius of the stabilizing agent in comparison to the ions making up the host lattice). It has to be emphasized however, that a simple prediction of the elastic modulus e.g. of t-ZrO2 with 3 mol% Y2O3 by a simple mixture rule or the like is principally not possible, although the room temperature elastic constants of pure polycrystalline yttria (density 5.0 g/cm3) are well known ( E 0 = 179 GPa, G0 = 69 GPa, K 0 = 149 GPa, ν 0 = 0.30, cf. [Munro 2002, Phani & Niyogi 1987b]). Note that micromechanical approaches can generally only be applied to multiphase mixtures or materials with microstructure, but not to one-phase mixtures like solid solutions (mixed crystals) or glasses, even if the constituent properties are known.

4.2

Effective Elastic Moduli of Dense Alumina-Zirconia Composite Ceramics

First let us focus on two-phase (i.e. pore-free, densely sintered) mixtures of alumina and tetragonal zirconia (t-ZrO2 with 3 mol.% Y2O3). We confine ourselves to classical microcomposites, i.e. the grain size is assumed to be sufficiently large (> 100 nm) for the phase boundary contribution to the volume fractions to be negligible. Moreover we confine ourselves to isometric particles in random orientation resulting in a macroscopically isotropic composite, to which the Voigt-Reuss bounds and the Hashin-Shtrikman bounds apply, and use the (effective) elastic moduli G0 and K 0 recommended in Table 8 as an input information for the following calculations. From now on let us consider the elastic properties of the “pure“ (i.e. single-phase) polycrystalline end members to be given once and for all and reserve the term “effective“ exclusively for the composite properties. In general (except for the Reuss bound) the effective tensile moduli E have to be calculated from the effective shear and bulk moduli G and K via the elasticity standard relation (30). Fig. 3 shows the Voigt bound of the tensile modulus EV (as a function of the zirconia volume fraction calculated exactly from the Voigt values of the shear modulus GV = GV

φZ )

(φ Z ) and the bulk

(φ Z ) (crosses slightly above the upper solid line) and the approximate = EV± (φ Z ) calculated via the mixture rule, i.e. as the arithmetic average

modulus K V = K V ±

Voigt bound EV

EV± ≈ E A = (1 − φ Z ) E1 + φ Z E 2 ,

(125)

Willi Pabst and Eva Gregorová

78

where E1 = 400 GPa (alumina) and E 2 = 210 GPa (zirconia). In Fig. 3 the latter is drawn as the upper solid line.

Effective tensile modulus [GPa]

500

400

300

200

100

0 0

0.2

0.4

0.6

0.8

1

Zirconia volume fraction [1] Figure 3. Effective tensile modulus of dense alumina-zirconia composite ceramics; Voigt bound (crosses slightly above the upper solid line calculated from the Voigt values of the effective shear and bulk moduli), approximate Voigt bound (according to the mixture rule, upper solid line), Reuss bound (results of both calculations identical, crosses and lower solid curve), upper and lower HashinShtrikman bounds (dashed curves) and values measured by the resonant frequency method for dense (porosity < 3 %) alumina-zirconia composite ceramics prepared by slip casting. ±

It is evident that in practice the difference between EV

(φ Z ) and EV (φ Z ) is negligibly

small for the alumina-zirconia system (< 0.6 %), so that the mixture rule can well be used to calculate the Voigt bound for the effective tensile modulus of alumina-zirconia composite ceramics. The Reuss bound E R = E R (φ Z ) calculated directly as a harmonic average via Equation (80) is of course identical with that calculated indirectly from the Voigt values of the shear modulus G R = G R (φ Z ) and the bulk modulus K R = K R (φ Z ) via the elasticity standard relations (30). Apart from the Voigt and Reuss bounds Fig. 3 shows the upper and lower Hashin-Shtrikman bounds (dashed curves in between the Voigt and Reuss bounds) and values measured by the resonant frequency technique for dense (porosity < 3 %) aluminazirconia composite ceramics prepared by slip casting. It is evident that in the case of the alumina-zirconia system the Hashin-Shtrikman bounds are sufficiently close together for their arithmetic average to be an excellent prediction of the effective tensile moduli, that will be precise enough for most practical purposes. The underlying reason is of course that the ratio between the tensile moduli of the end members (alumina and zirconia) is “only“ 2, a contrast that must be considered as small from the viewpoint of micromechanics. The excellent agreement of the HS prediction with the experimentally measured values fully confirms the adequacy of the approach chosen and gives a feeling for the magnitude of errors (statistical scatter) that can be expected in practice. Similar conclusions hold in principle for the effective shear modulus and the effective bulk modulus. In order to fully exploit the usefulness of the

Effective Elastic Moduli of Alumina, Zirconia and Alumina-Zirconia ...

79

HS prediction in practical applications the HS averages in dependence of the zirconia volume fraction φ Z have been fitted by second-order polynomials [Pabst et al. 2005]. This results in the following handy formulae (with all terms in GPa):

E = 400 − 245.9 φ Z + 56.3 φ Z2 ,

(126)

G = 163 − 110.4 φ Z + 28.3 φ Z2 ,

(127)

K = 247 − 73.6 φ Z + 10.8 φ Z2 .

(128)

For the calculation of the effective Poisson ratio the mixture rule is usually recommended. In the case of alumina-zirconia composite ceramics it takes the form (dimensionless)

ν = 0.23 + 0.08 φ Z .

(129)

The indirect calculation via the elasticity standard relation (33) using the HS averages of E and G , Equations (126) and (127), results in effective Poisson ratios which can be expressed as

ν = 0.23 + 0.0669 φ Z + 0.0111 φ Z2 .

(130)

The difference between the two formulae is small (< 1.4 %) and affects only the third decimal of the effective Poisson ratio: at zirconia volume fractions of approx. 50 % the linear formula yields values which are higher by approx. 0.004. With respect to the difficulty of experimental measurements of ν this difference should be negligible. All the above formulae allow the direct and easy calculation of the effective elastic moduli and Poisson ratios of dense alumina-zirconia composites for a given zirconia volume fraction φ Z . Since usually the composition of alumina-zirconia composites (in the sequel abbreviated as “AZ composites“) is given in weight percent (weight fractions wZ ) the volume fractions

φ Z have usually to be calculated via the (theoretical) density. In the

extreme case of AZ composites with only tetragonal zirconia (density 6.1 g/cm3) the effective (theoretical) density is (with all terms in g/cm3)

ρ 0 (wZ ) = 4.0 + 1.41 wZ + 0.3 wZ2 + 0.4 wZ3 ,

(131)

while in the other extreme case of AZ composites with only monoclinic zirconia (density 5.6 g/cm3) the effective density is

ρ 0 (wZ ) = 4.0 + 1.10 wZ + 0.3 wZ2 + 0.2 wZ3 ,

(132)

80

Willi Pabst and Eva Gregorová

cf. [Pabst et al. 2005]. These handy formulae can be used to calculate the theoretical density of classical AZ microcomposites when the zirconia weight fraction is given. We emphasize, however, that they may be inadequate to predict the theoretical density of AZ nanocomposites (prepared e.g. via sol-gel techniques). Trivially, the corresponding dependences on the zirconia volume fraction φ Z are given by simple linear mixture rules,

ρ 0 (φ Z ) = 4.0 + 2.1 φ Z ,

(133)

ρ 0 (φ Z ) = 4.0 + 1.6 φ Z

(134)

for the extreme cases of AZ composites with only t-ZrO2 and only m-ZrO2, respectively. With the help of these formulae it can easily be verified that under the assumption of 100 % prevalence of t-ZrO2 the zirconia volume fraction φ Z is 0.105 for zirconia-toughened alumina with a zirconia weight fraction wZ of 0.15 (in the sequel this special composition with 15 wt.% of zirconia will be abbreviated “ZTA“) and 0.726 for alumina-containing tetragonal zirconia polycrystals with a zirconia weight fraction wZ of 0.80 (in the sequel this special composition with 80 wt.% of zirconia will be abbreviated “ATZ“). Both ZTA and ATZ are typical and commercially successful types of AZ composites. When the aforementioned formulae for the effective elastic moduli are applied to these ZTA and ATZ composites one obtains E = 375 GPa, G = 151 GPa, K = 239 GPa for the ZTA composite and E = 251 GPa, G = 97 GPa, K = 199 GPa for the ATZ composite.

4.3

Effective Elastic Moduli of Porous Ceramics in the Alumina-Zirconia System

In the following let us reserve the term “effective“ for the macroscopic properties of porous materials, since, based on the findings of the preceding sections, it is justified to assume that the elastic moduli of the pore-free (i.e. densely sintered) polycrystalline ceramics (alumina, zirconia and alumina-zirconia composite ceramics) can be reliably predicted. Experimental evidence confirms that the predictions are reliable and in good agreement with reality. In particular, the Hashin-Shtrikman average yields a prediction for the elastic moduli of dense AZ composites which should be sufficiently precise for most practical purposes. This success of the prediction based on the HS average must be ascribed to the fact that the contrast between the phase moduli of the end members (alumina and zirconia) is small from the viewpoint of micromechanics, as mentioned above. The application of the simple HS bounds in the form given by Equations (88) and (89) is of course restricted to macroscopically isotropic two-phase composites. Although generalizations of these bounds to anisotropic multiphase (number of phases > 2) composites exist [Torquato 2002, Walpole 1966a, 1966b, 1969], in many cases it is not necessary to complicate the calculations with such a generalization. Doubtlessly, porous AZ composites would strictly have to be modeled by a three-phase model, particularly when interstitial porosity is considered, i.e. porosity that results from the

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81

packing of particles during the shaping process [German 1989]. Naturally this interstitial porosity, remnants of which may remain after partial sintering [De Jonghe & Rahaman 2003, German 1996, Rahaman 1995, Rice 1998, 2003], is characterized by a typical pore size smaller than (but of the same order of magnitude as) the typical grain size. On the other hand, the usual way to produce ceramics with controlled porosity is not via partial sintering [Hardy & Green 1995, Lam et al. 1994, Nanjangud et al. 1995] but via introducing artificial porosity characterized by a pore size significantly larger than the grain size. This can principally be done in at least four ways [Rice 1998, 2003]: first by replication of polymeric, especially polyurethane or carbon-based, foams (which leaves after drying, burnout and firing highly porous open-cell ceramic foams or networks with hollow struts), second by blowing agents ensuring gas development during due to heating and / or chemical reactions (e.g. calcium sulfate or gypsum, leaving CaO, which is a harmless or even benign additive in many zirconia ceramics), third by ceramic balloons (hollow spheres), which are commercially available for alumina and zirconia (primarily used for thermal insulation purposes) and fourth by introducing pore-forming agents which burn out during firing (e.g. carbon / graphite and various polymers of organic and bio-organic origin, like saw dust, crushed nut shells and starch). The latter three result usually in porous materials with essentially closed pores (which of course excludes their application for catalysis, filtering and implants designed for bone tissue ingrowth), although possibly interconnected by narrow throats. In contrast to open-cell ceramic foams [Gibson & Ashby 1997] materials with a closed-pore microstructure are more corrosion-resistant, combine desirable mechanical properties with relatively light weight and can be useful e.g. for thermal or electrical insulation purposes. However different the processing details may be, all four techniques of porosity control just mentioned have one feature in common: the porosity introduced in this artificial way is characterized by a pore size which is generally at least 1−2 orders of magnitude larger (typically tens of micrometers) than the grain size (typically around 1 micrometer). Pores of such a size, embedded in a finegrained matrix of host material, will not contribute to overall shrinkage during firing (even at temperatures and in time schedules where the matrix becomes densely sintered, i.e. attains almost theoretical density) and will remain as void inclusions in the final ceramic. For such materials it is possible to consider the matrix (dense host material, e.g. alumina, zirconia or AZ composites) as a homogenized medium and to invoke the Voigt bounds and the Hashin-Shtrikman upper bounds in their simple form for two-phase materials. Of course, in the case of porous materials (which can be considered as materials with an extremely, substantially infinitely, high contrast in the phase properties from the viewpoint of micromechanics) both the Reuss bound and the lower HS bound of the effective elastic moduli degenerate to zero (in mathematical terms “almost everywhere“, i.e. everwhere except at the singular point φ = 0 where they are equal to the matrix values M (0 ) = M 0 ), cf. Fig. 4 for the relative tensile modulus. Note that the HS upper bound has been drawn as one single curve in Fig. 4, corresponding to Equation (91), which is identical to the Mori-Tanaka prediction and the Kuster-Toksöz relation (112). Indeed, a comparison of the values obtained by this simple +

+

+

expression with the values obtained when E HS is calculated from G HS and K HS via Equation (90) reveals that the difference is smaller than 0.1 %, i.e. absolutely negligible. Also for the shear modulus the difference is small enough (< 2.6 %) to be negligible and thus Equation (112) may certainly be used to replace the HS upper bound for alumina, zirconia

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and AZ composites in practice, cf. Fig. 5. This is not the case for the bulk modulus, however, cf. Fig. 6 (error up to 14.3 %). 1

Relative tensile modulus [1]

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Porosity [1] Figure 4. Voigt bound (solid straight line) and Hashin-Shtrikman upper bound (solid curve) for the relative tensile modulus of porous ceramics (e.g. alumina, zirconia or alumina-zirconia composites); the Reuss bound and the lower Hashin-Shtrikman bound (dashed lines along the axes) degenerate to zero for all porosities φ > 0. 1

Relative shear modulus [1]

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Porosity [1] Figure 5. HS upper bounds for the relative shear modulus of alumina (lower curve) and zirconia (upper curve); ZTA and ATZ composite ceramics are in between (not shown).

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1

Relative bulk modulus [1]

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Porosity [1] Figure 6. HS upper bounds for the relative bulk modulus of alumina (upper solid curve), ZTA (upper dashed curve), ATZ (lower dashed curve) and zirconia (lower solid curve).

The HS upper bound for the relative bulk modulus of alumina (with ν 0 = 0.23) and zirconia (with ν 0 = 0.31) can be approximately expressed by the second-order polynomials + (ν 0 = 0.23) ≈ 1 − 1.78 φ + 0.78 φ 2 , K HS

(135 a)

+ (ν 0 = 0.31) ≈ 1 − 2.00 φ + 1.00 φ 2 , K HS

(135 b)

respectively, while for the tensile and shear modulus the corresponding expressions are much closer to (i.e. almost identical with) Equation (91). Now the question arises to what degree it is possible to predict the porosity dependence of the elastic moduli, in particular of those elastic moduli (e.g. K ) which have not been or cannot be easily measured for the material in question. This question, although of an immense practical interest, cannot be answered in a unique way, because in principle microstructures can be imagined that ensure finite moduli up to porosities close to 100 %, e.g. in the open-cell ceramic foams mentioned earlier [Gibson & Ashby 1997], while on the other hand closedpore structures formed with blowing agents can exhibit the possibility of overlapping (i.e. consist of penetrable pores, which can easily form large connected pore spaces), can loose integrity (corresponding to zero elastic moduli) at very small porosities. With regard to this fact the experimental measurement of the porosity dependence of the elastic moduli is in the first instance an indispensable tool to gain experience with a certain type of material microstructures. This experience may then, with due caution, be used to make certain statements about or give rough estimates concerning the porosity dependence of the same or other elastic moduli for materials with the same or a similar type of microstructure.

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Relative tensile modulus [1]

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.2

0.4

0.6

0.8

1

Porosity [1] Figure 7. Relative tensile modulus of porous ceramics (measured values, predictions and master fit); HS upper bound (thin solid curve), predictions for spherical pores (thin dotted: special case of the Spriggs relation Eq. (110), thin dashed: Coble-Kingery Eq. (117), thick solid: modified exponential relation Eq. (114)), experimentally measured values (squares: alumina, diamonds: ZTA, triangles: ATZ, circles: zirconia, empty: potato starch as a pore-forming agent, full: corn starch as a pore-forming agent) and master curve (thick dotted curve obtained by fitting with the Pabst-Gregorová relation, Eq. (121), critical porosity 0.729).

Fig. 7 shows the porosity dependence of the tensile moduli experimentally measured (by the resonant frequency technique) for alumina, zirconia and alumina-zirconia composites (ZTA and ATZ) prepared from submicron oxide powders using starch as a pore-forming agent (corn starch with a median size of approx. 14 micrometers and potato starch with a median size of approx. 50 micrometers). The processing method is starch consolidation casting, cf. the details given and references cited in [Pabst et al. 2004b, 2005]. Due to its organic character, the pore-forming agent (starch) is burnt out and vanishes completely during heating up to the sintering temperature of the ceramics. The microstructure is essentially of matrix-inclusion type: there are large pore bodies, interconnected by small and narrow pore throats. The scatter of the measured values is relatively large, obviously mainly due to errors in the exact determination of porosity (because the resonant frequency technique used for measuring is rather precise, cf. Fig. 3, and there can be no doubt that also the reference elastic moduli, used to calculate the relative moduli, are reliably known). Nevertheless, within experimental scatter, there is a common trend for all six materials, as expected. All values shown in Fig. 7 obey the HS upper bound. Evidently, the special case of the Spriggs relation (110) is useless for prediction, because it violates even the HS bounds. The Coble-Kingery relation is evidently better: it satisfies at least the HS bounds and is clearly an improvement over the latter when a rough estimate is needed. However, when all data are fitted with one master curve using the Archie relation (116) the resulting intrinsic tensile modulus obtained from fitting is [ E ] = 2.51, cf. Table 9. Similarly, when the Phani-

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85

Niyogi relation (119) is used for fitting the intrinsic tensile modulus is [E ] = 2.50 (the exponent [ E ] φ C being 2.45 with the critical volume fraction being close to unity, i.e.

φC =

0.979), cf. Table 9. This indicates, that when the exponent 2 in the Coble-Kingery relation (117) is allowed to vary freely (as an adjustable fit parameter), the fitted curve corresponds to a value significantly larger than 2, and is clear evidence of the fact that the Coble-Kingery prediction is not satisfactory in this case. Table 9. Fit parameters determined for the master curve of the porosity dependence of porous ceramics (alumina, zirconia and AZ composites) with matrix-inclusion type microstructure prepared by starch consolidation casting with starch as a pore-forming agent. Fit model

[E ]

φC

Modified Exponential

 − [E] φ   E r = exp   1−φ 

2.09

−

Mooney-type with [E ] = 2

 −2φ E r = exp   1 − φ φC

  

−

1.006

Mooney-type

 − [E] φ E r = exp   1 − φ φC

  

2.48

1.737

Archie

E r = (1 − φ )

2.51

−

2.50

0.979

−

0.729

Phani-Niyogi Pabst-Gregorová

[E]

E r = (1 − φ φ C )

[ E ] φC

E r = (1 − φ ) (1 − φ φ C )

By far the best prediction is achieved with the modified exponential relation (114). We emphasize that this is, similar to the HS upper bound and the Coble-Kingery prediction, an unbiased a priori prediction without the need for fitting or input parameters of any kind. It is solely based on the assumption that the pores are spherical or isometric. In this case, if the intrinsic tensile modulus is allowed to vary, i.e. if we consider [E ] as an adjustable parameter to be determined by fitting according to relation (113), the result is [ E ] = 2.09, which is a value very close to 2. Similarly, when a critical porosity is introduced as a fit parameter, i.e. the Mooney-type relation (115) is used with [E ] = 2, this critical porosity turns out to be

φ C = 1.006, i.e. φ C ≈ 1. Both findings confirm the quality of the prediction

via the modified exponential relation (114). Note, however, that the two-parameter fit via the general Mooney-type relation (115) can lead to physically unreasonable results (typically φ C > 1), cf. Table 9 and [Pabst et al. 2004b, 2005]. In concluding this section we address the most intricate question: assuming the porosity dependence of one effective elastic modulus (say E ) to be known and analyzed (e.g. by fitting), can an approximate estimate be given for the porosity dependence of another

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effective elastic modulus (say K ) which has not been measured ? To be concrete, our task is the following: Given the porosity dependence of the tensile modulus as

−2φ   E r = exp   1−φ 

(114)

φ   E r = (1 − φ ) 1 − ,  0.729 

(136)

or, slightly more precisely, as

what is the porosity dependence of the effective bulk modulus ? A first guideline to the tentative answer is the fact that for porous materials with spherical pores and a matrix Poisson ratio of ν 0 = 0.2 the intrinsic elastic modulus should be equal to 2 for all moduli, i.e. also

[ K ] = 2, cf. Equations (102) and (105). Therefore, for such materials also the porosity dependence of the bulk modulus should be given by the modified exponential relation

−2φ   . K r (ν 0 = 0.2) = exp   1−φ 

(137)

Comparison with the approximate expressions (135 a) and (135 b) shows, that this relation does not violate the HS upper bound even for zirconia, as can be seen from the initial tangent slope of this relation (which is equal to −2 at φ = 0), although of course relation (137) must be expected to be a better approximation for alumina (with ν 0 = 0.23) than for zirconia (with ν 0 = 0.31). However, as mentioned several times before, the intrinsic bulk modulus [K ] is a much more sensitive function of the matrix Poisson ratio

ν 0 than the

intrinsic tensile modulus [E ] . For alumina and zirconia the intrinsic bulk modulus [K ] should be 2.139 and 2.724, respectively, cf. Table 1. Therefore it might be justified to replace the modified exponential relation (137) by

 − 2.139 φ   , K r (ν 0 = 0.23) = exp   1−φ 

(138 a)

 − 2.724 φ   K r (ν 0 = 0.31) = exp   1−φ 

(138 b)

in the case of alumina and zirconia, respectively. But the problem may be tackled from another side as well: if a reliable estimate can be given for the porosity dependence of the effective Poisson ratio ν , then it should be possible to calculate the effective bulk modulus K from E and ν via the elasticity standard relation (32). In order to estimate the porosity

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dependence of the effective Poisson ratio of alumina and zirconia we invoke the asymptotic values ν for porosities close to 100 %, given by Equations (83) and (84). According to the differential approach this value is invariably 0.2 (independent of the matrix Poisson ratio), ∗

while according to the Mori-Tanaka approach, Equation (84), the asymptotic value of ν is 0.21 for alumina and 0.24 for zirconia. This leads to the following estimates for the porosity dependence of the effective Poisson ratios (with ξ = 0.03 and 0.02 for alumina and ξ = 0.11 ∗

and 0.07 for zirconia according to the differential and the Mori-Tanaka approach, respectively),

ν =ν0 −ξ φ .

(139)

The various predictions for the porosity dependence of the relative bulk modulus K of alumina and zirconia are compared in Fig. 8. Alumina-zirconia composite ceramics of any composition will show intermediate behavior. 1

Relative bulk modulus [1]

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Porosity [1] Figure 8. Relative bulk modulus of porous alumina (red) and zirconia (blue); modified exponential relation for spherical pores and matrix Poisson ratio 0.2 (green solid curve), predictions for alumina with intrinsic bulk modulus 2.139 (red solid curve) and for zirconia with intrinsic bulk modulus 2.724 (blue solid curve), dashed and dotted curves are relative bulk moduli calculated via elasticity standard relations using the differential and Mori-Tanaka prediction, respectively.

5

Outlook

In this chapter we have treated the effective elastic moduli of alumina, zirconia and aluminazirconia composite ceramics. We have invoked the theory of linear elasticity [Gurtin 1972], which is the appropriate framework for the macroscopic mechanical behavior of alumina and

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zirconia at room temperature and above (up to a few hundred °C, say). Polycrystalline ceramics in general can be considered as completely brittle at these temperatures, because dislocation motion is impeded by the absence of appropriate slip systems (in contrast to single crystals) and the fact that grain boundary sliding, the principal mechanism of plastic deformation and creep in polycrystalline ceramics, leads to crack nucleation and opening [Davidge 1979, Green 1998, Munz & Fett 1999, Wachtman 1996]. It is a common experience that at these temperatures, when a critical stress value (the “strength”) is exceeded, polycrystalline ceramics respond by brittle fracture long before inelastic modes of behavior can operate and even before elasticity can become nonlinear. This is also the deeper reason why linear fracture mechanics [Irwin 1958] can be readily applied to these materials, why the mechanical strength shows − in dependence of accidental microstructural defects − considerable scatter (more than e.g. the strength values of metals or polymers) and why Weibull statistics (with its “weakest link” assumption) [Weibull 1939, 1947, 1951] is ideally suited to describe this scatter. However, at higher temperatures (still well below the sintering temperature) both alumina and zirconia exhibit plastic deformation and creep, and especially zirconia is well known for its so-called superplastic behavior [Nieh et al. 1997]. Based on rational thermomechanics [Truesdell & Toupin 1960, Truesdell 1984, Truesdell & Noll 2003] it is possible today to extend the theory of linear elasticity presented in this chapter in a rigorous and well-defined way to linear thermoelasticity [Carlson 1972, Šilhavý 1997] and to use this theory as the basis for a detailed investigation of thermo-mechanical properties (i.e. apart from the elastic constants thermal conductivity, specific heats and coefficient of thermal expansion). For alumina, zirconia and alumina-zirconia composite ceramics this working programme has been launched by the authors recently. Similarly, based on [Eringen & Maugin 1989, 1990], an extension of this research programme to electrical, magnetic and electromagnetic (including optical) properties is thinkable and can be performed in the future. It goes without saying that plastic deformation and high-temperature creep of alumina and zirconia are major challenges to mathematical modeling, because the appropriate constitutive equations are in general nonlinear, but in principle the basic apparatus for solving this task in a rational way is available in the literature on rational thermomechanics and theory of materials [Haupt 2000, Krawietz 1986]. Rational theory, originating from the work of Gibbs [Gibbs 1961], can also tackle in a rigorous way with phase boundaries and phase transformations [Grinfeld 1991, Šilhavý 1997], the latter of which would be of primary importance in the case of zirconia. This is what can be modeled from the viewpoint of rational theory without invoking mixture theories. Note that chemical interactions like stress-corrosion cracking and subcritical grain growth of alumina (e.g. in the presence of water vapor), closely related to “fatigue”, as well as hydrothermal ageing (lowtemperature surface degradation) of zirconia (also in the presence of water vapor, e.g. at elevated temperatures in the range 150-250 °C), including the accompanying diffusion phenomena (e.g. yttria diffusion inside and between the zirconia grains) and surface chemical reactions (e.g. with oxygen bonds linking to OH groups from the ambient atmosphere) can principally only be attacked by mixture theories. So far, rational mixture theories have not been applied to these complex phenomena; they are commonly modeled by more traditional concepts (equilibrium thermodynamics and kinetics). We would like to emphasize at this point that strength, fracture toughness, hardness and wear behavior, although important mechanical parameters for materials selection and engineering design, are not well-defined material properties, because they are intimately

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related to (and physically meaningful only for) a certain (and often very special) test geometry and procedure and (what is worse) specimen and probe geometry. The same holds true for several other quantities, e.g. thermal shock parameters. All these parameters arise in the context of very specific model situations and should be used only for the purpose of relative comparison (i.e. in order to compare the behavior of other materials in the same situation as the material tested). For a refreshing account of brittle fracture, largely untouched by engineering tradition but inspired by percolation theory, the reader may consult [Sahimi 2003b]. Of course, during the second half of the 20th century the engineering literature on these properties for alumina, zirconia and alumina-zirconia composite ceramics has grown immensely. Elastic properties, without doubt, are the perfect playground for micromechanics. Together with viscosity they belong to a class of properties which arises in the context of “class B problems” in the sense of Torquato [Torquato 2002] and are in the general case characterized by fourth-order tensors. Hooke’s law is the constitutive equation of linearly elastic solids in the same way as Newton’s law is the constitutive equation of linearly viscous fluids. Note that in general, i.e. in the case of anisotropic fluids, also viscosity is a fourthorder tensor, similar to the stiffness tensor used in this chapter, cf. e.g. [Šilhavý 1997, Truesdell 1984] (in the case of isotropic fluids we obtain again two constants, e.g. the shear viscosity and the bulk viscosity, in complete analogy with the two elastic constants needed for isotropic solids, cf. e.g. [Billington & Tate 1981]). Other properties (related to Torquato’s “class A problems”) are generally described by linear constitutive equations involving second-order tensors, e.g. thermal conductivity (Fourier’s law), electrical conductivity (Ohm’s law), dielectric constant and magnetic permeability, which in the case of isotropic materials reduce to only one constant. The coefficient of thermal expansion is a similar property (symmetric second-order tensor, i.e. three constants in general and scalar, i.e. one constant, for isotropic materials). Of course, the micromechanical relations (bounds, approximations or fit models) presented in this chapter for the effective elastic properties are by no means restricted to the alumina-zirconia system but can be applied to many types of ceramics and ceramic composites. On the other hand they cannot be expected to be automatically applicable to matrix-inclusion type composites in cases where the matrix consists of nonlinearly elastic materials (polymers), viscoelastic materials (glasses or porcelain at high temperature) or elastoplastic materials (metals). In particular, they cannot be a priori expected to be justified for materials of biological origin, although their application to many of these materials, e.g. bone, might be seductive and dictated by practical needs. With respect to the inherent anisotropy and the hierarchical microstructure of these materials [Ontañón et al. 2000], however, any mathematical modeling or description of their composition-structure-property relationships has to be performed with due caution. All micromechanical relations presented in this chapter are based on the volume fractions of the constituent phases (i.e. one-point correlation functions and a special part of the twopoint correlation functions) as the only input information. If complete two-point and threepoint information would be available (e.g. from stereological analysis or 3D tomography), the Hashin-Shtrikman bounds (i.e. the two-point bounds) could be improved in principle, which leads to the three-point bounds due to [Beran & Molyneux 1966] and [Milton & Phan-Thien 1982], cf. also [Jeulin 2001, 2002, Markov 2000, Milton 2002, Sobczyk & Kirkner 2001, Torquato 2000, 2002]. Of course, in the case of dense alumina-zirconia composites it is quite

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evident, that (due to the relatively small contrast in the phase properties) already the HashinShtrikman bounds are precise enough to permit property prediction to a sufficient degree. In the case of porous materials, however, the three-point bounds could in principle improve the upper bound. Throughout this chapter macroscopic isotropy was assumed. In the case of anisometric constituent phases (inclusions), e.g. alumina platelets (tabular alumina) or fibers (e.g. the socalled Saffil fibers), this assumption implies of course statistical random orientation. In the case of preferred orientation of anisometric inclusions (e.g. flow-induced fiber or platelet alignment) the resulting composites are usually anisotropic (e.g. transversely isotropic). The generalization of the Hashin-Sthrikman bounds to anisotropic materials is due to Willis [Willis 1977], and exact series expansions for anisotropic composites have been given by Torquato [Torquato 1997, 1998]. Anisometric inclusions and voids play a major role in partially stabilized zirconia (PSZ) ceramics, viz. first as (oriented) lenticular lamellae of tZrO2 in a matrix of c-ZrO2 and second as microcracks around those lamellae which have transformed to m-ZrO2. Both can be modeled as oblate spheroids or flat disks, the first of finite thickness, the latter infinitely thin (in this exceptional limiting case the volume fraction has to be replaced by the so-called crack-density parameter). Explicit expressions for spheroidal (disk-like and needle-like) inclusions and voids are given by [Markov 2000, Milton 2002, Torquato 2002]. The authors’ current work in progress concerns the application of analogous micromechanical relations (bounds, approximations and fit models) to the thermal conductivity and the thermal expansion coefficient of alumina, zirconia and alumina-zirconia composites, both dense and porous and the subsequent description of experimentally determined microstructure-property relationships, i.e. effective properties and their dependence on the zirconia volume fraction and porosity, respectively. In complete analogy to the elasticity context of this chapter, it is possible to derive a Coble-Kingery-type relation for the effective thermal conductivity [Pabst 2005] of porous ceramics and, based on this, a Pabst-Gregorová-type relation allowing for a critical porosity (percolation threshold) [Pabst & Gregorová 2005]. Based on the effective bulk modulus predicted in this chapter it will also be possible to calculate the effective thermal expansion coefficient of alumina-zirconia composites via the Levin formula [Levin 1967], a so-called cross-property relation. In a similar spirit, cross-property bounds have been constructed between bulk or shear modulus and thermal conductivity [Berryman & Milton 1988, Gibiansky & Torquato 1996, Sahimi 2003a, Torquato 1992, 2002]. Note that the micromechanical approach chosen throughout this chapter can be expected to be realistic only for classical microcomposites with particle sizes not much smaller than 100 nm, i.e. in cases where the phase boundaries (interfaces) can be considered and modeled as singular discontinuities [Grinfeld 1991, Šilhavý 1997]. In other words, the phase boundaries are considered as sharp interfaces between two bulk phases, without structure and properties and without contribution to the volume fraction. In the framework of rational mixture theories this corresponds to the case of so-called multiphase mixture theories, cf. e.g. [Dobran 1991, Passman et al. 1984]. Such a conception will certainly not be realistic for nanocrystalline materials and nanocomposites, where the volume fraction of atoms in the grain boundaries can be as high as 30 vol.% in the case of 10 nm grains, compared to only 3 vol.% for 100 nm grains, cf. e.g. [Ajayan 2003, Gleiter 1989, Niihara 1991, Suryanarayana & Koch 1999, Winterer 2002]. We emphasize, however, that the apparent grain size dependence

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of elastic properties which can develop as a consequence of the growing importance of phase boundaries has nothing to do with the ubiquitous and well-known influence of grain size on strength and other mechanical properties. All available experience indicates that the influence of grain size on the elastic properties is negligible unless the grain size becomes extremely small (< 10 nm, say) [Krstic et al. 1993, Poole & Owens 2003]. From an atomistic point of view, phase boundaries (e.g. grain boundaries or surfaces in general) are always characterized by a higher concentration of defects (e.g. due to curvature) or a lower degree of crystallinity in comparison to the adjacent bulk phase. Obviously, in contrast to the micromechanical conception of a sharp interface, in reality there must be a gradient in structure (degree of crystallinity) and properties which is more adequately modeled by considering the interfaces as diffuse. The effective thickness of such a diffuse interface can be expected to be of the order of 1 nm up to a few nm. In nanocystalline materials of two constituents the interface thus plays the role of a third phase (i.e. with a finite volume fraction) but with diffuse structure and undefined properties, which are usually not accessible to measurement. Although it might be seductive for such cases to invoke the socalled mesophase concept [Theocaris 1987] which treats the phase boundary as a separate phase, this and similar models occurring in the micromechanical literature are complicated without really solving anything, since the properties of the third (meso-) phase are a priori unknown. This situation is somewhat similar to the situation in rational mixture theories [Samohýl 1999, 1997, Truesdell 1984, Truesdell & Toupin 1960], where the partial quantities are introduced as primitive concepts and only the mixture quantities but not the partial quantities (which are of course different from the constituent quantities “before mixing“ or “outside the mixture“) are accessible to measurement [Samohýl & Šilhavý 1990], with one remarkable exception: the chemical potential (i.e. the specific Gibbs free energy). Note that in the case of solids the chemical potential is not a scalar quantity but a second-order tensor (Eshelby tensor), cf. [Samohýl & Pabst 1997, 2004]. Glass scientists and engineers are very familiar with this situation as well: usually the effective (e.g. elastic) properties of glasses cannot sensu strictu be predicted a priori from the constituent properties but have in principle always to be determined by measurement. The underlying deeper reason is the absence of microstructure or, more precisely, the fact that microstructure is lost in the smeared-out phase boundaries, which represent in a certain sense a new phase and make up the whole volume of the glass body (amorphous material). From this point of view nanocrystalline materials and nanocomposites can be considered as intermediate between classical (micro-) composites (which can be called “immiscible mixtures“) and glasses (which can be called “miscible mixtures“ or “solid solutions“). In the first case (heterogeneous materials), volume fractions of the constituent phases are the appropriate concentration measures and the effective properties can be predicted when the properties of the constituent bulk phases (i.e. the components “before mixing“ or “outside the mixture“) and the microstructural details are known. In the latter case (homogeneous materials), weight (mass) or molar fractions are the appropriate concentration measures (since volume fractions are undefined), microstructural details are irrelevant (homogeneous materials do not have microstructure in this sense) and the effective properties cannot be predicted from the properties of the components. This statement holds true for elastic properties as well as other well-defined properties (e.g. thermal conductivity and thermal expansion coefficient). In glass science and technology a pragmatic empirical solution is usually preferred: from a large amount of data, experimentally measured for glasses of

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various composition (mixture properties), one can deduce partial properties, which can (a posteriori) be used for tentatively “predicting“ (interpolating or extrapolating or at least approximately estimating) the effective properties of yet unmeasured glass compositions with the help of empirical relations, cf. the rich literature on this subject cited in [Volf 1988]. A first step to rationalize the modeling of nanomaterials could be the elaboration of a theoretically sound model of phase boundaries and phase transitions based on modern continuum thermodynamics [Grinfeld 1991, Šilhavý 1997]. A second step towards the prediction of effective properties of nanomaterials would then be the elaboration of a rational mixture theory for solids in analogy to the so-called “chemical thermodynamics“ for fluids, which is essentially based on Gibbs‘ key paper [Gibbs 1961], cf e.g. [Guggenheim 1957, Samohýl 1982, 1999]. Partial quantities will then be available for the constituents of solid mixtures in a similar way as the molar partial quantities available for fluid mixtures (i.e. gas mixtures and liquid solutions). The chemical potential tensor [Samohýl & Pabst 1997, 2004] might play a key role in this step.

Acknowledgement: This work was part of the frame research programme “Preparation and Research of Functional Materials and Material Technologies using Micro- and Nanoscopic Methods“, supported by the Ministry of Education, Youth and Sports of the Czech Republic (Grant No. MSM 6046137302). It is based on fundamental research carried out within the project “Mechanics and Thermomechanics of Disperse Systems, Porous Materials and Composites“, supported by the Czech Science Foundation (Grant No. 106/00/D086). The authors would like to thank Dr. M. Černý (Prague) for Young’s modulus measurements and Prof. D. Jeulin (Fontainebleau), Prof. I. Samohýl (Prague), Prof. S. Torquato (Princeton) and Prof. R. W. Zimmerman (London) for valuable hints and their encouraging interest in our work.

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Sobczyk, K.; Kirkner, D. J. Stochastic Modeling of Microstructures; Birkhäuser: Basel, Switzerland, 2001, pp 149-245. Soltesz, U.; Richter, H. In Metal and Ceramic Biomaterials, Vol. II: Strength and Surface; Ducheyne, P.; Hastings, G. W.; Eds.; CRC Press: Boca Raton, FL, 1984; pp 23-61. Spriggs, R. M. J. Am. Ceram. Soc. 1961, 44, 628-629. Stauffer, D.; Aharony, A. Introduction to Percolation Theory; Routledge / Taylor and Francis: London, UK, 1985 (revised second edition); pp 1-181. Sumner, D. R.; Turner, T. M.; Igloria, R.; Urban, R. M.; Galante, J. O. J. Biomech. 1998, 31, 909-917. Suryanarayana, C.; Koch, C. C. In Non-Equilibrium Processing of Materials; Suryanarayana, C.; Ed.; Pergamon Press: Amsterdam, The Netherlands, 1999; pp. 311-344. Takebayashi, H. In Handbook of Advanced Ceramics Vol. I: Materials Science; Somiya, S.; Aldinger, F.; Claussen, N.; Spriggs, R. M.; Uchino, K.; Koumoto, K.; Kaneno, M. Elsevier Academic Press: Amsterdam, The Netherlands, 2003; pp 313-331. Theocaris, P. S. The Mesophase Concept in Composites; Akademie-Verlag: Berlin, Germany 1987. Timoshenko, S.; Goodier J. N. Theory of Elasticity; McGraw Hill: New York, NY, 1951 (second edition); pp 359-362. Torquato, S. In Macroscopic Behavior of Heterogeneous Materials from the Microstructure; Torquato, S.; Krajcinovic; Eds.; ASME: New York, NY, 1992; Vol. 147; pp 53-65. Torquato, S. J. Mech. Phys. Solids 1997, 45, 1421-1448. Torquato, S. J. Mech. Phys. Solids 1998, 46, 1411-1440. Torquato, S. Int. J. Solids Struct. 2000, 37, 411-422. Torquato, S. Random Heterogeneous Materials – Microstructure and Macroscopic Properties; Springer: New York, NY, 2002; pp 1-701. Truesdell, C. Rational Themodynamics; Springer: New York, NY, 1984 (second edition); pp 219-395. Truesdell, C.; Noll, W. The Non-Linear Field Theories of Mechanics; Springer: New York, NY, 2003 (third edition); pp 1-426. Truesdell, C.; Toupin, R. A. In Encyclopedia of Physics; Flügge, S.; Ed.; Springer-Verlag: Berlin, 1960; Vol. III/1, pp 226-793. Tsuru, T. In Handbook of Advanced Ceramics Vol. I: Materials Science; Somiya, S.; Aldinger, F.; Claussen, N.; Spriggs, R. M.; Uchino, K.; Koumoto, K.; Kaneno, M. Elsevier Academic Press: Amsterdam, The Netherlands, 2003; pp 291-312. Voigt, W. Wiedemann’s Ann. Physik Leipzig 1889, 38, 573-587. Voigt, W. Lehrbuch der Kristallphysik; Teubner: Leipzig, 1910 (reprint 1928); pp 954-964. Volf, M. B. Mathematical Approach to Glass; Elsevier: Amsterdam, 1988; pp 87-103, 266275. Wachtman, J. B. Mechanical Properties of Ceramics; Wiley: New York, NY, 1996; pp 23-35. Wachtman, J. B.; Tefft W. E.; Lam D. G.; Stinchfield, R. P. J. Res. Natl. Bureau of Standards – A. Physics and Chemistry 1960, 64A, 213-227. Wagh, A. S.; Poeppel, R. B.; Singh, J. P. J. Mater. Sci. 1991, 26, 3862-3868. Walpole, L. J. J. Mech. Phys. Solids 1966, 14, 151-162. Walpole, L. J. J. Mech. Phys. Solids 1966, 14, 289-301. Walpole, L. J. J. Mech. Phys. Solids 1969, 17, 235-251. Warren, W. E.; Kraynik, A. M. J. Appl. Mech. 1988, 55, 341-346.

