Environmental Effects on the Fracture of Oxide Ceramics

UITNODIGING

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Environmental effects on the fracture of oxide ceramicsoxide of fracture the on effects Environmental verdeding van mijn Environmental effects proefschrift Environmental on the fracture of oxide ceramics effects on the fracture of oxide ceramics

De promotieplechtigheid zal plaatshebben op

woensdag 18 december 2002 om 16.00 uur

in het Auditorium van de Technische Universiteit Eindhoven.

Aansluitend aan deze plechtigheid zal een receptie plaatsvinden waarvoor U ook van harte bent uitgenodigd.

Niels van der Laag der van Niels Niels van der Laag Niels van der Laag F.D. Rooseveltlaan 239 5625 AZ Eindhoven 040-2486540 [email protected]

± 170 pag. = 11mm rug f.c. glanslaminaat

Environmental effects on the fracture of oxide ceramics

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnificus, prof.dr. R.A. van Santen, voor een commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op woensdag 18 december 2002 om 16.00 uur

door

Niels Johan van der Laag

geboren te Boxmeer Dit proefschrift is goedgekeurd door de promotoren:

prof.dr. G. de With en prof.dr. R.C. Bradt

Copromotor: dr.ir. L.J.M.G. Dortmans

Druk: Universiteitsdrukkerij, Technische Universiteit Eindhoven

CIP-DATA LIBRARY TECHNISCHE UNIVERSITEIT EINDHOVEN

Laag, Niels J. van der

Environmental effects on the fracture of oxide ceramics / by Niels J. van der Laag. Eindhoven : Technische Universiteit Eindhoven, 2002. Proefschrift. ISBN 90-386-2754-8 NUR 913

Trefwoorden: keramische materialen; breuk / vastestofchemie / spinel; magnesium aluminaat / gahniet / zink aluminaat / breukmechanica / fysisch-chemische simulatie en modellering; thermische eigenschappen Subject headings: ceramics; fracture / solid state chemistry / spinel; magnesium aluminate / gahnite / zinc aluminate / fracture mechanics / physicochemical simulation and modeling; thermal properties Aan mijn ouders

Table of Contents

1 Introduction ...... 1

1.1 Galloping Gertie...... 1 1.2 Ceramics and failure...... 2 1.3 Aim and lay-out of this thesis...... 3 1.4 References ...... 6

2 Theoretical and computational background ...... 7

2.1 Introduction ...... 7 2.2 Fracture mechanics: Stress intensity approach...... 8 2.2.1 Stress concentrators 2.2.2 Stress intensity factor 2.3 Fracture mechanics: Energy approach ...... 11 2.3.1 Griffith energy balance 2.3.2 Mechanical energy release rate 2.3.3 Intrinsic work of fracture 2.3.4 Failure criterion 2.4 Adsorption ...... 15 2.4.1 Adsorption isotherm 2.4.2 Adsorption and fracture energy 2.4.3 Adsorption and fracture toughness 2.5 Subcritical crack growth...... 18 2.5.1 General 2.5.2 Constitutive model 2.5.3 Empirical power law 2.5.4 Combined kinetic and adsorption model

v Table of contents

2.6 Experimental mechanical tests ...... 23 2.6.1 Fracture toughness: SENB 2.6.2 Effect SCG in SENB test 2.6.3 Strength: Three-point bending 2.6.4 Fracture toughness - strength ratio 2.7 Computer simulations...... 28 2.7.1 General 2.7.2 Bulk and surface properties 2.7.3 Potentials 2.7.3.1 Pair potentials (PP) 2.7.3.2 Density Functional Theory 2.8 References ...... 33

3 Water vapour influence on the fracture of commercially applied oxide ceramics...... 35

3.1 Introduction ...... 35 3.2 Experimental ...... 36 3.2.1 Materials and specimens 3.2.2 Fracture toughness and strength 3.2.3 Measurement 3.3 Results and discussion ...... 39 3.3.1 Fractography 3.3.2 Fracture toughness 3.3.3 Relation between fracture strength and fracture toughness for Wesgo995 3.4 Conclusions ...... 47 3.5 Acknowledgements ...... 48 3.6 References ...... 49

4 Subcritical crack growth in calcium hydroxyapatite ...... 51

4.1 Introduction ...... 51 4.2 Experimental ...... 53 4.2.1 Synthesis 4.2.2 Characterisation 4.2.3 Fracture 4.3 Results and discussion ...... 54 4.3.1 Synthesis and characterisation 4.3.2 Fractography 4.3.3 Fracture toughness 4.3.4 Analysis

vi Table of contents

4.4 Conclusions ...... 60 4.5 Acknowledgements ...... 61 4.6 References ...... 61

5 Fracture of magnesium aluminate spinel (MgAl2O4) ...... 63 5.1 Introduction ...... 63 5.2 Experimental ...... 65 5.2.1 Material preparation and characterisation 5.2.2 Mechanical testing 5.2.3 Surface investigation preparation 5.3 Results and discussion of bulk properties ...... 68 5.3.1 Characterisation 5.3.2 Mechanical characterisation 5.4 Results and discussion of surface properties ...... 73 5.4.1 EBSD 5.4.2 LEIS 5.4.3 DRIFTS 5.5 Conclusions ...... 79 5.6 Appendix: Fracture and EBSD...... 80 5.7 Acknowledgements ...... 81 5.8 References ...... 82

6 Structural, mechanical, thermophysical and dielectric properties of zinc aluminate (ZnAl2O4) ...... 85 6.1 Introduction ...... 85 6.2 Crystallographic aspects of zinc aluminate ...... 86 6.3 Instrumentation...... 87 6.4 Preparation ...... 88 6.4.1 Powder synthesis 6.4.2 Compaction and sintering 6.5 Results and discussion ...... 89 6.5.1 Structural properties 6.5.1.1 XRD 6.5.1.2 Composition 6.5.1.3 MAS-NMR 6.5.2 Mechanical properties

vii Table of contents

6.5.3 Thermophysical properties 6.5.3.1 Heat capacity 6.5.3.2 Thermal diffusivity 6.5.3.3 Thermal conductivity 6.5.4 Dielectric properties 6.6 Conclusions ...... 100 6.7 Acknowledgements ...... 101 6.8 References ...... 101

7 Computational investigation of bulk and surface properties of zinc aluminate (ZnAl2O4) ...... 105 7.1 Introduction ...... 105 7.2 Experimental ...... 106 7.3 Computational methods ...... 106 7.3.1 Potentials 7.3.2 Surface energy 7.4 Results and discussion ...... 108 7.4.1 Structural and bulk properties 7.4.2 Surfaces (simulation) 7.4.3 Surfaces (experimental) 7.5 Conclusions ...... 115 7.6 Acknowledgements ...... 116 7.7 References ...... 116

8 Predictive calculation of intrinsic thermophysical properties from ab-initio phonon density of states: a case study for oxide ceramics ...... 119

8.1 Introduction ...... 119 8.2 Theory ...... 120 8.2.1 Phonons 8.2.2 Thermophysical properties 8.2.3 Debye model 8.2.4 Moment representation of phonon density of states 8.3 Criteria for comparison ...... 124 8.4 Computer simulation ...... 125

viii Table of contents

8.5 Results and discussion ...... 125 8.5.1 Density of states 8.5.2 Heat capacity 8.5.3 Entropy 8.5.4 Debye temperature 8.5.5 Moments of density of states 8.5.6 Debye temperature from elastic constants 8.6 Conclusions ...... 131 8.7 Acknowledgements ...... 131 8.8 References ...... 132

9 Epilogue: overview, conclusions & future work ...... 133

9.1 Introduction ...... 133 9.2 Overview and conclusions ...... 134 9.2.1 Fracture and adsorption 9.2.2 Surface and computer modelling 9.3 Thoughts for future work ...... 136 9.3.1 Toughening 9.3.2 Wake field 9.4 References ...... 138

Summary ...... 139

Samenvatting...... 143

Samenvatting voor niet-wetenschappers ...... 147

Dankwoord ...... 157

Curriculum Vitae ...... 159

ix Table of contents

x - 1 -

Introduction

1.1 Galloping Gertie Large man-made structures like dykes, bridges and buildings catch the people’s imagination for its shear intransigence and are therefore symbols of eternity. An important point for all such large and costly constructions is its (economical) lifespan, which is for modern bridges at least 50 years and for the Delta Project (improvement of the dutch coastal defence systems) at least 200 years. The construction has to be designed by engineers in a way preventing early collapse. Therefore, they have to understand how the constructions will behave under all applied external and internal forces under extreme local conditions.