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Wei, G. J. Chem. Phys. 1992, 96, 3226-3233. Weibull, W. Swed. Inst. Eng. Res. Proc. 1939, 151, 1-45. Weibull, W. J. Appl. Mech. 1947, 14, 63. Weibull, W. J. Appl. Mech. 1951, 18, 293. Weinans, H.; Sumner, D. R.; Igloria, R.; Natarajan, R. N. J. Biomech. 2000, 33, 809-817. Weng, G. J. Int. J. Eng. Sci. 1984, 22, 845-856. Willis, J. R. J. Mech. Phys. Solids 1977, 25, 185-202. Winterer, M. Nanocrystalline Ceramics – Synthesis and Structure; Springer: Berlin, 2002; pp 1-260. Yamada, T. In Handbook of Advanced Ceramics Vol. I: Materials Science; Somiya, S.; Aldinger, F.; Claussen, N.; Spriggs, R. M.; Uchino, K.; Koumoto, K.; Kaneno, M. Elsevier Academic Press: Amsterdam, The Netherlands, 2003; pp 37-58. Zallen, R. The Physics of Amorphous Solids; Wiley: New York, NY, 1983; pp 49-59, 135204. Zeng, K.; Breder, K.; Rowcliffe, D. J.; Herrstrom, C. J. Mater. Sci. 1992, 27, 3789-3792. Zimmerman, R. W. Int. J. Rock Mech. Mining Sci. 1984, 21, 339-343. Zimmerman, R. W. Mech. Mater. 1991, 12, 17-24. Zimmerman, R. W. Compressibility of Sandstones; Elsevier: Amsterdam, The Netherlands, 1991; pp 110-127. Zimmerman, R. W. Appl. Mech. Rev. 1994, 47, S38-S44.

In: Ceramics and Composite Materials: New Research ISBN: 1-59454-370-4 Editor: B.M. Caruta, pp. 101-132 © 2006 Nova Science Publishers, Inc.

Chapter 3

PROGRESS IN BIOCERAMIC MATERIALS P. N. De Aza* Instituto de Bioingeniería, Universidad Miguel Hernández de Elche, Edificio Torrevaillo. Avda. de la Universidad s/n. 03202- Elche, Alicante, Spain.

Abstract A strong interest in the use of ceramics for biomedical engineering applications developed in the late 1960’s. Used initially as alternatives to metallic materials in order to increase the biocompatibility of implants, bioceramics have become a diverse class of biomaterials presently including three basic types: relatively bioinert ceramics maintain their physical and mechanical properties in the host and form a fibrous tissue of variable thickness; surface reactive bioceramics which form a direct chemical bonds with the host; and bioresorbable ceramics that are dissolved with the time and the surrounding tissue replaces it. A review of the composition, physicochemical properties and biological behaviour of the principal types of bioceramics is given, based on the literature and some of our own data. The materials include, in addition to bioceramics, bioglasses and bio-glass-ceramics. Special attention is given to structure as the main physical parameter determining nor only the properties of the materials, but also their reaction with the surrounding tissue. In addition, biomaterials with appropriate three-dimensional geometry (appropriate porous structure) are highly recommended because are able to become osteoinductive (capable of osteogeneis), and can be effective carriers of bone cell seeds. A new way of preparing “in situ” porous bioactive ceramics with interconnected porosity from a dense bioactive ceramic is presented. To this purpose the binary system wollastonite- tricalcium phosphate is selected, taking into account the different bioactivity behavior of both phases. The present status of research and development of bioceramic is characterized as first step in the solution of complex problems at the confluence of materials science and engineering, biology and medicine.

Introduction Ceramics, glasses, and glass-ceramics include a broad range of inorganic/non-metallic materials whose chemical compositions, chemical bond types and properties vary over a very *

E-mail address: [email protected]

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wide range. Bioceramic are those engineered materials that find their applications in the field of medicine [1]. Traditionally, ceramics materials have been limited as far as their applications by their brittleness, their low mechanical fracture toughness and their low resistance to the impact. Ceramics are generally hard; in fact the measurement of hardness is calibrated against ceramic materials. Diamond is the hardest, with a harness index on Mohs of 10, and talc is the softest ceramic (Mohs hardness 1), while ceramic such as alumina (Mohs hardness 9), quartz (Mohs hardness 8), and apatite (Mohs hardness 5), are in the middle range. Other characteristics of ceramic materials are their high melting temperatures and low conductivity of electricity and heat. Nevertheless, a strong interest in the use of ceramics for biomedical engineering applications was developed in the late of sixties. New ceramics, with very improved properties were contributing to increased possibilities for use of ceramic in biomedicine and their use has extended considerably since then [2,3]. The great chemical inertia of the ceramics, their high compression strength and their aesthetic appearance, caused that these materials were begun to use in dentistry, mainly in dental crowns. Later their use extended to orthopaedic applications [4-6]. A basic reason for the high demand for “spare body parts” is the damage to organic tissues which occurs with age. Bone tissue is particularly subject to age related changes. Women are vulnerable to this to a greater degree, from 30 to 65 years of age the bone strength of women decreases by approximately 40% (men by 20%). Bone density decreases with age because the osteobleasts (cells which generate bone tissue) became less productive in new tissue formation. Decreasing density substantially lowers the strength of spongy tissue located at the end of long bones and in vertebrae. A consequence of this are numerous breaks of the hip bone stem in old people, or deformation of the vertebrae and pain in the spin column. A basic problem in the use of bioceramics is replacement of damage bone with a material which can function successfully throughout the entire lifetime of a patient. Since the average lifetime in developed countries is now more than 80 years, the need for implants begins at about 60, the desired service life of a bioceramic is 20 years or more. The requirements of life long functionality must be satisfied under conditions which are extreme for a bioceramics: corrosive body fluids at 37°C, simultaneous action of ferments and complex cell systems under the conditions of variable multiaxial and cylindrical loading. The great efficacy of specially created ceramics for biomedical applications under such conditions is one of the most important achievements of research and manufacturing in the present century.

Classification of Bioceramics Used initially as alternatives to metallic materials in order to increase the biocompatibility of implants, bioceramics can be classified from different points of views [7-10]: 1. 2. 3.

According to the type of answer of the living host According to the application to which they are destined According to the characteristics of the material

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According to the Type of Answer of the Living Host

Bioinert: Relatively bioinert ceramics maintain their physical and mechanical properties while in the host. They are those stable bioceramics that do not react appreciably when they are implanted in the body. The implant does not form a bond with bone. Alumina (α-Al2O3) is a typical example of ceramic bioinert. Other examples, as we see next, are the zirconia ceramics (ZrO2) and pirolitics carbon ceramics. Bioactive or surface reactive: The concept of bioactive material is intermediate between a bioinert material and biodegradable or resorbable material. Upon implantation in the host, surface reactive ceramics form a strong bond with adjacent tissue. The bioactive materials (ceramics, glasses and glass-ceramics) bone to living bone through a carbo-hydroxyapatite layer (CHA) biologically active, which provides the interface union with the host. This phase is chemical and structurally equivalent to the mineral phase of the bone, and the responsible of the interface union. Biodegradable or resorbable: As the name implies, the ceramic dissolves with time and are gradually replaced by the natural tissues. A very thin or non-existent interfacial thickness is the final results. This type of bioceramics would be the ideals, since only remain in the body while their function is necessary and disappear as the tissue regenerates. Their greater disadvantage is that their mechanical strength diminishes during the reabsorption process. One of the few bioceramics that fulfil partially these requirements is the tricalcium phosphate (TCP).

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According to the Application to which They Are Destined

Structural ceramics: They are bioceramics of high mechanical strength and generally bioinerts. Typical examples are alumina (α-Al2O3) and zirconia (ZrO2). Non-structural ceramics: They are generally biodegradable or bioactive bioceramics, dense or porous, with low mechanical strength, since they do not have to support great loads. Typical examples, the hydroxyapatite (HA) and the tricalcium phosphate (TCP).

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According to the Characteristics of the Material

Dense and inert bioceramics: They are materials without porosity. The bond to the bone is morphologic; by growth of the tissue in the superficial irregularities of the implants or bond through acrylic cement or by fit the implant in a defect by pressure. Typical example of this group is monocrystalline as much as polycrystalline alumina. Porous and inert bioceramic: The bond to the bone is mechanical and the fixation is biological. The growth of the bone takes place through the pores of implants. Porous polycrystalline alumina is also a typical example of the group.

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Dense or porous bioactives bioceramics: The bond to the tissue is of chemical type and the fixation bioactive. Typical example of the group is HA, bioglasses and bioactive glassceramics Dense and resorbable bioceramics: The implant is replaced slowly by the bone. Belong to this group the tricalcium phosphate (TCP) and other phosphates as well as the calcium sulphate (CaSO4½H2O = plaster of Paris).

Bioinert Ceramics Bioinert ceramics undergo little or no chemical changes when they are exposed to physiological environments. The answer of the host to these bioceramics is the formation of a very fine fibrous tissue capsule of varying thickness, several micrometers or less, that surround implant materials. The fixation of implants in the body is made through a strong mechanical interlocking, by tissue ingrowth into undulating surfaces [11]. When high strength is required, the bond is made by perforations in the implants using threads, cements, etc. When so high strength are not required can be used porous inert bioceramics, with sizes of pore between 100 and 150 µm, which guarantee the growth of the tissue towards within implants assuring its fixation [12-14].

Alumina (α-Al2O3) Alumina of high density and purity (99.5 % in weight of α-Al2O3), with an average grain size < 4 µm is probably, the bioinert ceramic material of greater biological interest. This material was developed, as alternative to used metallic alloys in load-bearing hip prosthesis and in dental implants, to display an excellent biocompatibility, good resistance to the corrosion, to form a very fine fibrous capsule, to have a low coefficient of friction and good mechanical properties as much high strength as wear resistance [15]. The main source of high-purity alumina is bauxite and it is obtained by Bayer process. After that, the material is putting under a purification process and heat treatment to obtain the phase α-Al2O3 and milling until sizes < 0,3 µm. Pieces of alumina are obtained by cold isostatic press of these high purity powder and sintering at temperatures ranging from 1600 to 1700°C depending the properties of the raw material. Nowadays also alumina is obtained by hot isostatic press (HIP) achieving grain sizes < 4 µm. The obtained material polishes with diamond until superficial roughness of ≤ 0.02 µm. A very small amount of MgO (approximately a 0.5 % in weight) is added to the alumina in order to inhibit the growth of the grain [16,17]. According to the International Standards Organization (ISO), the purity of the alumina that is used in biomedical applications has to be over 99.5 %, being the rest of the impurities (SiO2, Na2O, K2O, CaO, etc.) below 0.1 % in weight, in order to avoid the formation of a liquid phase during the sintering and consequence a large grain size increase. An increase of average grain size to ∼7 µm can decrease mechanical properties by about 20 %. Table I shows the requirements for alumina implants according to ISO 6474-81.

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Table I. Physical Characteristics of Al2O3 Implants Property % by weight of Al2O3 Density (g/cm3) Average grain size (µm) Hardness (Vickers) Compressive strength (MPa) Bending strength (MPa) Young modulus (GPa)

ISO 6474-81 ≥ 99.50 ≥ 3.90 <7 > 2000 4000 400 380

The most common application of alumina bioceramics is in the area of orthopaedics as component in hip and knee prostheses. The first clinical use of total hip prosthesis with an alumina head and alumina socket was in 1971. The long term coefficient of the pair alumina/alumina and the wearing down index decrease with time and approaches the value of a normal healthy joint [18]. The main problem with total hip system is the loosening of the acetabular component, which is caused by wear debris. For this reason alumina is used only in the head of femur since numerous clinical studies indicates that alumina/ultrahigh molecular weigh polyethylene (UHMWPE) pair reduces wear debris by a factor of 10 or greater [19]. The main limitations of the alumina/UHMWPE pair are their sensitivity to the watery means, which can cause static fatigue, and their low resistance to the fracture, that can get to be critical in zones of stress concentrations. These conditions are present in the hip implants (watery atmosphere, severe cycles of tensions, overloads, impacts, etc.)[20,21]. Nevertheless, the main problem is the relaxation of the acetabular component with emission of particles. These particles are accumulated and been able to induce very adverse tissue reactions. Other clinical applications of alumina are the used for screws, tooth-root implants, orbital wall, implants for maxillofacial reconstructions, dental implants etc. Between these last ones, the most well-known application is Tübingen implant [22-24] employee like substitute dental immediately after the extraction or in edentulous regions. The effectiveness of these implants is of 92.5 %. The cross-sectional fracture is the most frequent failure of these implants, due to relatively low resistance to the flexion, being the main cause of failure. Due to it, the use of monocrystalline alumina began to used as dental implant [25,26] with resistance to the flexion 3 times over polycrystalline alumina. However, the use of monocrystals does no eliminate the basic disadvantage of alumina bioceramics (low fracture toughness), which for monocrystalline alumina is equal to 4-5 MPa m1/2. These monocrystals are obtained generally by grown and crystallization from the melt or by a process of fusion in flame with germ by the Verneuil method [27].

Zirconia (ZrO2) Zirconia, has become a popular alternative to alumina as structural ceramics because, properly treated, it has a greater resistance to the fracture (greater fracture toughness) of any monolithic ceramic. On the other hand, zirconia is also exceptionally inert in the physiological environment and presents very good static fatigue strength. In addition, the

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zirconia/UHMWPE pair displays lower coefficient of friction than the alumina/UHMWPE pair, with the consequent diminution of particles to the physiological environment [28,29]. The oxide of pure zirconium (Zirconia = ZrO2) is obtained from sands of zircon (ZrSiO4) or baddeleyite (ZrO2) by a chemical processes via of chlorination and thermal decomposition, alkali oxide decomposition or lime fusion or by plasma decomposition [30]. Zirconia presents three different crystalline forms: monoclinic, from room temperature to ∼ 1100 ºC, tetragonal, from this temperature to ∼2372 ºC, and cubic, from this last temperature to the melt at ∼2680 ºC. The monoclinic ↔ tetragonal transformation is of martensitic type, and therefore totally reversible when warming up or to cool through the transition temperature (∼1100 ºC), with an increase of volume of the order of 3 to 5 %, which entails the cracking or fracture of the pieces. In order to avoid this transformation, and even increase the toughness of the zirconia materials, is recommended the use of total or partial stabilization of the tetragonal or cubic phase by suitable additives. Zirconia of higher toughness usually is obtained stabilizing its tetragonal phase of high temperature by doping with yttria (∼3 mol %), followed on the suitable heat treatments, considering the binary diagram ZrO2 - Y2O3 [31]. By this way the zirconia bioceramic is a totally a tetragonal material with a microstructure formed by very small grains of the order of 0.2 to 0.5 µm. The additives most used in biomedical applications are the yttrium oxide (Y2O3 = yttria) and the magnesium oxide (MgO = the magnesia). Between all the ceramic materials, zirconia are those where their mechanical properties depend more by the sintering process, because is necessary to have a balance between the density and the average grain size of the bioceramic [32]. Table II shows the physical characteristic of tetragonal zirconia stabilized with yttria (TZP = tetragonal circonia polycrystals) and magnesium oxide partially stabilized zirconia (Mg-PSZ = magnesia partially stabilized zirconia [33]. Table II.- Physical Characteristic of Two Types of Zirconia Materials Characteristic Purity % Y2O3 % MgO Density(g/cm3) Average grain size (µm) Bending strength (MPa) Compressive strength (MPa) Young Modulus (GPa) Hardness(Vickers) Fracture toughness (KIc) (MPam½)

TZP ∼ 97 % 3 mol % 6.05 0.2 – 0.4 1000 2000 150 1200 7–8

Mg-PSZ ∼ 96.5 3.4 mol % 5.72 0.42 800 1850 208 1120 ∼8

Nevertheless, recently studies of new materials of TZP, doped with yttria tested in simulated body fluids and in animals, have shown slight decreases in fracture toughness and Young Modulus [34-36]. The observed strength, after two years, is sill much higher that the strength of alumina bioceramic tested under similar conditions. Because the zirconia/zirconia

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pair display a wear rate 5000 times greater than the alumina/alumina pair, those do not have to be used in articulated surfaces. As far as the potential radioactivity of the zirconia prosthesis, although the detected activity has been small, the long term effects of the alpha radioactivity still must be evaluated. French government, very recently, has prohibited the manufacture, distribution, export and use of these prosthesis in France by decision of the July 22, 2003, because of having detected frequent breakage of femoral heads made in zirconia [37].

Composite Materials of Al2O3-ZrO2 Recently, researches have been made with materials composed of alumina reinforced with zirconia (until 15 % in volume of zirconia) with the purpose of improving the reliability of the single-phase alumina and zirconia ceramic implants. De Aza et. at [38] has put in evidence that, in general, these new composite materials can display not only a greater toughness (KIC) that the monolithic materials previously mentioned, otherwise, and what it is more important, a greater tension threshold for the stress intensity factor (KI0), below which some of crack does not take place propagation (Table III). Therefore, in the case of the ceramics prostheses, this tension threshold provides a rank of intensity of tensions of total security for the use of the composite material under mechanical efforts. Table III Material

Alumina (Al2O3) Zirconia (3Y-TZP) Composite (Al2O3-10% ZrO2)

Fracture threshold KI0(MPam½) 2.5 ± 0.2 3.5 ± 0.2 4.0 ± 0.2

Tougnhess KIc(MPam½) 4.2 ± 0.2 6.1 ± 0.2 5.9 ± 0.2

Hardness H (Vickers) 1800 1290 1530

On the other hand, since the hardness and the chemical stability are equally important in the field of the prosthesis, these composite materials, with relatively low contents of zirconia (10 % in volume), display values of hardness similar to those of alumina and they are not susceptible to the hydrothermal instability observed in some cases of the stabilised zirconia bioceramics. Therefore, these materials appear like an alternative, to consider in the future, for the production of ceramics prostheses. Figure 1 shows a microstructure of one of these composite alumina-zirconia materials.

Carbons Carbon presents a great variety of forms: amorphous carbon, graphite, diamond, vitreous carbon and pyrolitic carbon. Some of them display the most excellent properties of biocompatibility, chemical inertia and thromboresistance that any other bioceramic. On the other hand, another advantage of these materials is that their physical characteristics are next to those of the bone [40]. Thus, their densities, according to the type carbon, change between 1.5 - 2.2 g/cm3, and their elastic modules between 4 - 35 GPa. In spite of all the mentioned

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varieties, only three types of carbon are used for biomedical devices: the pyrolitic carbon, in its two varieties of low temperature isotropic (LTI) and ultra-low temperature isotropic (ULTI), and the vitreous carbon. The three types of carbon have disordered lattice structures and are collectively referred to as turbostratic carbons. While the microstructure of turbostratic carbon might seem very complicated due to its disordered nature, it is in fact quite closely analogous to the structure of graphite, but with at random oriented layers [41].

Figure 1. Microstructure of a material composed of alumina (dark phase) with a 10% in volume of zirconia (clear phase) [39].

Pyrolitic carbon is widely used for implant fabrication. It is normally used as surface coating. It is also possible to coat surfaces with diamond. Although the techniques of coating with diamond have the potential to revolutionize medical device manufacturing, they are not yet commercially available. Pyrolitic carbon devices are typically made via a chemical vapour-deposition, from a hydrocarbon gas in a fluidized-bed using at a controlled temperature and pressure. Its great cellular biocompatibility with the blood and the soft tissue as well as its excellent thromboresistance, does carbon material to be used fundamentally in applications of the circulatory apparatus, blood vessel and mechanical cardiac-valve prosthetic devices, being this last the most extended application. Nowadays, most of the modern heart valves are made with a coating of LTI on a polycrystalline graphite substrate or like a monolithic material [42]. In both cases is frequently added up to 10 % in weight of silicon, often in the form of discrete sub-micron β-SiC particles randomly dispersed in a matrix of roughly spherical micron- size subgrain of pyrolitic carbon. The doping with silicon improve the mechanical properties of pyrolitic carbons, issue of great importance in the heart valves, where the joint is subject to degradation by cyclic fatigue or stress corrosion and possible cavitations by erosion that can happened during the life of the patient. Whereas the pyrolitic carbons coating have been applied in zones in contact with the blood, due to their excellent thromboresistance, the vitreous carbons have been studied mainly to bond to soft and hard tissues, without inflammatory answer in the adjacent tissues. Similar behaviour has been registered for pyrolitic carbons LTI and ULTI.

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Apart from of the mistral and aortic heart valves, there are applications in dental field [43] and middle ear reconstruction and in devices of LTI coating on titanium to make easy the circulation of the blood. Recently, success was achieved in coating ULTI onto the surfaces of blood vessel implants made of polymers. The coating has excellent compatibility with blood and is thin enough not to interfere with the flexibility of the grafts [44].

Bioactive Ceramics The appearance of this type of bioceramics born of the need to eliminate the interface movement that takes place with the implantation of bioinert ceramics. Consequently, L. L. Hench proposes in 1967 to the U.S.A. Army Medical Research and Development Command, a research based on the modification of the chemical composition of ceramics and glasses so that they have chemical reactivity with the physiological system and form chemical bond between the adjacent tissue and the surface of implant materials. Hench and collaborators show for the first time in 1971 that a material done by the man can be bond to the bone. This material was called “Bioglass®”; beginning therefore the field of the bioactive materials [45]. Materials that are bioactive develop an adherent interface with tissue that resist substantial mechanical forces. In many cases, the interfacial strength of adhesion is equivalent to or greater than the cohesive strength of the implant material or the tissue bonded to the bioactive implant. Generally, the break takes place in the implant or in the bone but almost never in the interface. Hench and Wilson [46] show that the surface chemical reactions result in the formation of a carbo-hydroxyapatite (CHA) layer to which bone can bond. This CHA layer is biologically activates and it is chemical and structurally equivalent to the constituent mineral phase of the bone [47-49].

Hydroxyapatite (Ca10(PO4)6(OH)2) Hydroxyapatite (HA) is the primary mineral content of bone representing ∼ 43 % by weight. HA is a calcium phosphate whose stoichiometric formula corresponds to a: Ca10(PO4)6(OH)2, with a Ca/P molar ratio = 1.67. HA belong the mineral family of Apatites whose name derived from Greek “απαταω”means deception or deceit, due to the facility it was confused with other mineral species like the beryl or the tourmaline [50]. Apatites form an important series of minerals. They occur as a minor constituent of many igneous rocks, although a few large igneous deposits are known, like the Kola peninsular. Apatite is also present in most metamorphic rocks, especial crystalline limestone. Less well crystallised deposits of rather variable composition, usually referred to as rock phosphates or phosphorite, occur in large deposits, some of which were formed by the reaction between phosphatic solutions from guano and calcareous rock, or precipitated from sea water. HA displays ionic character, and its crystalline structure can be describe like a compact hexagonal packing of oxygen atom with metals occupying the tetrahedral and octahedral holes of the periodic network. The basic apatite structure is hexagonal with space group P63/m and approximate lattice parameters a = 9.4 and c = 6.9 Å, being the fluorapatite

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(Ca10(PO4)6F2) the representative example of this structure. Nevertheless, HA presents a low symmetry, monoclinic, due to the distortion of ion OH- with respect to the ideal model that would represent the position of spherical ion F- in the fluorapatite. Nevertheless, in most of the works in the field of biomaterials it assumed that HA has a fluorapatite structure but with lattice parameter a = b = 9.418 Å and c = 6.884 Å. α = ß = 90° and γ = 120°, Z = 2. The unit cell contains 10 Ca+2, 6 PO43- and 2 groups OH- [51]. HA allow the substitutions of many ions in its structure. Change for calcium ions (Ca2+) or phosphoate groups (PO4)3− or hydroxyls groups (OH)−. Consequences of these substitutions are the change in its properties like lattice parameters, morphology, solubility etc., without significant change in the symmetry. Thus for example, the substitution of F− by OH− involve a contraction in the a axis without change in the c axis, associated to an increase in the crystalllinity and imparts greater stability to the structure. On the other hand, this imply that the fluorapatite is less soluble than the HA and that all F- free apatites including biological apatites. Many other ions can enter in the HA network, affecting the properties, crystallyinity, thermal stability, speed of dissolution etc. The mechanical properties of the HA are similar to those of the most resistant components of the bone. HA has an elastic modulus of 40-100 GPa, dental enamel 74 GPa, the dentine 21 GPa, the compact bone18-12 GPa. Nevertheless, dense bulk compact of HA has a mechanical resistance of the order of 100 MPa in front of to the 300 MPa of the human bone, diminishing drastically this resistance in the case of porous bulk compact. Biological apatites are those components of the enamel and dentine of the teeth as well as of the human bones and some pathological calcifications (salivary and urinary stones). These biological apatites differ from pure HA in physical and mechanical properties, stoichiometry, composition, crystallinity and other properties, but in general terms they are possible to be assimilated more appropriately to the calcium deficient HA and carbohydroyiapatite, where group (CO3)2- replace group (PO4)3- and where the Ca2+ is replaced by Na+ to balance the charges [52,53]. The most current methods of obtaining pure HA are by hydrotermal and solid state reactions. However, when prepared from aqueous reactions either by precipitation or hydrolysis methods, the HA obtained is usually calcium deficient (Ca/P molar ratio lower than the stoichiometric value of 1.69). When the precipitation reaction is carried out under very basic conditions, the precipitate will contain carbonate, which will make the Ca/P molar ratio higher than the stoichiometric value. The interest of the HA as biomaterial comes clearly by its similarity with the mineral phase of the bone tissue. In principle, HA would be the suitable material as much as for restoration as for bone substitution. However, the relatively low strength and toughness of HA, produced little interest among researchers when the focus of attention is on dense structural samples. Therefore, its use is restricted to all those applications where mechanical efforts are not required, finding its application concentrated as coating on metallic substrate with the object to accelerate and to increase the fixation of the prosthesis to the bone [54,55]. Industrial and laboratory techniques, used for HA coating and other ceramics materials onto metallic substrates, include plasma spraying, laser ablation, electrophoretic deposition, sputtering and hot isostatic pressing. The most extended is first, mainly in the coating of hip prosthesis, obtaining what has been denominated biological fixation of the prosthesis [56,57]. Among other applications are: a) coating of dental and maxillofacial prosthesis; b) dental

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implants; c) middle ear reconstruction; d) augmentation of alveolar ridge; e) periodontal defects; f) spinal surgery; g) pulp-capping materials [58].

SiO2-Based Ceramic: Wollastonite (CaSiO3) Silicates represent the dominant category of the traditional ceramics and glass industries. These materials are economical due to the abundant availability of raw materials. Also, silicates provide adequate mechanical, thermal and optical properties for a wide range of traditional and advanced materials applications. Ceramics and glasses are distinguished primary by the presence respectively of crystalline or noncrystalline structure on the atomic scale. For noncrystalline silicate glass, biomedical applications have been a significant demand because the development of Bioglass®. Since the development of Bioglass®, many bioactive glasses and glass-ceramics in the complex systems containing CaO-SiO2-P2O5 as the main components have been produced [45,59-61]. Hench et al. [62,63], Gross et al. [64,65], Karlson et al. [66,67] and Kokubo et al. [68,69] pointed out that these glasses and glass-ceramics bond to a living bone through hydroxyapatite (HA) layer which is formed on their surfaces. Conversely, Ohura et al. [70] have observed that CaO-SiO2 glasses, free of P2O5 as well as those containing very small amounts of P2O5, form the HA layer on their surfaces (bioactivity), when they are soaked in simulate body fluid (SBF), whereas CaO-P2O5 glasses free of SiO2 do not form HA layer in SBF. This seems to indicate that bioactive materials can be obtained with compositions based on the CaO-SiO2 system rather than in the CaO-P2O5. Others researchers still claim that to show bioactivity, glasses and glass-ceramics must contain both CaO and P2O5, which are the main components of the hydroxyapatite [71,72]. On these bases, two polycrystalline chain silicate minerals have been developed synthetically for use as bioactive ceramic materials. Diopside (CaMgSi2O6) [73,74] and wollastonite (CaSiO3) [75,85]. Pseudowollastonite (CaSiO3), the high temperature form of wollastonite, displays in its structure calcium ion chains easily removals by protons, as it was shown by Bailey and Reesman [86] in the study of the kinetic of dissolution of the wollastonite in H2O-CO2. system. This fact suggests the possibility of extraction of calcium ions, from wollastonite structure by protons in an appropriate medium, favouring therefore the precipitation and formation of a layer of HA on the surface of the material. De Aza et al. have demonstrated the formation of a HA-like layer on the surface of pseudowollastonite ceramic both in vitro” [75,76,79,83-85] and “in vivo” [80,81]. Experiments “in vitro” involved suspension of the material in simulated body fluid with an ion concentration, pH and temperature virtually identical to human blood plasma, and in human parotid saliva. Figure 2 shows the overall microstructure of the polished cross-section of the pseudowollastonite pellet after one month immersion in simulated body fluid and its relevant X-ray maps for silicon, calcium and phosphorous elements. This compositional microcharacterisation of the interface showed that the reaction zone was composed of two chemically dissimilar layers formed on the pseudowollastonite surface. The outer layer, with an average thickness of about 10 µm, was composed of a CaO/P2O5-rich phase, identified as

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HA-like phase by thin film X-ray diffraction [77], while the much thinner underlayer (about 1/10 of the thickness of the outer layer) in direct contact with the pseudowollastonite substrate was rich in silicon but depleted in calcium.

Figure 2. X-ray elemental maps of Si, Ca, P, and a relevant SEM image of the cross-section of the pseudowollastonite ceramic after 1 month immersion in SBF.

The reaction begins with an ionic exchange of Ca2+ from the pseudowollastonite network for 2H3O+ from the surrounding fluid, which progressively transforms the pseudowollastonite crystals into an amorphous silica phase. As the reaction proceeds, the calcium concentration and the pH of the surrounding fluid increase, creating appropriate conditions for partial dissolution of the amorphous silica and subsequent HA-like precipitation. This type of HAlike layer has not been observed for non-bioactive materials, indicating that the HA-like layer plays an essential role in forming the tight chemical bond between the bioactive material and the bone [7,69,110]. The authors also have evaluated the cytotoxicity of the pseudowollastonite and the suitability of the material as a substratum for cell attachment and the ability to effect osteoblast at a distance from the material surface [83,85]. These experiments demonstrate that extracts of pseudowollastonite do not show any significant cytotoxic effects and confirm the biocompatibily of this material. On the other hand, osteoblastic cell can rapidly adhere to the material surface and that process is aided by the binding of serum proteins to the surface HA layer. In addition, osteoblastic cells at a distance form the material exhibited a dramatic alteration on their appearance. This reinforces the suggestion that the release of soluble factors as well as direct interaction of osteoblactic cell with the surface of the material may mediate the osteogenic effects of bioactive materials. “In vivo” experiments were consisted of implanting small cylinders of pseudowollastonite into rat tibias [80,81]. Histological observations twelve weeks after implant show that the bone in contact with the surface of the pseudowollastonite appeared to

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be progressively replaced by bone with laminar arrangement. The new bone was growing in direct contact with the peudowollastonite implants after only 3 weeks [80].

Figure 3 SEM image of the polished cross-sections of the pseudowollastonite ceramic implant after twelve weeks of implantation and its individual X-ray maps of the silicon, calcium and phosphorous distribution.

Figure 3 shows SEM micrograph of the polished cross-sections of the pseudowollastonite ceramic implants after twelve weeks of implantation. The individual X-ray maps of the silicon, calcium and phosphorous distribution are also included. This compositional microcharacterisation of the interface showed that there are two chemically separate layers formed on the implant surface. The outer layer contained Ca and P, while the underlayer was rich in Si and depleted in Ca. Overrall, these results suggest that pseudowollastonite is biocompatible and osteoinductive and it can be used for substitution or repair of bone tissues in places where mechanical solicitations are not high.

Composite Bioeutectic® Material (CaSiO3-Ca3P2O8) Natural and synthetic materials have been used clinically for many years to reconfigure anatomic structures for aesthetic and therapeutic reasons in several different surgical situations, however many outstanding problems still remain unsolved. The most important is the process of total osteointegration of ceramic implants in the human body. When bioactive materials are implanted in a living body, the interaction between the bone tissue and these materials usually take place on their surfaces, with the remaining bulk of the material unchanged, often causing a harmful shear stress. To improve the ingrowth of new bone into implants (osteointegration), and therefore the strength of the bonding with time, the use of materials with an appropriate interconnected

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porous structure has been recommended [87-89]. The design of a porous ceramic implant material has the potential of controlling the bone ingrowth, but its initial mechanical properties are usually inferior to the more compact material. To overcome this problem the idea of creating artificial bone, making use of coral, has been developed [89,90-92]. At present, however, there is no evidence “in vivo” to suggest that the highly organised macropore structure of the coral-derived HA is better, with regard to bone ingrowth and bioresorption, than the random pore structures of the more easily prepared synthetic porous implants. A new way of making strong porous ceramics with interconnected porosity has been developed by De Aza et al. [93-98]. This is based on designing dense bioactive ceramic materials with the ability to develop an in situ porous hydroxyapatite-like (HA) structure when they are implanted into a living body. The material was composed of two phases, pseudowollastonite (α-CaSiO3 = psW) of bioactive properties, and α-tricalcium phosphate (α-TCP) of resorbable properties. Taking into account the bone structure and the eutectic nature of the wollastonite - tricalcium phosphate system [99], the microstructure of the material was developed by slow solidification (0.5ºC/h) of the eutectic composition (60 wt% psW and 40 wt% α-TCP) through the eutectic temperature region (1402º ± 3ºC). Due to the volume fraction of the two phases at eutectic point (psW = 0.61 and a-TCP = 0.38) and their high dimensionless entropy of fusion α >2, where α is defined as α = Sf /R ; Sf = entropy of fusion, R = gas constant; αpsW = 3.71 and αα-TCP = 9.66 [100-103] the obtained microstructure was of irregular eutectic structure. Figure 4 shows a SEM image of the microstrutrure of the eutectic material (Bioeutectic®), which consists of quasispherical colonies formed by alternating lamellae of pseudowollastonite and α-TCP.

Figure 4. SEM image of the microstructure of the eutectic material

The material of this composition was initially exposed to simulated body fluid (SBF) for one month and the cross section of the specimen is shown in Figure 5. Two HA-like layers with different morphologies were found to form below and on the surface of the material, by two different and consecutive mechanisms. At first the material reacts with the surrounding SBF by dissolving the psW phase and forming a porous structure of HA-like phase that mimic porous bone, in a pseudomorphic transformation of the α-TCP, according to the reaction:

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3[Ca3(PO4)2] + Ca2+ + 2(OH)- → Ca10(PO4)6(OH)2 During a second stage a dense HA-like layer is formed by precipitation on the outer surface of the material according to the reaction: 6[HPO4=] + 10 Ca2+ + 8(OH)- → Ca10(PO4)6(OH)2+ 6(H2O)

Figure 5. SEM image of a polished cross-section of the eutectic material after three weeks immersed in SBF

The first stage of the reaction mechanism, which at the beginning is very fast, is slowed down due to the combined effect of the difficulty of diffusion of the H3O+ from the SBF to reach the reaction interface as the distance is increased and due to the closure of the channels as HA is formed on the surface of the simple. Conversely, the last stage of the reaction does no occur when the eutectic sample is immersed in slow stream of SBF (50 cc/h) instead of using a static solution. Then the complete transformation of the eutectic material takes place given rise to a HA artificial porous bone. Figure 6A show a SEM image of a fresh fracture surface of the eutectic material after immersion in a slow stream of SBF for three weeks and for comparison in figure 6B shows the SEM picture of a human bone. In parallel to this study, “in vivo” experiments are being carried out and it is expected that this material might behave in a similar way as “in vitro” tests, facilitating the osteointegration of the implant. The procedure developed by the authors opens the opportunity to obtain a new family of bioactive materials for which the general name of Bioeuctectics® is proposed by the authors. The material may present an interesting alternative in biomedical applications. Bioeutectic® material is the only bioactive material, at present, which is totally colonized in simulated body fluid.

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A

B

Figure 6. (A) SEM image of a fresh fracture surface of the Bioeutectic® material after immersion in a slow stream of SBF for three weeks. (B) SEM image of a human bone.

Bioactive Glasses Since the discovery by Hench, at the beginning of the Seventies, of which some glasses belonging to the SiO2-CaO-NaO-P2O5 system do not induce the formation of fibrous tissue capsule that surround implant materials [45], but the bioactive glasses have chemical reactivity with the physiological system and form chemical bond between the adjacent tissue and the surface of implant materials various types of glasses and glass-ceramics have been developed and used as bone replacement. Probably the divided point, to select the glass, is that it would not have to contain strange or poisonous elements to the organism and between the components would have to be calcium and the phosphorus, because both are the greater components of the mineral phase of the bone tissue. The first bioactive glass (Bioglass®) developed by Hench and col., was the denominated Bioglass 45S5, which presents a great bioactivity, and bond to soft and hard tissues. It is also the composition of bioglass more studied. Its composition in % in weight is the following: 45% SiO2, 24.4% of CaO, 24.5 NaO and 6% P2O5 [45]. The composition of the Bioglass 45S5 was studied in "in vitro" and "in vivo" with three different microstructures: amorphous, partially crystalline and totally crystalline. All the implants were bond to the cortical bone, in a model of rat femur after 6 weeks. There was no difference between amorphous implants and crystalline implants. [46]. Taking the Bioglass 45S5 as the base composition, Hench and col., developed and studied a great series of glasses within the quaternary system SiO2-CaO-NaO-P2O5, maintaining in all of them a constant content of 6% in weight of P2O5. Some of these compositions are in Table IV. In the initial stages of the development of the bioglasses, one thought that the presence of P2O5 in the composition was essential so that the glasses were bioactive. Nevertheless, Li and col [104] have observed that the SiO2-CaO-NaO glasses free of P2O5, obtained by sol-gel method, were bioactive even until 85 mol % in SiO2. Also, Ohura and col. [70] have observed that a binary glass of SiO2-CaO named CS, with a composition in weight percent of: 48.3 CaO and 51.7 SiO2, formed with the bone tissue a stable bond. Later, De Aza and col. [7577,79] demonstrated, for the first time, that a crystalline material, different from the

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hydroxyapatite and free of P2O5, the wollastonite (CaSiO3), in their high and low temperature phases, display a high bioactivity in simulated body fluid and in parotidea human saliva, and even bond to bone tissue [80,81]. Table IV- Glass Compositions in Weight % Biovidrio 45S5 45S5F 45B15S5 45B5S5 KCP1 45S5-N 45S5-C

SiO2 45 43 30 40 45 50 50

CaO 24.5 12 24.5 24.5 24.5 24.5 19.5

Na2O 24.5 23 24.5 24.5 19.5 24.5

P2O5 6.0 6.0 6.0 6.0 6.0 6.0 6.0

B2O3

F2Ca

K2O

16 15 5 24.5

From the point of view of their composition, it can be say that, most of glasses that are able to bond to the “in vivo” tissue contain phosphates, and the totality includes silicates with the purpose of providing a matrix with low solubility, since the single phosphate glasses are excessively soluble. All of them can be divided in two great groups: a) glasses rich in alkaline (glasses of Hench and col.), with alkaline oxide contents greater to 20% in weight and b) poor in alkaline, containing not more of 5% in weight of these oxides. Of the all exposed is deduced that exists a generous rank of compositions of bioactive glasses. This allows, in principle, to adapt the reactivity of glasses for different clinical applications. This versatility of the glass materials is a remarkable characteristic that difference them of the other biomaterials. Bioactive glasses are produces by conventional glass manufacturing methods. Purity of raw materials must be assured, since its quality affects greatly the properties of the obtained glass. Thus, usually quartzes or silica sands of great purity are used, sodium or potassium carbonate reagents, in some cases wollastonite (CaSiO3) of high purity, calcium phosphates without adsorption of water etc. The materials weighed and mixed are melts in Pt or Pt/Rh crucibles, homogenizing the melt and paying attention to the possible loss by volatilisation of some components of high steam tension such as NaO2 or P2O5. Depending on the composition, the temperatures of melting can oscillate between 1200 and 1450 ºC. The melt is cast in graphite or steel molds, since they do not contaminate and not adhere to the glass. The annealing of the glass is fundamental, since all the bioglasses usually have high coefficients of thermal expansion, changing the temperature of annealing, according to the composition of the glass, between 400 and 550 ºC approximately. In general, the bioglasses are relatively soft glasses, reason why they can be easy get the required dimensions by machining, using generally, diamond discs or dental handpieces with abundant refrigeration. When powdered or granulated bioglasses are required, the glass can be ground to the wished sizes or the melt can be quenched directly in water, obtaining what it is denominated glass frit and facility the milling. The glass frit must be dry quickly to avoid the possible hydrolytic corrosion. Since it has been exposed previously, the essential requirement for an artificial material to bond to living bone is believed to be the formation of a biologically active apatite-like

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layer on their surfaces in the body. The surface chemical reactions results in the formation of a carbo-hydroxyapatite (CHA) being this, perhaps, the only common characteristic of all the bioactive materials that are known. The mechanism by which the CHA layer forms on the exposed bioglass to physiological surrounding medium was proposed for the first time by Hench [7] and it consists of the following stages: 1) Leaching by ionic interchange of proton of the physiological medium by labile ions (Na+, K+, Ca2+, Mg2+, etc.). Si – O – Na+ + H+ + OH− → Si – OH+ + Na+ + OH− The rate of reaction, as well as the formation of the silanol groups (Si – OH) in the interface bioglass/solution is controlled by a diffusion process and exhibits a t-½ dependence. The cationic interchange increases the concentration of hydroxyl (OH-) groups in the interface bioglass/solution, giving rise to an increase of pH until levels of the order of 10.5, as it has been demonstrated by De Aza and col. by measures of interfacial pH with a microsensor (ISFET) in different bioactive materials [105]. 2) The increase of pH facilitates the dissolution of the network and formation of additional silanol groups, according to the reaction: - Si – O – Si - + H – O – H → 2 [Si – OH+] As well as the release of silica into the solution in the form of Si(OH)4. 3) Polymerization of the silica (SiO2) rich layer by condensation the neighbouring groups of Si-OH, giving rise to a rich amorphous silica layer O O O O | | | | O - Si - OH + HO - Si - O → O - Si - O - Si - O + H2O | | | | O O O O

4) In this stage, one takes place an ion migration Ca2+ and PO43- to the surface of the rich silica layer, forming an amorphous layer rich in CaO-P2O5 followed by the growth of this layer due to the incorporation of Ca2+ and PO43- coming from the solution. 5) Cystallization of the amorphous CaO-P2O5 layer by incorporation of OH−, CO32−, or F− anions from the solution to form a carbonateapatite or fluorapatite layer. As opposed to the mechanism described by Hench, nowadays seems to admit the unnecessary migration of calcium and phosphate ions through the amorphous silica layer (stage 4ª) to facilitate the nucleation of calcium ions and phosphate ions coming from physiological surrounding medium, especially when Ohura and col [70] and De Aza and col. [75-77,79] have demonstrated the formation of the CHA layer on CaO-SiO2 glasses and wollastonite (CaSiO3) respectively. Both materials are exempts of phosphate ions, and form a chemical bond with the bone. Finally, another hypothesis, perhaps simpler, suggested by De Aza and col. [105] is that pH interface is the responsible of the precipitation, from the solution, of Ca2+ ions and PO43-

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ions. This hypothesis seems to justify itself by the fact that the amorphous silica layer between the layer of CHA and the vitrocermic of A/W developed by Kokubo and col. [68,69] neither in the Bioeutectics materials of psW/α-TCP obtained by De Aza and col. [93-98] has never been found.