However, sometimes engineers are facing the unpleasant sight of a collapsing construction as e.g. the spectacular collapse of the 1800 m long suspension bridge over the Tacoma Narrows on 7 november 1940, see figure 1.1. This bridge was meant to be used for over 50 years, but already collapsed within 19 weeks after its opening and within two years after the start of its construction. The suspension bridge was nicknamed Galloping Gertie after the unusual large undulation of 1.5 m of the central span, which increased prior to the collapse to 8.4 m with an addition of a twist.

The video images of the collapsing Galloping Gertie are widely used in science classes to show the possible devastating effects of the resonance principle. The engineers of Galloping Gertie were not unaware of these devastating effects since they knew all about the resonance principle, but they were simply unaware of the impact of the wind on the bridge which causes turbulence

1 Chapter 1

Figure 1.1: Collapsing Tacoma Narrows Bridge (Galloping Gertie). leading to the resonance. The failure of Galloping Gertie initiated the research to the aerody- namic effects on constructions resulting in the present standard procedure to check the effects of the wind on constructions as bridges and large buildings during the design phase. Galloping Gertie showed dramatically that failure is actually a good teacher and a very important source of knowledge.

1.2 Ceramics and failure Ceramics are made by firing inorganic non-metallic-metallic compounds, such as carbides, ni- trides and oxides in consolidated powder form explaining the greek origin of the name, kera- mos, meaning "burned stuff". Ceramics have several favourable properties as e.g. chemical stability, wear resistance, thermal and electrical insulation and thermal shock resistance. In general, ceramics are divided in two major groups: traditional and advanced (or technical) ceramics. Traditional ceramics are materials like bricks, roof tiles and china, which have easy- quarrable (clay-) minerals as their source. Advanced ceramics like e.g. light-bulbs in cars or magnetic cores in transformers are high-tech products, where the special properties of ceramics are being enhanced by means of special processing. Advanced ceramics are less visible to the general public than traditional ceramics, but are widely present in many applications as e.g. in a Nokia mobile phone (without battery), where 16 % is made of ceramics or glass [1]. The total ceramic industry in the United States alone has a market worth over $35 billion [2] of which 10 % are advanced ceramics [3].

Ceramics have very favourable mechanical properties as, in general, high Young’s modulus, strength, fracture toughness and hardness. However, ceramic components can fail, like Gallop- ing Gertie did. Moreover, if they fail then they do it in a brittle way showing hardly any defor-

2 Introduction mation, which could act as a warning. Therefore, it is required for ceramic components prior to application to have an idea of its mechanical reliability or, in other words, to predict when they will fail.

The primary requirement for mechanical reliability is that already in the design phase the component is designed so that it will not be subjected to applied stresses higher than the strength of the ceramic. The mechanical reliability of future components can be further improved by a careful examination of components which have failed revealing the origin of their fracture. Since the fracture originates generally on microstructural imperfections as e.g. pores or inclusions, the mechanical reliability can then be improved by improving processing as e.g. better mixing of raw materials.

Another aspect influencing the mechanical reliability is the decrease in strength and fracture toughness due to application conditions limiting the lifespan of the components. Ceramics are vulnerable to the process of subcritical crack growth, where reactive molecules from the environment enter the (always present) cracks and react with the stretched bonds at the crack tip resulting in crack growth below the critical applied stress for fracture. The extent of subcritical crack growth is dependent on the susceptibility of the material towards subcritical crack growth and the partial pressure of the gaseous molecules.

However, reactive molecules do not only interact with the ceramic in a reaction with the stretched bonds at the crack tip. The molecules can also adsorb on the newly formed surfaces by the extension of the crack and lower the surface energy and with that also the strength and fracture toughness, since surface energy is an opposing force to crack extension. This decrease in strength and fracture toughness limits the life span and the mechanical reliability.