Bioactive Glass-Ceramics Glass-ceramics materials are those material obtained from a glass by suitable heat treatments, with the objective to produce the nucleation and growth of certain crystalline phases that are immersed in the residual vitreous matrix. There are two fundamentals reason why there is a great interest by the bioactive glassceramics materials for biomedical applications. First, it can be obtained very complex forms of products or devices made by glass-ceramics, since the initial stage is to obtain a glass, and like so, the well-known technologies of the glass production, economic and precise, such as quenching, blowing, pressed or laminated can be used. Second it is that, as a result of the crystallization, the glass-ceramics products usually display a very fine microstructure with very small or null residual porosity. Such microstructures tend to provide better mechanical properties of the final product or device. Nevertheless, the crystallization process requires a strict knowledge of the mechanisms of nucleation and growth of the crystalline phases. Figure 7 represents a scheme of the nucleation rates and growth of the crystalline nuclei based on the temperature. The rate of growth of the crystalline nuclei (length by time unit) shows a maximum, normally about 100 ºC below the temperature of liquidus (TL). As it can be observed, there is normally a big margin of temperatures below the temperature of liquidus (TL) where does not detect.

Vn

Vn or Vc

Vc

Tg

Temperature

TL

Figure 7.- Scheme of the nucleation rate (Vn) and growth (Vc) respectively, based on the temperature. TL = temperature of liquidus; Tg = vitreous transition temperature

After obtained the glass to room temperature, it is heated up until the optimal temperature of nucleation. Figure 8 shows a typical scheme of a process in two stages to produce a glassceramics. The rate of heating is not critical, but gradients of temperatures must be avoided. The normal rates usually are 2 to 5 ºC/minute, but it depends of the product of start with. It is advisable a high speed of nucleation, preferably near the maximum, whose temperature stays

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Temperature

constant by a certain time. Alternatively, after the glass is conformed, it cans cold directly until the temperature of nucleation. The following stage is critical. The temperature must be increased slowly, normally not more than 5 ºC/minute, trying to avoid the generation of tensions caused by the changes of volume that take place during the crystallization, which could even originate cracks or the breakage of the material. Slow rates of heating in this stage, give rise that the possible tensions generated by the volumetric changes, are eliminated by viscous flow of the residual glass, and as the crystalline fraction increases the material becomes progressively more rigid. Reached the temperature of "growth", this stays during a time until reaching the wished degree of crystallinity. Finally the glass-ceramic is cold until room temperature.

growth

nucleation

Time Figure 8. - Simplified scheme of the process of heat treatment in two stages for the preparation of a glass-ceramic material.

Depending on the composition of the glass, the crystallization sometimes takes place from the surface and not distributed homogenous in the volume, giving rise to a glassceramics of low resistance. In order to help the nucleation in volume and to obtain an appropriate microstructure it is frequently the use of additives. They are generally metallic particles (Cu, Ag, Au, Pt, etc.) and other times metallic oxides (SnO2, Ce2O3, TiO2, etc.). In the field of the bioactives glass-ceramics for biomedical applications, the P2O5, component of all products and devices developed commercially, acts like nucleate agent in the same way that the TiO2. Most of the glass-ceramics for biomedical applications are based on compositions similar to the bioglasses of Hench (Bioglass®), although in all of them present very low contents in alkaline oxide. Table V shows the analyses and characteristics of the bioactive glass-ceramics with clinical applications, compared with the Bioglass® 45S5.

Ceravital The first glass-ceramics material for clinical applications was developed by H. Brömer and E. Pfeil in 1973 [108,109] called Ceravital. This name includes a number of compositions. In its origins Ceravital were considered very optimistically, to even replace bones in zones of load and teeth. Nevertheless, like it is possible to be observed, in mentioned Table V, its mechanical properties (flexural, tensile strength) are below of the 160 MPa that presents the cortical human bone. On the other hand, long in vivo experiments put in doubt the stability of these materials, although, Gross and Strunz [48,64] developed other compositions with the intention to

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diminish the solubility of the glass-ceramics, using metal oxides like nucleates agents. Ceravital bioactivity index (BI = 5.6) is the half of the Bioglass® 45S5 (BI = 12.5). Nowadays, the only field in which Ceravital are used is in the replacement of the ossicular chain in the middle ear, where the loads are minimal and therefore the mechanical properties of material are sufficient. Table V - Characteristics of the Bioactives Glass-Ceramics with Clinical Applications, Compared with the Bioglass® 45S5. Compositión (% weight) Na2O K2O MgO CaO Al2O3 SiO2 P2O5 CaF2 Phases

Bioglass 45S5. 24.5 0 0 24.5 0 45.0 6.0 0 Glass

Ceravital

Ilmaplant

Bioverit

4.6 0.2 2.8 31.9 0 443 11.2 5.0 Apatite + βwollastonite + Glass 170

3-8 0 2-21 10-34 8-15 19-54 2-10 3-23 Apatite + Flogopite + Glass

100 – 150

Cerabone A/W 0 0 4.6 44.7 0 34,0 6.2 0.5 Apatite + βwollastonite + Glass 220

Flexural, tensile strength. (MPa)

42

Compressive strength (MPa)

n.d

500

1060

n.d

500

Young Modulus (GPa)

35

n.d

117

n.d

70 - 88

5-10 0.5-3.0 2.5-5 30-35 0 40-50 10-50 0 Apatite + Glass

100 – 160

Cerabone A/W One of the glass-ceramics with more clinical success, is probably the denominated A/W on the basis of their constituent crystalline phases: apatite (oxyfluorapatite = Ca10(PO4)6(O,F2) and wollastonite (β-CaSiO3), developed by Kokubo and col. [60,68,69,110]. This material is commercially known as Cerabone® A/W. After several insolvent attempts to obtain a glass-ceramics in the pseudoternary system Ca3(PO4)2 – CaSiO3 – MgCa(SiO3)2, starting off a monolithic glass and even from powder glass, the A/W glass-ceramic could be obtained adding small amounts of CaF2 to the composition of the parent glass (Table V). The glass was ground until 5µm in average size, isostaticaly pressed (200 MPa) in the wished forms and heated up to 1050°C; precipitating successively at 870 and 900 ºC the phases oxyfluorapatite and β-wollastonite respectively. Obtaining therefore a glass-ceramic with a very fine microstructure, free of cracks and pores. The special developed microstructure causes that this glass-ceramic presents the best mechanical properties (σf = 220 MPa) of all the materials exposed in Table V. Almost the

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twice that (115 MPa) of dense hydroxyapatite and even offers a resistance greater than the human cortical bone (160 MPa). Its fracture toughness is of 2.0 MPa m½ and its Vickers hardness is approximately 680 [68,111,112]. Like in other bioactive glass-ceramics, when glass-ceramic A/W is immersed in simulated body fluid the same type of carbo-hydroxyapatite (CHA) layer is formed on the surface. The compositional and structural characteristics of this CHA layer are similar to those of the apatite in the natural bone and it’s expected that osteoblast would proliferate on the surface of the CHA layer. This suggests that this CHA layer plays and essential role in forming the chemical bond of the glass-ceramics to the bone. Nevertheless, in contrast to the bioglasses, in glass-ceramics A/W it has not been possible to find the amorphous silica layer, between the layer of CHA and glass-ceramics A/W, even not at level of high resolution transmission electronic microscopy. Nevertheless Kokubo and col., consider that the silanol groups formed on the surface of the glass-ceramic are the responsible for the CHA formation, because provides favourable sites for apatite nucleation and growth. Due to its good properties of biocompatibility, bioactivity and mechanical properties glass-ceramics A/W is used from the Eighties, for the reconstructions of the iliac crests, artificial vertebrae, intervertebral discs. Glass-ceramics A/W can easily machined into various shapes using discs and reels of diamond, even into screws and in powder form can be used to repair and fill bone defects.

Ilmaplant-L1 Berger and col [61] prepared another type of glass-ceramics of apatite/wollastonite called Ilmaplant-L1. As it is possible to appreciate in Table V, the fundamental difference with the glass-ceramics A/W is the alkaline contents and the greater proportions of CaF2, SiO2 and P2O5 and minor content of CaO. Due to its smaller mechanical resistance to flexion (Table V) its use has been restricted to maxilofacials implants. Bioverit In 1983 Holand and col. [113] at the University of Jena, developed a series of bioactives glass-ceramics that they denominated under the name of Bioverit® type I (Table V). These glass-ceramics are easily machined into various shapes with standard tools and even retouched in the operating theatre. They have a very complex composition and they are obtained from of a silico-phosphate glass type within the SiO2-(Al2O3)-MgO-Na2O-K2O-FCaO-P2O5 system. Its process of obtaining consists of generating a phase separation glass by means of the control of the nucleation, crystallization and later growth of the crystalline phases by heat treatments between 610 and 1050 ºC. The product thus obtained is formed by a residual vitreous phase, and a mixture of apatite crystal (1 - 2µm) and mica of the fluoroflogopite type (Na/K Mg3[AlSi3O10F2]) that is the responsible of the easy mechanized of the glass-ceramic. Later, such authors developed another family of glass-ceramic, also machineable, called Bioverit® II consisting of a new type of curved mica of fluorophlogopite-type as the main crystal phase and secondary crystals of cordierite (Mg2[Si5Al4O4]). The basis for the nucleation and crystallization is also the phase separation of the glasses which has been caused by Na2O, K2O and F- additions to the glass.

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Finally, Vogel and col.[114,115] developed another new family of Bioverit, called type III from a phosphate glass without any silica content. The processing starts form phosphate invert glasses of the P2O5-Al2O3-CaO-Na2O system, which structure is formed only by mono and diphosphate units doping it with Fe2O3 o ZrO2. The glass-ceramic thus obtained is formed by a residual vitreous phase, and a mixture of apatite crystal as main phase, berlinite (AlPO4) and a complex phosphate structures of varulite type (phosphate of Na-Ca-Fe). More that 1000 implants of Bioverit® have been successfully used as bone substitution by the middle of the Nineties. In these case we include orthopaedic surgery (replacement of vertebrae, reconstruction of the root of the acetabulum in a dislocated hip, osteotomy of the tibial head, partial replacement of vertebrae, etc) and head and neck surgery (reconstruction of the posterior wall of the auditory canal, reconstruction of the skull base, rhinoplasty, etc).

Bioactive Composites As it has been exposed, the fundamental limitation by the application of the ceramic, glass and glass-ceramics are their low mechanical properties. In order to try to avoid these limitations was proposed the use of these materials as part of a composite material. Bioceramic composite synthesis serves the purpose of enhancing the mechanical behaviour of the parent ceramic, glass or glass-ceramic, while maintaining its excellent biocompativility and bioactivity. Bioceramic composites can be divided in bioiner, bioactive and biodegradable and the ceramic phase can be the reinforcing material, the matrix or both. There are several bioceramic composite which have been manufactures and used in clinical applications, the first bioactive composite was stainless steel/bioactive glass. The preparation of a metal fiber/bioactive glass composite includes preparation of the fiber, impregnating the perform with glass matrix, and heat treating the composite. For a effective stress transfer between the matrix and the reinforced metal fibers, there must be a good bond between the glass and the metal fibers. This is achieved through the oxidation of the metal fibers before immersion in the molten glass. The titanium fiber bioactive glass composite represents a fundamental advance in biomaterials, which derives from consideration of the failure mode of current implants with bioactive coatings as bonding vehicles. Dental implants or hip prostheses with plasma sprayed calcium phosphate coatings are known to fail at the ceramic to metal interface. The advent to the titanium fiber-glass composite represents a new era in that continuity between metal implant and bioactive coating may be achieved, using fibers that are sintered onto the implant substrate and the run throughout the coating up the surface [116,117]. Ducheyne and Hench [118] developed a composite made by Bioglass® 45S5 and stainless steel AISI 316L metal fibers. This type of biomaterial is obtained by a process of immersion of metallic fibers in the melted glass. The composite material thus obtained has a greater mechanical resistance, better ductility and a similar Young modulus to the human the cortical bone. Other composite material composed with a glass-ceramic is Ceravital reinforced with titanium particles [119] and A/W glass-ceramic reinforced with particles of zirconia partially stabilized [120].

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Some of the properties of these composite materials that make them apt for their use as dental implant or in orthopaedic surgery are: low modulus of elasticity; good resistance to the stress; good resistance to the impact and easily machineable.

Biodegradable or Resorbable Ceramics The resorbibles ceramics began to be used in 1969. These types of bioceramics dissolve with time and are gradually replaced by the natural tissues. A very thin or non-existent interfacial thickness is the final results. They would be the ideals implants, since only remain in the body while its function is necessary and disappear as the tissue regenerates. Their greater disadvantage is that their mechanical strength diminishes during the reabsorcion process. Consequently, the function of these materials are to participate in the dynamic process of formation and resorpcion that takes place in bone tissues; so they are used like scaffolding or filling spaces allowing to the tissues their infiltration and substitution [121]. All the resorbibles ceramics, except plaster (CaSO4½H2O), are based on calcium phosphates, varying their biodegradability in the sense: α-TCP > β-TCP >>>> HA The rate of biodegradation is increased, as it is logical, as: a) the specific surface increases (the powders are more quickly biodegraded that porous solids and these more than dense solids); b) when the crystallinity decreases; c) when the grain and crystal size decrease; d) when there are ionic substitutions of CO32-, Mg2+ and Sr2+ in the HA. The factors that tend to decrease the rate of biodegradation include: a) substitution of ion F- in the HA; b) Mg2+ substitution in the β- TCP and c) lower β- TCP/HA ratios in two-phase compounds. The biodegradation or resorption of calcium phosphates is caused by three factors: 1) Physicochemical dissolution, which depends on the solubility product of the material and pH of its surroundings. New phases can be formed in the surface, such as amorphous calcium phosphates, dihydrated dicalcium phosphate, octacalcium phosphate and even HA replaced in an anionic way. 2) Physical disintegration in small particles as result of a preferential attack to the grain boundaries. 3) Biological factors, such as phagocytosis, which gives rise to a local diminution of pH, the cellular activity and the site of implantation. One of the few bioceramics that satisfy partially these requirements is the tricalcium phosphate (TCP).

Tricalcium Phosphate (Ca3(PO4)2) The tricalcium phosphate (TCP) is the biodegradable phosphate par excellence. TCP has a stoichiometric formula Ca3(PO4)2 with a Ca/P molar ratio = 1.15. The TCP is a neutral

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compound in which the 6 positive charges of the Ca+2 ions are compensated by 6 negative charges of the anions PO43-. It belongs to the family of the Whitlockites that respond to general formulate (Ca,Mg)3 (PO4)2, so that the calcium can be replaced partial or totally by magnesium. The TCP is a material that displays polymorphism. Exist three polymorphic stated that are known like the phases: β, α, α`, from low to high temperature of stability respectively [122,123]. The β phase crystallizes in the hexagonal system, and their characteristic parameters of network are: a = 10.429Å and c = 37.38Å. α=β= 90° and γ=120°. The α phase crystallizes in the orthorhombic system, and their characteristic parameters of network are: a = 15.22 Å, b = 20.71 Å and c = 9.109Å. α=β= γ= 90°. The α´phase crystallizes in the monoclinic system, with lattice parameters: a = 12.887Å, b = 27.280Å and c = 15.219Å. α=γ= 90° and β=126.2° [124]. The phase transition α → α' at 1475 ± 5ºC is totally reversible in both senses. Also it happens to the transition β →α at 1150± 10ºC. When cooling below this temperature the inverse transformation is due to produce α→β, nevertheless if the cooling in very fast the phase α is metastably preserved at room temperature. Like the synthesis of HA, the synthesis of β-TCP, in aqueous solution, is affected by numerous physical and chemical parameters, which originates great difficulty to obtain a product 100% pure. However, pure β-TCP can be easy obtained by solid state reaction of calcium and phosphorous compounds with relationships Ca/P = 1.5. Applications are the temporary fillings and periodontal defects.

Acknowledgment The author thanks to CICYT the financial support of the Project MAT2003-08331-C02-02

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P. N. De Aza Handbook of Bioactive Ceramics, Vol I, Bioactive Glasses and Glass Ceramics, (18391). Boca Raton, FL. CRC Press. Sakamoto, O. & Ito, S. (1990). Mechanical properties of bioactive glass-ceramic composite by SiC whiskers. In T. Yamamuro, L.L. Hench & J. Wilson. Handbook of Bioactive Ceramics, Vol. I, Bioactive Glasses and Glass Ceramics, (155-60). Boca Raton, FL. CRC Press. Ducheyne, P. & Hench, L.L. (1982). The processing of static mechanical properties of metal fibre-reinforced bioglass. J. Mater. Sci. 17, 595-606. Müller-Mai, Ch., Schmitz, H.J., Strunz, V., Fuhrmann, G., Fritz, Th. & Gross, U.M. (1989). Tissues at the surface of the new composite material titanium / glass-ceramic for replacement of bone and teeth. J. Biomed. Mater. Res. 23, 1149-61. Kasuga, T., Nakajima, K., Uno, T. & Yoshida, M. (1990). Bioactive glass-ceramic composite toughened by tetragonal zircona. In T. Yamamuro, L.L. Hench & J. Wilson. Handbook of Bioactive Ceramics, Vol. I, Bioactive Glasses and Glass Ceramics, (13742). Boca Raton, FL. CRC Press. Neo, M., Kotani, S., Fujita, Y., et al. (1992). Differences in ceramic bone interface between surface-active ceramics and resorbable ceramics - a study by scanning and transmission electron-microscopy. J. Biomed. Mater. Res. 26 (2), 255-267. Welch, J.H., Gutt, W. (1961). High-temperature studies of the system calcium oxidephosphorous pentoxide. J. Chem. Soc. 4442-4444. Nurse, R.W., Welch, J.H. & Gutt, W. (1969). High-temperature equibria in the system dicalcium silicate-tricalcium phosphate J. Chem. Soc.1077-1083. Frondel, C. (1941). Whitlockite: a new calcium phosphate Ca3(PO4)2. Am. Miner; 26, 145-152.

In: Ceramics and Composite Materials: New Research ISBN: 1-59454-370-4 Editor: B.M. Caruta, pp. 133-146 © 2006 Nova Science Publishers, Inc.

Chapter 4

NOVEL BIOACTIVE HYDROXYAPATITE-BASE CERAMICS AND GELATIN-IMPREGNATED COMPOSITES V.S. Komlev1,2 and S.M. Barinov1 1

Institute for Physical Chemistry of Ceramics, Russian Academy of Sciences, Ozernaya 48, Moscow 119361, Russia; 2 Department of Sciences Applied to Complex Systems, Polytechnic University of Marche, Via Breccee Bianche, I60131 Ancona, Italy.

Abstract This study was aimed at the development of novel ceramic and composite materials intended for application in bone tissue engineering. A method to produce porous spherical hydroxyapatite (HA) granules was proposed. The method is based on liquids immiscibility effect using the HA/gelatin suspension and oil as liquids. The granules of 50 to 2000 µm diameter contain open pores of 10 nm to 10 µm size in amount up to 45 vol.%. A route for the fabrication of porous HA ceramics having two population of open pores was developed. Ceramics contain intragranular, up to 10 µm size pores and intergranular, hundreds micrometer size interconnected pores were prepared. The interconnections in the intergranular pores are the pathway to conduct cells and vessels for the bone ingrowth. The gelatin solution impregnation into the pores was demonstrated to enhance strength significantly. The strength of HA/gelatin composites was increased by 6 to 10 times compared to that of initial ceramics. The ceramics can further be filled with a drug resulting in a drug delivery system of slow drug release rate.

1

Introduction

The use of synthetic materials to repair of traumatically or pathologically damaged bone tissue offers certain advantages compared to the traditional biological methods of bone-defect management, e.g. autografting and allografting of cancellous bone [1]. Since bone grafts are avascular and dependent on diffusion, the size of the defect and the viability of the host bed

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can limit their application. Furthermore, the new bone maintenance can be problematic due to unpredictable bone resorption. There can be significant donor site morbidity associated with infection, pain, and hematoma. Allografting introduces the risk of disease and infection causing lessening of the bone inductive factors, etc. During the last decade, the tissue engineering approach becomes very important allowing the diminish these risks. Tissue engineering method is based on combining bone-forming cells and a supportive scaffold before implantation in bony site [2, 3]. Progenitor cells are seeded onto resorbable scaffold. The cells grow outside the body and become differentiated and mimic naturally occurring tissues. These tissue-engineered constructs are then implanted into the patients to replace diseased or damaged tissues. With time the scaffolds are resorbed and replaced by host tissues that include a viable blood supply and nerves. The living tissue-engineered constructs adapt to the physiological environment and should provide long-lasting repair [2]. The particular requirements for the bone tissue engineering determined the scaffold to be in a porous form, because in this form large number of cells can be delivered. The materials commonly used for this purpose are polymers, ceramics or composites. Ceramic phosphate scaffolds guarantee clear advantages. Functional scaffolding devices need to sustain physiologically applied loads, and ceramics - although fragile - possess an intrinsic strength. Calcium phosphate phase, hydroxyapatite (HA), is the major mineral constituent of human bone. HA-based ceramics are of a limited number of materials which form strong chemical bonds with bone in vivo. Biocompatibility of HA is based on chemical resemblance to bone mineral. A number of porous HA ceramics were developed for application in tissue engineering [4-10]. The pores are known to have a direct influence on bone formation. Porous ceramics body function is to serve as a substrate for proliferation and differentiation of cells followed by the bone ingrowth into the pores. These processes are affected by pore size, morphology, volume content and connectivity [9-11]. In particular, it has been reported that the volume of bone ingrowth increases with an increase of pore size [12]. An optimal pore size occurs for cells infiltration and host tissue ingrowth: 5-15 µm for fibroblasts, 20-125 µm for adult mammalian skin tissue, and 100-135 µm for bone tissue cells [13, 14]. No unanimous opinion exists about optimal pores content in an implanted device. However, it becomes obvious that the pores’ degree of interconnections directly influences the biological fluids, especially in cells and vessels, that favor tissue nutrition and condition of new bone formation [11]. As a consequence, some contradictions appeared concerning an optimal pore size if their degree of interconnection is not taken into account [9]. Generally, a minimum pore size of 100-135 µm for sustained healthy bone ingrowth is accepted [9]. However, flexural strength, strain-to-failure, and fracture toughness values of HAceramics are significantly less than those of bone, whereas the elastic modulus is much higher [8, 9]. These mechanical mismatches influence the reliability of ceramics when implanted into the bone tissue. To improve the mechanical compatibility, a composite approach may be promising. The combination of different materials within a composite structure may lead to a composite material that reveals specific physical, chemical and/or mechanical properties, particularly resulted from a synergy principal. However, current applications of composite biomaterials in medicine are still remarkably less than expected a few years ago. In many biomedical applications, the research and the testing of composites has been introduced and highly developed, but only in a very few cases an industrial production and commercial distribution of medical devices partially or entirely

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made of composites has started [15]. The main critical issues is that the available fabrication methods may limit the possible reinforcement configurations, may be time consuming, expensive, highly skilled. Generally, there are two main types of HA-containing composites under development: (i) ceramics reinforced with small particles, discrete or continuous fibres and (ii) biocompatible polymers reinforced with small ceramic particles and satisfy to the aim of the tissue engineering approach, e.g. to be of porous microstructure and of load-bearing capacity [16]. An alternative approach is to introduce a polymer into a porous ceramic matrix to obtain composites with a continuous ceramic skeleton, in contrast to those reported in [16]. The biological and mechanical properties of such ceramic/polymer composites can differ markedly from those of the polymer/ceramic materials. This paper presents some results of the study aimed at the development of a technique to fabricate a porous HA matrix followed by the vacuum impregnation of the matrix with a biopolymer solution to obtain a porous HA/biopolymer composite usable in the bone tissue engineering applications. The important aspect of the tissue engineering approach is a combination of a porous scaffold with a controlled-in-time drug release system. The pores of ceramics scaffold could be impregnated with a drug like antibiotic, growth factor, bone morphgenetic proteins, etc. to hinder the bacterial colonization of the scaffold or to reach the osteoinductive behaviour.

2 2.1

Materials and Methods Materials

As the starting materials, fine HA powder, gelatine and a vegetable oil were used. The powder of HA was fabricated by precipitation from aqueous solution described in detail elsewhere [17]. The analytical grade Ca(NO3)2, (NH4)2HPO4 and NH4OH were used as the starting reagents. The synthesis reaction was 10Ca(NO3)2 + 6(NH4)2HPO4 + 8NH4OH ⇒ Ca10(PO4)6(OH)2 + 20NH4NO3 + 6H2O (1) Raw product of the reaction, white powder, was calcined at 9000C for 6 hours in an air furnace. The characteristics of the calcined product were as follows: Ca/P = 1.67 (wet chemical analysis); BET specific area as determined by nitrogen gas adsorption (a Quantachrome Autosorb) is about 5.2 m2/g; the median powder agglomerates size is about 2.2 µm (a Coulter Counter); the density is 2.16 g/cm3. Food-grade bovine gelatin P-11 (Russian Standard GOST 11293-89) was supplied by Gelatin Factory, Moscow, Russia. Vegetable oil (Rape oil, Russian Standard GOST 8988), as a dispersion media, was of food-grade refined oil supplied by Cereol Magyarorszag RT (Hungary). The relative density of the oil is 0.917 g/cm3, its viscosity is 0.585 Pa s at 25 0C. Other solvents and reactants were of analytical grade.

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Porous Spherical Granules Preparation

Aqueous solution of 10% gelatin content was prepared by dissolution of 10 g gelatin in 100 ml distilled water at 39 0C. Fine HA powder was added to the above solution in amounts of 1 g HA powder per 1.8 –3.0 ml of gelatin solution. After that, the HA/gelatin slurry was dispersed in 500 ml of oil in a reaction flask by stirring with a glass paddle stirrer at 200 –500 rpm. The oil temperature was from 15 0C to 30 0C. The stirring results in the beads formation due to the surface tension forces. The precipitated beads were washed in acetone followed by ethanol, filtered onto a Buechner cone filter and dried in air [18].

2.3

Porous Ceramics and Composites Preparation

Porous hydroxyapatite blocks were prepared by sintering of porous HA/gelatin microgranules. Disc samples of 10 mm diameter and about 4-6 mm thickness were uniaxial pressed at 10-100 MPa pressure at a room temperature. Green bodies were further sintered at 12000C for 1 hour in air [19]. To fabricate the composites, 1, 4, 7 and 10 vol % food grade bovine gelatin P-11 solutions in distilled water were used. The samples of porous HA ceramics were immersed in the polymer solution under a vacuum of 1.33 Pa for 10 or 30 min, and without vacuum for 30 min. The temperature of the solution was 400C. After vacuuming at this temperature, the samples were dried for 24 h in an air furnace [20].

2.4

Testing of Materials

Microstructure was monitored with a Neophot-32 microscope (Karl Zeiss, Jena) equipped with a Videolabs Desktop Color Video Camera (Videolabs Inc.), as well as by scanning electron microscopy (SEM) (a JEOL JSM-35-CF microscope). Open pores size distribution (an Autoscan Qunatachrome Porometer) was measured; their content was estimated by a common hydrostatic weightening in distilled water. The disc specimens were tested for diametral compression loading to estimate the rupture tensile stress (Brazilian test) [21]. To calculate the strength, the following equation was used

σ = 2P/πdh

(2)

where P is rupture load; d is diameter, and h is thickness of the disc. The samples were loaded in an UTS-100 stiff testing machine (UTS Testesysteme GmbH) at a cross-head speed of loading device of 1.0 mm/min. An error of the rupture load measurement was kept at ±1%.

3

Results and Discussion

The method described allows the production of spherical HA granules of wide dimension range, from <50 µm to above 2000 µm depending on the route. In Fig. 1 the size distribution

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plots for the granules prepared at a stirrer rotation rate of 200 –500 rpm at oil temperatures 15 C to 30 0C are shown. The granules size distribution obeys the normal distribution function. The mathematic expectation of granule diameter is equal to 1735 µm at a stirring rate of 200 rpm and an oil temperature of 15 0C, while at a stirring rate of 500 rpm and an oil temperature of 30 0C it falls down to 670 µm. Thus, content of the coarser granules increases generally with a decrease in the stirring rate and the oil temperature. The temperature effect is due to both the increased viscosity and surface tension of the oil. It is revealed by the direct measurements of the granule size that the granulation process is practically completing during the first 1–2 min from the beginning of the suspension stirring, as it can be seen from Fig. 2 as an example. The granules were fractionated using standard test sieves, and these of 400 µm to 600 µm in diameter were used in further experiments. 0

50

50

45

a Relative content (vol. %)

Relative content (vol. %)

45 40 35 30 25 20 15 10 5

b

-200 r.p.m.

40

-300 r.p.m.

35

-400 r.p.m.

30

-500 r.p.m.

25 20 15 10 5 0

0 100500

5001000

10001500

100500

1500- > 2000 2000

5001000

10001500

1500- > 2000 2000

Granules size ( µm) (µm)

Granules size ( m) (µm)

Fig.1. Size distribution of microgranules produced at oil temperatures 15 0C (a) and 30 0C (b). 800

Mean granules size (mm)

700 600 500 400 300 200 100 0 0

2

4

6

8

10

Stirring time (min)

Fig.2. Mean granule size versus stirring time (stirring rate 500 rpm and oil temperature 25 0C).

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HA granule Intergranular pore HA block Intragranular pore

Fig. 3. A structure model of the body sintered of porous spherical granules. 45 Open pores content (vol. %)

d 40 35 30

c

b

25 a 20 15 950

1000

1050

1100

1150

1200

1250

Sintering temperrature ( C)

Fig. 4. Open pores content versus heat treatment temperature of the granules: gelatin content in starting slurry 2.0 ml/g (a); 2.5 ml/g (b); 3.0 ml/g (c), and 3.5 ml/g (d).

Fig. 3 represents a structure model of porous bi-modal pore-size distribution ceramics. The bi-modality is due to the intergranular and intragranular pores in the body. Intragranular pores size and content are dependent strongly on the preparation route, e.g. on the gelatin/HA ratio in the initial suspension and on the heat treatment regime. Fig. 4 shows the mercury porometry data on open pores content versus the heat treatment temperature for the granules prepared of varied initial gelatin/HA content in the initial suspension. The porosity in the range from about 43 to 17 vol.% can be obtained. Dominant pores size (effective diameter) is in the range 4 to 10 nm, the pores size increases within above range with an increase in the heat treatment temperature. These pores occupy from 34 to 68 vol.% of the total open pores within the granules. Other pores are these of from 1 to 10 µm in diameter. The existence of the latest pores population is confirmed by microscopy observations. It can be supposed, according to [22], that the pores of tens micrometers diameter are connected either directly or by narrow slit-like channels. In the last case the mercury porometry data are accounted for

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these channels size only, because the penetration of the mercury under the pressure is limited by these channels size. Intergranular pores result from the granules packing features. Monosize particles can be free packed into a green body having the relative open pores content 0.3 to 0.4 of the total body volume [23], if the packing is uniform and no arch effect of packing exists. Such packing density corresponds to the coordination number, N, of about 7 in the particles package N = 11.6 (1 – Θ)

(3)

where Θ is the pores content [24]. This N value is close to the coordination number of a body centered cubic (b.c.c.) structure model. In the frame of this model, minimum pore crosssection can be evaluated from a simple geometric consideration of the packing density in the most dense-packed (110) plane of the b.c.c. structure. This minimum size is equal to about 0.2 x 0.4 of the spherical granule diameter. Therefore, the packing of monosize spherical granules of 500 µm in diameter provides the open interconnecting pores of up to 100 x 200 µm2 cross-section. This size is in accordance to the requirement of a minimum pores dimension of 100-135 µm essential for sustained healthy bone ingrowth into implanted device [9]. To increase the pores size, coarser granules must obviously be used. The situation becomes more complicated in the case of the polydisperse granules packing. A model of spherical granules packing for bi-size granules mixtures has been devised in [25]. It has been shown that the open intergranular pores percentage varies from 12.5 to about 35 vol.%, the value being dependent on the granules dimension ratio and their relative contents in the mixture. If the granules of two diameters, d1 and d2, d1≥ 5d2, are packed, so a minimum of the intragranular pores content can be predicted as

Θmin = kV2/[(1 – k)V1 + V2]

(4)

where k is a coefficient in the range from 0.3 to 0.4; V1 and V2 are the volumes occupied by coarse and fine spherical granules, respectively [25].

Fig. 5. Scanning electron microscopy picture of sintered at 12000C ceramics being compacted at 40 MPa.

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Microstructure of ceramics produced by sintering the granules depends on the green body compacting pressure. The higher is the compaction pressure; the lower is the interpores size and content. Shown in Fig. 5 is the pores view near an individual granule in the sintered ceramics, the green body being compacted at a pressure of 40 MPa. The pores size is up to 100 µm. Fig. 6 gives microstructure of the surface of the granule in the sintered ceramics. The pores of from 1 to 10 µm dimensions are clearly evidenced.

Fig. 6. Scanning electron microscopy picture of the granule surface in the sintered ceramics.

Open pores content (vol. %)

70 60 50 40 30 20 10 0 0

20

40

60

80

100

Pressure (MPa)

Fig. 7. Total open pores content in sintered bodies versus compaction pressure.

Mercury porometry measurements performed on individual 500 µm diameter granules after the heat-treatment at 12000C for 1 hour revealed that the open pores content equals to about 18 vol.%. Shown in Fig. 7 is the total open pores content in sintered at 12000C body versus the compaction pressure plot. The open pores content decreases gradually with an increase of the compaction pressure until the pressure of about 60 MPa is reached. This is due to the granules rearrangement and deformation, just because the granules contain gelatin making the granules plastic. Measured for the samples prepared at very low compaction

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pressure, the porosity is as high as 65 vol.%. That value is higher than the sum of the intragranular porosity (18 vol.%) and the predicted intergranular porosity for the uniform packing case (30 to 45 vol.%). Therefore, a non-uniformity of the packing exists, probably resulting from the sticking the granules together due to the gelatin. If the pressure exceeds 60 MPa, so the pores content in sintered body becomes independent on the compaction pressure being equals to about 18 vol.%. This value corresponds to the open pores content in the granules themselves. Therefore, open porosity in the sintered bodies which are compacted at a pressure of 60 MPa and above is the intragranular porosity. Bodies prepared at the compacting pressure of 10 and 20 MPa are weak and may be crushed by hand easy. So, the ceramics obtained at 30 to 50 MPa densification pressure can only be considered as being of interest for practice. Given in Table 1 are the mercury porometry measurements data on the open pores content and distribution within the sintered at 12000C blocks in dependence on the green body unaxial compacting pressure. The ceramics containing intragranular fine pores and intergranular coarse these, of about 100 µm effective diameter, can be produced with the use of the 500 µm size spherical granules batch compacted at 30 MPa pressure. Table 1. Open pores characteristics of ceramics sintered at 12000C.

Pressure, MPa

Total open pores content, vol.%

10

67

30

51

50

28

Dominant open pores size and content Intergranular pores Intragranular pores Content, Content, Size, nm Size, µm vol.% vol.% 2-10 and 17-18 >100 48-49 5000-14000 2-10 and 17-18 about 100 33-34 5000-14000 2-10 and 17-18 about 30 9-10 5000-14000

Tensile strength (MPa)

6 5 4 3 2 1 0 30

35

40

45

50

Open pores content (vol. %)

Fig. 8. Tensile strength estimated by diametral compression test versus open pores content in ceramics.

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Fig. 8 gives the dependence of the rupture tensile stress of ceramics on the open pores content. The strength reduces drastically with an increase of porosity in the range from about 30 to 40 vol.% reaching approximately constant level with the further increase of porosity. The strength is known influenced strongly by pores. Generally, an exponential dependence is used,

σ = σ0 exp (-ηΘ)

(5)

where σ0 is the strength of a non-porous body, η = 7-8 is a constant, to predict the pores effect [23]. The other common equation is

σ = σ0(1 – Θ)m

(6)

where exponent m equals to 3-10 for ceramics [26, 27]. Estimations give that the pores content of about 30 vol.% decreases the strength by factor about 8-12. Therefore, the measured strength value seems to be no contradictory one, because the tensile strength of dense hydroxyapatite ceramics varies from 10 to 100 MPa [28]. The relative weight increment of the gelatin-impregnated samples increased with an increase in the concentration of solution, initial porosity of ceramics, and impregnation time. Maximum weight increment of about 7.3 % was reached by using 10 % gelatin solution, respectively, under a vacuum degree of 1.33 Pa for 30 min impregnation duration. SEM investigation indicates high compatibility between HA and gelatin. The polymer formed thin layers on the surface of the ceramic phase. Fig. 9 illustrates the microstructure of HA/gelatin composite, which have been fabricated using 7 % gelatin solution, under a vacuum of 1.33 Pa for 30 min. Consequently, about 80 % of the pore space remains free of gelatin that is useful for the osteointegration process.

Fig. 9. Scanning electron microscopy picture of the microstructures of HA/gelatin composites.

Novel Bioactive Hydroxyapatite-Base Ceramics … 14

14

b

a

12 Tensile strength (MPa)

12 Tensile strength (MPa)

143

10 8 6 4

10 8 6 4 2

2

0

0 1

4

10

1

Concentration of solution (%)

4

10

Concentration of solution (%)

Fig. 10. Tensile strength estimated by diametral compression test of HA/gelatin composite materials using ceramics obtained at (a) 50 and (b) 100 MPa pressure, as a function of concentration of solution ( —vacuum 1.33 Pa for 30 min, —vacuum 1.33 Pa for 10 min, and —without vacuum for 30 min).

Fig. 10 shows the tensile strength of the composites. The increase in strength is proportional to the polymer concentration in solution. The samples infiltrated with 10% gelatin solutions exhibit the maximum tensile strengths of 9.8 and 13.7 MPa; these values were reached if the infiltration was performed under vacuum 1.33 Pa for 30 min for ceramics obtained at 50 and 100 MPa, respectively. 8 7

Value of η

6 5 4 3 2 1 0 0

2

4

6

8

10

12

Concentration of gelatin solution (%)

Fig. 11. The η value as a function of the gelatin solution concentration.

The value of η in Equation 5 is a measure of the porosity effect on the strength. A wide range of change in η resulted from its dependence on the pore geometry: an increase of the stress concentration factor created by pore increases the η value. Shown in Fig. 11 is the η value as a function of the gelatin solution concentration. The value of η was calculated as η =

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ln(σ0/σ)/Θ, according to Equation 5, where σ is strength of the body of open pores content Θ being fabricated with the use of certain gelatin solution. The impregnation can be seen to reduce η value, e.g. strength sensitivity to the stress concentration created by pores. The impregnation is supposed to fill the pore volume at the necks between sintered granules reducing the stress concentration factor due to both the geometry of pore and the elastic module mismatch. Preliminary in vitro and in vivo testing of porous (non-impregnated) scaffolds for antibiotics release and biocompatibility demonstrated high antibiotic adsorption efficiency (46±5 %) and prolonged (up to 42 days) its release [29]. The gelatin impregnation, even being at maximum reached of 7.3 wt. % level, is not expected to influence the pharmokinetics significantly. The ceramics were demonstrated to possess excellent osteoconduction and could be useful for the treatment of chronic osteomyelitis as well as infected joint arthroplasty [29].

4

Conclusion

Novel HA ceramics of bimodal pore size distribution were developed. The strength of ceramics was demonstrated to enhance significantly by means of gelatin solution impregnation. The impregnation degree as low as 7.3 wt. % resulted in strength increasing from about 2 MPa for initial ceramics to 13.7 MPa for composite, mainly due to the stress concentration decrease at the necks between the sintered ceramics granules. These porous HA/gelatin composite materials can further be filled with a drug and are expected to be useful as drug delivery systems for the treatment of osteomyelitis and as scaffolds in bone tissue engineering.