Although, the adsorption of molecules must be taken into account, this is hardly done in contrast to subcritical crack growth despite the known role of adsorption in the decrease of fracture toughness of hexaferrites [4] and MnZn ferrites [5]. The question is raised whether this adsorption is in general unimportant so that it not necessary be taken into account, or that it is an unknown important factor like e.g. the wind factor in the Galloping Gertie case.

1.3 Aim and lay-out of this thesis The fracture process is a complex phenomenon where among others intrinsic material properties, environment and microstructure interact in an unrevealed manner. Present theoretical approaches generally ignore several aspects and include only the crack tip itself [6], but can not

3 Chapter 1 explain significantly the influence of phenomena as e.g. adsorption and microstructure. A better understanding of the mechanisms involved in the process zone in brittle (e.g. ceramics) and quasi-brittle (ceramic refractories) materials is therefore required.

The aim of this thesis is to understand the role of adsorption in the fracture process of brittle materials. A better understanding of the associated chemomechanical processes would enable material improvement by e.g. chemical composition changes. Therefore, a three-way approach was followed: general survey, model material experiments and computational simulations.

Firstly, the influence of water vapour on the fracture of several oxide ceramics is investigated in order to find out whether the adsorption is a general influencing feature or that it is only limited to the iron oxides as hexaferrite and MnZn ferrites. Subsequently, a model oxide ceramic is used for an experimental and computational investigation on an atomistic level of the influence of adsorption. Spinel (magnesium aluminate, MgAl2O4) was chosen since it is a prototype of the ternary oxides class. It does not show plasticity and has a much less complex stoichiom- 2+ 2+ 2+ 3+ 3+ etry than the spinel-type MnZn ferrites (Mn Fe Zn , Mn Fe O4). However, it still exhibits cation inversion, which is difficult to simulate in structural computational investigations. Therefore, it is attempted to eliminate this inversion by synthesizing ceramics of spinels with just a small inversion naturally present. Zinc aluminate (ZnAl2O4), manganese aluminate

(MnAl2O4) and cobalt aluminate (CoAl2O4) are suitable candidates, but the latter two show also oxidative transfer of Mn2+ and Co2+ to Mn3+ and Co3+, respectively. Accordingly, zinc aluminate ceramics have been made for further experiments and validation of computational investigations. Finally, computers simulations have been performed to investigate bulk and surface properties of zinc aluminate.

Furthermore, as fracture is often treated on an atomistic (micro-)level, a number of phenomena, as e.g. microstructure, are generally not taken into account. Therefore, the process-zone must be scaled up from this level to a meso-level requiring, among others, thermophysical properties to be taken into account. As a first attempt, a comparison was made between thermophysical properties determined from phonon spectra calculated using density functional theory methods, from the Debye model and from experimental data. In later work, this should be incorporated into process zone processes.

The set-up of this thesis is aimed to facilitate the reader to read it in sections. Therefore, the required background is brought together in a single chapter and the other chapters can be read in-

4 Introduction dividually. As a consequence, repetitions of some parts (notably descriptions of experimental set-ups) is unavoidable.

In chapter 2, some theoretical background is treated. Firstly, fracture mechanics is dealt with using the works of Inglis, Irwin and Griffith. Subsequently, the role of adsorption in fracture mechanics is introduced and incorporated in a combined kinetic-adsorption model for subcritical crack growth. Finally, the theoretical (quantum mechanical) background of the used computer simulations is explained briefly.

In chapters 3, 4 and 5, the influence of adsorption on the fracture of several commercially applied oxide ceramics, a calcium hydroxyapatite ceramic and magnesium aluminate ceramics (porous and dense), respectively, is investigated. Additionally, the synthesis of the calcium hydroxyapatite ceramic is also described. Furthermore in chapter 5, the influence of water vapour on the fracture of single crystal spinel is investigated. The resulting fracture surfaces of single crystals are studied by Low Energy Ion Scattering (LEIS) to obtain information about the atomistic composition of the fracture planes for comparison with computer simulation results. Fi- nally, the orientations of the grains on and next to fracture path in a magnesium aluminate ceramic are investigated using Electron Backscattering Diffraction (EBSD).

In chapter 6, three different synthesis routes of zinc aluminate powder are described and its effect of the inversion of zinc and aluminium cations. Furthermore, zinc aluminate ceramics are made in order to find out whether inversion and atom type also have an influence on the adsorption effect during fracture. Although fully dense zinc aluminate ceramics could not be generated, several mechanical, thermophysical and dielectric properties are characterised and, subsequently, extrapolated to fully dense values.

In chapter 7, the bulk and surface properties of zinc aluminate are investigated. The bulk properties obtained using computer simulations are compared to literature and experimental data. Furthermore, the possible fracture surfaces along several low-index planes are simulated for later verification with experiments. Finally, experimental surface investigation of hydroxyl groups on zinc aluminate using Diffuse Reflectance Infrared Fourier Transform Spectroscopy (DRIFTS) are performed to gather information about their location on the surface.

In chapter 8, the predictions of thermophysical properties of ceramics in general by the Debye model and from the phonon spectra, calculated using density functional theory (DFT), are compared to experimental data.

5 Chapter 1

Finally, the epilogue contains some general considerations and ideas about the effect of adsorption on the fracture of oxide ceramics. Furthermore, some ideas are presented for future work.

1.4 References [1] Website of Nokia, www.nokia.com [2] Website of the American Ceramic Society, www.ceramics.org [3] Critical Technology Assessment of the U.S. Advanced Ceramics Industry (003-009- 00687-2), Bureau of Industry and Security, USA (1993) [4] G. de With, Anisotropy in mechanical properties of sintered Sr-hexaferrite, Silicates Industrielles, 295 (1984), 185-189 [5] M.A.H. Donners, L.J.M.G. Dortmans and G. de With, Adsorption and kinetic effects on MnZn ferrites, J. Mater. Res., 15 (2000), 1377-1388 [6] G. de With, Environment induced failure of brittle and quasi-brittle materials, Mat. Chem. Phys., 75 (2002), 229-234

6 - 2 -

Theoretical and computational background

2.1 Introduction The failure of ceramic materials typically originates at defects as large grains, inclusions, pores and (micro) cracks, which can be formed during the synthesis of the material (e.g. pores) or formed during processing and storing (e.g. surface scratches). This requires the understanding of the role of defects in the fracture process. However, also reactive gaseous molecules take part in the fracture process via e.g. subcritical crack growth and adsorption phenomena. Therefore, they must also be incorporated into the theoretical treatise.

This chapter is devoted to the theoretical background required in the following chapters and can be divided in two parts: fracture mechanics and computational methods. The first paragraphs will be dealing with the concepts of stress concentrations leading to stress intensity factors. The next paragraphs will focus on the energies involved in the crack extension. Subsequently, adsorption is treated since it affects the energy balance in the fracture process. Next, after an introduction of subcritical crack growth due to reactive gaseous molecules kinetically interacting with the crack, a combined model of kinetic and adsorption effects is derived [1,2]. Finally, this combined model is applied in single edge notched beam (SENB) and three-point bending experiments for the influence of reactive gaseous molecules on the fracture toughness and strength, respectively.

In the second part, computational methods are treated, which provide insight into and predict mechanical parameters involved in the fracture process. Firstly, some compact theoretical back-

7 Chapter 2

2b C

σ A Figure 2.1: Plate containing an elliptical cavity with semi-axes b and c, subjected to an σ applied uniform stress ( A). C denotes the crack tip. ground of several mechanical properties is provided, but the thermophysical properties are explained in more detail in chapter 8. Finally, the potentials and their theoretical backgrounds of the pair-potential and density functional theory methods are briefly discussed.

2.2 Fracture mechanics: Stress intensity approach

2.2.1 Stress concentrators The stress distribution in a plate with an elliptical cavity when subjected to an uniform applied σ stress ( A) was studied by Inglis [3]. The elliptical cavity in the plate, with semi-axes b and c as shown in figure 2.1, can be described by:

2 2 ----x- +1----y- = . (2.1) c2 b2 The cavity has a minimum radius of curvature ρ at point C (assuming b < c):

2 ρ = ----b- , (2.2) c σ which is also the location of the maximum stress C, given by:

8 Theory

c c σ ==σ 12+ --- σ 12+ --- . (2.3) C Ab Aρ

The analysis of Inglis showed that a stress concentration of several times the applied stress oc- σ σ curs in the plate. The stress-concentration factor ( C / A) is dependent on both the size and shape of the cavity.