Acknowledgments The supports available from the RFBR grant N 03-03-32230 and the Moscow Government grant N 1.1.58 are deeply appreciated. Partially supported by funds from the KMM NoE European project and INTAS Nr 0484-289.

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6. Hing, K.A., Best, S.M. & Bonfield, W. (1999). Characterization of porous hydroxyapatite. J. Mater. Sci.: Mater. Med., 10, 135-145. 7. Kim, H.W., Knowles, J.C. & Kim, H.E. (2004). Hydroxyapatite/poly( -caprolactone) composite coatings on hydroxyapatite porous bone scaffold for drug delivery. Biomaterials, 25, 1279-1287. 8. Uchida, A., Nade, S.M., McCartney, E.R. & Ching, W. (1984). The use of ceramics for bone replacement. A comparative study of three different porous ceramics. J. Bone Joint. Surg., 66B, 269-275. 9. Hing, K.A., Best, S.M., Tanner, K.E., Bonfield, W. & Revell, P.A. (1999). Quantification of bone ingrowth within bone-derived porous hydroxyapatite implants of varying density. J. Mater. Sci.: Mater. Med., 10, 663-670. 10. Eggli, P.S., Müller, W. & Schenk, R.K. (1987). Porous hydroxyapatite and tricalcium phosphate cylinders with two different size ranges implanted in the cancellous bone of rabbits. Clin. Orthoped., 232, 127-138. 11. Liu, D.M. (1996). Fabrication and characterization of porous hydroxyapatite granules. Biomaterials, 17, 1955-1957. 12. Jarcho, M. (1981). Calcium phosphate ceramics as hard tissue prosthetics. Clin. Orthop. Rel. Res., 157, 259-278. 13. Yamamoto, M., Tabata, Y., Kawasaki, H. & Ikada, Y. (2000). Promotion of fibrovascular tissue ingrowth into porous sponges by basic fibroblast growth factor. J. Mater. Sci.: Mater. Med., 11, 213-218. 14. Hulbert, S.F. & Klawitter J.J. (1972). Application of porous ceramics for the attachment of load-bearing internal orthopedic applications. Biomed. Mater. Symp., 2, 161-229. 15. Salernitano, E. & Migliaresi C. (2003). Composite materials for biomedical applications: a review. J. Appl. Biomater. & Biomechanics, 1, 3-18. 16. Cao, W. & Hench, L.L. (1996). Bioactive materials. Ceramics Int., 22, 493-507. 17. Orlovskii, V.P., Komlev, V.S. & Barinov, S.M. (2002). Hydroxyapatite and hydroxyapatite-based ceramics. Inorg. Mater. (Russia), 38, 973-984. 18. Komlev, V.S., Barinov, S.M. & Koplik, E.V. (2002). A method to fabricate porous spherical hydroxyapatite granules intended for time-controlled drug release. Biomaterials, 23, 3449-3454. 19. Komlev, V.S. & Barinov, S.M. (2002). Porous hydroxyapatite ceramics of bi-modal pore size distribution. J. Mater. Sci.: Mater. Med., 13, 295-299. 20. Komlev, V.S., Barinov, S.M. & Rustichelli F. (2003). Strength enhancement of porous hydroxyapatite ceramics by polymer impregnation. J. Mater. Sci. Lett., 22, 1215-1217. 21. Barinov, S.M., & Shevchenko, V.Ja. Strength of Engineering Ceramics. Moscow: Science; 1996. 22. Strelov, KK. Structure and Properties of Refractories. Moscow: Metallurgy; 1972. 23. Dulnev, G.N., & Zarichnjak, Yu. P. Heat Conductivity of Mixtures and Composite Materials. Leningra: Energy; 1974. 24. Kaganer, M.G. Heat Insulation in Low Temperatures Technology. Moscow: Mashinostrojenije; 1966. 25. Krasulin, Yu. L., Timofeev, V. N., Barinov, S.M., Ivanov, A. B., Asonov, A. N., & Shnyrev, G. D. Porous Structural Ceramics. Moscow: Metallurgy; 1980. 26. Andrievski, R. A. (1982). Strength of sintered bodies. Powder Metal. (Russia) 1, 37-43.

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27. Andrievski, R. A., & Spivak, I. I. Strength of Refractory Compounds and Related Materials. Tcheljabins: Metallurgy; 1989. 28. Suchanek, W. & Yoshimura, M. (1998). Processing and Properties of HydroxyapatiteBased Biomaterials for Use as Hard Tissue Replacement Implants. J. Mater. Res., 13, 94117. 29. Hasegawa, M., Sudo, A., Komlev, V.S., Barinov, S.M. & Uchida, A. (2004). High release of antibiotic from a novel hydroxyapatite with bimodal pore size distribution. J. Biomed. Mater. Res. Part B: Appl. Biomater., 70B, 332-339.

In: Ceramics and Composite Materials: New Research ISBN 1-59454-370-4 c 2005 Nova Science Publishers, Inc. Editor: B.M. Caruta, pp. 147-195

Chapter 5

T HERMAL S TRESSES IN PARTICLE-M ATRIX S YSTEM AND R ELATED P HENOMENA . A PPLICATION TO S I C-S I3N 4 CERAMICS Ladislav Ceniga∗ Institute of Materials Research Slovak Academy of Sciences Watsonova 47 043 53 Koˇsice Slovak Republic Phone: +421 55 63 381 15 Fax: +421 55 63 371 08

Abstract The paper deals with elastic thermal stresses in an isotropic particle-matrix system of homogeneously distributed spherical particles in an infinite matrix, divided into cubic cells containing a central spherical particle embedded in a matrix of a dimension equal to an inter-particle distance. Originating during a cooling process as a consequence of the difference in thermal expansion coefficients between the matrix and the particle, and investigated within the cubic cell, the thermal stresses of the isotropic multi-particle-matrix system are thus functions of the spherical particle volume fraction, v, and are transformed for v = 0 to those of the isotropic one-particle-matrix system. With regard to the particle-matrix system yield stress, the thermal stresses are derived for such temperature range within which the particle-matrix system exhibit elastic deformations. Similarly, the particle-matrix boundary adhesion strength is also considered. In addition to the thermal-stress-induced elastic energy of the cubic cell, the elastic energy gradient within the cubic cell, calculated by two equivalent mathematical techniques and representing a surface integral of the thermal-stress-induced elastic energy density, is presented to derive the thermal-stress strengthening in the spherical particle and the cubic cell matrix. Considering a curve integral of the thermal-stress-induced elastic energy density, the critical particle radii related to crack formations in ideal-brittle particle and matrix, ∗ E-mail

address: [email protected], [email protected]

148

Ladislav Ceniga functions describing crack shapes in a plane perpendicular to a direction of the crack formation in the particle and the matrix, and consequently particle and matrix crack dimensions are derived along with the condition concerning a direction of the particle and matrix crack formation. The former parameters for v = 0 are derived using a spherical cell model for the spherical cell radius, Rc → ∞. The derived formulae are applied to the SiC-Si 3N4 multi-particle-matrix system, and calculated values of investigated parameters are in an excellent agreement with those from published experimental results.

Keywords: particle-matrix system, thermal stress, strengthening, crack formation, SiC, Si3 N4 .

1

Introduction

Investigated usually by approximate computational and experimental methods [1]-[3], thermal stresses represent an important phenomenon observed in materials. With regard to material science, thermal stresses originate as a consequence of the difference in thermal expansion coefficients between individual material components, consequently influencing mechanical properties [4]-[8], superconductivity [9]-[11], and diffusion processes [12]-[17]. Resulting from an analytical model of elastic thermal stresses in isotropic multi-particlematrix system of homogeneously distributed spherical particle embedded in an infinite matrix [18], the paper is devoted to the calculation of the elastic energy determining the thermal-stress equilibrium state, to the calculation of the elastic energy gradient influencing a dislocation motion and being a reason of the thermal-stress strengthening, and along with an analyse of conditions for particle and matrix crack formations, the results, presented in the section Theoretical background and consequently applied to the SiC-Si 3 N4 multi-particle-matrix system, accordingly enable to predict and design materials properties, as additionally confirmed by an excellent agreement of calculated values of investigated parameters with those from experimental results [19]-[21].

2

Theoretical Background

2.1 Isotropic Multi-Particle-Matrix System 2.1.1 Cell Shape The thermal radial and tangential stresses, σr and σϕ , σν , derived in the system of the spherical coordinates (r, ϕ, ν) related to the Cartesian system (Ox1 x2 x3 ) (Fig. 1), are investigated
in the point P of a continuum along the axes xr and xϕ , xν of the Cartesian system Pxr xϕ xν , where r is the distance from the spherical particle centre, O. To derive the thermal stresses acting in the system of the homogeneously distributed spherical particles embedded in the infinite matrix, the multi-particle-matrix system is di-

Thermal Stresses in Particle-Matrix System and Related Phenomena...

149

Figure 1: The axes xr , xϕ , xν , and the point P of the position determined by the coordinates

(r = |OP| , ϕ, ν), where O is the spherical particle centre. The radial and tangential stresses in the particle (q = p) and in the matrix (q = m), σrq and σϕq , σνq , act in the point P along the axes xr and xϕ , xν , respectively.

vided into cells of the shape depending on the particle distribution, and the thermal stresses are thus investigated within the cell [22]. Depending on the distribution of the spherical particles of the radius R (Fig. 2), the infinite matrix is divided into cubic cells of the dimension d, representing an inter-particle distance, and consequently the spherical particle volume fraction of the multi-particle-matrix system, v, as the ratio of the spherical particle volume to the cubic cell volume, is derived as [18] 4π v= 3

 3  R π , ∈ 0, d 6

(1)

where π/6 results from d = 2R. Due to the isotropy of the multi-particle-matrix system, the shape symmetry of the spherical particle and the cubic cell, the symmetrical distribution of the cubic cells resulting from the matrix infinity, the thermal stresses are thus sufficient to be investigated within one twenty-fourth of the cubic cell (Fig. 3), then for r ∈ h0, rci, ϕ ∈ h0, π/4i, ν ∈ hν34 , π/2i, where rc = |OC|, ν34 = ∠ (x2 , OC34), C is a point on the cubic cell surface C1C2C3C4 , and then [18] rc =

d , 2 cos ϕ sin ν

(2)

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Ladislav Ceniga

Figure 2: The multi-particle-matrix system of the homogeneously distributed spherical particles of the radius, R, embedded in the infinite matrix divided into the cubic cells of the dimension, d, representing an inter-particle distance, and the Cartesian system (Ox1 x2 x3 ) in the point O in the spherical particle centre.

Figure 3: The surface C1C2C3C4 of one eighth of the cubic cell and the point C on the surface C1C2C3C4 of the position determined by the coordinates (rc , ϕ, ν), where rc = |OC|, O is the spherical particle centre, d is the cubic cell dimension (Fig. 2). The thermal stresses are thus sufficient to be investigated within one twenty-fourth of the cubic cell, then for r ∈ hR, rc i, ϕ ∈ h0, π/4i, ν ∈ hν34 , π/2i.

Thermal Stresses in Particle-Matrix System and Related Phenomena... 

|OC12 | ν34 = arctan |C12C34 |





 1 = arctan . cos ϕ

151

(3)

2.1.2 Thermal Stresses The radial and tangential stresses of the cubic cell, acting in the spherical particle ( q = p) for r ∈ h0, Ri and in the cubic matrix (q = m) for r ∈ hR, rci (see Eq. (2)), σrq and σϕq , σνq , respectively, have the forms [18] σrp = σϕp = σνp = −pb , (

(4)

"

 3 #) R , σrm = −pb 1 + c6 1 − r ( "   #) 1 R 3 , σϕm = σνm = −pb 1 + c6 1 + 2 r

(5)

(6)

where σ > 0 and σ < 0 represent the tensile and compressive stresses, respectively. The compressive or tensile particle-matrix boundary radial stress, pb > 0 or pb < 0, respectively, is derived as [18] pb = c7

ZTi

(αm − α p )dT,

(7)

T

where the coefficient ci (i = 1 − 19) is presented in the section Appendix (see Eqs. (112)(127), (130)-(133)); Ti and T are the initial and final temperatures of a cooling process, respectively; αq is the thermal expansion coefficient of the particle ( q = p) and the matrix (q = m). Considering αq to be temperature-independent, the integral αq (Ti − T ).

RTi

αq dT is replaced by

T

2.1.3 Temperature Range of Cooling Process With regard to a real particle-matrix system exhibiting plastic deformation, the initial temperature, Ti ≤ 0.5 × Tmq , represents the homologous temperature below which the stress relaxation does not occur as a consequence of thermal-activated processes [20, 21], where Tmq is the melting temperature of the particle (q = p) or the matrix (q = m), and if either Tmp < Tmm or Tmp > Tmm then either Tmp or Tmm is considered, respectively. Slowly cooling the real particle-matrix system at temperature T > Ti , the thermal stresses are completely released by plastic deformations [20, 21]. If at least one of individual components of the particle-matrix system represents a ceramic material, the initial temperature, Ti ≈ Tmq , where Tmq is the melting temperature of the ceramic component [21].

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The thermal stresses exhibit an maximal value on the particle-matrix boundary, acted by the radial stress, pb (see Eq. (7)). Accordingly, the presented calculation is considerable in the temperature range T ∈ hTc , Tii, where Tc is the critical final temperature of a cooling process at which the stress, |pb | = σyq , where σyq = σycq or σyq = σytq is a yield stress in compression or tension related to the compressive or tensile stress, pb > 0 or pb < 0, respectively, and if σyp < σym or σyp > σym then σyq = σyp or σyq = σym , respectively. With regard to Eq. (7), the critical temperature, Tc, can be derived from ZTi c7 (αm − α p ) dT − σyq = 0, (8) T c

for a concrete isotropic multi-particle-matrix system by a numerical method, and considering αq (q = p, m) to be temperature-independent, Tc has the form Tc = Ti −

σyq . |c7 (αm − α p )|

(9)

If the particle-matrix boundary bond is characterized by the adhesion radial stress, pa , and loaded by the tensile radial stress, pb < 0, the presented calculation is considerable for |pb | ≤ pa , the critical temperature, Ta , related to the condition, |pb | = pa , can be derived from Eq. (8), or has the form given by Eq. (9), both for σyq → pa . Finally, if Ta > Tc, the presented calculation is considerable in the temperature range T ∈ hTa , Ti i. 2.1.4 Elastic Energy Applying the Hooke laws for an isotropic continuum [23] ε11 = s11 σ11 + s12 (σ22 + σ33 ) ,

(10)

ε22 = s11 σ22 + s12 (σ11 + σ33 ) ,

(11)

ε33 = s11 σ33 + s12 (σ11 + σ22 ) ,

(12)

into the formula for the elastic energy density [23] w=

1 3 ∑ εi j σi j , 2 i, j=1−3

(13)

and with regard to εi j , σi j = 0 (i 6= j) and the subscript transformations, 11 → r, 22 → ϕ, 33 → ν, the elastic energy density of the thermal stresses in the spherical particle and in the cubic cell matrix, w p and wm , respectively, are derived as wp =

3p2b (s11p + 2s12p ) , 2

(14)

Thermal Stresses in Particle-Matrix System and Related Phenomena...

153

"  6 # 3p2b R . 2c28 (s11m + 2s12m) + c26 (s11m − s12m) wm = 4 r

(15)

where the elastic moduli, s11q, s12q, for the particle (q = p) and the matrix (q = m) are presented in the section Appendix (see Eqs. (110), (111)) [23]. Consequently, the elastic energy of the thermal stresses in the spherical particle and in the cubic cell matrix, Wp and Wm, respectively, have the forms Z

Wp =

w p dVp =

Wm = +

w p r2 drdΦ = 2πR3 p2b (s11p + 2s12p ) ,

(16)

Φ 0

Vp

Z

Z ZR

wm dVm = 24

Z Zrc

2

wm r drdΦ =

πp2b R3



Vm Φ R
 2 c6 (s11m − s12m) π2 − 36vc3 ,

2c28 (s11m + 2s12m)



1 −1 v

 (17)

where dVq = r2 drdΦ represent an infinitesimal volume of the spherical particle ( q = p) or the cubic cell matrix (q = m) for r ∈ h0, Ri, Φ ∈ h0, 4πi or r ∈ hR, rci, Φ ∈ h0, π/6i, respectively. With regard to the term r−6 in Eq. (15), dΦ ≡ dϕdν for ϕ ∈ h0, π/4i, ν ∈ hν34 , π/2i (Fig. 3), and consequently rc and ν34 are given by Eqs. (2) and (3), respectively. The elastic energy of the cubic cell is then W = Wp +Wm . 2.1.5 Surface Integral of Elastic Energy Density The surface integral, Wsp and Wsm1 , of w p and wm (see Egs. (14), (15)) over the surfaces P2 P3 P4 k x2 x3 and P1 P2 P4 P5 k x2 x3 (Fig. 4), representing the elastic energy gradient within the spherical particle and the cubic cell matrix along the axis x1 ∈ h0, Ri (Figs. 2-4), Wsp = ∂Wp /∂x1 and Wsm1 = ∂Wm /∂x1 , respectively, as the elastic energy ‘surface’ density, equivalent to those along the axes x2 , x3 due to the isotropy of the multi-particle-matrix system, has the form

Wsp =

Z

w p dS p = 4

Sp

Zπ/2ZR2 0

w p r23 dr23 dξ =


3πp2b (s11p + 2s12p ) R2 − x21 , 2

0

x1 ∈ h0, Ri,

Wsm1 =

Z Sm

wm dSm = 8

(18)

Zπ/4ZR23

wm r23 dr23 dξ

0 R2

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Ladislav Ceniga

Figure 4: The planes P1 P2 P3 P4 P5 k x2 x3 and P6 P7 P8 k x2 x3 in the positions x1 ∈ h0, Ri and x1 ∈ hR, d/2i, respectively, in the cubic cell of the dimension d, containing the spherical particle of the radius R (Fig. 2).

= +

(

"

# 2/3
4π 4c28 (s11m + 2s12m) R2 − π R2 − x21 3v
R2 c26 (s11m − s12m) π − 4c9 R4 , x1 ∈ h0, Ri, 3p2b 8



(19)

and then, for x1 ∈ h0, Ri, dSq = r23 dr23 dξ is an infinitesimal part in the point P of the spherical particle (q = p) or the cubic cell matrix (q = m) on the surfaces S p ≡ P2 P31 P32 k x2 x3 or Sm ≡ P11 P2 P32 P33 P12 k x2 x3 for r23 ∈ h0, R2 i or r23 ∈ hR2 , R23 i (Figs. 4, 5), respectively, where P2 P31 P32 , P11 P2 P32 P33 P12 ⊂ P1 P2 P3 P 4 P5 (Fig. 4); r23 = |P31 P|; P31 P k x2 x3 ; q

R2 − x21 ; R23 = |P31 P23 | = d/(2 cos ξ); q 2 + x2 . P31 P2 , P31 P23 k x2 x3 ; and the coordinate r in Eq. (15) is derived as r = |OP| = r23 1 Due to the isotropy of the multi-particle-matrix system, the angle ξ is sufficient to be varied in the interval ξ ∈ h0, π/4i. The surface integral, Wsm2 , of wm (see Eq. (15)) over the surface P6 P7 P8 k x2 x3 (Fig. 4), representing the elastic energy gradient within the cubic cell along the axis x1 ∈ hR, d/2i (Figs. 2-4), Wsm2 = ∂Wm /∂x1 , equivalent to those along the axes x2 , x3 , has the form R = |OP2 | = |OP32 |; R2 = |P31 P2 | = |P31 P32 | =

Wsm2 =

Z Sm

wm dSm = 8

Zπ/4ZR68

wm r68 dr68 dξ

0

0

Thermal Stresses in Particle-Matrix System and Related Phenomena...

155

Figure 5: The planes P2 P31 P32 k x2 x3 and P11 P2 P32 P33 P23 P12 k x2 x3 of the spherical particle of the radius R and of the cubic cell matrix of the dimension d (Fig. 2), included, along with the point P, in the plane P1 P2 P3 P4 P5 k x2 x3 in the position x1 ∈ h0, Ri (Fig. 4).

(



=

3p2b R2 8

+

c26 (s11m − s12m )

4c28

4π 3v

2/3

(s11m + 2s12m)

π − 4c9 x41






R x1

4 )

 d , , x1 ∈ R, 2 

(20)

and then, for x1 ∈ hR, d/2i, dSm = r68 dr68 dξ is an infinitesimal part in the point P of the cubic cell matrix on the surface Sm ≡ P62 P72 P73 P63 k x2 x3 (Figs. 4, 6) for r68 = |P72 P| ∈ h0, R68i, respectively, where P62 P72 P73 P63 ⊂ P6 P7 P8 (Fig. 4); P72 P k x2 x3 ; R68 = |P q72 P68 | =

2 + x2 . d/(2 cos ξ); P72 P68 k x2 x3 ; and the coordinate r in Eq. (15) is derived as r = |OP| = r68 1 Due to the isotropy of the multi-particle-matrix system, the angle ξ is sufficient to be varied in the interval ξ ∈ h0, π/4i Alternatively, the surface integral, Wsm2 , of wm (see Eq. (15)) over the surface P6 P7 P8 k x2 x3 (Fig. 4), is also derived as

Wsm2

"   2 2 3p R 4π 2/3 4c28 = wm dSm = 4 wm r2 dϕdν = b (s11m + 2s12m ) 8 3v 0 ν6 Sm   4 #  R d 2 , (21) , x1 ∈ R, + c6 c15 (s11m − s12m ) x1 2 Z

Zϕ73 Zπ/2

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Ladislav Ceniga

Figure 6: The plane P62 P72 P73 P68 P63 k x2 x3 of the cubic cell matrix of the dimension d (Fig. 2), included, along with the point P, in the plane P6 P7 P8 k x2 x3 in the position x1 ∈ hR, d/2i (Fig. 4).

and then, for x1 ∈ hR, d/2i, dSm = r2 dϕdν represents an infinitesimal part in the point P of the cubic cell matrix on the surface Sm ≡ P62 P72 P73 P63 k x2 x3 (Figs. 4, 7) for r = |OP| = x1 / (cos ϕ sin ν), ϕ ∈ h0, ϕ73 i, ν ∈ hν6 , π/2i, where P62 P72 P73 P63 ⊂ P6 P7 P8 (Fig. 4); and the angles ϕ73 = ∠ (OP72 , OP73 ), ν6 = ∠ (x3 , OP6 ) are presented in the section Appendix (see Eqs. (128), (129)). The x1 -dependence of Wsp + Wsm1 and Wsm2 exhibits concave and convex courses in the intervals x1 ∈ h0, Ri and x1 ∈ hR, d/2i, respectively. As a reason of material properties (e.g. dislocation motion, coercivity) [20, 21], the maximum, Wsmax = (Wsp +Wsm1 )x1 =0 (see Eq. (18), (19)), has the form

Wsmax = +

3πp2b R2 3p2 R2 (s11p + 2s12p ) + b 2 8
c26 (s11m − s12m ) π − 4c9 R4 .

(

4c28 (s11m + 2s12m )

"

4π 3v

2/3

−π

# (22)

2.1.6 Particle and Matrix Thermal-Stress Strengthening With regard to (Eq. (13)) and σ11 = σ, σii = 0 (i = 2, 3), considering the planes P1 P2 P3 P4 P5 and P6 P7 P8 to be loaded by the stress, σ, constant over the individual plane in the positions x1 h0, Ri and x1 hR, d/2i, respectively, the elastic energy density, wσq , induced by the stress,

Thermal Stresses in Particle-Matrix System and Related Phenomena...

157

Figure 7: The plane P62 P72 P73 P63 k x2 x3 of the cubic cell matrix of the dimension d (Fig. 2), included, along with the point P, in the plane P6 P7 P8 k x2 x3 in the position x1 ∈ hR, d/2i (Fig. 4).

σq , acting along the axis x1 in the particle (q=p) and the cubic cell matrix (q = m), is derived as s11q σ2q . (23) 2 The elastic energy gradient within the spherical particle and the cubic cell along the axis R x1 (Figs. 2-4) is Wsσq = wσq Sq , where S p ≡ P2 P3 P4 , Sm ≡ P1 P2 P4 P5 and Sm ≡ P6 P7 P8 for wσq =

Sq

x1 h0, Ri and x1 hR, d/2i, respectively. The stresses, σ p and σmi , derived from the conditions, Wsσ p = Wsp and Wsσmi = Wsmi (i = 1, 2) (see Eqs. (18)-(21)), and accordingly inducing the same influence as induced by the thermal stresses and thus representing the thermal-stress strengthening within the spherical particle and the cubic cell matrix, as a resistance against compressive, tensile and tensile, compressive mechanical loading for α p > αm , α p < αm , respectively, have the forms s   2s12p , (24) σ p = pb 3 1 + s11p r Wsm1 , (25) σm1 = c16 r Wsm2 , (26) σm2 = c17

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Ladislav Ceniga

where Eqs. (25) and (26) are functions of x1 ∈ h0, Ri (i = 1) and x1 ∈ hR, d/2i (i = 2), respectively, and (σm1 )x1 =R = (σm2 )x1 =R (see Eqs. (19)-(21), (130), (131)). With regard to Eq. (1), the average of thermal-stress strengthening in the cubic cell matrix, σm , is defined as [24]

σm =

=



ZR

2 d 2 R



σm1 dx1 +

0

3v 4π

1/3

Zd/2 R



ZR r



0



σm2 dx1 

Wsm1 dx1 + c16

Zd/2r R

 Wsm2 dx1  , c17

(27)

where the integrals can be derived for a concrete isotropic particle-matrix system by a numerical method. 2.1.7 Curve Integral of Elastic Energy Density The curve integrals within the cubic cell, Wcp , Wcm1 and Wcm2 , as integrals of w p , wm and wm along the abscissae P1 P2 , P2 P3 and P4 P5 in the plane x1 x3 in the positions x1 ∈ h0, Ri and x1 ∈ hR, d/ (2 cos ϕ)i (Fig. 8), respectively, as the elastic energy ‘curve’ density, equivalent to those along the abscissae in the planes x1 x2 , x2 x3 due to the isotropy of the multi-particlematrix system, are derived as

Figure 8: The abscissae P1 P2 P3 and P4 P5 in the plane x12 x3 (Fig. 3) in the positions x1 ∈ h0, Ri and x1 ∈ h0, d/ (2 cosϕ)i, respectively, perpendicular to the axis x12 , where x12 ⊂ x1 x2 and ϕ = C (x12 , x1 ) ∈ h0, π/4i (Fig. 3).

Thermal Stresses in Particle-Matrix System and Related Phenomena... Z

Wcp =

w p dx3 =

P1 P2

Wcm1

3p2b (s11p + 2s12p ) 2

q

R2 − x21 , x1 ∈ h0, Ri,

159

(28)

"   # q 1/3 4π 5c28 (s11m + 2s12m) R = − 2 R2 − x21 3v P2 P3   5      2R (3v)1/3  2   + c6 R (s11m − s12m ) 1 − q ,    R2 (4π)2/3 + 4x21 (3v)2/3  Z

3p2 wm dx3 = b 20

x1 ∈ h0, Ri,

Wcm2 =

Z P4 P5

+

(29)

3p2 R wm dx3 = b 20

c26 (s11m − s12m) 

x1 ∈ R,





R x1

5c28 (s11m + 2s12m)  5  



4π 3v

1/3

5    2x1 (3v)    1− q ,    R2 (4π)2/3 + 4x2 (3v)2/3  

d , 2

1/3

1

(30)

q where ϕ = ∠ (x1 x12 ) ∈ h0, 2πi and the axis x12 ⊂ x1 x2 (Fig. 3); |P1 P2 | = R2 − x21 ; |P2 P3 | = q d/2 − R2 − x21 and |P4 P5 | = d/2 for x1 ∈ h0, Ri and x1 ∈ hR, d/2i, respectively. With regard to the term r−6 in Eq. (15), and due to the isotropy of the multi-particle-matrix system, the elastic energy accumulated in the cubic cell matrix between the points P2 and is equal to that accumulated between the points at P3 , and between the points P4 and P5 , q

radii r = |OP2 | = R and r = |OP3 | = (d/2)2 + x21 , and at the radii r = |OP4 | = x1 and q r = |OP5 | = (d/2)2 + x21 , for x1 ∈ h0, Ri and x1 ∈ hR, d/2i, respectively. Accordingly, the term r−6 dx3 (see Eqs. (15), (28), (29)) is replaced by r−6 dr, where r ∈ hR, |OP3 |i and r ∈ hx1 , |OP5 |i for x1 ∈ h0, Ri and x1 ∈ hR, d/2i, respectively. 2.1.8 Particle and Matrix Crack Formation Provided that pb < 0 and then for αm < α p , or pb > 0 and then for αm > α p (see Eqs. (4)(7)), the spherical particle or the cubic cell matrix are acted by the tensile thermal stresses, or the tensile radial thermal stress and the compressive tangential thermal stresses, respectively, and consequently, as a consequence of releasing of the elastic energy of the thermal stresses, equal circular cracks are assumed to be equivalently formed in the planes x1 x2 ,

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Ladislav Ceniga

x1 x3 , x2 x3 of the spherical particle or the cubic cell matrix (Figs. 9-14), due to the isotropy of the multi-particle-matrix system. Resulting from the isotropy of the multi-particle-matrix system, the formulae (28)-(30) are also related to the plane x01 x3 , and accordingly the particle or matrix cracks, symmetrical with respect to the formation plane x1 x2 (Figs. 9-14), exhibit the same shape in all planes x12 x3 , where the axis x12 ⊂ x1 x2 and ϕ = ∠ (x1 x12 ) ∈ h0, 2πi (Fig. 3). The curve integrals, Wcp and Wcm1 (see Eqs. (28), (29)), represent decreasing and increasing functions of x1 ∈ h0, Ri, respectively, both depending on the parameter R, where (Wcp )x1 =R = 0, and accordingly the sum, Wcp + Wcm1 , is a decreasing, increasing or decreasing-increasing dependence on x1 ∈ h0, Ri, exhibiting a minimum as a function of the parameter R. With regard to the decreasing, increasing or decreasing-increasing x1 dependence of the sum, the circular particle crack is thus formed from the particle centre, O, to the particle surface (Fig. 9), vice versa (Fig. 10), or simultaneously from the particle centre and the particle surface (Fig. 11), resulting from the conditions, (Wcp +Wcm1 )x1 =0 > (Wcm1)x1 =R , (Wcp +Wcm1 )x1 =0 < (Wcm1 )x1 =R or (Wcp +Wcm1 )x1 =0 = (Wcm1 )x1 =R , from which the coefficient, c18 (see Eq. (129)), is determined as c18 > 0, c18 < 0 or c18 = 0, and consequently, the function, f p , represents a decreasing, increasing or decreasing-increasing x1 -dependence, respectively. Similarly, with regard to c18 > 0, c18 < 0 or c18 = 0, the matrix crack is formed from the position x1 = 0 to the position x1 = R (Fig. 12), vice versa (Fig. 13), or simultaneously from the positions x1 = 0 and x1 = R (Fig. 14), respectively. Considering c18 > 0, c18 < 0 and c18 = 0, critical particle radii, as reasons of the formation of the particle and matrix cracks of the infinitesimal length, are determined with respect to positions of the maximum value of Wcp + Wcm1 , then for x1 = 0, x1 = R and x1 = 0 or x1 = R, respectively. The coefficient c18 depends on the spherical particle volume fraction, v, and on the material parameters of the particle (q = p) and the matrix (q = m), s11q , s12q , and considering c18 as a function of v, the critical spherical particle volume fraction, v0 ∈ (0, π/6), related to determination of a direction of the particle cracking, can be derived from the condition, c18 = 0, for a concrete isotropic particle-matrix system by a numerical method. Particle crack formation I After the particle crack formation, the elastic energy, dWp = (Wcp +Wcm1 ) x1 dϕdx1 /3, accumulated in the cubic cell infinitesimal volume of the dimensions x1 dϕ × dx1 × d/2, and being a consequence of the particle cracking of the infinitesimal surface area, dS p = x1 dϕds p , is in the equilibrium state with the energy of an infinitesimal crack surface of the particle, dWcsp = γ p ds p dx1 dϕ, and then dWp = dWcsp [19]-[21], where the multiplication factor 1/3 results from the formation of equal circular particle cracks in the planes x1 x2 , x1 x3 , 2 [19]-[21] is the particle crack surface energy per unit surface x2 x3 (Fig. 1); γ p = s11p KICp q area; KICp is the particle fracture toughness; ds p = dx1 1 + (∂ f p /∂x1 )2 [24] is infinitesimal length of the curve, f p , describing the particle crack shape in the interval x1 ∈ h0, Ri (Figs. 9-11), representing a function of x1 and the parameter R, corresponding to the x1 - and R-dependence of Wcp +Wcm1 . With regard to dWp = dWcsp , we get

Thermal Stresses in Particle-Matrix System and Related Phenomena... 1 ∂ fp =± 2 ∂x1 3s11p KICp

r

2

(Wcp +Wcm1 ) −



2 3s11p KICp

2

,

161

(31)

and accordingly, the energy condition for the particle crack formation, is derived as 2 > 0, Wcp +Wcm1 − 3s11p KICp

(32)

fulfilled for the particle radius greater than critical, as a reason of the particle cracking.

Figure 9: The particle crack formed in the plane x12 x3 for R > R p1 from the particle centre, O, to the particle surface; the particle crack radius, r p , for R ∈ hR p1 , R p2 i. Considering such v when c18 > 0 (see Eq. (129)), the critical particle radii, R p1 and R p2 , as a reason of the particle crack formation from the particle centre, O, to the particle surface, and as a reason of the particle crack tip on the particle surface (Fig. 9), represent roots of Eqs. (33) and (34) in the forms 2 = 0, (Wcp +Wcm1 )x1 =0 − 3s11p KICp

(33)

2 = 0, (Wcp +Wcm1 )x1 =R − 3s11p KICp

(34)

as functions of the variable R, resulting from the condition for the particle crack tip in the particle centre in the position x1 = 0 and on the particle surface in the position x1 = R, (∂ f p /∂x1 )x1 =0 = 0 and (∂ f p /∂x1 )x1 =R = 0, respectively, related to an ideal-brittle particle [19]-[21], where R p1 < R p2 for c18 > 0. With regard to Eqs. (28), (29), (33), (34), R p1 and R p2 are derived as

162

Ladislav Ceniga

1 R p1

= +

1 R p2

( "  # 1/3 1 4π 5c28 (s11m + 2s12m) −2 s11p + 2s12p + 10 3v "  5/3 #)! 3v , c26 (s11m − s12m ) 1 − 32 4π

p2b 2 2s11pKICp

=

+

p2b 2 20s11p KICp

5c28 (s11m + 2s12m)   



4π 3v

(35)

1/3

5    2 (3v)1/3  2   c6 (s11m − s12m ) 1 − q .    (4π)2/3 + 4 (3v)2/3  

(36)

The particle crack radius (Fig. 9), r p , determining the particle crack tip position, representing a root of the equation 2 = 0, Wcp +Wcm1 − 3s11p KICp

(37)

as a function of the variable x1 and the parameter R ∈ hR p1 , R p2 i, resulting from the condition for the particle crack tip in the position x1 = r p , (∂ f p /∂x1 )x1 =r p = 0 (see Eq. (31)), can be derived from Eq. (37) for a concrete isotropic multi-particle-matrix system by a numerical method, where r p = 0 and r p = R p2 for R = R p1 and R = R p2 , respectively. With regard to Eq. (31), the function, f p , describing the particle crack shape in the intervals x1 ∈ h0, r p i and x1 ∈ h0, Ri for R ∈ hR p1 , R p2 i and R ≥ R p2 (Fig. 9), respectively, has the form " # Z r  2 1 2 2 (Wcp +Wcm1 ) − 3s11p KICp dx1 , βp − fp = 2 3s11p KICp

(38)

representing a decreasing function of x1 , and accordingly the sign - in Eq. (31) is considered. The integral can be derived for a concrete isotropic particle-matrix system by a numerical method, and with respect to the boundary conditions given by Eqs. (39) and (40) in the forms x1 = r p ,

f p = 0,

R = R p2 , x1 = R p2 ,

f p = 0,

(39) (40)

related to the particle radius R ∈ hR p1 , R p2 i and R ≥ R p2 , respectively, the integration constant, β p , is derived for R ∈ hR p1 , R p2 i as

Thermal Stresses in Particle-Matrix System and Related Phenomena...

βp =

"Z r



2 (Wcp +Wcm1 )2 − 3s11p KICp

2

dx1

#

,

163

(41)

x1 =r p

and for R ≥ R p2 as βp =

"Z r



2 (Wcp +Wcm1 )2 − 3s11p KICp

2

dx1

#

.

(42)

x1 =R p2 ; R=R p2

Particle crack formation II Considering such v when c18 < 0 (see Eq. (129)), the critical particle radii, R p2 and R p1 (see Eqs. (35), (36)), are a reason of the particle crack formation from the particle surface in the position x1 = R to the particle centre, O, and a reason of the particle crack tip in the particle centre in the position x1 = 0 (Fig. 10), respectively, where R p2 < R p1 for c18 < 0.

Figure 10: The particle crack formed in the plane x12 x3 (Fig. 8) for R > R p2 from the particle surface to the particle centre, O; the particle crack radius, r p , for R ∈ hR p2 , R p1 i. With regard to Eq. (31), the function, f p , describing the particle crack shape in the intervals x1 ∈ hr p , Ri and x1 ∈ h0, Ri for R ∈ hR p2 , R p1 i and R ≥ R p1 (Fig. 10), respectively, has the form " # Z r  2 1 2 2 (Wcp +Wcm1 ) − 3s11p KICp dx1 , βp + (43) fp = 2 3s11p KICp representing an increasing function of x1 , and accordingly the sign + in Eq. (31) is considered. With respect to the boundary conditions given by Eqs. (44) and (45) in the forms

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Ladislav Ceniga

x1 = r p ,

f p = 0,

(44)

R = R p1 , x1 = 0, f p = 0,

(45)

related to the particle radius R ∈ hR p2 , R p1 i and R ≥ R p1 , respectively, the integration constant, β p , is derived for R ∈ hR p2 , R p1 i as "Z r #  2 2 (Wcp +Wcm1 )2 − 3s11p KICp dx1 , (46) βp = − x1 =r p

and for R ≥ R p1 as βp = −

"Z r

2



2 (Wcp +Wcm1 ) − 3s11pKICp

2

dx1

#

.

(47)

x1 =0; R=R p1

The particle crack radius (Fig. 11), r p (see Eq. (46)), represents a root of Eq. (37), as a function of the variable x1 and the parameter R ∈ hR p2 , R p1 i, and can be derived from Eq. (37), along with the integral in Eq. (43), for a concrete isotropic multi-particle-matrix system by a numerical method, where r p = 0 and r p = R p1 for R = R p2 and R = R p1 , respectively. Particle crack formation III Considering such v when c18 = 0 (see Eq. (129)), the critical particle radii, R p1 , R p2 and R p1/2 , are a reason of the particle crack formation simultaneously from the particle centre in the position x1 = 0 and from the particle surface in the position x1 = R, and a reason of the particle crack tip in the position x1 = R/2 (Fig. 11), where R p1 = R p2 for c18 = 0, and R p1/2 > R p1 represents a root of Eqs (33) as a function of the variable R on the condition x1 = R/2 in the form 1 R p1/2

=

+

p2b 2 20s11pKICp

5c28 (s11m + 2s12m)   

"

4π 3v

1/3

√ − 3

#

5    2 (3v)  2   q c6 (s11m − s12m) 1 − .    (4π)2/3 + (3v)2/3  

1/3

(48)

The particle crack is described by the functions, f p1 and f p2 , decreasing and increasing in the intervals x1 ∈ h0, r p1 i, x1 ∈ h0, r p i and x1 ∈ hr p2 , Ri, x1 ∈ hr p , Ri (Fig. 11), respectively, related to the disconnected, interconnected particle cracks, as depended on the parameter R. The particle crack radii, r p1 , r p2 , and the position in which the function, f p = f p1 + f p2 , describing the interconnected particle cracks exhibits a minimum, r p , represent roots of

Thermal Stresses in Particle-Matrix System and Related Phenomena... 165

 Eq. (37), as a function of the variable x1 and the parameter R, for R ∈ R p1 , R p1/2 and R ≥ R p1/2 , respectively, and can be derived from Eq. (37) for a concrete isotropic multiparticle-matrix system by a numerical method, where r p1 = r p2 = 0 and r p1 = r p2 = R p1/2 /2 for R = R p1 and R = R p1/2 , respectively.

Figure 11: The particle crack simultaneously formed in the plane x12 x3 (Fig. 8) for R > R p1 = R p2 from the particle centre, O, to the particle surface, and vice versa; the particle crack radii, r p1 

and r p2 , of the disconnected cracks for R ∈ R p1 , R p1/2 ; the position, r p , of the minimum of the interconnected cracks for R ≥ R p1/2 .

With respect to the boundary conditions given by Eqs. (49), (50) and (51) in the forms x1 = r p1 ,

f p1 = 0,

(49)

x1 = r p2 ,

f p2 = 0,

(50)

R p1/2 , f p1 = f p2 = 0, (51) R = R p1/2 , x1 = 2 

related to the particle radius R ∈ R p1 , R p1/2 and R ≥ R p1/2 , respectively, the functions, f p1 and f p2 , are given by Eqs. (38), (41), (42), and Eqs.

(43), (46),  (47), on the conditions x1 = r p1 and x1 = r p2 in Eqs. (41) and (46) for R ∈ R p1 , R p1/2 , respectively, and on the condition x1 = R p1/2 , R = R p1/2 in Eqs. (42), (47) for R ≥ R p1/2 .

166

Ladislav Ceniga

Matrix crack formation I Considering the particle-matrix boundary adhesion radial stress, pa , provided that pb > 0 and then for αm < α p , the ca -multiple of σrm , σϕm , σνm , and c2a -multiple of w p , wm , the latter resulted from the dependence, w ∝ σ2 (see Eq. (13)), are concerned in the matrix crack formation, where the multiplier, ca , is derived as ca = 1 −

pa . pb

(52)

After the matrix crack formation, the elastic energy related to the positions x1 ∈ h0, Ri and x1 ∈ h0, d/2i, dWm = c2a (Wcp +Wcm1 ) x1 dϕdx1 /3 and dWm = c2aWcm2 x1 dϕdx1 /3, respectively, accumulated in the cubic cell infinitesimal volume of the dimensions x1 dϕ × dx1 × d/2, and being a consequence of the matrix cracking of the infinitesimal surface area, dSm = x1 dϕdsm , is in the equilibrium state with the energy of an infinitesimal crack surface of the matrix, dWcsmi = γm dsmi dx1 dϕ (i = 1, 2), and then dWm = dWcsmi [19]-[21], where the multiplication factor 1/3 results from the formation of equal circular particle cracks in the 2 [19]-[21] is the matrix crack planes x1 x2 , x1 x3 , x2 x3 (Fig. 1); γm = s11mKICm q surface energy

per unit surface area; KICm is the matrix fracture toughness; dsmi = dx1 1 + (∂ fmi /∂x1 )2 [24] is infinitesimal length of the curves, fm1 and fm2 , describing the matrix crack shape in the intervals x1 ∈ h0, Ri and x1 ∈ hR, d/ (2 cos ϕ)i (Fig. 12), respectively, representing functions of x1 and the parameter R, corresponding to the x1 - and R-dependence of Wcm2 , where ϕ = ∠ (x1 x12 ) ∈ h0, 2πi and the axis x12 ⊂ x1 x2 (Fig. 3). With regard to dWm = dWcsmi, we get 1 ∂ f m1 =± 2 ∂x1 3s11mKICm

q


2 2 c2a (Wcp +Wcm1 )2 − 3s11mKICm ,

1 ∂ f m2 =− 2 ∂x1 3s11mKICm

q


2 2 − 3s 2 c2aWcm2 11m KICm ,

(53)

(54)

and accordingly, the energy condition for the particle crack formation, is derived in the interval x1 ∈ h0, Ri as 2 > 0, ca (Wcp +Wcm1 ) − 3s11mKICm

(55)

and in the interval x1 ∈ hR, d/ (2 cos ϕ)i as 2 > 0, caWcm2 − 3s11m KICm

(56)

both fulfilled for the particle radius greater than critical, as a reason of the matrix cracking. Considering such v when c18 > 0 (see Eq. (129)), the critical particle radii, Rm1 and Rm2 , as a reason of the matrix crack formation from the position x1 = 0 to the position x1 > 0, and as a reason of the matrix crack tip in the position x1 = R (Fig. 12), represent roots of Eqs. (57) and (58) in the forms

Thermal Stresses in Particle-Matrix System and Related Phenomena...

167

2 ca (Wcp +Wcm1 )x1 =0 − 3s11mKICm = 0,

(57)

2 = 0, ca (Wcm2 )x1 =R − 3s11mKICm

(58)

as functions of the variable R, resulting from the conditions for the matrix crack tip, (∂ f m1 /∂x1 )x1 =0 = 0 and (∂ fm1 /∂x1 )x1 =R = 0, respectively, related to an ideal-brittle particle [19]-[21], where Rm1 < Rm2 for c18 > 0. With regard to Eqs. (28), (29), (57), (58), Rm1 and Rm2 are derived as

Figure 12: The matrix cracks formed in the plane x12 x3 (Fig. 8) from the position x1 = 0 to the position x1 ∈ h0, rm1 i, x1 ∈ h0, Ri and from the position x1 = R to the position x1 ∈ hR, rm2 i, x1 ∈ hR, d/ (2 cosϕ)i for R ∈ hRm1 , Rm2 i, R ≥ Rm2 and R ∈ hRm2 , Rm3 i, R ≥ Rm3 , respectively; the matrix crack radii, rm1 , rm2 .

1 Rm1

= +

( "  # 1 4π 1/3 2 5c8 (s11m + 2s12m ) −2 s11p + 2s12p + 10 3v "  5/3 #)! 3v 2 . (59) c6 (s11m − s12m ) 1 − 32 4π

ca p2b 2 2s11mKICm

1 Rm2

=

+

ca p2b 2 20s11mKICm

5c28 (s11m + 2s12m)   



4π 3v

1/3

5    2 (3v) 2   c6 (s11m − s12m) 1 −  q ,   2/3 2/3   (4π) + 4 (3v) 

1/3

(60)

168

Ladislav Ceniga

The matrix crack radius (Fig. 12), rm1 , determining the matrix crack tip position, representing a root of the equation 2 ca (Wcp +Wcm1 ) − 3s11mKICm = 0,

(61)

as a function of the variable x1 and the parameter R ∈ hRm1 , Rm2 i, resulting from the condition for the particle crack tip in the position x1 = rm1 , (∂ fm1 /∂x1 )x1 =rm1 = 0 (see Eq. (53)), can be derived from Eq. (61) for a concrete isotropic multi-particle-matrix system by a numerical method, where rm1 = 0 and rm1 = Rm2 for R = Rm1 and R = Rm2 , respectively. With regard to Eq. (53), the function, f m , describing the matrix crack shape in the intervals x1 ∈ h0, rm1 i and x1 ∈ h0, Ri for R ∈ hRm1 , Rm2 i and R ≥ Rm2 (Fig. 12), respectively, has the form   Z q
2 1 2 2 2 ca (Wcp +Wcm1 ) − 3s11mKICm dx1 , βm1 − (62) f m1 = 2 3s11mKICm representing a decreasing function of x1 , and accordingly the sign - in Eq. (53) is considered. The integral can be derived for a concrete isotropic particle-matrix system by a numerical method, and with respect to the boundary conditions given by Eqs. (63) and (64) in the forms q 2 , (63) x1 = rm1 , fm1 = R2 − rm1 x1 = R, ( fm1 )x1 =R = ( fm2 )x1 =R ,

(64)

related to the particle radius R ∈ hRm1 , Rm2 i and R ≥ Rm2 , respectively, the integration constant, βm1 , is derived for R ∈ hRm1 , Rm2 i as βm1 = +

2 3s11mKICm

Z q

q

2 R2 − rm1

c2a (Wcp +Wcm1 )2 −


2 2 3s11mKICm dx1



,

(65)

x1 =rm1

and for R ≥ Rm2 as Z q


2 2 − 3s 2 c2aWcm2 βm1 = βm2 − 11m KICm   q
2 2 2 2 − ca (Wcp +Wcm1 ) − 3s11mKICm dx1

,

(66)

x1 =R

where the function, f m2 , and the integration constant, βm2 , are given by Eqs. (81) and (84), (85), respectively.

Thermal Stresses in Particle-Matrix System and Related Phenomena...

169

Matrix crack formation II Considering such v when c18 < 0 (see Eq. (129)), the critical particle radii, Rm2 and Rm1 , are a reason of the matrix crack formation from the position x1 = R to the position x1 < R, and a reason of the matrix crack tip in the position x1 = 0 (Fig. 13), respectively, where Rm2 < Rm1 for c18 < 0. With regard to Eq. (53), the function, fm1 , describing the matrix crack shape in the intervals x1 ∈ h0, rm1 i and x1 ∈ h0, Ri for R ∈ (Rm2 , Rm1 ) and R ≥ Rm1 (Fig. 13), respectively, has the form f m1 =

  Z q
2 1 2 2 2 (W +W + c ) − 3s K dx β m1 cp cm1 11m ICm 1 , a 2 3s11mKICm

(67)

representing an increasing function of x1 , and accordingly the sign + in Eq. (53) is considered. With respect to the boundary conditions given by Eqs. (63) and (64) related to the particle radius R ∈ hRm2 , Rm1 i and R ≥ Rm1 , respectively, the integration constant, βm1 , is derived for R ∈ hRm2 , Rm1 i as βm1 = −

2 3s11mKICm

Z q

q

2 R2 − rm1

c2a (Wcp +Wcm1 )2 −


2 2 3s11mKICm dx1



,

(68)

x1 =rm1

and for R ≥ Rm1 as Z q


2 2 − 3s 2 c2aWcm2 βm1 = βm2 − 11m KICm   q
2 2 2 2 + ca (Wcp +Wcm1 ) − 3s11mKICm dx1

,

(69)

x1 =R

where the function, f m2 , and the integration constant, βm2 , are given by Eqs. (81) and (84), (85), respectively. The matrix crack radius (Fig. 13), rm1 (see Eq. (68)), represents a root of Eq. (61), as a function of the variable x1 and the parameter R ∈ hRm2 , Rm1 i, and can be derived from Eq. (61), along with the integral in Eq. (67), for a concrete isotropic multiparticle-matrix system by a numerical method, where rm1 = 0 and rm1 = Rm1 for R = Rm2 and R = Rm1 , respectively. Matrix crack formation III Considering such v when c18 = 0 (see Eq. (129)), the critical particle radii, Rm1 , Rm2 and Rm1/2 , are a reason of the matrix crack formation simultaneously from the positions x1 = 0, x1 = R to the positions x1 > 0, x1 < R, respectively, and a reason of the matrix crack tip in the position x1 = R/2 (Fig. 14), respectively, where Rm1 = Rm2 for c18 = 0, and Rm1/2>Rm1

170

Ladislav Ceniga

Figure 13: The matrix cracks formed in the plane x12 x3 (Fig. 8) from the position x1 = R to the positions x1 ∈ hrm1 , Ri, x1 ∈ h0, Ri and x1 ∈ hR, rm2 i, x1 ∈ hR, d/ (2 cosϕ)i for R ∈ hRm2 , Rm1 i, R ≥ Rm1 and R ∈ hRm2 , Rm3 i, R ≥ Rm3 , respectively; the matrix crack radii, rm1 , rm2 .

represents a root of Eq. (57) as a function of the variable R on the condition x1 = R/2 in the form 1 Rm1/2

=

+

ca p2b 2 4s11mKICm

√ 3(s11p + 2s12p ) + c28 (s11m + 2s12m)   

"

5    2 (3v)   (s11m − s12m) 1 −  q .   5 2/3 2/3   (4π) + (3v)

c26



4π 3v

1/3

√ − 3

#

1/3

(70)

The matrix crack is described by the functions, f m11 and fm12 , decreasing and increasing in the intervals x1 ∈ h0, rm11 i, x1 ∈ h0, rm1 i and x1 ∈ hrm12 , Ri, x1 ∈ hrm1 , Ri (Fig. 14), respectively, related to the disconnected, interconnected matrix cracks, as depended on the parameter R. The matrix crack radii, rm11 , rm12 , and the position in which the function, f m1 = fm11 + fm12 , describing the interconnected particle cracks exhibits a minimum, rm1 , represent roots  of Eq. (61), as a function of the variable x1 and the parameter R, for R ∈ Rm1 , Rm1/2 and R ≥ Rm1/2 , respectively, and can be derived from Eq. (61) for a concrete isotropic multi-particle-matrix system by a numerical method, where rm11 = rm12 = 0 and rm11 = rm12 = Rm1/2 /2 for R = Rm1 and R = Rm1/2 , respectively. With regard to Eq. (53), the functions, f m11 and fm12 , have the forms   Z q
2 1 2 2 2 ca (Wcp +Wcm1 ) − 3s11mKICm dx1 , βm11 − (71) f m11 = 2 3s11mKICm

Thermal Stresses in Particle-Matrix System and Related Phenomena...

171

Figure 14: The matrix cracks simultaneously formed in the plane x12 x3 (Fig. 8) from the positions x1 = 0 and x1 = R to the positions x1 ∈ h0, rm11 i, x1 ∈ h0, rm1 i and x1 ∈ hrm12 , Ri, x1 ∈ hrm1 , Ri for

 R ∈ Rm1 , Rm1/2 , R ≥ Rm1/2 , respectively, and from the position x1 = R to the position x1 ∈ hR, rm2 i, x1 ∈ hR, d/ (2 cos ϕ)i for R ∈ hRm2 , Rm3 i, R ≥ Rm3 , respectively; the matrix crack radii, rm11 , rm12 , of the disconnected cracks; the position, rm1 , of the minimum of the interconnected cracks.

  Z q
2 1 2 2 2 fm12 = ca (Wcp +Wcm1 ) − 3s11mKICm dx1 , βm12 + 2 3s11mKICm

(72)

representing increasing and decreasing function of x1 , and accordingly the sign - and + in Eqs. (71) and (75) are considered, respectively. The integrals can be derived for a concrete isotropic particle-matrix system by a numerical method, and with respect to the boundary conditions given by Eqs. (73), (74) and (75), (79) in the forms q 2 , (73) x1 = rm11 , fm11 = R2 − rm11 x1 = rm12 ,

fm12 =

q 2 , R2 − rm12

x1 = rm1 , fm11 = fm12 ,

(74) (75)

(76) x1 = rm1 , fm11 = fm12 , 

related to the particle radius R ∈ Rm1 , Rm1/2 and R ≥ Rm1/2 , respectively, the integration constants, βm11 and βm12 , are given by Eqs.

(38) and (47), replacing the term rm1 by the terms rm11 and rm12 , respectively, for R ∈ Rm1 , Rm1/2 , and derived for R ≥ Rm1/2 as

172

Ladislav Ceniga Z q


2 2 3s11mKICm dx1

β11m = βm2 + 2 Z q 
2 2 2 2 − ca (Wcp +Wcm1 ) − 3s11mKICm dx1 x1 =R Z q 
2 2 − 3s 2 − c2aWcm2 , 11m KICm dx1 c2a (Wcp +Wcm1 )2 −

 x1 =rm1

(77)

x1 =R

Z q


2 2 3s11mKICm dx1

βm12 = βm2 − Z q 
2 2 2 2 − caWcm2 − 3s11mKICm dx1 c2a (Wcp +Wcm1 )2 −

 x1 =R

,

(78)

x1 =R

where the function, f m2 , and the integration constant, βm2 , are given by Eqs. (81) and (84), (85), respectively. Matrix crack formation IV With regard to the function, f m2 , describing the matrix crack shape in the interval x1 ∈ hR, d/ (2 cos ϕ)i (Figs. 3, 13, 14), the critical particle radius, Rm2 , given by Eq. (60), is also a reason of the matrix crack formation from the position x1 = R to the position x1 > R. Consequently, the critical particle radius, Rm3 , resulting from the condition for the matrix crack tip on the cell surface in the position x1 = d/ (2 cos ϕ), (∂ fm2 /∂x1 )x1 =d/(2 cosϕ) = 0, related to an ideal-brittle matrix [19]-[21], corresponding to a connection of the matrix cracks in neighbouring cells between the points C1 and C12 (Fig. 3), where Rm2 < Rm3 , represents a root of Eq. (58) as a function of the variable R on the condition x1 = d/ (2 cos ϕ), and regarding Eq. (1) is derived as an increasing function of the angle ϕ in the form 1 Rm3

= +

ca p2b 2 20s11mKICm

(

5c28 (s11m + 2s12m) 

3v 32c26 cos5 ϕ (s11m − s12m ) 4π



5/3

4π 3v "

1/3

1−

1 (1 + cos2 ϕ)5/2

#)

.

(79)

The matrix crack radius (Figs. 13, 14), rm2 , determining the matrix crack tip position, representing a root of the equation 2 = 0, caWcm2 − 3s11m KICm

(80)

as a function of the variable x1 and the parameter R ∈ hRm2 , Rm3 i, resulting from the condition for the particle crack tip in the position x1 = rm2 , (∂ fm2 /∂x1 )x1 =rm2 = 0 (see Eq. (54)),

Thermal Stresses in Particle-Matrix System and Related Phenomena...

173

can be derived from Eq. (80) for a concrete isotropic multi-particle-matrix system by a numerical method. With regard to Eq. (54), the function, fm2 , describing the matrix crack shape in the intervals x1 ∈ hR, rm2i and x1 ∈ hR, d/ (2 cos ϕ)i for R ∈ hRm2 , Rm3 i and R ≥ Rm3 (Figs. 13, 14), respectively, has the form   Z q
2 1 2 2W 2 − 3s − c K dx β (81) f m2 = m2 11m ICm 1 . a cm2 2 3s11mKICm The integral can be derived for a concrete isotropic particle-matrix system by a numerical method, and with respect to the boundary conditions given by Eqs. (82) and (83) in the forms x1 = rm2 ,

R = Rm3 , x1 =

fm2 = 0,

d , 2 cos ϕ

(82)

fm2 = 0,

(83)

related to the particle radius R ∈ hRm2 , Rm3 i and R ≥ Rm3 , respectively, the integration constant, βm1 , is derived for R ∈ hRm2 , Rm3 i as Z q 
2 2 2 2 caWcm2 − 3s11mKICm dx1 , (84) βm2 = x1 =rm2

and for R ≥ Rm3 as βm2 =

Z q

2 − c2aWcm2


2 2 3s11mKICm dx1



.

(85)

x1 =d/(2 cos ϕ); R=Rm3

2.2 Isotropic One-Particle-Matrix System As presented in the section 3, concerning the v-dependences of thermal-stress related parameters of the SiC-Si3 N4 particle-matrix system in the interval v ∈ h0, π/6), the formulae related to the isotropic one-particle-matrix system represented by one spherical particle of the radius R embedded in the infinite matrix, as transformation of those related to the isotropic multi-particle-matrix system are for v = 0, are required to be derived. 2.2.1 Thermal Stresses The radial and tangential stresses, acting in the spherical particle ( q = p) for r ∈ h0, Ri and in the infinite matrix (q = m) for r ∈ hR, ∞), σrq and σϕq , σνq , respectively, have the forms [25] σrp = σϕp = σνp = −pb ,

(86)

174

Ladislav Ceniga  3 R σrm = −2σϕm = −2σνm = −pb , r

(87)

where σ > 0 and σ < 0 represent the tensile and compressive stresses, respectively. The compressive or tensile particle-matrix boundary radial stress, pb > 0 or pb < 0, respectively, is derived as [25] 1 pb = c19

ZTi

(αm − α p ) dT,

(88)

T

and Eqs. (4)-(7), (112)-(115) are transformed for v = 0 to Eqs. (86)-(88), (130), for RTi

(pc )v=0 = 0. Considering αq to be temperature-independent, the integral αq dT is replaced T

by αq (Ti − T ). 2.2.2 Temperature Range of Cooling Process

With regard to the section 2.1.3, the presented calculation is considerable in the temperature range T ∈ hTc , Ti i, and the critical final temperature of a cooling process, Tc , can be derived from Ti 1 Z − σyq = 0, (α − α )dT (89) m p c19 T c

for a concrete isotropic multi-particle-matrix system by a numerical method, and considering αq (q = p, m) to be temperature-independent, Tc has the form c19 . (90) Tc = Ti − σyq (αm − α p ) 2.2.3 Elastic Energy With regard to Eq. (13), and consequently to σi j = 0 (i 6= j) and the subscript transformations, 11 → r, 22 → ϕ, 33 → ν, the elastic energy density and the elastic energy of the thermal stresses in the infinite matrix, wm and Wm , respectively, are derived as  6 3p2b R (s11m − s12m) , wm = 4 r Wm =

Z Vm

wm dVm =

Z2πZ∞

wm r2 drdΦ = πp2b R3 (s11m − s12m) ,

(91)

(92)

0 R

where dVm = r2 drdΦ represent an infinitesimal volume of the matrix, and Φ ∈ h0, 4πi.

Thermal Stresses in Particle-Matrix System and Related Phenomena...

175

The elastic energy density and the elastic energy of the thermal stresses in the spherical particle, w p and Wp , respectively, are given by Eqs. (14), (16), considering the particlematrix boundary radial stress, pb , derived by Eqs. (88), (130). The elastic energy of the cubic cell is then W = Wp +Wm . 2.2.4 Surface Integral of Elastic Energy Density The surface integral, Wsp , of w p (see Eg. (14)) over the surface P2 P3 P4 k x2 x3 (Fig. 15) is given by Eq. (18), considering the particle-matrix boundary radial stress, pb , derived by Eqs. (88), (130). Determining the surface integral of the elastic energy density, the spherical particle is considered to be embedded in the spherical cell of the radius Rc (Fig. 15), and derived formulae then represent transformations of the integrals for Rc → ∞.

Figure 15: The planes P1 P2 P3 P4 P5 k x2 x3 and P6 P7 P8 k x2 x3 in the positions x1 ∈ h0, Ri and x1 ∈ hR, Rc i, respectively, in the spherical cell of the radius Rc → ∞, containing the spherical particle of the radius R.

The surface integral, Wsm1 , of wm (see Egs. (88), (91), (130)) over the surface P1 P2 P4 P5 k x2 x3 (Fig. 15), representing the elastic energy gradient within the spherical cell matrix of the radius Rc → ∞ along the axis x1 ∈ h0, Ri, Wsm1 = ∂Wm/∂x1 , as the elastic energy ‘surface’ density, equivalent to that along the axes x2 , x3 due to the isotropy of the one-particle-matrix system, has the form

Wsm1 =

Z Sm

wm dSm = 4

Zπ/2ZR1 0 R2

wm r23 dr23 dξ

176 =

Ladislav Ceniga "  4 # 2 2 3πp2b R2 R Rc →∞ 3πpb R (s11m − s12m ) 1 − (s11m − s12m) , = 8 Rc 8 x1 ∈ h0, Ri,

(93)

and then, for x1 ∈ h0, Ri, dSq = r23 dr23 dξ is an infinitesimal part in the point P of the spherical particle (q = p) or the spherical cell matrix (q = m) on the surfaces S p ≡ P2 P31 P32 k x2 x3 or Sm ≡ P1 P2 P32 P33 k x2 x3 , P2 P31 P32 , P1 P2 P32 P33 ⊂ P1 P2 P3 P4 P5 (Fig. 15), for r23 ∈ h0, R2i or r23 ∈ hR2 , R1 i (Fig. 16), respectively, where r23 = |P31 P|,qP31 P k x2 x3 , R = |OP2 | =

|OP32 |, Rc = |OP1 | = |OP33 | (Fig. 5), R1 = |P31 P1 | = |P31 P33 | = R2c − x21 , R2 = |P31 P2 | = q |P31 P32 | = R2p − x21 , P31 P1 , P31 P2 k x2 x3 ; and the coordinate r in Eq. (91) is derived as q 2 + x2 . Due to the isotropy of the particle-matrix system, the angle ξ is r = |OP| = r23 1 sufficient to be varied in the interval ξ ∈ h0, π/2i.

Figure 16: The planes P2 P31 P32 k x2 x3 and P1 P2 P32 P33 P23 k x2 x3 of the spherical particle of the radius R and of the spherical cell matrix of the radius Rc → ∞, included, along with the point P, in the plane P1 P2 P3 P4 P5 k x2 x3 in the position x1 ∈ h0, Ri (Fig. 15).

The surface integral, Wsm2 , of wm (see Eqs. (88), (91)) over the surface P6 P7 P8 k x2 x3 (Fig. 15), representing the elastic energy gradient within the spherical cell of the radius Rc → ∞ along the axis x1 ∈ hR, Rci (Fig. 1), Wsm2 = ∂Wm /∂x1 , equivalent to those along the axes x2 , x3 , has the form

Wsm2

=

Z Sm

wsm dSm = 4

Zπ/2ZR68

wm r68 dr68 dξ

0

0

Thermal Stresses in Particle-Matrix System and Related Phenomena...  4 #  4 " 3R2 πp2b R R 1− (s11m − s12m ) = 8 x1 Rc   2 2 R 4 Rc →∞ 3πpb R (s11m − s12m ) = , x1 ∈ hR, Rci , Rc → ∞, 8 x1

177

(94)

and then, for x1 ∈ hR, Rci, dSm = r68 dr68 dξ is an infinitesimal part in the point P of the spherical cell matrix on the surface Sm ≡ P62 P72 P73 P63 k x2 x3 (Fig. 17) for r68 = |P72 P| ∈ |P72 P68 | = h0, q R68i, respectively, where P62 P72 P73 P63 ⊂ P6 P7 P8 (Fig. 15); P72 P k x2 x3 ; R68 =q

2 + x2 . R2c − x21 ; P72 P68 k x2 x3 ; and the coordinate r in Eq. (91) is derived as r = |OP| = r68 1 Due to the isotropy of the particle-matrix system, the angle ξ is sufficient to be varied in the interval ξ ∈ h0, π/2i.

Figure 17: The plane P6 P72 P73 k x2 x3 of the spherical cell matrix of the radius Rc → ∞, included, along with the point P, in the plane P6 P7 P8 k x2 x3 in the position x1 ∈ hR, Rc i (Fig. 15).

Consequently, the maximum, Wsmax = (Wsp +Wsm1 )x1 =0 , derived from Eqs. (18), (93) for x1 = 0, has the form Wsmax =

3R2 πp2b [4 (s11p + 2s12p ) + s11m − s12m ]. 8

(95)

2.2.5 Particle and Matrix Thermal-Stress Strengthening The elastic energy gradients within the spherical cell matrix along the axis x1 , Wsσm1 and Wsσm2 , equivalent to those along the axes x2 , x3 due to the isotropy of the one-particle-matrix system, induced by the stresses, σm1 and σm2 , constant on the plane perpendicular to the

178

Ladislav Ceniga

axis x1 , representing integrals of wσm over the surfaces P2 P3 P4 and P1 P2 P4 P5 for x1 h0, Ri and x1 hR, Rci, respectively, have the forms Wsσm1 =

Z

wσm dSm =


πs11mσ2 2 Rc − R2 , x1 h0, Ri, Rc → ∞, 2

(96)

wσm dSm =


πs11mσ2 2 Rc − x21 , x1 hR, Rci , Rc → ∞. 2

(97)

Sm

Z

Wsσm2 =

Sm

With regard to Rc → ∞, the thermal-stress strengthening in the infinite matrix of the isotropic one-particle-matrix system, σmi (i = 1, 2), derived from the condition, Wsσmi = Wsmi , is derived as σmi = 0,

(98)

and the thermal-stress strengthening in the spherical particle of the isotropic one-particlematrix system, σ p , is given by Eq. (24), considering the particle-matrix boundary radial stress, pb , derived by Eqs. (88), (130). 2.2.6 Curve Integral of Elastic Energy Density The curve integrals within the spherical cell of the radius Rc → ∞, Wcm1 and Wcm2 , as integrals of wm along the abscissae P4 P5 and P7 P8 in the plane x1 x3 in the positions x1 ∈ h0, Ri and x1 ∈ hR, Rci (Fig. 15), respectively, as the elastic energy ‘curve’ density, equivalent to those along the abscissae in the planes x1 x2 , x2 x3 due to the isotropy of the multi-particlematrix system, are derived as Z

=

Wcm1

P4 P5 Rc →∞

=

Wcm2

=

Z P7 P8

Rc →∞

=

"  5 # 3Rp2b R (s11m − s12m) 1 − wm dx3 = 20 Rc

3Rp2b (s11m − s12m) , 20

(99)

 5 "  5 # 3Rp2b R x1 (s11m − s12m) wm dx3 = 1− 20 x1 Rc

 5 3Rp2b R (s11m − s12m) . 20 x1

(100)

With regard to the term r−6 in Eq. (91), and due to the isotropy of the one-particle-matrix system, the elastic energy accumulated in the cubic cell matrix between the points P2 and P3 , and between the points P4 and P5 , is equal to that accumulated between the points at

Thermal Stresses in Particle-Matrix System and Related Phenomena...

179

radii r = |OP2 | = R and r = |OP3 | = Rc, and at the radii r = |OP4 | = x1 and r = |OP5 | = Rc , for x1 ∈ h0, Ri and x1 ∈ hR, Rci, respectively. Accordingly, the term r−6 dx3 (see Eqs. (91), (99), (100)) is replaced by r−6 dr, where r ∈ hR, |OP3 |i and r ∈ hx1 , |OP4 |i for x1 ∈ h0, Ri and x1 ∈ hR, Rci, respectively. The curve integral within the spherical particle, Wcp , as an integral of w p along the abscissae P3 P4 in the plane x1 x3 in the position x1 ∈ h0, Ri (Fig. 15), is given by Eq. (28), considering the particle-matrix boundary radial stress, pb , derived by Eqs. (88), (130). 2.2.7 Particle and Matrix Crack Formation With regard to Eqs. (28), (99), Wcp + Wcm1 and Wcm1 represent decreasing function of x1 , depending on the parameter R, and consequently the particle and matrix cracks exhibit shapes as shown in Figs. 9, 12. Particle crack formation The critical particle radii, R p1 and R p2 , where R p2 > R p1 , as a reason of the particle crack formation from the particle centre, O, in the position x1 = 0, to the particle surface (Fig. 9), and as a reason of the particle crack tip on the particle surface in the position x1 = R, represent roots of Eqs. (33) and (34) in the forms R p1 =

2 20s11p KICp

p2b [10 (s11p + 2s12p ) + s11m − s12m] R p2 =

2 20s11p KICp

p2b (s11m − s12m)

,

.

(101)

(102)

The particle crack radius (Fig. 9), r p , determining the particle crack tip position, representing a root of Eq. (37), as a function of the variable x1 and the parameter R ∈ hR p1 , R p2 i, is derived as

rp =

  

R2 −

1 (s11p + 2s12p )2

"

2 2s11p KICp p2b

#2 1/2  R − (s11m − s12m ) ,  10

(103)

and the function, f p , describing the particle crack shape in the intervals x1 ∈ h0, r p i and x1 ∈ h0, Ri for R ∈ hR p1 , R p2 i and R ≥ R p2 (Fig. 9), is given by Eqs. (38), (41) and (42), respectively. Matrix crack formation The critical particle radii, Rm1 and Rm2 , as a reason of the matrix crack formation from the position x1 = 0 to the position x1 = R, and as a reason of the matrix crack tip in the position x1 = R to the position x1 > R (Fig. 12), represent roots of Eqs. (57) and (58) in the forms Rm1 =

2 20s11mKICm , ca p2b (s11m − s12m )

(104)

180

Ladislav Ceniga

Rm2 =

2 20s11mKICm . ca p2b [10 (s11p + 2s12p ) + s11m − s12m]

(105)

The matrix crack radii (Fig. 12), rm1 ∈ h0, Ri and rm2 > R, both for R > Rm , representing roots of Eqs. (61) and (80), as functions of the variable x1 and the parameter R, for R ∈ hRm1 , Rm2 i and R ≥ Rm2 , respectively, are derived as rm1 =

(

R2 −



1 (s11p + 2s12p )2 

2 )1/2

2 2s11mKICm R − (s11m − s12m) 2 10 ca pb

ca R6 p2b (s11m − s12m ) rm2 = 2 20s11mKICm

1/5

,

,

(106)

(107)

and the matrix crack (Fig. 12) is thus described by Eqs. (62), (65), (66) and (81), (84) for x1 ∈ h0, rm1 i and x1 ∈ hR, rm2i, related to R ≥ Rm1 and R ≥ Rm2 , respectively.

3

Thermal Stresses and Related Phenomena in SiC-Si3 N4 Particle-Matrix System

Compared to high strength and wear resistance, ceramic materials are characterized by low fracture toughness, usually strengthened by addition of particles of higher thermal expansion coefficient than that of a matrix. Increasing fracture toughness of the Si 3 N4 matrix by addition of the SiC particles of higher thermal expansion coefficient than that of the Si 3 N4 matrix, α p > αm (Tab. 1), the SiC particle and the Si3 N4 matrix are thus acted by the tensile thermal stresses, σrp = σϕp = σνp = −pb > 0 (see Eqs. (4), (7)) (Fig. 18), and by tensile and compressive radial and tangential thermal stresses, σrm > 0 and σϕm = σνm < 0 (see Eqs. (5)-(7)) (Fig. 19), respectively. Representing a resistance against tensile mechanical loading, the tensile radial thermal stress in the Si 3 N4 cubic cell matrix, as a reason of the Si 3 N4 fracture toughness increase, and the compressive tangential radial stresses, both decrease within the cubic cell matrix (Fig. 19), the former from the maximal value on the particle-matrix boundary, (σrm )r=R = −pb , where the particle-matrix boundary radial stress, −pb , as a function of the spherical particle volume fraction, v ∈ h0, π/6), exhibits the maximum for the critical particle volume fraction, vc = 0.132 (Fig. 18). Similarly, the tensile thermal stresses in the SiC particle represents a resistance against compressive mechanical loading. Resulting from experimental results [19]-[21], the SiC-Si 3 N4 multi-particle-matrix system exhibits a maximal fracture toughness for the SiC particle volume fraction, v ≈ 0.15, corresponding to the calculated value, vc = 0.132. With regard to the yield stress in tension of the SiC particle, σyt p = 1000 MPa, and to the initial temperature of a cooling process, Ti = 1700◦C, the v-dependence of the critical final temperature of a cooling process, Tc (see Eq. (9)), exhibits values < -273.15◦C in the

Thermal Stresses in Particle-Matrix System and Related Phenomena... 1.2

-200 1-4: T = 20, 400, 800, 1200°C Ti = 1700°C σ = σϕ = σν = -pb p p rp

0.8

1: -pb

0.6

2: -pb

0.4

3: -pb

0.2

4: -pb

-225 Tc [C]

1.0

-pb [GPa]

181

-250

Tc 0.0

-275 0.0

0.1

0.2

0.3

0.4

0.5

v

Figure 18: The tensile radial and tangential thermal stresses in the SiC spherical particle, σrp > 0 and σϕp = σνp > 0, respectively, equal to the particle-matrix boundary radial stress, −pb , and the critical final temperature of a cooling process, Tc , as functions of the spherical particle volume fraction, v ∈ h0, π/6). intervals v ∈ h0, 0.07) ∪ h0.193, π/6) and the maximum for v = vc = 0.132 (Fig. 18), and the presented calculation is thus considerable in the temperature range T ∈ hTc, Ti i. Table 1: The material constants of the SiC particle and the Si 3 N4 matrix [19]-[21]: the Young’s modulus, E; the Poisson’s number, µ; the thermal expansion coefficient, α; the yield stress in tension and compression, σyt and σyc , respectively; the fracture toughness, KIC ; the SiC particle radius and volume fraction, R and v, respectively.

E [GPa] SiC 360 310 Si3 N4 ∗ ◦ 20 C/1100◦C

µ 0.19 0.235

α∗ [10−6 K−1 ] 4.14/5.05 2.35/3.75

σyt [GPa] 1 1

σyc [GPa] 5 5

KIC [MPam1/2 ] 3.25 5.25

R [nm] 10-500 -

v 0.05-0.3 -

With regard to the material constants listed in Tab. 1 [19]-[21], the dependences in Figs. 18, 19 and the following ones are generated on the condition of the linear T -dependence of the thermal expansion coefficient of the particle ( q = p) and the matrix (q = m), αq , derived the form αq =

αq2 (T − T1 ) − αq1 (T − T2 ) , T2 − T1

(108)

where αq1 and αq2 are related to the temperature T1 = 20◦C and T2 = 1100◦C, respectively,

182

Ladislav Ceniga

1.0 1

1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

0.6

2

v = 0.132 R = 250 nm d = 794 nm

0.4

3

0.2

4

σ rm [GPa]

0.8

a

0.0 250

275

300

325 350 x1[nm]

375

400

0.0 4

σϕ m= σν m [GPa]

-0.1

-0.2

-0.3

b

3 2 1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

1

-0.4 250

v = 0.132 R = 250 nm d = 794 nm 275

300

325 350 x1[nm]

375

400

Figure 19: The tensile and compressive radial (a) and tangential (b) thermal stresses in the Si 3 N4 cubic cell matrix, σrm < 0 and σϕm = σνm > 0, respectively, as functions of the position r = x1 ∈ hR, d/2i (Figs. 1-3).

Thermal Stresses in Particle-Matrix System and Related Phenomena...

183

for the spherical particle volume fraction equal to a critical, v = vc = 0.132 (Fig. 18), for the average SiC particle radius, R = 250 nm, consequently for the inter-particle distance, d = 794 nm, for the initial and different temperature of a cooling process, Ti = 1700◦C and T = 20, 400, 800, 1200◦C, respectively, and regarding the interval v ∈ h0, π/6), the formulae presented in the section 2.2 are considered for v = 0. Furthermore, the v-dependences of the investigated parameters also result from the fact that a real multi-particle-matrix system is characterized by different local v, as a consequence of non-homogeneously distributed particles of different dimensions. 10 1

6 4

2

2

3

1

8

Wm/ R3 [MJ]

Wp/ R3 [MJ]

8

1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

6 2

4

2

3 4

4

0

0

a

1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

0.0

0.1

0.2

0.3

0.4

0.5

b

v

0.0

0.1

0.2

0.3

0.4

0.5

v

20 1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

1 W/ R3 [MJ]

15

10 2 5 3 4 0

c

0.0

0.1

0.2

0.3

0.4

0.5

v

Figure 20: The elastic energy of the thermal stresses in the SiC spherical particle ( a), Wp /R3 , in the Si3 N4 cubic cell matrix (b), Wm /R3 , and in the SiC-Si 3 N4 cubic cell (c), W = (Wp +Wm ) /R3 , as functions of the spherical particle volume fraction, v ∈ h0, π/6).

The v-dependences of the elastic energy of the thermal stresses acting in the SiC spherical particle, Wp /R3 (see Eq. (16)), in the Si3 N4 cubic cell matrix, Wm/R3 (see Eq. (17)), and

184

Ladislav Ceniga

in the SiC-Si3 N4 cubic cell, W = (Wp +Wm )/R3 , exhibit the maximum for v = vc = 0.132, the minimum for v = 0.451 and the maximum for v = 0.057, respectively (Fig. 20). Although the function, W = W (v), does not exhibit the minimum, v = 0.451 may be assumed to represent a value corresponding to a thermal-stress equilibrium state of the isotropic SiC-Si3 N4 multi-particle-matrix system. With regard to the parameters, Wp ,Wm,W ∝ R3 , the v-dependences of Wp , Wm , W are related to the spherical particle radius R = 1. 250 1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

1

300

2

200

100

1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

200

v = 0.132 R = 250 nm d = 794 nm

Wsm [10-9Jm-1]

Wsp [10-9Jm-1]

400

150 100

2

50

3

v = 0.132 R = 250 nm d = 794 nm

1

3

4

4

0

0 0

50

100 150 x1[nm]

600

200

250

1

100

200 x1[nm]

300

400

1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

500 Ws [10-9Jm-1]

0

v = 0.132 R = 250 nm d = 794 nm

400 300

2

200 3

100

4 0 0

100

200 x1[nm]

300

400

Figure 21: The elastic energy gradient of the thermal stresses in the SiC spherical particle and the Si3 N4 cubic cell matrix, Wsp (a) and Wsm1 , Wsm2 (b), as functions of the position x1 ∈ h0, Ri and x1 ∈ h0, Ri, x1 ∈ hR, d/2i (Figs. 1-3), respectively, along with the elastic energy gradient of the thermal stresses in the SiC-Si 3N4 cubic cell, Ws = Wsp +Wsm1 +Wsm2 (c). If a dislocation moves from the cubic cell surface to the particle-matrix boundary, the elastic energy gradient within the SiC-Si 3 N4 cubic cell, Ws = Wsp + Wsm1 + Wsm2 (see Eqs. (18)-(21)), as an increasing function of x1 h0, d/2i (Fig. 21), and the maximum for

Thermal Stresses in Particle-Matrix System and Related Phenomena...

185

x1 = 0, Wsmax (see Eq. (22), (95)), as a function of the spherical particle volume fraction v ∈ h0, π/6), exhibiting the maximum for v = 0.096 (Fig. 22), represent an energy barrier and its height, respectively, influencing a dislocation motion [20, 21]. The elastic energy gradient, Wsm2 , derived in the interval x1 hR, d/2i by Eq. (20) and (21) exhibits identical courses. With regard to Wsmax ∝ R2 , the v-dependence of Wsmax is related to the spherical particle radius R = 1. 10 1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

Wsmax / R2 [MJm-1]

1 8 6 2 4 2

3 4

0 0.0

0.1

0.2

0.3

0.4

0.5

v

Figure 22: The maximum of the elastic energy gradient of the thermal stresses in the SiC-Si 3 N4 cubic cell, Wsmax /R2 , as a function of the spherical particle volume fraction, v ∈ h0, π/6).

The elastic energy gradients, Wsm1 and Wsm2 , correspond to thermal-stress strengthening within the Si 3 N4 cubic cell matrix, σm1 and σm2 , in the intervals x1 ∈ h0, Ri and x1 ∈ hR, d/2i (see Eqs. (24)-(26)) (Fig. 23), respectively. The thermal-stress strengthening in the SiC spherical particle, σ p (see Eq. (24)), and the average thermal-stress strengthening in the Si3 N4 cubic cell matrix, σm (see Eq. (27)), as functions of the spherical particle volume fraction, v ∈ h0, π/6), are presented in Fig. 24, where |σ p | exhibits the maximum for v = vc = 0.132, and σm is independent on the parameter R, as resulted from the integration in Eq. (27) by a numerical method. With regard to α p > αm , the parameter, σm , represents the strengthening related to tensile mechanical loading. The σm − v dependence for the SiCSi3 N4 multi-particle-matrix system is in an excellent agreement with experimental results adverting to the tensile strength increase in the range ∆Rm = 0.4 − 0.45 GPa for v ≈ 0.15 and T = 20◦C [19]-[21]. Resulting from the tensile thermal stresses in the SiC particle and consequently with regard to the particle crack formation, the SiC particle radius can be varied within the intervals, R ∈ (0, R p1 ) and R ∈ (0, R p2 ), for v ∈ h0, v0 i and v ∈ hv0 , π/6i (Fig. 25a), where R p1 and R p2 , are reasons of the particle crack formation from the particle centre, O, to the particle surface, and vice versa (Figs. 9, 10), respectively, and v0 = 0.487. In addition, Fig. 25b shows the T -dependence of R p0 = (R p1 )v=v0 = (R p2 )v=v0 for v = v0 , as a reason

186

Ladislav Ceniga 0.6 1-4: T = 20, 400, 800, 1200°C Ti = 1700°C

0.5

v = 0.132 R = 250 nm d = 794 nm

1

σm[GPa]

0.4 2

0.3 0.2

3

0.1

4

0.0 0

100

200 x1[nm]

300

400

Figure 23: The thermal-stress strengthening in the Si 3 N4 cubic cell matrix, σm1 and σm2 , as functions of the position x1 ∈ h0, Ri and x1 ∈ hR, d/2i (Figs. 1-3), respectively.

of the particle crack formation simultaneously from the particle centre, O, and the particle surface (Fig. 11). Although, the critical particle radii, R p1 , R p2 , are extremely high compared to the SiC particle radius of the interval, R ∈ h10, 500i [nm ], as an illustration, Fig. 26 shows the function, f p (see Eq. (38)), describing the particle crack shape in the interval x1 ∈ h0, r p i, in the form f p = 1.183 × 10−5 − 0.598x1 + 4.256 × 103 x21 − 2.591 × 109 x31 + 8.383 × 1014 x41 − 1.418 × 1020x51 + 1.452 × 1025x61 − 8.815 × 1029 x71 + 3.246 × 1034 x81 − 6.569 × 1038x91 + 5.624 × 1042x10 1 −...

(109)

and accordingly the contribution of the term x21 is considerable for x1 ≈ r p , and the contribution of the terms xn1 for n ≥ 3 is neglecting, where r p = 24.966 µm for v = vc = 0.132, R = 45 µm. The quasi-linear shape corresponds to that from experimental results [9]-[11], [19]-[21].

4

Conclusions

The paper is the continuation of the calculation published in [18] and [25], presenting the thermal stresses in the isotropic multi- and one-particle-matrix systems, respectively, originating during a cooling process as a consequence of the difference of thermal expansion coefficients between the particle and the matrix. The isotropic multi- and one-particle-matrix systems are represented by the homogeneously distributed spherical particles embedded in

Thermal Stresses in Particle-Matrix System and Related Phenomena...

0.0 -0.2

1-4: T = 20, 400, 800, 1200ºC Ti= 1700ºC

187

4 3 2

σp [GPa]

-0.4 1 -0.6 -0.8 -1.0 -1.2 0.0

a

0.1

0.2

0.3

0.4

0.5

v

1.0 0.8

1

1-4: T = 20, 400, 800, 1200ºC Ti= 1700ºC

− σ

m [GPa]

2 0.6 3

0.4

4

0.2 0.0

b

0.0

0.1

0.2

0.3

0.4

0.5

v

Figure 24: The thermal-stress strengthening in the SiC spherical particle, σ p , and the average thermal-stress strengthening in the Si 3 N4 cubic cell matrix, σm , as functions of the spherical particle volume fraction, v ∈ h0, π/6).

188

Ladislav Ceniga

10.00

1-4: T = 20, 400, 800, 1200ºC Ti= 1700ºC

4

1.00 Rp [mm]

3 2 1

0.10

0.01

a

0.0

0.1

0.2

0.3

0.4

0.5

v

1E+3

Ti= 1700ºC

Rp0 [mm]

1E+2

1E+1

1E+0

1E-1

b

0

500

1000 T [°C]

1500

Figure 25: The critical SiC particle radii, R p1 and R p2 (a), as functions of the spherical particle volume fraction, v ∈ h0, v0 i and v ∈ hv0 , π/6), respectively, and the critical SiC particle radius, R p0 = (R p1 )v=v0 = (R p2 )v=v0 (b), as a function of the final temperature of a cooling process, T ∈ h20, 1700i [◦ C ] for v0 = 0.487.

Thermal Stresses in Particle-Matrix System and Related Phenomena...

189

fp [ µ m]

12 10

T = 20°C Ti = 1700°C

8

v = 0.132 R = 45 µ m

6 4 2 0 0

5

10 15 x1[ µ m]

20

25

Figure 26: The function, f p , describing the SiC particle crack for the spherical particle volume fraction, v = vc = 0.132, and for the SiC particle radius, R > (R p1 )v=vc = 38.6 µm (Fig. 25a).

the infinite matrix divided into cubic cells, containing one central particle [18], and by one spherical particle embedded in the infinite matrix [25], respectively. Contributed to the results published in [18] and [25] (see sections 2.1.1, 2.1.2, 2.2.1), the results of the presented calculation related to isotropic multi- and one-particle-matrix systems are as follows: • The initial and critical final temperature of a cooling process between which the isotropic particle-matrix systems are acted by elastic thermal stresses are derived (see sections 2.1.3, 2.2.2). • The thermal-stress induced elastic energy density in a point, consequently on a surface, and along a curve in the spherical particle, in the cubic cell matrix and in the infinite matrix of the isotropic particle-matrix systems are derived (see sections 2.1.4, 2.1.5, 2.1.7, 2.2.3, 2.2.4, 2.2.6). • Resulting from the ‘surface’ elastic energy density, the thermal-stress strengthening in the spherical particle, the cubic cell matrix and the infinite matrix of the isotropic particle-matrix systems are derived (see sections 2.1.6, 2.2.5). • Resulting from the ‘curve’ elastic energy density, the sections 2.1.8, 2.2.7 devoted to the particle and matrix crack formation include – the condition concerning a direction of the crack formation in the spherical particle, the cubic cell matrix and the infinite matrix of the isotropic particlematrix systems is derived,

190

Ladislav Ceniga – the critical particle radii, as a reason of the crack formation in the spherical particle, the cubic cell matrix and the infinite matrix of the isotropic particlematrix system are derived, – the functions describing cracks in the spherical particle, the cubic cell matrix and the infinite matrix of the isotropic particle-matrix systems are derived.

Applying the derived formulae to the SiC-Si 3 N4 multi-and one particle-matrix systems, the main results included in the section 3 are as follows: • The SiC spherical particle thermal stresses, the particle-matrix boundary radial stress and the critical final temperature of a cooling process, as functions of the spherical particle volume fraction, v, are presented (Fig. 18). • The critical SiC particle volume fraction, vc = 0.132 (Fig. 18), corresponding to maximal calculated thermal stresses, is in an excellent agreement with the SiC particle volume fraction, v ≈ 0.15, corresponding to a maximal fracture toughness as obtained from published experimental results [19]-[21]. • The thermal stresses, as functions of the position in the Si 3 N4 cubic cell matrix, are presented for the v = vc = 0.132 and for the average SiC particle radius, R = 250 nm (Fig. 19). • The thermal-stress induced elastic energy in the SiC spherical particle, the Si 3 N4 cubic cell matrix and SiC-Si3 N4 cubic cell, as functions v, are presented (Fig. 20). • The thermal-stress induced elastic energy gradient, as a function of the position in the SiC-Si3 N4 cubic cell, is presented (Fig. 21) along with the v-dependence of the gradient maximum (Fig. 22). • The thermal-stress strengthening, as a function of the position in the Si 3 N4 cubic cell matrix, is presented (Fig. 23) along with the v-dependences of the thermal-stress strengthening in the SiC spherical particle and the average thermal-stress strengthening in the Si 3 N4 cubic cell matrix (Fig. 24), being in an excellent agreement with published experimental results [19]-[21]. • The critical SiC particle radii, as reasons of the SiC particle crack formation and as functions of v, are presented along with the temperature dependence of the SiC critical particle radius (Fig. 25). • The function describing the particle crack for the critical particle volume fraction and for the SiC particle radius greater than critical is presented (Fig. 26).

Thermal Stresses in Particle-Matrix System and Related Phenomena...

191

Acknowledgement This work was supported by the Slovak Grant Agency VEGA (2/4173/04, 2/4175/04, 2/4062/04, 1/1111/04), by NANOSMART, Centrum of Excellence, Slovak Academy of Sciences, by Science and Technology Assistance Agency under the contract No. APVT-51049702, by EU 5th FP project GRD1-2000-25352 “SmartWeld”, and by COST Action 536 and COST Action 538. The author is thankful to his dearest parents for their support.

5

Appendix

The elastic modulus of the particle (q = p) and the matrix (q = m), sq , has the forms s11q =

1 , Eq

s12q = −

(110)

µq , Eq

(111)

where Eq and µq are the Young’s modulus and Poisson’s number, respectively. The coefficient ci (i = 1 − 19) is as follows c1 =

3π (s11m + s12m) , 2 (π − 6v)

(112)

c2 = c1 + s11p + 2s12p − (s11m + 2s12m) , 

c3 =

Zπ/4 0

cos4 ϕ

1 p − 1 + cos2 ϕ 3

cos2 ϕ

p 1 + cos 2 ϕ

!3   dϕ,

(113)

(114)

where the coefficient c3 can be derived by the Taylor series for the integrated function. Consequently, after integration of the Taylor series in terms of (ϕ − ϕ0 )n about the point ϕ0 = 0, the coefficient c3 = 0.337497 for n = 100;

c4



2 c1 = 2v (s11p + 2s12p ) c2 "   1 6vc1 2 2 + 2π (1 − v) (s11m + 2s12m) 1 − πc2 (π − 6v)2  2 #
c1 + v (s11m − s12m ) π2 − 36vc3 1 − , c2

(115)

192

Ladislav Ceniga    2c1 (s11p + 2s12p ) 1 6vc1 12π (1 − v) (s11m + 2s12m ) 1 − + c2 πc2 (π − 6v)2  
c1 (s11m − s12m ) π2 − 36vc3 1 − , (116) c2

c5 = +

π [c2 c4 + vc5 (c1 − c2 )] , (π − 6v) (c2 c4 + vc1 c5 )

(117)

c2 c4 + vc1 c5 , c22 c4

(118)

c8 =

v [πc2 c5 − 6 (c2 c4 + vc1 c5 )] , (π − 6v) (c2 c4 + vc1 c5 )

(119)

c9 =

3π + 8 2R4

c6 =

c7 =

c9 =

"

Zπ/4 0



3v 4π

4/3

,

x1 = 0, #2

4 (3v)2/3 cos2 ξ R2 (4π)2/3 + 4x1 (3v)2/3 cos2 ξ

(120)

 d , dξ, x1 ∈ 0, 2 

(121)

where the coefficient c9 for x1 ∈ (0, d/2i can be derived for a concrete isotropic particlematrix system by the Taylor series for the integrated function in terms of (ξ − ξ0 )n about the point ξ0 = 0; h i 2x1 R (12πv)1/3 3R2 (4π)2/3 + 20x21 (3v)2/3 , (122) c10 = 3ϕ73 + h i2 R2 (4π)2/3 + 4x21 (3v)2/3 c11 =

Zϕ73

ν6 cos4 ϕdϕ,

(123)

0

c12 =

Zϕ22 0

×

−

(

R4 (4π)4/3 cos5 ϕ

R (4π)1/3 i2 dϕ = q R2 (4π)2/3 + 4x21 (3v)2/3 R2 (4π)2/3 cos2 ϕ + 4x21 (3v)2/3

h

R2 (4π)2/3 + 3x21 (3v)2/3

R2 (4π)2/3 + 2x21 (3v)2/3 "        #)  2x1 2 3v 2/3 R2 (4π)2/3 + 3x21 (3v)2/3 1 R 2 4π 2/3 ln 1 + , R 4π 2 x1 3v R2 (4π)2/3 + 4x2 (3v)2/3 1

(124)

Thermal Stresses in Particle-Matrix System and Related Phenomena...

Zϕ73

c13 =

0

(

×

c14 =

Zϕ73

−

+

(3v)1/3 q dϕ = R2 (4π)2/3 cos2 ϕ + 4x21 (3v)2/3 R2 (4π)2/3 + 4x21 (3v)2/3

R2 (4π)2/3 − 4x21 (3v)2/3

4πR3

h

−

h

(4π)4/3 R4 cos7 ϕ R2 (4π)2/3 cos2 ϕ + 4x21 (3v)2/3

0

×

R2 (4π)2/3 cos5 ϕ

R (12πv)1/3 3 (3v)1/3 R2 (4π)2/3 + 4x21 (3v)2/3 "     #) 1 R 2 4π 2/3 6vx41 ln 1 + , πR3 2 x1 3v

+

  R2 (4π)2/3 − 8x2 (3v)1/3 

h

193

R (4π)

1 1/3

+

i2 dϕ = q

i (125)

1 R2 (4π)2/3 + 4x21 (3v)2/3

3R4 (4π)4/3 − 16x41 (3v)4/3 h i 4R (4π)1/3 R2 (4π)2/3 + 2x21 (3v)2/3

4πR3

i 3 R2 (4π)2/3 + 4x21 (3v)2/3  h i " # 5/3 4/3 4/3   5 4 4   R (4π) − 6x1 (3v) 2R (4π) 1 R 2 4π 2/3  ln 1 + ,(126) h i4  2 x1 3v  R2 (4π)2/3 + 4x21 (3v)2/3

3πc10 2x1 − 3c11 + c15 = 16 R



3v 4π

1/3 "

c12



2x1 R

2 

3v 4π

2/3

#

+ 4c13 − c14 ,

(127)

where the coefficient c11 can be derived for a concrete isotropic particle-matrix system by the Taylor series for the integrated function in terms of (ϕ − ϕ0 )n about the point ϕ0 = 0; "  1/3 # R 4π , (128) ϕ73 = arctan 2x1 3v "  1/3 # 3v 2x1 , (129) ν6 = arctan R cos ϕ 4π "   #
s11m 2 4π 2/3 R − π R2 − x21 , x1 ∈ h0, Ri, (130) c16 = 2 3v s11mR2 c17 = 2



4π 3v

2/3

,

(131)

194

Ladislav Ceniga n

5v2 (s11m + 2s12m )[c5 (6vc1 − πc2 ) + 6c2 c4 ]2 o − (s11p + 2s12p ) (π − 6v)2 (c2 c4 + vc1 c5 )2  5/3 2 3v 2 + 16π (s11m − s12m) [vc5 (c1 − c2 ) + c2 c4 ] 4π   5    1/3   (4π)  , × 1−q    (4π)2/3 + 4 (3v)2/3 

c18 =

1 c19 = s11p + 2s12p + (s11m − s12m ) . 2

(132)

(133)

References [1] Siv´ak, P.; Ostertag, O.; Ivanˇco, V.; Kostoln´y, K. Acta Mech. Slovaca 1998, vol. 2, 201-204. [2] Peˇsek, L.; Novotn´y, L.; Ivanˇco, V. Acta Mech. Slovaca 1999, vol. 3, 171-174. [3] Ivanˇco, V.; Kub´ın, K.; Kostoln´y, K. Finite element method I; ISBN 80-96731-4-0; Elfa: Koˇsice, SK, 1994; Vol. 1, pp 225-243 (in Slovak). [4] Hvizdoˇs, P.; Lofaj, F.; Dusza, J. Metall. Mater. 1995, vol. 6, 473-483. [5] Lofaj. F.; Dusza J.; Richarz, B. Metall. Mater. 1997, vol. 35, 247-256. [6] Lofaj, F.; Dorˇca´ kov´a, F. Metalurgija 2003, vol. 42, 229-233. [7] Lofaj, F.; Hvizdoˇs, P.; Dorˇca´ kov´a, F.; Satet, R.; Hoffmann, M.J.; Arellano L´opez, A.R. Mater. Sci. Eng. 2003, vol. A357, 181-187. [8] Lofaj, F.; Dorˇca´ kov´a, F.; Kovalˇc´ık, J.; Hoffmann, M.J.; Arellano L´opez, A.R. Metall. Mater. 2003, vol. 41, 145-157. [9] Diko, P. Supercond. Sci. Technol. 1998, vol. 11, 68-72. ˇ c´ıkov´a, M.; Zmorayov´a, K.; Diko, P.; Babu, H.; Cardwell, D. Acta Metallur. Slo[10] Sefˇ vaca 2004, vol. 10, 889-891. ˇ c´ıkov´a, M.; Diko, P.; Babu, H.; Cardwell, D. Acta Metallur. Slo[11] Zmorayov´a, K.; Sefˇ vaca 2004, vol. 10, 892-894. ˇ [12] Janovec, J.; G¨uth, A.; V´yrostkov´a, A.; Stefan, B. Metall. Mater. 1988, vol. 26, 714724.

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195

[13] Janovec, J.; V´yrostkov´a, A.; Hol´y, A. J. Mater. Sci. 1992, vol. 27, 6564-6572. [14] Janovec, J.; V´yrostkov´a, A.; Svoboda, M. Metall. Mater. Trans. 1994, vol. 25A, 267275. [15] Janovec, J.; V´yrostkov´a, A.; Hol´y, A. Canadian. Metall. Quart. 1994, vol. 33, 227232. [16] Kov´acˇ , F.; Dˇzubinsk´y, M.; Sidor, Y. J. Mag. Mag. Mater. 2004, vol 269, 333-340. [17] Sidor, Y.; Dˇzubinsk´y, M.; Kov´acˇ F. Mater. Character. 2003, vol. 51, 109-116. [18] Ceniga, L. J. Therm. Stress. 2004, vol. 27, 425-432. [19] Skoˇcovsk´y, P.; Bok˚uvka, O.; Koneˇcn´a, R.; Tillov´a, E. Materials Science for Mechanˇ ˇ ical Engineers; ISBN 80-7100-831-1; EDIS Technical University in Zilina: Zilina, SK, 2001; pp 36-63 (in Slovak). [20] Skoˇcovsk´y, P.; Palˇcek, P.; Koneˇcn´a, R.; V´arkoly, L. Structural materials; ISBN 80ˇ ˇ 7100-608-4; EDIS Technical University in Zilina: Zilina, SK, 2000; pp 41-45 (in Slovak). [21] Hidv´eghy. J.; Dusza, J. Nonmetallic Structural Materials. Plastic and Structural Ceramics; ISBN 80-7090-363-4; Technical University in Koˇsice: Koˇsice, SK, 1998; pp 126-144 (in Slovak). [22] Mizutani, T. J. Mater. Res. 1996, vol. 11, 483-494. ˇ ca´ k, F. Theory of elasticity; ISBN 80-7099-478-9; Vien[23] Trebuˇna, F.; Jurica, V.; Simˇ ala: Koˇsice, SK, 2000; Vol. 2, pp 72-81 (in Slovak). ˇ [24] Kluv´anek, I.; Miˇs´ık, L.; Svec, M. Mathematics; Alfa: Bratislava, SK, 1959; Vol. 1, pp 685-687 (in Slovak). [25] Ceniga, L.; Kov´acˇ , F. Mater. Sci. Eng. 2001, vol. B86, 178-181.

In: Ceramics and Composite Materials: New Research ISBN: 1-59454-370-4 Editor: B.M. Caruta, pp. 197-214 © 2006 Nova Science Publishers, Inc.

Chapter 6

NOVEL BONE-REPAIRING MATERIALS: BIOACTIVE ORGANIC-INORGANIC HYBRIDS Masanobu Kamitakahara* Graduate School of Materials Science, Nara Institute of Science and Technology, 8916-5, Takayama-cho, Ikoma-shi, Nara 630-0192, Japan

Abstract In general, artificial materials implanted into bone defects are encapsulated by a fibrous tissue. Some ceramics, such as Bioglass®, sintered hydroxyapatite and glass-ceramic A-W, however, form a bone-like apatite layer on their surfaces in the living body and bond to living bone through this apatite layer, i.e. they show bioactivity. Although these bioactive ceramics are used clinically as important bone-repairing materials, they are essentially brittle and hence limited in their applications. It is desirable to develop new types of deformable bioactive materials. Organic–inorganic hybrids prepared by a sol-gel method are expected to exhibit bioactivity as well as deformability. Polydimethylsiloxane (PDMS)-CaO-SiO2-based hybrids were prepared. To evaluate their potential of bone-bonding property, their ability of apatite formation was examined in a simulated body fluid (SBF). Since the apatite formation on bioactive materials in the body can be reproduced even in SBF, we can estimate the bioactivity of the material by using SBF. Mechanical properties were also examined. Some of them showed apatite-forming ability and mechanical properties analogous to those of human cancellous bone. These CaO-containing hybrids, however, showed a decrease in mechanical strength in SBF. These hybrids containing no CaO do not form apatite on their surfaces in SBF. We then prepared CaO-free PDMS-TiO2 and poly(tetramethylene oxide) (PTMO)-TiO2 hybrids in which anatase was precipitated by hot-water treatments. These hybrids also showed apatite-forming ability on their surfaces in SBF and flexibility, and they showed little decrease in mechanical strength in SBF. In conclusion, it was revealed that we can design new bioactive materials for bone-repair by controlling the compositions and structures in organicinorganic hybrid systems.

*

E-mail address: [email protected]. Tel: +81-743-72-6122, Fax: +81-743-72-6129

Masanobu Kamitakahara

198

1

Introduction

Bone supports our body, protects our organs, such as brain and heart against external force, and enables us to do various kinds of motion. Once even one of the bones is damaged by a disease or an accident, quality of our lives is much decreased. Defects of bones have been repaired by autogenous or allogenic bones. In the case of autogenous bones, however, a healthy part of the patient is damaged and available amount of bone is limited. In the case of allogenic bones, infections are sometimes caused through viruses, bacteria, etc. Therefore, artificial materials, which are free from virus and bacteria and not limited in its available amount, are needed to repair the defects of bones. Artificial materials implanted into bone defects, however, are generally encapsulated by a non-calcified fibrous tissue to be isolated from the surrounding bone [1]. In early 1970s, Hench et al. showed that some glasses in the system Na2O-CaO-SiO2P2O5 spontaneously bond to living bone without forming the fibrous tissue around them and named them Bioglass® [2-4]. Since the discovery of Bioglass®, various kinds of ceramics, such as glass-ceramic Ceravital® containing crystalline apatite [5, 6], sintered hydroxyapatite (Ca10(PO4)6(OH)2) [7, 8], glass-ceramic A-W containing crystalline apatite and wollastonite [9, 10], glass-ceramic Bioverit® containing crystalline apatite and fluorophlogopite ((Na, K)Mg3(AlSi3O10)F2) [11, 12], MgO-CaO-SiO2-P2O5 glasses [13] and CaO-SiO2 glasses [14], have been found to bond to living bone. These materials are called bioactive materials, and some of them are already clinically used as important bone substitutes, such as artificial iliac crests, artificial vertebrae, artificial intervertebral discs, bone fillers, etc [15-24]. They, however, occupy only 10 to 30 % of whole the bone grafts needed, because they are brittle, poor in fracture toughness and deformability, and too high in elastic modulus. Among the problems described above, the low fracture toughness has been solved by development of bioactive metals with high fracture toughness. For example, Kim et al. showed that titanium metal and its alloys bond to living bone, if they are subjected to prior NaOH and heat treatments to form a sodium titanate layer on their surfaces [25-34]. Miyazaki et al. showed that tantalum metal also bonds to living bone, if it is subjected to prior NaOH and heat treatments to form a sodium tantalate layer on its surface [35-37]. These metallic materials, however, have higher elastic moduli than that of human cortical bones, and hence are liable to induce resorption of the surrounding bone due to stress shielding by them. Bioactive materials with low elastic moduli and high deformability are desired to be developed. Living bone is a composite of 70 wt% inorganic component, apatite, and 30 wt% organic component, collagen, as shown in Fig. 1 [38, 39]. Due to its unique composition and structure, natural bone can show not only high strength and high fracture toughness, but also deformability and low elastic modulus. Therefore, it is expected that a bioactive material with lower elastic modulus and deformability could be obtained if a bioactive inorganic substance is combined with some organic substance. Bonfield et al. early developed a composite of hydroxyapatite granules with polyethylene [40, 41]. These composites, however, lose deformability when the hydroxyapatite content exceeds 40 vol%. When the hydroxyapatite content is limited to less than 40 vol%, the composite can not show high bioactivity [3].

Novel Bone-Repairing Materials: Bioactive Organic-Inorganic Hybrids

2

199

Bone-Bonding Mechanism

It has been shown that the bioactive materials bond to living bone through an apatite layer which is formed on their surfaces in the body [3, 31, 42-49]. The schematic representation is shown in Fig. 2. Therefore, it is believed that the essential requirement for an artificial material to bond to living bone is formation of an apatite layer on their surfaces in the living body. This apatite layer consists of calcium-deficient and carbonate-containing nano-sized hydroxyapatite with low crystallinity and defect structure [50, 51]. This apatite is similar to the bone mineral in composition and structure, and hence bone-producing cell called osteoblast can proliferate preferentially on this surface apatite layer, and differentiate to produce bone matrix composed of apatite and collagen [4, 52, 53]. As a result, the surrounding bone can come into direct contact with the surface apatite layer. When this occurs, a tight chemical integration is formed between the bone apatite and the surface apatite on bioactive materials in order to decrease interfacial energy between them.

200

Masanobu Kamitakahara

Kokubo et al. [54] previously showed that the apatite formation on the surfaces of bioactive materials in the living body can be reproduced even in an acellular protein-free simulated body fluid (SBF) with ion concentrations nearly equal to those of human blood plasma. Ion concentrations of human blood plasma and SBF are shown in Fig. 3. This indicates that bioactivity of a material can be evaluated even in vitro by examining the apatite-formation on its surface in SBF.

3

Design of Bioactive Hybrids

Currently, organically modified sol-gel-derived metal oxides are paid attention because of their unique properties, such as low elastic modulus and deformability [55-59]. In these hybrid materials, organic components are chemically incorporated into inorganic networks at nanometer level. It has been revealed that some sol-gel-derived metal oxide gels, such as silicon oxide [60-63], titanium oxide [63-65], tantalum oxide [66], zirconium oxide [67, 68] and niobium oxide [69], form a bone-like apatite layer on their surfaces in SBF. Therefore, it is expected that if some organic component is incorporated into a network of these metal oxide gels by a sol-gel method, the obtained hybrids could show not only apatite-forming ability but also deformability. Moreover, if calcium ions are incorporated into these hybrids, the obtained hybrid could show higher apatite-forming ability, because the incorporated calcium ions are released into the surrounding body fluid to accelerate apatite nucleation by increasing the ionic activity product of apatite. [70].

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Polydimethylsiloxane (PDMS)-CaO-SiO2-Based Hybrids

Modification of CaO-SiO2 system with organic polymer may provide bioactive organicinorganic hybrids. Tsuru et al. [71] prepared polydimethylsiloxane (PDMS)-CaO-SiO2 hybrids by a sol-gel method. Tetraethoxysilane (TEOS) was hydrolyzed and reacted with PDMS, and calcium nitrate (Ca(NO3)2) was incorporated. The model structure of the PDMS-

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CaO-SiO2 hybrid is shown in Fig. 4. Some of the resultant hybrids formed an apatite layer on their surfaces in SBF. This indicates that these hybrids are expected to form a bone-like apatite layer on their surfaces in the living body and bond to living bone through this apatite layer [72]. However, they did not show their mechanical properties because they could not prepare large enough specimens for mechanical test. Recently, Chen et al. [73-77] and we [78, 79] successfully synthesized large samples enough for mechanical test in PDMS-CaO-SiO2TiO2 and PDMS-CaO-SiO2 hybrids and revealed that some of the hybrids show both apatiteforming ability on their surfaces in SBF and deformability.

We prepared the PDMS-CaO-SiO2 hybrids whose starting compositions are given in Table 1 [78, 79]. The surfaces of the hybrids before and after soaking in SBF were observed by scanning electron microscopy (SEM). Figure 5 shows SEM photographs of the surfaces of the PDMS-CaO-SiO2 hybrids before and after soaking in SBF for 7 days. Apatite-forming ability of the PDMS-CaO-SiO2 hybrids increased with decreasing PDMS content. When PDMS content increases, the amount of silanol groups which induce apatite nucleation decreases, accordingly the apatite-forming ability decreases. The CaO-free hybrid (Ca0) did not form apatite on its surface. The calcium-containing hybrids release calcium ions and the released calcium ions accelerated apatite nucleation of apatite [70]. Once the apatite nuclei are formed, they can grow spontaneously by consuming calcium and phosphate ions from the surrounding SBF, since SBF is highly supersaturated with respect to apatite. Figure 6 shows stress-strain curve of one of the PDMS-CaO-SiO2 hybrids, in comparison with that reported for human cancellous bone [80]. Unlike usual brittle ceramics, the hybrid was deformable and showed mechanical properties analogous to those of human cancellous bone. Modification of inorganic component CaO-SiO2 with organic component PDMS gives a hybrid material with apatite-forming ability and mechanical properties analogous to those of human cancellous bone.

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Degradation of PDMS-CaO-SiO2-Based Hybrids in SBF

For the clinical application of these hybrids, to know their changes in mechanical properties in the body environment is important. The mechanical properties of PDMS-CaO-SiO2-TiO2 hybrids were examined before and after soaking in SBF [81]. The starting compositions of PDMS-CaO-SiO2-TiO2 hybrids are given in Table 2. The changes in mechanical strength and strain at failure of the PDMS-CaO-SiO2-TiO2 hybrids before and after soaking in SBF are shown in Fig. 7. After soaking in SBF, the bending strength and strain to failure of hybrids Ti0Ca0 and Ti10Ca0 only slightly decreased, whereas those of hybrids Ti0Ca15, Ti10Ca15, Ti30Ca0, and Ti30Ca15 significantly decreased after soaking in SBF. The decreases in both the bending strength and the strain to failure of the calcium-containing hybrids due to soaking in SBF were larger than those of the corresponding calcium-free hybrids. It has been reported that calcium ions inhibit the polycondensation of TEOS in the preparation of a CaO-SiO2 glass by a sol-gel method [82]. Therefore, it is considered that calcium ions inhibit the formation of a continuous three-dimensional network of the hybrids in the calcium-containing hybrids. In such cases, calcium ions are easily released via exchange with hydronium ions in SBF and water molecules also easily enter into the network of calcium-containing hybrids to break the Si-O-Si bonds of the hybrids. These may be the reasons why the deterioration of the calcium-containing hybrids was larger than that of the calcium-free hybrids in SBF.

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It can be concluded that incorporation of a large amount of CaO into the PDMS-CaOSiO2-based hybrid system results in deterioration of the hybrids in the body environment.

6

PDMS-TiO2 Hybrids

Preparation of bioactive CaO-free PDMS-modified hybrids that can form apatite on their surfaces was attempted. Recently, it has been shown that apatite formation on titania gel significantly depends on its structure. The Ti-OH groups in the anatase structure are most effectively induce apatite nucleation, whereas Ti-OH groups in the amorphous and rutile structure are not so effective [65]. It is expected that even a CaO-free hybrid can show high apatite-forming ability if the structure of a TiO2-based hybrid is controlled. A PDMS-TiO2 hybrid was synthesized and its structure was controlled by hot-water treatment [83]. A polydimethylsiloxane (PDMS)-TiO2 hybrid was prepared by a sol-gel method from PDMS (M=550) and tetraethylorthotitanate with PDMS/tetraethylorthotitanate molar ratios at 1.35. The as-prepared hybrid was amorphous and did not form apatite on its surface after soaking in SBF for 7 days. The PDMS-TiO2 hybrid was then soaked in hot water at 60ºC or 80ºC for various periods. The structural change was examined by thin film X-ray diffraction (TF-XRD). Figure 8 shows TF-XRD patterns of the surfaces of the PDMS-TiO2 hybrid subjected to the hot-water treatment at 60°C or 80°C for various periods. Anatase peaks were observed for the PDMS-TiO2 hybrid treated with hot water at 60°C within 7 d and at 80 °C within 1 d. The peak intensities of anatase increased with increasing period and temperature of the hot-water treatment. Transmission electron microscopic (TEM) observation showed that the hybrid took a homogeneous amorphous structure before the hotwater treatment, and precipitated anatase particles 10-20 nm in size after the hot-water treatment at 80°C for 7 days (Fig. 9). The hot-water-treated hybrids were soaked in SBF. The

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SEM photographs of surfaces of the hot-water-treated PDMS-TiO2 hybrids after soaking in SBF are shown in Fig. 10. The apatite was formed on the hybrid treated with hot water at 60°C for the periods longer than 3 days and at 80°C for the periods longer than 1 day. The amount of apatite formed on them increased with increasing period and temperature of the hot-water treatment. This indicates that the anatase precipitated in the hybrid induces apatite formation on the surface of the hybrid. It has been shown that apatite formation on titania depends mainly on its crystalline phase, and anatase is more effective for apatite formation than amorphous and rutile-type titania [65].

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The obtained hybrid showed rubber elasticity even after the hot water treatment, as shown in Fig. 11. The representative stress-strain curves of the PDMS-TiO2 hybrid before and after the hot-water treatment, and after both hot-water treatment and SBF treatment are shown in Fig. 12. Both tensile strength and Young’s modulus of the hybrid decreased, and its strain to failure increased after the hot-water treatment. It is considered that the matrix of the hybrid was enriched with PDMS due to the segregation of TiO2 by the hot-water treatment and hence the mechanical properties of the hybrid subjected to the hot-water treatment became close to the matrix rich in PDMS, since the continuous phase predominantly governs the overall mechanical behavior of composite materials. The observation that the strain energies stored until failure were not significantly changed by the hot-water treatment might indicate that rearrangement of the hybrid by the hot-water treatment does not cause deterioration of the hybrid. After soaking in SBF, its strain to failure decreased, but its tensile strength did not change and its Young’s modulus a little increased. It is considered that an increase in its Young’s modulus after soaking in SBF is due to apatite formation on its surface, because apatite shows much higher Young’s modulus than the hybrid. In this case, its strain to failure decreases when its tensile strength does not change. Since the environment in hot water is more severe than in the body environment, the unreacted reagents or solvents remaining in the hybrid might be released into hot water, and therefore the resultant hybrid is expected to be non-toxic and stable in the living body at around 36.5 °C. A highly deformable PDMS-TiO2 hybrid with apatite-forming ability and durability in SBF was obtained by the present method. Besides deformability, rubbery elasticity of the hybrid would also be an advantage, because the hybrid can return to the original shape, even after a large deformation.

7

Poly(Tetramethylene Oxide) (PTMO)-TiO2 Hybrids

The PDMS-TiO2 hybrid treated with hot water showed apatite-forming ability and high deformability. The mechanical strength (≈0.4 MPa) and Young’s modulus (≈0.2 MPa) of the obtained hybrid, however, were much lower than those of human bones. It is expected that if PDMS is replaced with poly(tetramethylene oxide) PTMO, a deformable bioactive material with higher mechanical strength might be obtained [84]. Poly(tetramethylene oxide) (PTMO)-TiO2 hybrids were prepared by a sol-gel method from triethoxysilane functionalized PTMO (Si-PTMO) and tetraisopropyltitanate with weight ratios of 30/70, 40/60 and 50/50 (hybrids PT30, PT40 and PT50, respectively), and subsequently subjected to a hot-water treatment at 95 ºC for 2 d [85]. The structural model of the PTMO-TiO2 hybrid is shown in Fig. 13. All of the obtained hybrids were amorphous before the hot-water treatment, and precipitated nanosized anatase after the hot-water treatment. The amount of precipitated anatase increased with decreasing PTMO content. Apatite was not formed on the surfaces of the hybrids in SBF before the hot-water treatment, but was formed after the hot-water treatment, and its amount increased with decreasing PTMO content, as shown in Fig. 14. This indicates that apatite-formation was induced by the anatase precipitated by hot-water treatment, as the case of the PDMS-TiO2 hybrid. Figure 15 shows representative stress-strain curves of hybrid PT40 before and the hot-water treatment, and that after both the hot-water treatment and SBF treatment for 28 days, in comparison with

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that reported for human cancellous bone. Hybrid PT40 showed strength and Young’s modulus analogous to those of human cancellous bones, and high deformability after the hotwater treatment. The PDMS-TiO2 hybrid treated with hot water showed strength of only 0.4 MPa and Young’s modulus of 0.2 MPa. The mechanical properties of the hybrid were improved by changing the organic component from PDMS to PTMO, from the point of view of strength and Young’s modulus. Although the PDMS-TiO2 hybrid treated with hot water can deform up to 200%, such extremely high deformability is not needed and 10% deformability of the PTMO-TiO2 hybrid treated with hot water is considered to be enough for a bone-repairing material. After soaking in SBF, it’s Young’s modulus increased, and it’s bending strength and strain to failure decreased. The increase in its Young’s modulus after soaking in SBF might be due to the apatite-formation on its surface because apatite shows much higher Young’s modulus than the hybrid. The decreases in its bending strength and strain to failure after soaking in SBF might be due to the degradation of the hybrid. These decreases were, however, much lower than those of a CaO-containing PTMO-CaO-TiO2 hybrid [81].

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Summary

Development of new types of deformable bioactive materials is desired for bone repair. A bioactive material which shows mechanical properties analogous to those of human bones and high durability in the body environment can be obtained by controlling not only the compositions but also the structures of sol-gel-derived organic-inorganic hybrids. The hybrid material can be a candidate for a novel bone-repairing material.

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[71] Tsuru, K; Ohtsuki, C; Osaka, A; Iwamoto T; Mackenzie, JD. Bioactivity of sol-gel derived organically modified silicates, part I: in vitro examination. J. Mater. Sci.: Mater. Med., 1997, 8, 157-161. [72] Kokubo, T. Bioactive glass ceramics; Properties and applications. Biomaterials, 1991, 12, 155-163. [73] Chen, Q; Miyaji, F; Kokubo, T; Nakamura, T. Apatite formation on PDMS-modified CaO-SiO2-TiO2 hybrids prepared by sol-gel method. Biomaterials, 1999, 20, 1127-1132. [74] Chen, Q; Miyata, N; Kokubo, T; Nakamura, T. Bioactivity and mechanical properties of PDMS-modified CaO-SiO2-TiO2 hybrids prepared by sol-gel process. J. Biomed. Mater. Res., 2000, 51, 605-611. [75] Chen, Q; Miyata, N; Kokubo, T; Nakamura, T. Bioactivity and mechanical properties of PDMS-modified calcia-silica hybrids with added titania. J. Am. Ceram. Soc., 2003, 86, 806-810. [76] Chen, Q; Kamitakahara, M; Miyata, N; Kokubo, T; Nakamura, T. Preparation of bioactive PDMS-modified CaO-SiO2-TiO2 hybrids by the sol-gel method. J. Sol-Gel Sci. Tech., 2000, 19, 101-105. [77] Chen, Q; Miyata, N; Kokubo, T; Nakamura, T. Effect of heat treatment on bioactivity and mechanical properties of PDMS-modified CaO-SiO2-TiO2 hybrids via sol-gel process. J. Mater. Sci.: Mater. Med., 2001, 12, 515-522. [78] Kamitakahara, M; Kawashita, M; Miyata, N; Kokubo, T; Nakamura, T. Bioactivity and mechanical properties of polydimethylsiloxane(PDMS)-CaO-SiO2 hybrids with different PDMS contents. J. Sol-Gel Sci. Tech., 2001, 21, 75-81. [79] Kamitakahara, M; Kawashita, M; Miyata, N; Kokubo, T; Nakamura, T. Bioactivity and mechanical properties of polydimethylsiloxane(PDMS)-CaO-SiO2 hybrids with different calcium contents. J. Mater. Sci.: Mater. Med., 2002, 13, 1015-1020. [80] Hench, LL; Wilson, J. In: Hench LL, Wilson J. editors. An Introduction to Bioceramics. Singapore: World Scientific; 1993; 12. [81] Kamitakahara, M; Kawashita, M; Miyata, N; Kokubo, T; Nakamura, T. Degradation of bioactive polydimethylxiloxane-CaO-SiO2-TiO2 and poly(tetramethylene oxide)-CaOTiO2 hybrids in a simulated body fluid", J. Am. Ceram. Soc., 2004, 87, 235-239. [82] Hayashi, T; Saito, H. Preparation of CaO-SiO2 glasses by the gel method. J. Mater. Sci., 1980, 15, 1971-1977. [83] Kamitakahara, M; Kawashita, M; Miyata, N; Kokubo, T; Nakamura, T. Apatite formation on CaO-free polydimethylsiloxane(PDMS)-TiO2 hybrids", J. Mater. Sci.: Mater. Med., 2003, 14, 1067-1072. [84] Wen, J; Wilkes, GL. Organic/inorganic hybrid network materials by the sol-gel approach. Chem. Mater., 1996, 8, 1667-1681. [85] Kamitakahara, M; Kawashita, M; Miyata, N; Kokubo, T; Nakamura, T. Apatite-forming ability and mechanical properties of CaO-free poly(tetramethylene oxide) (PTMO)-TiO2 hybrids treated with hot water. Biomaterials, 2003, 24, 1357-1363.

In: Ceramics and Composite Materials: New Research ISBN: 1-59454-370-4 Editor: B.M. Caruta, pp. 215-236 © 2006 Nova Science Publishers, Inc.

Chapter 7

EFFECT OF SAMPLE SIZE AND DISTRIBUTION PARAMETERS IN ESTIMATION OF CONFIDENCE LOWER BOUNDS FOR WEIBULL PERCENTILES Burak Birgoren* Associate Professor Kırıkkale University, Faculty of Engineering, Department of Industrial Engineering, 71451-Kırıkkale, Turkey

Abstract The Weibull distribution is widely used for modelling fracture strength of composite materials. Estimating lower percentiles of the Weibull distribution has been a major concern, because small sizes of fracture strength experiments leads to unreliable estimates. Therefore, several recent studies have focused on estimating confidence lower bounds on the lower percentiles, namely A-basis and B-basis material properties. The general linear regression methods with different probability indices and weight factors, and the maximum likelihood method have been used for the estimation; the performance of these methods were compared in recent studies and different methods were proposed using different criteria for comparison. Aside from this ambiguity, the criteria and comparison studies do not allow any practical interpretation. In particular, experimenters would like to know, in practical terms, how well a method performs with respect to others and how its performance is affected by the sample size. This chapter intends to clarify answers to these questions. Percent departures of the lower bounds from the true percentiles convey such a practical meaning. Simulating these departures, probabilistic upper bounds for maximum percent departures are determined; tables are provided for computing the upper bounds for a wide range of sample sizes. Using the upper bounds as a criterion for comparison, it is observed that the maximum likelihood method is the best as compared to several variations of the general linear regression methods. Its superiority is illustrated by comparing the sample sizes necessary for different methods to achieve the same level of performance. For the maximum likelihood method, the upper bound levels are shown to increase at a higher rate for sample sizes less than 30 as compared to larger sizes. Also, the Weibull modulus is observed to have a very serious effect on the upper bounds. Finally, a two-stage experimental setup is proposed for sample size determination; *

E-mail address: [email protected]. Fax: ++90 318 357 2459

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after running a pilot experiment, the setup enables the experimenter to calculate the number of additional experiments necessary for lowering the upper bounds to a certain level. Its implementation is illustrated on a practical example.

Introduction The Weibull distribution has been widely used in recent years to model the statistical variation in the fracture strength of many materials such as advanced ceramics, metallic matrix composites and ceramic matrix composites [1]; it is also used to describe the fracture toughness behavior of steels in ductile-brittle transition region [2]. In particular, it is in the ASTM standards for modeling fracture strength of ceramic materials [3]. The scatter is caused by various factors such as anisotropy, internal structure, and working environment of composites, which prevent designers from having a specific strength value to characterize their mechanical behavior. The two-parameter Weibull distribution function is given by

F (σ ) = 1 − exp  − (σ / σ 0 )  m

(1)

where m is the shape parameter or Weibull modulus, σ 0 is the scale parameter or characteristic strength of the distribution, and F is the fracture probability of the material at or below uniaxial tensile stress σ . Weibull modulus, m, is related to the scatter of the data: the higher the m the lower the dispersion of fracture stress. The scale parameter, on the other hand, is closely related to the mean fracture stress. The Weibull parameters m and σ 0 are estimated from a sample of strength measurements σ 1 , σ 2 ,..., σ n . The Maximum Likelihood (ML) and the General Linear Regression (GLR) methods are commonly employed for estimating these parameters [1, 4, 5]. In practice, it is only possible to perform a limited number of experiments which result in a small sample of measured strength data, hence low accuracy of estimates. Therefore, statistical properties of the estimators and confidence intervals for m and σ 0 , particularly for m, have been studied extensively [1, 2, 6-18]. The methods for estimating Weibull parameters are also used in the estimation of statistically-based design values, namely A-basis and B-basis material properties (or simply A-basis and B-basis values), which are the 95% confidence lower bounds on the first and tenth percentiles of a Weibull distribution, respectively. They are of great interest to the engineer in the design of structural and mechanical components; however, recent research on their estimation has remained limited [1, 4, 5, 19, 20]. The basis values are often regarded as constants which can be used to help characterize the material and processing. Since they will always change from one sample (data set) to the next, treating them as material constants is always an approximation [21]. Therefore, ‘enough data’ should be used to reduce the error in the approximation within a tolerable level. Investigating the effect of sample size on the basis values through the use of simulated data from the normal distribution, Ref. [21] demonstrates that as the sample size becomes larger, the mean of the basis values increases while the standard deviation decreases. These outcomes are normally expected according to the theory of confidence intervals.

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217

This chapter proposes a general approach to quantifying the effect of sample size on the error of the approximation. The proposed approach measures the error as the percent departures of the lower bounds from the true percentiles, and it is applied to the ML and GLR methods. A general formula is derived in expressing the largest amount of error as a function of both the sample size and the Weibull modulus. The formula allows studying the general behavior of the error for different parameters and sample sizes for a given method. It is also used to compare the performance of the methods. Based on this approach, an experimental setup is proposed for determining the size of an experiment that lowers the error to a level specified by the experimenter. The parameter estimation methods, the confidence lower bounds, and results of recent studies for comparing the methods are described in the following three sections. The proposed approach and its implementation are discussed later.

Parameter Estimation Methods Several methods are available for the estimation of Weibull parameters. Among them, the most commonly used are the ML and the GLR methods [1, 5, 19]. ˆ and σˆ 0 should satisfy the following set of equations [22]: The ML parameter estimates m  ∑ ln σ i (σ i )mˆ ln σ i − n  i =1 n ∑ ˆ m  i =1  ∑ i =1 (σ i ) n

n

 n + =0  m  ˆ

(2)

and ˆ 1/ m

ˆ m σˆ 0 =  ∑ (σ i ) / n   i =1  n

(3)

where σ i represents strength measurements. Although Equation 2 is non-linear, it can be solved by the Newton-Raphson algorithm using Menon’s estimate as a starting point [23]: The Fortran source code for this algorithm can be found in Ref. [24]. Also, Wasserman [25] explains the use of Microsoft Excel® spreadsheets for computing the ML estimates. Equation 1 becomes a straight line by a double logarithmic transformation: ln [ − ln (1 − F (σ ) )] = m ln σ − m ln σ 0

(4)

In order to apply the GLR method, the measurements are ranked from the smallest to the largest. F-values are assigned according to the rank i of a measurement, σ ( i ) denoting the ith smallest: σ ( 1 ) ≤ σ ( 2 ) ≤ ... ≤ σ ( n ) . The most commonly used estimators of F are F (σ ( i ) ) = ( i − 0.3) / ( n + 0.4 )

(5a)

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Burak Birgoren

and F (σ ( i ) ) = i / ( n + 1)

(5b)

Considering the familiar form of a regression equation, Y = aX + b , the left side of Equation 4 corresponds to Y, ln σ corresponds to X, m corresponds to a, and −mln σ 0 corresponds to b. Using σ ( i ) and F (σ ( i ) ) pairs in Equation 4, a and b are obtained by the

ˆ = a and least squares procedure. Then the Weibull parameter estimates are calculated as m ˆ ). σˆ 0 = exp ( −b / m The regression model in Equation 4 has non-constant error variance [6], then use of the weighted least squares procedure gives estimates with better statistical properties: Bergman [6] and Faucher and Tyson [7] proposed using the following weight factors in linear regression respectively: wi = [(1 − F (σ ( i ) ) ) ln (1 − F (σ ( i ) ) )]

2

wi = 3.3F (σ ( i ) ) − 27.5 1 − (1 − F (σ ( i ) ) )

(6a) 0.025



(6b)

Also, Hung [15] proposed the following factors where F (σ ( i ) ) values are estimated by Equation 5a. wi =

[(1 − F (σ ( i ) ) ) ln (1 − F (σ ( i ) ) )]2 2 ∑ [(1 − F (σ ( i ) ) ) ln (1 − F (σ ( i ) ) )]

(6c)

Least squares and weighted least squares computations are performed by various statistics packages such as MINITAB® and SPSS®. Microsoft Excel® spreadsheets can also be used for this purpose. While there is no closed form solution to Equation 2 for the ML method, explicit formulae exist for the GLR estimates as follows [1]: ˆ = m

( ∑ w )( ∑ w z y ) − ( ∑ w z )( ∑ w y ) (∑ w )(w z ) − ( w z ) i

i

i

i

i

i

i

2 i

i

i

2

i

i

(7a)

i

and 

∑w y

(7b)

zi = ln σ (i )

(7c)

σˆ 0 = exp  − 

i

ˆ ∑ wi zi  −m  ˆ ∑ wi m  i

where

Effect of Sample Size and Distribution Parameters …

219

yi = ln [ − ln (1 − F (σ ( i ) ) )]

(7d)

When there is no weighting, the estimates are obtained by setting the weight factors to 1 in Equations 7a and 7b.

Confidence Lower Bounds An important reason for the estimation of Weibull parameters is the need to determine the (100p)th percentile, σ p , for a predefined failure probability p: Pr [σ ≤ σ p ] = p

(8)

By setting F (σ ) = p in Equation 1, σ p is obtained as

σ p = σ 0 [ln (1 / (1 − p ) )]

1/ m

(9)

The estimate of σ p , σˆ p , can be obtained through the use of sample estimates of m and ˆ and σˆ 0 ) by the following equation: σ 0 (i.e. m

σˆ p = σˆ 0 [ln (1 / (1 − p ) )]

ˆ 1/ m

(10)

However, the σˆ p values can be quite unreliable, especially when they are estimated from small samples. Therefore, instead of σˆ p values, confidence lower bounds for σ p have been used to characterize a material. Combining Equations 9 and 10, the following relation is obtained [4]: ˆ ln (σˆ p / σ p ) = mln ˆ (σˆ 0 / σ 0 ) + (1 − m ˆ / m ) ln ( − ln (1 − p ) ) m

(11)

ˆ and σˆ 0 are estimated by the ML or the GLR method, the statistics When m ˆ / m are distributed independently of m and σ 0 , and hence pivotal ˆ ln (σˆ 0 / σ 0 ) and m m ˆ ln (σˆ p / σ p ) is also a pivotal statistic according to Equation 11. statistics [4, 22]. Therefore, m This property was first indicated by Thoman et al. [26] for the ML estimates, and then used by Bain [27] for the derivation of exact confidence lower bounds for σ p . ˆ ln (σˆ p / σ p ) ≤ c p ] = 1 − α , the following is obtained: By setting Pr [ m ˆ ) ≤ σ p ] =1−α Pr [σˆ p exp ( −c p / m

(12)

ˆ) l p = σˆ p exp ( −c p / m

(13)

Therefore,

Burak Birgoren

220

is the lower bound for (1 − α ) level one-sided confidence intervals for σ p . The l p values for

α = 0.05 and p = 0.01 and p = 0.10 are the A-basis and B-basis material properties, respectively. Computation of confidence lower bounds requires c p values, which can be obtained from ˆ ln (σˆ p / σ p ) . Results of a simulation study based Monte-Carlo simulations of the statistic m on the ML method were provided in tabular form [21]. In an effort to eliminate the use of tables, another study proposed closed-form functions, approximating simulated data, for computing the lower bounds [1]; the functions were provided for the ML method and the GLR method with two weighting schemes (F-values and weight factors are assigned according to Equations 5b, and 6a and 6b, respectively). Recently, it was shown that simulation-run time for the ML computation of the lower bounds does not exceed 30 seconds even for large simulations: The C++ source code for the computation can be obtained from the authors of Ref. [20].

Comparison of Lower Bound Estimation Methods In two recent studies [5, 19], the ML and GLR methods with various combinations of wi and Fi were chosen for comparison as shown in Table 1. In their simulation study, Barbero et al. ˆ ln (σˆ p / σ p ) . [5] compared the first seven of the methods with respect to the variance of m The authors found out that Method 6 has the smallest variance for sample sizes n ≤ 7, and Method 7 has the smallest variance for n > 7 . Table 1. Estimation methods investigated in the literature Method 1 2 3 4 5 6 7 8

Type Linear regr. Linear regr. (weighted) Linear regr. (weighted) Maximum Likelihood Linear regr. Linear regr. (weighted) Linear regr. (weighted) Linear regr. (weighted)

Equation for Fi 5a 5a 5a  5b 5b 5b 5a

Equation for wi wi = 1 6a 6b  wi = 1 6a 6b 6c

Variance is a general measure for comparing estimators, but it may be misleading when used for confidence lower bounds. There is an established theory for determining the best confidence intervals. A well-known measure for comparing one-sided confidence intervals is derived from statistical hypothesis testing: The best method is defined as the one with the smallest false coverage probabilities Pr [l p ≤ σ ′p ] for all σ ′p < σ p [28], which are analogous to Type-II errors in hypothesis testing. Based on this measure, Birgoren [19] chose three levels of σ ′p and compared all the methods in Table 1 using Monte-Carlo simulation for various sample sizes. He showed that Method 4 (ML) is the best method, while Method 7 is the second best. Also, the simulation results indicate that all the GLR methods with weighted

Effect of Sample Size and Distribution Parameters …

221

least squares (Methods 2, 3, 6-8) performed better than those with no weights (Methods 1 and 5). On the other hand, false coverage probabilities do not allow any practical interpretation. In particular, an experimenter would like to know, in practical terms, how well a method performs with respect to others and how its performance is affected by the sample size. Let’s take a closer look at the behavior of l p . A simulation of l p values, which will be explained later, yielded the probability density functions for the two methods illustrated in Fig. 1. For this simulation the parameters are set as m = 5, σ 0 = 100, p = 0.01, and the sample size is chosen as n = 20. The density function of l p estimated by the ML method is slightly to the right of the function estimated by the GLR method 1. Therefore, for any σ ′p < σ p , the ML method has smaller false coverage probabilities. However, it is not clear whether there is a significant difference from the viewpoint of an experimenter. Further, Method 1 is one of the worst methods in Table 1, so the difference between the ML method and the GLR methods with weighted least squares is expected to be smaller. One practical way for quantifying the difference is to compute the sample size necessary for the ML method to have approximately the same density function as the GLR method 1.

Figure 1. Effect of different estimation methods on the probability density function of l p

For this purpose, let’s examine the behavior of l p with respect to the sample size. In this case, the density functions of l p were estimated by the ML method for three sample sizes as depicted in Fig. 2. The function for n = 16 in Fig. 2 seems to be very close to the function for the GLR method in Fig. 1. This indicates that confidence lower bounds estimated by the ML method with a sample size of 16 have almost the same precision of those estimated by the GLR method 1 with a sample size of 20. The difference between the two sample sizes means saving four test specimens by preferring the former method.

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222

Figure 2. Effect of sample size on the probability density function of l p

Fig. 2 also shows that as n grows larger, the expected value of l p moves toward σ p and the variance decreases. This is a typical behavior for any m, σ 0 and p levels. Note that Pr [l p ≤ σ p ] = 0.95 for all n.

A Practical Measure for Comparison and Sample Size Selection The previous case study illustrated in Figs. 1 and 2 helps us understand the impact of method and sample size selection. The rest of this chapter will discuss the development and implementation of an approach that will serve as a practical tool for assessing this impact: the methods are compared, the effects of sample size and Weibull parameters are quantified, and the minimum sample size satisfying certain precision requirements is calculated. First note the behavior in Fig. 2: larger sample sizes increase the probability of observing l p values closer to the true percentile σ p . Conversely, for small samples, the distance between σ p and l p can become undesirably high while Pr [l p ≤ σ p ] = 0.95 holds true. Therefore, an experimenter would like to know the minimum number of measurements to obtain a desired closeness between σ p and l p : the ratio of (σ p − l p ) to σ p can be used as a measure of precision indicating the percent departure of l p from the true percentile σ p . The following discussion derives probabilistic upper bounds for the maximum value of this departure. After the necessary manipulations, l p in Equation 13 can be reformulated as ˆ (σˆ p / σ p ) − c p mln l p = σ p exp  m  ˆ /m m  

The following term

(14)

Effect of Sample Size and Distribution Parameters … Sp =

ˆ ln (σˆ p / σ p ) − c p m ˆ /m m

223

(15)

in Equation 14 is a pivotal statistic, that is, its distribution does not depend on m and σ 0 , ˆ ln (σˆ p / σ p ) and m ˆ / m , and the value of c p do not depend on because the distributions of m m and σ 0 [5]. Combining Equations 14 and 15 yields l p = σ p exp ( S p / m )

(16)

Let’s denote the 5th percentile of l p by l p (0.05) , which satisfies Pr [l p ≤ l p( 0.05 ) ] = 0.05 . It is a lower bound for l p as illustrated in Fig. 2. In Equation 16, l p is an increasing function of S p , then, l p (0.05) can be computed from the 5th percentile of S p , S p( 0.05 ) : l p( 0.05 ) = σ p exp ( S p( 0.05 ) / m )

(17)

According to the above relationship, once S p( 0.05 ) for a sample size n is known, l p (0.05) can be easily computed. Using the facts that Pr [l p ≤ σ p ] = 0.95 and Pr [l p ≤ l p( 0.05 ) ] = 0.05 , the following is obtained: Pr [l p( 0.05 ) ≤ l p ≤ σ p ] = 0.90

(18)

Finally, it follows from Equations 17 and 18 that Pr [ 0 ≤ (σ p − l p ) σ p ≤ 1 − exp ( S p( 0.05 ) / m )] = 0.90

(19)

Here, 1 − exp ( S p( 0.05 ) / m ) is a probabilistic upper bound for (σ p − l p ) σ p , which is the precision measure for l p in this study. The remaining task is the generation of the S p( 0.05 ) values for various sample sizes.

Monte-Carlo Simulation Methods 1, 4 and 7 in Table 1 are selected for generating S p( 0.05 ) values: the motivation behind this selection is that some experimenters prefer using a GLR method with weighted least squares; Method 7 is the best in this category as discussed previously. If ordinary least squares (no weights) is preferred, then the best in this category is Method 1 [19]. Finally, Method 4 (ML) is the best among all the methods in Table 1. A two-stage Monte-Carlo simulation study was performed for generating S p( 0.05 ) values. In the first part, the c p values were estimated following the procedure of Birgoren [19]: for specified values of α and p, a sample of n values are generated from a Weibull distribution ˆ ln (σˆ p / σ p ) is calculated for the three methods and with parameters m = 1 and σ 0 = 1; then m

224

Burak Birgoren

this process is repeated N times to find the estimates of c p . Throughout this study α = 0.05 and N = 200.000. This simulation was performed for each combination of p = 0.01 and 0.10, and n = 6 ,7 ,...,70 ,72 ,...,120 . The second part estimates S p( 0.05 ) using the simulated c p values: N Weibull samples of size n were generated with m = 1 and σ 0 = 1; for each sample, the S p values were estimated for the three methods by Equation 7. Using N simulated values of S p , an estimate of S p( 0.05 ) was obtained. This procedure was repeated for all combinations of p and n. The whole simulation procedure is illustrated in Fig. 3.

Figure 3. Flow diagram of the Monte-Carlo simulation procedure

Effect of Sample Size and Distribution Parameters …

225

The procedure is coded in the C++ language. Weibull variables with the parameters m = 1 and σ 0 = 1 are generated by the formula σ i = ln (1 / u ) , where u follows a uniform distribution on the interval [0,1]. These uniform variables are generated by L’Ecuyer’s multiple recursive random number generator [29]. The estimated S p( 0.05 ) values are presented in Tables A1, A2 and A3 in the appendix. Also, the l p values in Figs. 1 and 2 were generated using the S p values of the second part.

A Practical Comparison Based on Equivalent Sample Sizes The methods with the same S p( 0.05 ) values have the same upper bounds for (σ p − l p ) σ p according to Equation 19; this provides a basis for a more formal treatment of the discussion of Figs. 1 and 2. The S p( 0.05 ) values for Method 4 (ML) were taken from Table A2 for certain sample sizes, and the sample sizes giving the closest S p( 0.05 ) values for the other two methods were determined from Tables A1 and A3. The corresponding sample sizes are provided in Table 2. Table 2. Sample sizes with equivalent levels of precision for compared methods Equivalent Sample Sizes (n)

Method 4 10 20 30 40 50 60 70

p = 0.01 Method 7 10 21 32 43 54 65 76

Method 1 13 29 45 61 78 92 110

Method 4 10 20 30 40 50 60 70

p = 0.10 Method 7 10 21 31 42 53 64 74

Method 1 13 27 41 56 69 84 96

For sample sizes of 30 or less in Table 2, Method 7 achieves the level of precision of Method 4 with one or two additional test specimens, which can be interpreted as the cost of not using Method 4 (ML). As the sample size grows larger, however, this cost increases reaching to an additional 4-6 specimens for n = 70. The performance of Method 1 is much poorer requiring nearly an additional half the sample size of Method 4 (for p = 0.01). In general, the cost is less severe in the estimation of B-basis values (p = 0.10) than that of Abasis values (p = 0.01). These results strongly discourage the use of the GLR method with ordinary least squares. Use of weighted least squares significantly improves the performance making Method 7 an alternative to the ML method for small samples (n ≤ 30). Nevertheless, the ML method should be preferred for larger samples.

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226

Effects of Weibull Modulus and Sample Size The previous comparison based on sample sizes suggests that an experimenter should use the ML method for estimating the confidence lower bounds. Therefore, only this method will be considered for a detailed study of the effects of Weibull parameters and sample size. First, it would be very convenient to provide the S p( 0.05 ) values in Table A2 in a closedform expression. For each p, the S p( 0.05 ) values were fitted as a function of n by S p( 0.05 ) = b1 − b2 * exp ( ( n − b3 )

− b4

)

(20)

with the use of non-linear regression analysis. The empirical parameters b1 − b4 for p = 0.01 and 0.10 are presented separately in Table 3. When compared to the estimated S p( 0.05 ) values in Table A2, the maximum error of the empirical function in Equation 20 is less than 1% for both p = 0.01 and p = 0.10. Table 3. Parameters for the empirical function for S p( 0.05 ) in Equation 20

p = 0.01 p = 0.10

b1 23.57 12.0175

b2 23.8582 12.1435

b3 1.9259 1.8909

b4 0.6502 0.6171

Combining Equations 19 and 20, the upper bound for the precision measure (σ p − l p ) σ p can be modelled as 1 − exp ( S p( 0.05 ) / m ) = 1 − exp ({b1 − b2 * exp ( ( n − b3 )

− b4

)} m )

(21)

where the parameters b1 − b4 are as given in Table 3. A detailed examination of Equation 21 helps to give a comprehensive perspective about the effect of parameters and sample size on the precision of the confidence lower bounds. The percent departure of the lower bounds from the true percentile varies from sample to sample. Equation 21 forms an upper bound for this random quantity with a high probability. Note that it does not involve the scale parameter σ 0 , which is nice because the measure of precision is in the form of percent departure. Figs. 4 and 5 displays the plots of the function 1 − exp ( S p( 0.05 ) / m ) with respect to n for p = 0.01 and p = 0.10, respectively; m is varied in a wide range from 2 to 40, which are common Weibull modulus values for composite materials. It is clear from the plots that the function is decreasing in n and m. Marginal change in the function values are much more significant when n ≤ 30. Moreover, the higher the value of m is the larger the marginal change occurs. This is illustrated in Table 4 where the percent decrease in the function value is computed for 20 additional measurements. For example, for m = 20 and p = 0.01, an increase in the sample size from 10 to 30 decreases the value of 1 − exp ( S p( 0.05 ) / m ) by about 50 percent. However, an increase

Effect of Sample Size and Distribution Parameters …

227

from 30 to 50 will cause only a further 26 percent decrease. In other words, the gain is halved although the amount of experimental cost remained the same.

Figure 4. Plot of n versus 1 − exp ( S p( 0.05 ) / m ) for p = 0.01 (A-basis case)

Figure 5. Plot of n versus 1 − exp ( S p( 0.05 ) / m ) for p = 0.10 (B-basis case)

Figs. 4 and 5 also show that the function values for p = 0.10 are lower than the value for p = 0.01 for all n and m. Therefore, for the same level of precision, estimation of A-basis material property always requires a larger sample than that of B-basis material property. When 1 − exp ( S p( 0.05 ) / m ) and m were kept constant, the ratio of the sample size for p = 0.10 to the sample size for p = 0.01 was computed, provided that the sample sizes ranged between 10 and 100. This was repeated for various levels of 1 − exp ( S p( 0.05 ) / m ) between 0 and 1, and for m between 2 and 40. The ratios varied between 2.2 and 2.63. Therefore, it is reasonable to

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decide on the number of measurements as for B-basis and then multiply the resulting n by 2.4 to obtain an approximate A-basis sample size. Table 4. Percent change in 1 − exp ( S p( 0.05 ) / m ) with respect to n

1 − exp ( S p( 0.05 ) / m ) m p = 0.01

p = 0.10

Percent change from

n = 10

n = 30

n = 50

n = 70

2

10 to 30

30 to 50

50 to 70

0.9738

0.7960

0.6820

0.6079

18.3

14.3

10.9

20

0.3053

0.1470

0.1082

0.0894

51.9

26.3

17.4

40

0.1665

0.0764

0.0557

0.0457

54.1

27.1

17.8

2

0.8624

0.5891

0.4756

0.4107

31.7

19.3

13.6

20

0.1799

0.0851

0.0625

0.0515

52.7

26.6

17.6

40

0.0944

0.0435

0.0318

0.0261

53.9

27.0

17.8

The same plots also point out achievable levels of precision: for a wide range of m values, 30 measurements will suffice to reduce the upper bound for the percent error of A-basis and B-basis values below 80% and 60%, respectively. For 50 measurements, the upper bound values decrease below 70% and 50%. These limit values are obtained for m = 2; larger m results in much lower upper bounds. Finally, it should be noted that Figs. 4 and 5 can be easily reproduced for any m and n via a spreadsheet program.

A Two-Stage Experimental Setup for Sample Size Determination So far, one application of Equations 19 and 21 is demonstrated; they can also be used to determine the sample size of an experiment. In this case, the parameters should be assumed unknown, but Equation 21 involves the unknown Weibull modulus m. As a solution, a pilot experiment can be conducted to gain some information about m; this is the first stage of the experiment. Using this information, a decision can be reached about the number of additional measurements required in order to satisfy certain precision requirements. In other words, the information from the first stage is used to determine the size of the experiment to be run in the second stage. The two stages constitute the full experiment. In the pilot experiment, a confidence interval for m will be helpful because of a nice property of the function 1 − exp ( S p( 0.05 ) / m ) in Equation 21: it is decreasing in n and m as mentioned before. Let ml be the lower bound for a (1 − β ) level one-sided confidence interval for m, that is, Pr [ m ≥ ml ] = 1 − β . Then, if m is greater than ml , the value of the function at m will be less than the function value at m = ml . The procedure for computing ml at β = 0.05 level is briefly given along with the associated coefficients in Table A4 in the appendix. The lower bound value is used as follows: suppose the engineer requires that the precision measure (σ p − l p ) σ p be less than the level u = 0.50 for the full experiment, that is,

Effect of Sample Size and Distribution Parameters …

229

the maximum allowable departure of l p from σ p is set to 50%. Then, he places the lower bound value ml in Equation 21 and computes the smallest n that reduces the function value below u = 0.50. The computed value for n gives the size of the full experiment. This has a simple probabilistic interpretation: for the computed sample size, Pr [1 − exp ( S p( 0.05 ) / m ) ≤ u ] = 1 − β , because Pr [ m ≥ ml ] = 1 − β and the function 1 − exp ( S p( 0.05 ) / m ) is decreasing in m. Using this fact and Equation 19 in Bonferroni Inequality [23], a well-known inequality of probability, it can be shown that Pr [ 0 ≤ (σ p − l p ) σ p ≤ u ] ≥ 0.90 − β

(22)

Therefore, when the full experiment is run, the value of (σ p − l p ) σ p will be between 0 and u with at least 0.90 − β probability. The following section will illustrate the proposed approach on a practical example. The remaining important question is the size of the pilot experiment. Behavior of the confidence intervals for m provides an answer. Barbero et al. [1] plotted the standard ˆ / m , which is used to construct the confidence intervals, with deviation of the statistic m respect to n. The plot shows a sharp decrease in the standard deviation for small n, particularly for n < 10. This indicates that with each additional measurement, much larger confidence lower bounds for m - much closer to σ p - are obtained for n < 10, as compared to larger values of n. Larger bounds in Equation 21 result in smaller sample sizes for the full experiment, and, thus, sample sizes of at least 10 are recommended for the pilot experiment. The choice of a proper level for u will be discussed after the following example.

Practical Example We will consider the results of the following experiment [30]: 19 identical composite specimens were prepared from quasi-isotropic carbon-epoxy sheets with (0°)3 configuration, 0.89 mm thickness, and 295 gr/m2 weight. The tension experiments were carried out on an Instron 8516+ universal testing machine according to the ASTM D3039 standard. A crosshead speed of 1.33 mm/min was used and room temperature conditions existed during the tests. The fracture strength values measured are presented in Table 5. Table 5. Fracture strength values from tension experiments Test No. Fracture strength [MPa] Test No. Fracture strength [MPa]

1

2

532.7

502.5

11

12

522

439

3

4

5

442

473

519

13

14

15

513.6

497.5

521.6

6

7

8

9

10

477

510

522

552

16

17

18

19

450.9

476.5

507.3

463.5

502.7

The parameter estimates and confidence lower bounds computed by Methods 1, 4 and 7 are presented in Table 6. The values of σˆ p were calculated from Equation 10, and the values

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of l p (A-Basis and B-basis values) were calculated from Equation 13, where the c p values were taken from Tables A1-A3. Table 6. Weibull estimates from the experimental data

Method 1

Method 4

Method 7

ˆ m

17.57

18.86

16.03

σˆ 0

510.60

510.18

510.83

σˆ p

393.00

399.77

383.42

lp

354.43

353.23

355.04

σˆ p

449.22

452.80

443.94

lp

424.34

423.38

424.53

p = 0.01

p = 0.10

While Method 4 is better than the others in Table 6, it does not have the largest basis values, because, as illustrated in Fig. 1, Method 4 produces larger basis values on the average, which does not guarantee larger values from a single sample. It is also interesting to note that the difference between the basis values are about 1-2 MPa., which are not practically significant. Again, the difference may get much larger for another sample. Now, we will treat these observations as our pilot experiment and determine the size of a ˆ = 18.86 from Table 6, and full experiment based on the ML method (Method 4). Using m using Table A4, the value of the confidence lower bound for m is computed as ml = 12.82. As an alternative, the formulae provided by Barbero et al. [1] can be used for this computation. We will compute the size of a full experiment for each of the basis values – for p = 0.01 and p = 0.10. Suppose that we want to reduce the precision measure (σ p − l p ) σ p below u = 0 .20. First, ml = 12.82, is placed in Equation 21 with the empirical parameters for p = 0.01 in Table 3. Then, the values of the function in Equation 21 are computed for different n in order to find the smallest sample size that lowers the value of the function 1 − exp ( S p( 0.05 ) / m ) below 0.20. This sample size can be found with a spreadsheet program, for instance, by enumerating the function values for all n between 19 and 120. As an alternative, a scientific calculator can be used after examining Fig. 4: for m = 10, the function value for n = 20 is about 0.20. It is obvious from the plot that higher n values are needed for m = 12.82. For n = 20 and 30 the function values are greater than 0.20, but for n = 40 the value is lower. Next, n is increased successively starting from 31, and finally n = 36 is found to be the smallest sample size lowering 1 − exp ( S p( 0.05 ) / m ) below 0.20. Therefore, this computation can be performed using a calculator with a reasonable amount of trial and error. Let’s interpret this result according to Equation 22: when additional 17 tests are carried out in the second stage of the experiment, hence a total of n = 36 observations, the value of (σ p − l p ) σ p will be below 0.20 with at least 1 − β = 0.85 probability. In other words, we are at least 85% confident that, the percent deviation of the B-basis value from the true 10th percentile will be less than 20% after the second stage is run.

Effect of Sample Size and Distribution Parameters …

231

When this computation is repeated for p = 0.10, it is found that the function value for the pilot experiment (n = 19) is 0.17, which is less than u = 0 .20. Therefore, no second stage will be required. According to Equation 22, the value of (σ p − l p ) σ p is already below 0.17 with at least 1 − β = 0.85 probability. The choice of u = 0 .20 in this example is subjective. Further, as can be seen from Fig. 4, if the confidence lower bound for m were ml = 2, it would be impossible to achieve this level of precision with even 120 measurements. Experimenters need pre-specified values with respect to which they judge a basis value, for instance, as “precise” and “not precise”. Such values may be specified based on the observed ml values. Revisiting Figs. 4 and 5, for instance, for ml ≥ 10, the levels of u = 0 .20 and u = 0 .10 may be recommended for precise Abasis and B-basis computations.

Conclusions This chapter has focused on the estimation of A-basis and B-basis material properties for the two-parameter Weibull distribution. It has proposed percent departures of the basis values from true percentiles as a measure of precision for the estimation methods; the ML method and several variations of the GLR method proposed in the literature have been taken into consideration. Using Monte-Carlo simulations, probabilistic upper bounds for the maximum percent departures were determined. The formula and the associated tables proposed for this purpose allow an experimenter to calculate the upper bounds for a wide range of sample sizes and Weibull modulus values. Using the measure as a criterion for comparison, the ML method has been shown to be the best method, that is, it has the smallest upper bounds. Its superiority has been illustrated by comparing the sample sizes necessary for different methods to produce the same upper bounds. The ML method produces sample sizes that are significantly lower than those for other methods except for a certain variation of the GLR method; it may be considered as an alternative to the ML method for small samples. The effects of the sample size and the Weibull modulus have been investigated for the ML method. The upper bound levels have been shown to increase at higher rates for sample sizes less than 30 as compared to larger sizes. Also, the Weibull modulus has been observed to have a very serious effect on the upper bounds; the higher the Weibull modulus the lower upper bounds. Finally, a two-stage experimental setup has been proposed for sample size determination; after running a pilot experiment, the setup enables the experimenter to calculate the number of additional experiments necessary for lowering the upper bounds to a certain level. As illustrated on a practical example, the setup is an alternative to classic experiment plans with pre-determined sample sizes; it may help experimenters reduce the total size of an experiment as compared to classic experiment plans, conducting experiments only necessary for achieving the target upper bound levels, hence, cutting down experimental costs.

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Appendix Table A1. The values of cp and Sp(0.05) for Method 1 (GLR with no weights)

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44

p = 0.01 cp Sp(0.05) 4.2189 -13.8452 3.6209 -11.9068 3.2718 -10.5799 2.9766 -9.6085 2.7589 -8.8341 2.5644 -8.2253 2.4064 -7.6707 2.2781 -7.2460 2.1822 -6.8574 2.0759 -6.5046 2.0023 -6.2461 1.9336 -5.9865 1.8680 -5.7409 1.8150 -5.5408 1.7570 -5.3511 1.6972 -5.1799 1.6581 -5.0389 1.6123 -4.8710 1.5684 -4.7155 1.5333 -4.5933 1.4960 -4.4883 1.4750 -4.3902 1.4469 -4.2796 1.4164 -4.1917 1.3991 -4.0997 1.3669 -4.0160 1.3377 -3.9313 1.3191 -3.8522 1.2908 -3.7762 1.2767 -3.7097 1.2581 -3.6481 1.2473 -3.6035 1.2226 -3.5178 1.2055 -3.4646 1.1946 -3.4219 1.1781 -3.3730 1.1607 -3.3287 1.1444 -3.2774 1.1344 -3.2263

p = 0.10 cp Sp(0.05) 2.2632 -7.2668 1.9738 -6.3011 1.7698 -5.5730 1.6162 -5.0705 1.5012 -4.6529 1.4064 -4.3484 1.3337 -4.0745 1.2576 -3.8385 1.2098 -3.6469 1.1507 -3.4459 1.1026 -3.2986 1.0693 -3.1685 1.0368 -3.0451 1.0015 -2.9268 0.9652 -2.8227 0.9450 -2.7469 0.9138 -2.6582 0.8930 -2.5764 0.8619 -2.4845 0.8498 -2.4322 0.8307 -2.3729 0.8197 -2.3240 0.8006 -2.2633 0.7865 -2.2181 0.7719 -2.1632 0.7575 -2.1245 0.7427 -2.0769 0.7350 -2.0427 0.7190 -1.9988 0.7113 -1.9651 0.6989 -1.9284 0.6876 -1.9003 0.6771 -1.8618 0.6687 -1.8351 0.6620 -1.8084 0.6522 -1.7827 0.6430 -1.7592 0.6374 -1.7344 0.6288 -1.7050

n 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104 106 108

p = 0.01 cp Sp(0.05) 1.0451 -2.9519 1.0316 -2.9003 1.0326 -2.8807 1.0145 -2.8429 1.0067 -2.8224 0.9993 -2.7784 0.9862 -2.7476 0.9811 -2.7340 0.9699 -2.6950 0.9697 -2.6760 0.9587 -2.6475 0.9489 -2.6144 0.9432 -2.5861 0.9308 -2.5669 0.9255 -2.5395 0.9082 -2.5076 0.9092 -2.5050 0.9054 -2.4878 0.8975 -2.4582 0.8878 -2.4299 0.8723 -2.3996 0.8655 -2.3605 0.8518 -2.3263 0.8411 -2.2820 0.8269 -2.2498 0.8211 -2.2184 0.8121 -2.1886 0.8041 -2.1599 0.7965 -2.1277 0.7814 -2.0956 0.7742 -2.0656 0.7619 -2.0437 0.7633 -2.0274 0.7536 -1.9995 0.7476 -1.9831 0.7409 -1.9621 0.7263 -1.9320 0.7227 -1.9149 0.7140 -1.8873

p = 0.10 cp Sp(0.05) 0.5818 -1.5644 0.5740 -1.5374 0.5684 -1.5177 0.5616 -1.4994 0.5584 -1.4871 0.5528 -1.4695 0.5464 -1.4518 0.5425 -1.4436 0.5409 -1.4297 0.5333 -1.4113 0.5279 -1.3938 0.5205 -1.3749 0.5200 -1.3638 0.5158 -1.3546 0.5107 -1.3412 0.5076 -1.3294 0.5023 -1.3211 0.5000 -1.3118 0.4927 -1.2946 0.4895 -1.2817 0.4858 -1.2701 0.4795 -1.2483 0.4720 -1.2277 0.4630 -1.2006 0.4594 -1.1906 0.4521 -1.1699 0.4483 -1.1541 0.4433 -1.1380 0.4399 -1.1239 0.4314 -1.1064 0.4257 -1.0899 0.4194 -1.0756 0.4173 -1.0652 0.4144 -1.0540 0.4107 -1.0442 0.4072 -1.0334 0.4014 -1.0213 0.3995 -1.0108 0.3951 -0.9992

Effect of Sample Size and Distribution Parameters …

n 45 46 47 48 49 50

p = 0.01 cp Sp(0.05) 1.1146 -3.1676 1.1059 -3.1431 1.0938 -3.0943 1.0767 -3.0543 1.0651 -3.0127 1.0653 -2.9918

p = 0.10 cp Sp(0.05) 0.6205 -1.6817 0.6128 -1.6620 0.6025 -1.6295 0.5977 -1.6157 0.5948 -1.5973 0.5895 -1.5777

n 110 112 114 116 118 120

p = 0.01 cp Sp(0.05) 0.7157 -1.8791 0.7029 -1.8471 0.7000 -1.8357 0.6910 -1.8186 0.6840 -1.7925 0.6842 -1.7824

233

p = 0.10 cp Sp(0.05) 0.3922 -0.9900 0.3882 -0.9747 0.3843 -0.9668 0.3793 -0.9583 0.3778 -0.9473 0.3743 -0.9372

Table A2. The values of cp and Sp(0.05) for Method 4 (ML)

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38

p = 0.01 cp Sp(0.05) 6.7836 -12.0682 5.6423 -10.1547 4.9408 -8.9221 4.3946 -8.0170 3.9751 -7.2589 3.6480 -6.7111 3.3799 -6.2506 3.1326 -5.8236 2.9529 -5.5038 2.7966 -5.2031 2.6457 -4.9498 2.5476 -4.7535 2.4253 -4.5443 2.3344 -4.3717 2.2267 -4.1922 2.1483 -4.0569 2.0845 -3.9462 2.0341 -3.8400 1.9444 -3.6933 1.8924 -3.5921 1.8383 -3.4812 1.7913 -3.4027 1.7498 -3.3297 1.7118 -3.2561 1.6772 -3.1880 1.6359 -3.1251 1.5990 -3.0579 1.5652 -2.9914 1.5359 -2.9383 1.4993 -2.8664 1.4832 -2.8281 1.4523 -2.7773 1.4251 -2.7321

p = 0.10 cp Sp(0.05) 3.5389 -6.4241 2.9759 -5.4701 2.6024 -4.8051 2.3182 -4.3220 2.1097 -3.9395 1.9487 -3.6607 1.8208 -3.4318 1.6944 -3.2079 1.6032 -3.0404 1.5119 -2.8679 1.4312 -2.7343 1.3763 -2.6193 1.3168 -2.5170 1.2673 -2.4172 1.2183 -2.3323 1.1761 -2.2602 1.1327 -2.1867 1.1057 -2.1287 1.0631 -2.0561 1.0408 -2.0071 1.0140 -1.9535 0.9832 -1.9008 0.9587 -1.8556 0.9429 -1.8236 0.9199 -1.7805 0.8992 -1.7456 0.8783 -1.7071 0.8690 -1.6851 0.8487 -1.6462 0.8330 -1.6131 0.8167 -1.5834 0.8005 -1.5539 0.7878 -1.5316

n 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 72 74 76 78 80 82 84 86 88 90 92 94 96

p = 0.01 cp Sp(0.05) 1.1756 -2.2628 1.1604 -2.2433 1.1489 -2.2173 1.1360 -2.1885 1.1212 -2.1650 1.1090 -2.1428 1.0976 -2.1203 1.0868 -2.1007 1.0723 -2.0775 1.0673 -2.0600 1.0549 -2.0411 1.0438 -2.0184 1.0360 -2.0020 1.0251 -1.9778 1.0162 -1.9600 0.9990 -1.9406 0.9946 -1.9258 0.9834 -1.9085 0.9765 -1.8937 0.9674 -1.8803 0.9502 -1.8463 0.9363 -1.8170 0.9178 -1.7843 0.9049 -1.7554 0.8890 -1.7260 0.8829 -1.7144 0.8691 -1.6857 0.8509 -1.6606 0.8436 -1.6410 0.8281 -1.6134 0.8190 -1.5957 0.8103 -1.5775 0.8067 -1.5655

p = 0.10 cp Sp(0.05) 0.6544 -1.2751 0.6480 -1.2665 0.6386 -1.2494 0.6313 -1.2324 0.6258 -1.2216 0.6180 -1.2089 0.6099 -1.1935 0.6043 -1.1822 0.5983 -1.1724 0.5923 -1.1599 0.5868 -1.1488 0.5816 -1.1381 0.5746 -1.1259 0.5696 -1.1143 0.5637 -1.1029 0.5585 -1.0945 0.5548 -1.0870 0.5500 -1.0785 0.5446 -1.0692 0.5429 -1.0642 0.5302 -1.0422 0.5219 -1.0245 0.5141 -1.0106 0.5062 -0.9934 0.4979 -0.9766 0.4877 -0.9627 0.4851 -0.9521 0.4803 -0.9430 0.4736 -0.9297 0.4654 -0.9160 0.4586 -0.9047 0.4553 -0.8945 0.4487 -0.8822

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n 39 40 41 42 43 44 45 46 47 48 49 50

p = 0.01 cp Sp(0.05) 1.3956 -2.6834 1.3816 -2.6445 1.3544 -2.6011 1.3365 -2.5681 1.3123 -2.5195 1.2948 -2.4895 1.2727 -2.4542 1.2571 -2.4189 1.2387 -2.3837 1.2261 -2.3660 1.2108 -2.3269 1.1948 -2.2978

p = 0.10 cp Sp(0.05) 0.7730 -1.5089 0.7634 -1.4826 0.7476 -1.4587 0.7402 -1.4423 0.7247 -1.4132 0.7201 -1.4034 0.7075 -1.3828 0.6962 -1.3599 0.6904 -1.3439 0.6813 -1.3317 0.6717 -1.3092 0.6635 -1.2928

n 98 100 102 104 106 108 110 112 114 116 118 120

p = 0.01 cp Sp(0.05) 0.7948 -1.5419 0.7838 -1.5247 0.7748 -1.5091 0.7653 -1.4943 0.7526 -1.4716 0.7485 -1.4596 0.7402 -1.4478 0.7343 -1.4278 0.7248 -1.4137 0.7170 -1.3998 0.7087 -1.3841 0.7049 -1.3752

p = 0.10 cp Sp(0.05) 0.4423 -0.8696 0.4385 -0.8634 0.4340 -0.8535 0.4295 -0.8473 0.4251 -0.8371 0.4159 -0.8216 0.4129 -0.8172 0.4104 -0.8081 0.4059 -0.8001 0.4030 -0.7930 0.4006 -0.7864 0.3965 -0.7802

Table A3. The values of cp and Sp(0.05) for Method 7 (GLR with weights)

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

p = 0.01 cp Sp(0.05) 2.8149 -12.5985 2.3891 -10.6259 2.1382 -9.3048 1.9445 -8.3729 1.7938 -7.5679 1.6844 -6.9989 1.5822 -6.4786 1.4976 -6.0636 1.4354 -5.7152 1.3872 -5.3990 1.3302 -5.1357 1.3019 -4.9163 1.2601 -4.7087 1.2333 -4.5322 1.1917 -4.3498 1.1586 -4.2008 1.1444 -4.0928 1.1282 -3.9732 1.0833 -3.8175 1.0746 -3.7353 1.0532 -3.6180 1.0334 -3.5270 1.0219 -3.4509 1.0095 -3.3831 1.0012 -3.3067 0.9846 -3.2496 0.9651 -3.1691

p = 0.10 cp Sp(0.05) 1.5494 -6.6763 1.3385 -5.6819 1.1995 -4.9821 1.0981 -4.4894 1.0208 -4.0793 0.9688 -3.7953 0.9214 -3.5389 0.8684 -3.3050 0.8408 -3.1320 0.8049 -2.9466 0.7749 -2.8127 0.7587 -2.6959 0.7364 -2.5883 0.7168 -2.4859 0.6979 -2.3914 0.6801 -2.3163 0.6652 -2.2488 0.6550 -2.1868 0.6336 -2.1058 0.6278 -2.0565 0.6174 -2.0016 0.6046 -1.9513 0.5966 -1.9078 0.5928 -1.8715 0.5845 -1.8311 0.5732 -1.7920 0.5638 -1.7547

n 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 72 74 76 78 80 82 84

p = 0.01 cp Sp(0.05) 0.8049 -2.3743 0.7920 -2.3459 0.7934 -2.3221 0.7849 -2.2904 0.7782 -2.2720 0.7724 -2.2474 0.7684 -2.2225 0.7610 -2.2033 0.7559 -2.1765 0.7566 -2.1637 0.7509 -2.1392 0.7431 -2.1180 0.7434 -2.1033 0.7391 -2.0750 0.7321 -2.0599 0.7196 -2.0344 0.7205 -2.0189 0.7200 -2.0077 0.7125 -1.9887 0.7041 -1.9728 0.7038 -1.9461 0.6943 -1.9139 0.6840 -1.8840 0.6765 -1.8488 0.6708 -1.8268 0.6720 -1.8151 0.6602 -1.7793

p = 0.10 cp Sp(0.05) 0.4678 -1.3192 0.4637 -1.3065 0.4621 -1.2930 0.4551 -1.2722 0.4551 -1.2663 0.4529 -1.2544 0.4456 -1.2355 0.4431 -1.2254 0.4411 -1.2151 0.4354 -1.1995 0.4348 -1.1900 0.4293 -1.1762 0.4302 -1.1693 0.4263 -1.1528 0.4228 -1.1446 0.4211 -1.1362 0.4174 -1.1241 0.4156 -1.1170 0.4143 -1.1080 0.4098 -1.0999 0.4058 -1.0809 0.4009 -1.0628 0.3974 -1.0500 0.3905 -1.0297 0.3887 -1.0184 0.3825 -1.0027 0.3813 -0.9911

Effect of Sample Size and Distribution Parameters …

n 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

p = 0.01 cp Sp(0.05) 0.9598 -3.1206 0.9421 -3.0512 0.9298 -2.9883 0.9291 -2.9389 0.9135 -2.8907 0.9027 -2.8455 0.8915 -2.7971 0.8814 -2.7468 0.8763 -2.7139 0.8677 -2.6717 0.8594 -2.6356 0.8539 -2.6014 0.8363 -2.5532 0.8344 -2.5254 0.8298 -2.4951 0.8204 -2.4635 0.8193 -2.4351 0.8133 -2.4066

p = 0.10 cp Sp(0.05) 0.5598 -1.7261 0.5536 -1.6924 0.5473 -1.6620 0.5386 -1.6254 0.5321 -1.5967 0.5249 -1.5749 0.5211 -1.5528 0.5185 -1.5303 0.5078 -1.5019 0.5080 -1.4851 0.4984 -1.4593 0.4999 -1.4481 0.4915 -1.4235 0.4865 -1.4041 0.4846 -1.3873 0.4803 -1.3727 0.4768 -1.3522 0.4739 -1.3378

n 86 88 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120

p = 0.01 cp Sp(0.05) 0.6502 -1.7515 0.6505 -1.7349 0.6392 -1.7133 0.6326 -1.6883 0.6293 -1.6665 0.6291 -1.6544 0.6234 -1.6312 0.6166 -1.6150 0.6126 -1.5985 0.6025 -1.5758 0.6010 -1.5649 0.5965 -1.5443 0.5916 -1.5319 0.5904 -1.5190 0.5835 -1.5001 0.5791 -1.4889 0.5740 -1.4716 0.5738 -1.4624

235

p = 0.10 cp Sp(0.05) 0.3775 -0.9781 0.3743 -0.9656 0.3681 -0.9513 0.3649 -0.9425 0.3606 -0.9269 0.3577 -0.9165 0.3576 -0.9077 0.3549 -0.8988 0.3517 -0.8890 0.3507 -0.8820 0.3454 -0.8702 0.3421 -0.8590 0.3377 -0.8516 0.3369 -0.8438 0.3351 -0.8359 0.3304 -0.8261 0.3279 -0.8177 0.3281 -0.8137

Table A4. Coefficients for computing confidence lower bounds for m*

n 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 *

d 2.4300 2.1782 2.0272 1.9121 1.8198 1.7496 1.6917 1.6412 1.6020 1.5676 1.5394 1.5152 1.4908 1.4711 1.4506

n 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

d 1.4348 1.4206 1.4091 1.3908 1.3807 1.3699 1.3596 1.3496 1.3429 1.3358 1.3279 1.3199 1.3135 1.3063 1.3001

n 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

d 1.2961 1.2903 1.2840 1.2778 1.2750 1.2690 1.2662 1.2612 1.2576 1.2532 1.2501 1.2454 1.2436 1.2411 1.2374

n 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

d 1.2329 1.2299 1.2288 1.2251 1.2220 1.2195 1.2172 1.2160 1.2126 1.2109 1.2092 1.2067 1.2050 1.2034 1.2003

n 66 67 68 69 70 72 74 76 78 80 82 84 86 88 90

ˆ d , ie Pr [ m ≥ ml ] = 0.95 . 95% confidence lower bound for m is ml = m

d values were produced by using 200.000 simulated samples for each n. See Ref. [22] for details.

d 1.1973 1.1963 1.1948 1.1925 1.1916 1.1877 1.1849 1.1812 1.1783 1.1760 1.1746 1.1714 1.1679 1.1664 1.1631

n 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120

d 1.1620 1.1597 1.1589 1.1562 1.1544 1.1523 1.1499 1.1479 1.1471 1.1457 1.1441 1.1427 1.1412 1.1394 1.1379

236

Burak Birgoren

References [1] Barbero, E.; Fernandez-Saez, J.; Navarro, C. Compos. Part B-Eng. 2000, 31, 375-381. [2] Barbero, E.; Fernandez-Saez, J.; Navarro, C. J. Mater. Sci. Lett. 2001, 20, 847-849. [3] “Standard Practice for Reporting Uniaxial Strength Data and Estimating Weibull Parameters for Advanced Ceramics”, ASTM C1239-94a (American Society for Testing and Materials, Philadelphia, PA, 1994). [4] Fernandez-Saez, J.; Chao, J.; Duran, J.; Amo, J. M. J. Mater. Sci. Lett. 1993, 12, 14931496. [5] Barbero, E.; Fernandez-Saez., J.; Navarro, C. J. Mater. Sci. Lett. 1999, 18, 1441-1443. [6] Bergman, B. J. Mater. Sci. Lett. 1986, 5, 611-614. [7] Faucher, B.; Tyson, W. R. J. Mater. Sci. Lett. 1988, 7, 1199-1203. [8] Khalili, A.; Kromp, K. J. Mater. Sci. 1991, 26, 6741-6752. [9] Langlois, R. J. Mater. Sci. Lett. 1991, 10, 1049-1051. [10] Gurvich, M. R.; Dibenedetto, A. T.; Pegoretti, A. J. Mater. Sci. 1997, 32, 3711-3716. [11] Gong, J. J. Mater. Sci. Lett. 1999, 18, 1405-1407. [12] Absi, J.; Fournier, P.; Glandus, J. C. J. Mater. Sci. 1999, 34, 1219-1227. [13] Gong, J. J. Mater. Sci. Lett., 2000, 19, 827-829. [14] Davies, I. J. J. Mater. Sci. Lett., 2001, 20, 997-999. [15] Hung, W. L. Qual. Reliab. Eng. Int. 2001, 17, 467-469. [16] Griggs, J. A.; Zhang, Y. J. Mater. Sci. Lett. 2003, 22, 1771-1773. [17] Wu, D.; Jiang, H. J. Mater. Sci. Lett. 2003, 22, 1745-1746. [18] Davies, I. J. J. Mater. Sci. 2004, 39, 1441-1444. [19] Birgoren, B. J. Mater. Sci. Lett. 2003, 21, 1121-1124. [20] Birgoren, B.; Dirikolu, M. H. Compos. Part B-Eng. 2004, 35, 263-266. [21] U.S. Department of Defence, Military Handbook- MIL-HDBK-17-1F: Composite Materials Handbook, Vol. 1, 2002. Chp 8: pp 10-14. (available at: www.knovel.com/knovel2/Toc.jsp?BookID=721 ) [22] Thoman, D. R.; Bain, L. J.; Antle, C. E. Technometrics 1969, 11, 445-460. [23] Law, A. M.; Kelton, W. D. Simulation Modeling and Analysis; McGraw-Hill: New York, NY, 2000; pp 303. [24] Keats, J. B.; Lawrence, F. P.; Wand, F. K. J. Qual. Technol. 1997, 29, 105-110. [25] Wasserman, G. S. Qual. Engng. 2000, 12, 569-581. [26] Thoman, D. R.; Bain, L. J.; Antle, C. E. Technometrics 1970, 12, 363-371. [27] Bain, L. J. Statistical Analysis of Reliability and Life-Testing Models; Marcel-Dekker: New York, NY, 1978; pp 227-246. [28] Casella, G.; Berger, R. L. Statistical Inference; Wadsworth: Belmont, CA, 1990; pp 433. [29] L’Ecuyer, P.; Oper. Res. 1999, 47, 159-164. [30] Dirikolu, M. H.; Aktas, A; Birgoren, B. Turkish J. Engng. Environ. Sci. 2002, 26, 45-48.

INDEX A accuracy, 46, 216 acetabulum, 123 acetone, 136 acid, 213 activation, 12 activation energy, 12 additives, 6, 10, 106, 120 adhesion, ix, 109, 126, 147, 152, 166 adhesion strength, ix, 147 adsorption, 117, 135, 144 aerogels, 67 affect, 8, 35 age, 102 ageing, 35, 88 agent, 77, 84, 85, 120 Al2O3 particles, 6, 7, 8, 9 algorithm, 217 alloys, 104, 198, 211 alternative, 43, 57, 59, 64, 65, 104, 105, 107, 115, 135, 225, 230, 231 alternatives, viii, 101, 102 aluminium, 126 ambiguity, x, 215 anatase, x, 197, 204, 207 animals, 106 anisometry, 70 anisotropy, 38, 89, 216 annealing, 117 antibiotic, 135, 144, 146 appendix, 225, 228 arithmetic, 47, 54, 55, 56, 57, 77, 78 arthroplasty, 144 aseptic, 35 association, 2, 10 assumptions, 55, 60 atmospheric pressure, 71 atoms, 33, 90 attachment, 19, 112, 145

attention, viii, 101, 110, 117, 200 availability, 111 averaging, 47

B bacteria, 198 behavior, viii, 13, 25, 32, 35, 36, 42, 43, 47, 49, 62, 66, 71, 87, 88, 101, 207, 210, 216, 217, 221, 222 bending, 34, 44, 203, 208 benign, 81 binding, 112 biocompatibility, viii, 101, 102, 104, 107, 108, 122, 126, 127, 144 biodegradability, 124 biodegradation, 124 biomaterials, viii, 101, 110, 117, 123, 128, 129, 134 biomedical applications, 102, 104, 106, 111, 115, 119, 120, 129, 134, 145 biopolymer, 135 blocks, 136, 141 blood, 108, 109, 111, 127, 134, 200 blood plasma, 111, 200 blood supply, 134 blowing agent, 81, 83 body, ix, x, 24, 35, 49, 91, 102, 103, 104, 106, 111, 113, 114, 115, 117, 118, 122, 124, 131, 134, 138, 139, 140, 141, 142, 144, 197, 198, 199, 200, 201, 203, 204, 207, 209, 211, 212, 213, 214 body fluid, x, 102, 106, 111, 114, 115, 117, 122, 131, 197, 200, 213, 214 bonding, x, 33, 113, 123, 197, 210, 211 bonds, 33, 88, 198, 203 brain, 198 breakdown, 67 burnout, 81

238

Index

C C++, 220, 225 calcium, 81, 104, 109, 110, 111, 112, 113, 116, 117, 118, 123, 124, 125, 126, 128, 132, 199, 200, 201, 203, 214 capillary, 23 capsule, 104, 116 carbon, 81, 103, 107, 108, 127, 229 carrier, 54 case study, 222 cast, 117 catalyst, 34 cation, 2 cell, viii, ix, 2, 81, 83, 101, 102, 110, 112, 130, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 166, 172, 175, 176, 177, 178, 180, 182, 183, 184, 185, 186, 187, 189, 190, 199 cell culture, 130 cell surface, 149, 172, 184 channels, 115, 138 chemical bonds, viii, 46, 101, 134 chemical interaction, 88 chemical properties, 15 chemical reactions, 81, 88, 109, 118 chemical reactivity, 109, 116 chemical stability, 107 chlorination, 106 circulation, 109 closure, 115 CO2, 111 coagulation, 23, 25 coatings, 123, 127, 128, 130, 145 collagen, 198, 199 colonization, 135 combined effect, 115 compatibility, 67, 68, 109, 126, 134, 142 competition, 4 compliance, 37, 38, 47, 50, 52, 72, 73, 74 components, 37, 38, 47, 50, 51, 52, 72, 73, 74, 91, 110, 111, 116, 117, 148, 151, 200, 216 composites, vii, ix, 31, 33, 46, 47, 48, 50, 51, 56, 57, 58, 79, 80, 82, 84, 89, 90, 91, 123, 131, 133, 134, 135, 216 composition, viii, 16, 18, 22, 23, 32, 33, 36, 48, 79, 80, 87, 89, 92, 101, 109, 110, 114, 116, 117, 120, 121, 122, 198, 199 compounds, 124, 125 compressibility, 45 computation, 220, 230, 231 computer simulations, 48, 66 computing, x, 215, 217, 220, 228, 235

concentration, 2, 6, 8, 13, 14, 21, 25, 91, 111, 112, 118, 142, 143, 144 conception, 90, 91 concrete, 86, 152, 158, 160, 162, 164, 165, 168, 169, 170, 171, 173, 174, 192, 193 condensation, 118 conduct, ix, 133 conduction, vii, 1, 2, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 18, 20, 21, 27 conductivity, 2, 7, 10, 13, 15, 21, 27, 33, 88, 89, 90, 91, 102 conductor, 2 confidence, x, 9, 215, 216, 217, 219, 220, 221, 226, 228, 229, 230, 231, 235 confidence interval, 216, 220, 228, 229 configuration, vii, 1, 3, 6, 21, 22, 24, 27, 229 confusion, 57 Congress, 4 connectivity, 49, 69, 134 consolidation, viii, 32, 84, 85 contamination, 2, 8, 9 context, 59, 63, 68, 89, 90 continuity, 123 control, 81, 122, 129 cooling, ix, 16, 24, 25, 71, 125, 147, 151, 152, 174, 180, 181, 183, 186, 188, 189, 190 cooling process, ix, 147, 151, 174, 180, 181, 183, 186, 188, 189, 190 Copyright, 4 corn, 84 correlation, 3, 48, 56, 89 correlation function, 48, 56, 89 corrosion, 35, 81, 88, 104, 108, 117 costs, 231 coverage, 220, 221 creep, 32, 88 critical value, 32, 36 crystallinity, 91, 110, 124, 199 crystallites, 46, 47, 52 crystallization, 16, 27, 105, 119, 120, 122 crystals, 50, 77, 112, 122, 128, 129 culture, 127 cycles, 105 cytotoxicity, 112, 130 Czech Republic, 31, 92, 98

D damage, 102 danger, 35 data set, 216 decomposition, 106 defects, ix, 32, 88, 91, 111, 122, 125, 197, 198

Index definition, 61 deformability, x, 33, 197, 198, 200, 201, 207, 208 deformation, 25, 36, 43, 44, 88, 102, 140, 207 degenerate, 56, 57, 59, 81, 82 degradation, 88, 108, 208 degree of crystallinity, 91, 120 density, 5, 13, 15, 23, 24, 25, 33, 34, 44, 65, 74, 77, 79, 80, 81, 90, 102, 104, 106, 135, 139, 145, 147, 153, 158, 175, 178, 210, 221 deposition, 108, 110, 128, 130 designers, 216 developed countries, 102 deviation, 56, 229, 230 diamonds, 84 dielectric constant, 89 differential approach, 57, 64, 87 differentiation, 134 diffraction, 17, 71 diffusion, 6, 88, 115, 118, 133, 148 diffusion process, 118, 148 diffusivity, 33 direct measure, 46, 137 discs, 117, 122 dislocation, 88, 148, 156, 184, 185 disorder, 47 dispersion, 6, 10, 135, 216 displacement, 37 distilled water, 136 distribution, vii, x, 1, 9, 14, 18, 19, 20, 21, 22, 25, 28, 60, 69, 107, 113, 134, 136, 137, 138, 141, 144, 145, 146, 149, 215, 216, 223, 225, 231 distribution function, 216 doping, 27, 106, 108, 123 drug delivery, ix, 133, 144, 145 drug release, ix, 133, 135, 145 drying, 81 ductility, 123 durability, 207, 209 duration, 13, 142

E elaboration, 47, 92 elastic deformation, ix, 147 elasticity, 32, 36, 37, 38, 39, 41, 43, 47, 49, 56, 57, 68, 74, 77, 78, 79, 86, 87, 88, 90, 124, 195, 207 electric field, 18 electrical conductivity, 28, 89 electrical properties, 18, 20, 21, 25 electricity, 102 electrodes, 18, 20, 21, 22, 23, 26 electron diffraction, 15

239 electron microscopy, 128, 129, 139, 140, 142, 212 emission, 105 energy density, ix, 147, 152, 156, 174, 175, 189 entropy, 43, 114 environment, 44, 105, 134, 203, 204, 207, 209, 211, 212, 216 epitaxial growth, 13 equating, 65 equilibrium, 44, 45, 88, 148, 160, 166, 184 erosion, 35, 108 estimating, vii, x, 1, 92, 215, 216, 226 ethanol, 27 EU, 191 Euro, 28, 29, 30, 127, 128 evidence, 8, 9, 13, 57, 80, 85, 107, 114 expectation, 137 expression, 55, 81 extraction, 105, 111

F fabrication, ix, 108, 133, 135 failure, 35, 105, 123, 134, 203, 207, 208, 219 family, 109, 115, 122, 123, 125 fatigue, 35, 88, 105, 108 femur, 105, 116 fibers, 32, 90, 123 fibrin, 210 fibroblast growth factor, 145 fibroblasts, 134 fibrous cap, 104 fibrous tissue, viii, ix, 101, 104, 116, 197, 198 fillers, 198 films, 9, 127 financial support, 125 fixation, 103, 104, 110 flame, 105 flexibility, x, 109, 197 fluid, 92, 112 foams, 57, 81, 83 focusing, 36 food, 135, 136 fracture stress, 216 France, 107 free energy, 91 friction, 104, 106 fuel, 2, 34 function values, 226, 227, 230

G gel, 200, 204, 213, 214

240

Index

gene, 80 generalization, 61, 68, 80, 90 generation, 120, 223 Germany, 92, 93, 94, 95, 96, 97, 98, 99, 131 glasses, 32, 33, 43, 46, 77, 89, 91, 101, 103, 109, 111, 116, 117, 118, 122, 123, 129, 131, 198, 210, 211, 213, 214 grain boundaries, 21, 90, 91, 124 grains, 4, 15, 17, 46, 88, 90, 106 granules, ix, 133, 136, 138, 139, 140, 141, 144, 145, 198 graphite, 81, 107, 108, 117 groups, 40, 88, 110, 117, 118, 204 growth, 13, 15, 23, 35, 88, 103, 104, 118, 119, 120, 122, 127, 128, 135 growth factor, 135

H hardness, 32, 33, 35, 88, 102, 107 heart valves, 108 heat, vii, 1, 3, 10, 11, 12, 13, 16, 21, 27, 44, 102, 104, 106, 119, 120, 122, 123, 138, 140, 198, 211, 212, 214 heating, 13, 14, 81, 84, 119, 120 heating rate, 13 height, 44, 185 hemisphere, 18 hemocompatibility, 127 hip, 102, 104, 105, 110, 123, 127 hip joint, 127 host, viii, 2, 77, 81, 101, 102, 103, 104, 133, 134 hybrid, x, 197, 200, 201, 204, 207, 209, 214 hydrolysis, 110 hydroxyapatite, ix, 103, 109, 111, 114, 117, 118, 122, 130, 131, 133, 134, 136, 142, 145, 146, 197, 198, 199, 210, 211, 213 hydroxyl, 118 hydroxylapatite, 128 hypothesis, 49, 70, 118, 220 hypothesis test, 220

I ideas, viii, 32, 36 image analysis, 48 immersion, 111, 112, 115, 116, 123 implementation, x, 216, 217, 222 impregnation, ix, 133, 135, 142, 144, 145 impurities, 6, 14, 104 in vitro, 111, 115, 116, 126, 129, 130, 144, 200, 214 inadmissible, 67

inclusion, viii, 32, 57, 59, 60, 84, 85, 89 indices, x, 215 induction, 213 inelastic, 88 inequality, 42, 229 inertia, 44, 102, 107 infinite, ix, 49, 61, 147, 148, 150, 173, 174, 178, 189, 190 influence, 14, 32, 33, 69, 70, 91, 128, 131, 134, 144, 157 input, 45, 48, 56, 77, 85, 89 instability, 107 Instron, 229 insulation, 81 integration, 162, 164, 168, 169, 171, 172, 173, 185, 191, 199 integrity, 36, 63, 67, 71, 83 intensity, 107 interaction, 6, 112, 113 interactions, 60, 62, 127 interest, viii, 65, 83, 92, 101, 102, 104, 110, 119, 141, 216 interface, 16, 18, 23, 26, 35, 91, 103, 109, 111, 113, 115, 118, 123, 128, 129, 131, 132, 210 interface energy, 16, 18, 23 interpretation, x, 42, 66, 68, 69, 70, 215, 221, 229 interrelations, 43 interval, 66, 154, 155, 160, 166, 172, 173, 176, 177, 183, 185, 186, 225, 228 intrinsic value, 70 intuition, 42 inversion, 37, 50, 72 ionic conduction, vii, 1, 4, 5, 6 ions, 33, 77, 110, 111, 118, 125, 129, 200, 201, 203 Italy, 133

J Japan, 126, 129, 197, 211, 213

K kinetics, 14, 88 knowledge, 43, 119 Korea, 1

L laminar, 113 language, 225 lanthanum, 26 laser ablation, 110, 128

Index lattice parameters, 109, 110, 125 laws, 152 lead, 62, 85, 134 Least squares, 218 legal, 4 liability, 4 lifetime, 35, 102, 127 likelihood, x, 215 limitation, 123 linear dependence, 60 liquid phase, vii, 1, 5, 7, 8, 17, 18, 19, 21, 23, 25, 28, 104 liquids, ix, 23, 133 location, 22

M macrophages, 127 magnesium, 106, 125, 126 management, 133 manufacturing, 102, 108, 117 mass, 9, 48, 91 material surface, 112 materials science, viii, 42, 101 matrix, viii, ix, 16, 32, 37, 38, 39, 40, 41, 47, 49, 50, 57, 59, 60, 61, 62, 63, 64, 65, 69, 70, 71, 72, 73, 74, 81, 84, 85, 86, 87, 89, 90, 108, 117, 119, 123, 135, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 162, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 189, 190, 191, 193, 199, 207, 216 measurement, 19, 20, 21, 44, 74, 83, 91, 102, 136, 217, 229 measures, 36, 91, 118, 217 mechanical properties, x, 35, 81, 88, 91, 104, 106, 108, 110, 114, 119, 120, 121, 122, 123, 131, 132, 134, 135, 148, 197, 201, 203, 207, 208, 209, 214 media, 135, 213 median, 84, 135 melt, 16, 105, 106, 117 melting, 16, 102, 117, 151 melting temperature, 16, 102, 151 melts, 117 membranes, 34 men, 102 mercury, 138, 141 metals, 35, 36, 88, 89, 109, 198, 211 micrometer, ix, 46, 81, 133 microscope, 21, 136

241 microstructure, viii, 6, 18, 25, 32, 36, 46, 47, 48, 60, 66, 77, 81, 83, 84, 85, 89, 90, 91, 106, 107, 108, 111, 114, 119, 120, 121, 135, 140, 142 migration, 26, 118 mixing, 6, 91 mode, 34, 44, 123 modeling, viii, 32, 36, 49, 88, 89, 92, 216 models, 68, 69, 70, 89, 90, 91 modules, 107 modulus, x, 32, 33, 34, 42, 43, 45, 46, 48, 52, 53, 54, 55, 56, 58, 59, 60, 61, 62, 64, 65, 66, 67, 68, 69, 70, 75, 76, 77, 78, 81, 82, 83, 84, 85, 86, 87, 90, 92, 105, 110, 123, 124, 134, 181, 191, 198, 200, 207, 208, 215, 216, 217, 226, 228, 231 molar ratios, 204 molar volume, 33, 45 mole, 48, 75 molecular weight, 126 molecules, 203 monolayer, 8 morbidity, 134 morphology, 46, 110, 128, 134 Moscow, 133, 135, 144, 145 motion, 88, 148, 156, 185, 198 motivation, 223 movement, 109 multiphase materials, vii, 31, 36, 46, 48, 55, 56 multiplication, 160, 166 multiplier, 166

N nanocomposites, 46, 80, 90, 91 nanomaterials, 46, 92 Nd, 128 needs, 4, 89 Netherlands, 94, 95, 96, 97, 99, 100, 125, 126 network, 109, 110, 112, 118, 125, 200, 203, 214 niobium, 200, 213 nitrogen, 135 nitrogen gas, 135 normal distribution, 137, 216 nucleation, 13, 16, 88, 118, 119, 120, 122, 200, 201, 204 nuclei, 119, 201 nucleus, 13 nutrition, 134

O observations, 21, 112, 138, 230 oil, ix, 133, 135, 136, 137

242

Index

optical properties, 35, 111 organism, 116 orientation, 46, 48, 52, 77, 90 osteomyelitis, 144 osteotomy, 123 oxidation, 123 oxides, 2, 71, 117, 120, 121, 200 oxygen, 2, 34, 88, 109 oxygen sensors, 34

P pain, 102, 134 parameter, viii, 48, 64, 66, 68, 69, 70, 85, 90, 101, 110, 160, 162, 164, 165, 166, 168, 169, 170, 172, 179, 180, 185, 216, 217, 218, 226, 229, 231 parameter estimates, 217, 218, 229 parameter estimation, 217 parents, 191 parotid, 111 particles, ix, 6, 7, 47, 77, 81, 105, 106, 108, 120, 123, 124, 127, 135, 139, 147, 148, 149, 150, 180, 183, 186, 204 percentile, 219, 222, 223, 226, 230 percolation, viii, 31, 63, 67, 68, 69, 89, 90 percolation theory, 63, 89 periodicity, 47 periodontal, 111, 125 permeability, 89 permit, 44, 90 perspective, 226 pH, 111, 112, 118, 124 phagocytosis, 124 phase boundaries, 46, 54, 88, 90, 91, 92 phase transformation, 88 phase transitions, 92 phosphates, 104, 109, 117, 124, 128 phosphorus, 116 photographs, 201, 205 physical and mechanical properties, viii, 101, 103, 110 physicochemical properties, viii, 101 physics, 34, 45, 77 plasma, 106, 110, 123, 200 plastic deformation, 88, 151 plasticity, 32 platelets, 90 plausibility, 65, 72 Poisson ratio, viii, 31, 33, 34, 42, 43, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 69, 70, 73, 74, 75, 79, 86, 87 polarization, 3

polycondensation, 203 polydimethylsiloxane, 200, 204, 214 polymer composites, 135, 212 polymer-based composites, 35 polymers, 81, 88, 89, 109, 134, 135 polymorphism, 125 polyurethane, 81 poor, 117, 198 population, ix, 133, 138 porosity, viii, 31, 35, 36, 49, 56, 57, 58, 60, 62, 63, 65, 66, 67, 68, 69, 70, 73, 78, 80, 83, 84, 85, 86, 87, 90, 101, 103, 114, 119, 138, 140, 142, 143 potassium, 117 power, viii, 31, 36, 54, 68 precipitation, 110, 111, 112, 115, 118, 135 prediction, 47, 54, 58, 64, 65, 66, 67, 77, 78, 80, 81, 84, 85, 87, 90, 92 preparation, 4, 120, 123, 138, 203 pressure, 44, 45, 71, 103, 108, 136, 139, 140, 141, 143 principle, 33, 47, 48, 57, 67, 68, 69, 78, 83, 88, 89, 91, 110, 117 probability, x, 215, 216, 219, 221, 222, 226, 229, 230, 231 probability density function, 221, 222 probe, 89 production, 107, 119, 134, 136 program, 228, 230 proliferation, 134 propagation, 107 proposition, 55 prostheses, 123 prosthesis, 104, 105, 107, 110 proteins, 112 protons, 111 purification, 104

Q quartz, 102

R radius, 33, 77, 149, 154, 155, 161, 162, 164, 165, 166, 168, 169, 171, 172, 173, 175, 176, 177, 178, 179, 181, 184, 185, 186, 190 range, ix, x, 35, 43, 46, 57, 62, 65, 73, 74, 88, 101, 111, 136, 138, 139, 142, 143, 147, 152, 174, 181, 185, 215, 226, 228, 231 rare earth elements, 71 raw materials, 111, 117 reaction mechanism, 115

Index reaction zone, 111 reagents, 117, 135, 207 reality, 80, 91 recall, 69, 70 reconstruction, 109, 123 redistribution, 25 regression, x, 215, 218, 226 regression analysis, 226 regression equation, 218 reinforcement, 135 relationship, 223 relationships, 33, 89, 90, 125 relaxation, 3, 105, 151 relevance, 213 reliability, 34, 35, 107, 134 repair, x, 113, 122, 133, 134, 197, 198 replacement, 35, 102, 116, 121, 123, 126, 130, 132, 145 replication, 81 reputation, 36 resistance, 2, 3, 5, 8, 12, 20, 22, 24, 33, 102, 104, 105, 110, 120, 122, 123, 124, 127, 157, 180 resolution, 122, 128, 129 responsibility, 4 retrieval, 4 rheology, 61, 62, 69 rights, 4 risk, 134 room temperature, 2, 45, 46, 71, 74, 77, 88, 106, 119, 120, 125, 136, 229 roughness, 104 rubber, 207 Russia, 133, 135, 145 rutile, 204, 205

S saliva, 111, 117, 129 sample, x, 13, 19, 70, 115, 215, 216, 217, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231 scanning electron microscopy, 3, 136, 201 scatter, 78, 84, 88, 216 scattering, 34 schema, 10 security, 60, 107 seeding, 13 segregation, 5, 8, 9, 14, 27, 207 self, viii, 31, 48, 57, 62, 63 semicircle, 10, 15, 21 sensitivity, 105, 144 sensors, vii, 1 separation, 22, 23, 25, 122

243 series, 4, 19, 62, 66, 68, 70, 90, 109, 116, 122, 191, 192, 193 serum, 112 services, 4 shape, 3, 4, 7, 21, 48, 49, 62, 65, 69, 70, 149, 160, 162, 163, 166, 168, 169, 172, 173, 179, 186, 207, 216 shaping, 81 shear, 34, 42, 43, 44, 45, 48, 52, 53, 55, 56, 58, 59, 61, 62, 64, 70, 71, 77, 78, 81, 82, 83, 89, 90, 113 sheep, 211, 212 shock, 33, 89 Si3N4, v, ix, 148, 173, 180, 181, 182, 183, 184, 185, 186, 190 SIC, 147 sign, 61, 162, 163, 168, 169, 171 silanol groups, 118, 122, 201 silica, 112, 117, 118, 119, 122, 123, 213, 214 silicon, 108, 111, 113, 200 similarity, 110 simulation, 20, 220, 221, 223, 224 Singapore, 95, 129, 210, 211, 212, 214 single crystals, 71, 88 SiO2, x, 2, 6, 7, 12, 13, 14, 25, 26, 104, 111, 116, 117, 118, 121, 122, 131, 197, 198, 200, 201, 203, 204, 210, 213, 214 sites, 122 skeleton, 59, 60, 135 skin, 134 sodium, 117, 198 sol-gel, ix, 13, 80, 116, 131, 197, 200, 203, 204, 207, 209, 213, 214 solid matrix, 56, 58 solid oxide fuel cells, vii, 1 solid solutions, 77, 91 solid state, 34, 45, 77, 110, 125 solubility, 9, 110, 117, 121, 124 solvents, 135, 207 South Korea, 1 Spain, 101 species, 109, 213 specific surface, 32, 124 spectroscopy, vii, 1, 9, 18, 21, 26, 28 spectrum, 3 speed, 110, 119, 136, 229 spin, 102 spine, 210 spongy tissue, 102 spreadsheets, 217, 218 stability, 26, 110, 120, 125 stabilization, 106 stages, 116, 118, 119, 120, 228

244

Index

standard deviation, 216, 229 standards, 216 starch, viii, 32, 81, 84, 85 statistics, 88, 218, 219 steel, 36, 117, 123 stoichiometry, 110 strain, 32, 34, 36, 37, 47, 50, 52, 71, 201, 203, 207, 208 strength, ix, x, 32, 33, 34, 35, 77, 88, 91, 102, 103, 104, 105, 106, 109, 110, 113, 121, 124, 133, 134, 136, 141, 142, 143, 144, 180, 197, 198, 203, 207, 208, 211, 212, 215, 216, 217, 229 stress, ix, 32, 35, 36, 37, 38, 44, 47, 50, 52, 88, 105, 107, 108, 113, 123, 124, 136, 142, 143, 144, 147, 148, 151, 152, 156, 157, 158, 159, 166, 173, 174, 175, 178, 179, 180, 181, 184, 185, 186, 187, 189, 190, 198, 201, 207, 216 stress intensity factor, 32, 107 stress-strain curves, 207 strong interaction, 46 structural characteristics, 122 substitutes, 198 substitution, 38, 110, 113, 123, 124 substrates, 110 superconductivity, 148 superiority, x, 215, 231 surface area, 160, 166 surface energy, 32, 160, 166 surface region, 25, 26 surface structure, 211 surface tension, 32, 136, 137 Switzerland, 99, 131 symmetry, 37, 38, 39, 47, 50, 52, 149 synthesis, 71, 123, 125, 135 systems, x, 13, 36, 88, 102, 111, 144, 186, 189, 190, 197

T talc, 102 tantalum, 198, 200, 211, 212, 213 targets, vii, 31, 35 technology, 91 teeth, 110, 120, 132 temperature, vii, ix, 1, 2, 7, 8, 11, 12, 13, 15, 16, 21, 22, 23, 24, 25, 26, 34, 35, 43, 44, 45, 71, 74, 76, 84, 88, 89, 106, 108, 111, 114, 117, 119, 120, 125, 132, 136, 137, 138, 147, 151, 152, 174, 180, 181, 183, 188, 189, 190, 204 temperature dependence, 34, 43, 76, 190 tensile strength, 120, 121, 142, 143, 185, 207 tension, 107, 117, 152, 180, 181, 229 TEOS, 27, 200, 203

textbooks, 40, 42 theory, viii, 32, 36, 43, 44, 46, 47, 48, 57, 87, 88, 92, 216, 220 thermal decomposition, 106 thermal stability, 13, 110 thermal treatment, 212 thermodynamics, 42, 45, 76, 88, 92 threshold, viii, 31, 63, 67, 68, 69, 90, 107 thromboresistance, 107, 108 time, viii, 3, 6, 12, 15, 24, 32, 34, 35, 42, 70, 81, 101, 103, 105, 109, 113, 116, 118, 119, 120, 124, 134, 135, 137, 142, 145, 220 tissue, viii, 35, 81, 101, 102, 103, 104, 105, 108, 109, 110, 113, 116, 117, 124, 133, 134, 135, 144, 145 titanium, 35, 109, 123, 127, 132, 198, 200, 211, 213 topology, 60, 66 tradition, 89 transformation, 35, 36, 37, 71, 106, 114, 115, 125, 173, 217 Transformation curve, 16 transformations, 152, 174, 175 transition, 62, 106, 119, 125, 128, 216 transition temperature, 106, 119 transmission, 122, 132, 212 transport, 21, 26 trend, 84 trial, 230 Turkey, 215

U UK, 92, 93, 94, 95, 96, 97, 98, 99 uniform, 19, 21, 25, 46, 47, 52, 139, 141, 225

V vacuum, 135, 136, 142, 143 validity, vii, 1, 20, 39, 47, 49, 63 values, viii, ix, 6, 7, 12, 18, 21, 23, 26, 31, 33, 36, 42, 45, 47, 54, 56, 57, 62, 64, 65, 67, 70, 71, 73, 75, 76, 77, 78, 79, 81, 84, 87, 88, 107, 134, 143, 148, 180, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 228, 229, 230, 231, 232, 233, 234, 235 vapor, 18, 35, 88 variable, viii, 101, 102, 109, 161, 162, 164, 165, 167, 168, 169, 170, 172, 179, 180 variables, 23, 225 variance, 212, 218, 220, 222 variation, 10, 12, 15, 19, 20, 21, 23, 24, 25, 26, 69, 216, 231

Index vector, 37, 50 vehicles, 123 velocity, 34, 44 versatility, 117 vertebrae, 102, 211 vessels, ix, 133 Vickers hardness, 122 viruses, 198 viscosity, 32, 49, 62, 89, 135, 137 volumetric changes, 120

W water, x, 35, 49, 88, 109, 117, 197, 203, 204, 207, 214 water vapor, 35, 88 wear, 32, 35, 88, 104, 105, 107, 126, 127, 180 weight ratio, 207 wetting, 7, 18, 23, 24, 25 women, 102 words, 57, 62, 69, 90, 227, 228, 230 work, 48, 66, 88, 90, 92, 191

245

X X-ray diffraction, 71, 112, 204 XRD, 34, 204

Y yield, ix, 54, 57, 65, 147, 152, 180, 181 yttrium, 106

Z zirconia, vii, 1, 2, 3, 4, 6, 7, 8, 13, 14, 19, 21, 27, 31, 33, 34, 35, 36, 45, 46, 47, 49, 56, 58, 59, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 103, 105, 106, 107, 108, 123, 127, 210, 213 zirconium, 106, 200