2.2.2 Stress intensity factor Inglis showed that stress concentrations occur at the sharp end of the cavity. Therefore, the σ stress ( ij) and displacement (uj) fields around the crack tip are analysed by several scientists with linear elastic continuum approaches using Griffith’s idea to represent the crack as a infinitely sharp end of a cavity. The fields around the crack tip are described in closed analytical functions using polar coordinates with the assumption that the crack walls behind the tip are and remain traction free. These analytical functions are known as the Irwin "near-field" solutions (in general form):

σ 1 ()θ ij = K------fij and (2.4) 2πr

K r u = ------g ()θ , (2.5) j 2E 2π j where K is the stress intensity factor, E is the Young’s modulus, r and θ are the polar coordinates θ θ with the crack tip as origin and the functions fij( ) and gj( ) describe the angular part of the stress field and displacement field, respectively. It is important to realise that several cardinal components appear as separable factors in equation 2.4: the applied stress, size of the crack and specimen geometry are solely incorporated in the stress intensity factor K and the remaining factors only in the spatial distribution of the fields.

An (edge) crack can be loaded in three different ways known as the modes of loading as shown in figure 2.2: opening mode, sliding mode and tearing mode. Only mode I loading is used in this thesis1 and the entire set of expressions of equations 2.4 and 2.5 for mode I loading are given by:

1. Mode II and III loading will not be further discussed. For an extensive treatise see [2].

9 Chapter 2

I II III

Figure 2.2: The three modes of loading/fracture: I opening mode; II sliding mode and III tar- ing mode.

σ xx cos()θ ⁄ 2 []1 – sin()θ ⁄ 2 sin()3θ ⁄ 2  K  σ I yy = ------cos()θ ⁄ 2 []1 + sin()θ ⁄ 2 sin()3θ ⁄ 2  2πr σ ()θ ⁄ ()θ ⁄ ()θ ⁄ xy sin 2 cos 2 cos 3 2 and (2.6) σ ν′() σ σ zz = xx + yy σ σ xz ==yz 0

ux K ()1 + ν []()θ2κ – 1 cos()()⁄ 2 – cos()3θ ⁄ 2 = ------I------r  u 2E 2π()1 + ν []()θ2κ + 1 sin()()⁄ 2 – sin()3θ ⁄ 2 y with (2.7) ν″z u = – ------()σ + σ z E xx yy

κ = ()3 – ν ⁄ ()1 + ν ν′ = 0 ν″= ν (plane stress) , (2.8) κ = ()34– ν ν′= ν ν″ = 0 (plane strain) where ν is the Poisson’s ratio. For completeness, it must be noted that the various stress intensities are additive, i.e.:

Ktot = KI ++KII KIII . (2.9) The stress intensity factor in mode I with a uniform applied stress in mode I is given by:

σ KI = Y A c , (2.10) where c is the crack length and Y is the dimensionless geometry factor. At the moment of fracture equation 2.10 becomes:

10 Theory

σ KIc = Y f cc , (2.11) σ where KIc is the critical stress intensity, also called fracture toughness, f is the applied stress at fracture and cc is the critical crack length. The different crack geometries differ only in the geometry factor, as e.g. the geometry factor of a double-ended straight crack with a length of 2c in an infinite specimen is equal to Y = π1/2 (= 1.77), for an edge crack with length c to Y = 1.12π1/2 (= 1.99) and for a semi-circular penny shaped surface crack with radius c to Y = 2.24π-1/2 (= 1.26).

2.3 Fracture mechanics: Energy approach

2.3.1 Griffith energy balance Griffith extended the analysis of Inglis of a loaded plate with a crack with an energy based approach [4] using an infinitely sharp crack. Only two energy terms are dealing with the extension of the crack under constant displacement conditions: mechanical energy (UM) due to the loading

(i.e. strain energy) and fracture energy (UF) due to the increase of surface. The mechanical energy decreases in this case with crack growth assuming constant thickness of the plate:

d U < 0 (2.12) dc M and the fracture energy increases:

d U > 0 . (2.13) dc F The equilibrium for the crack system is given by:

d d U ==()U + U 0 . (2.14) dc dc M F Using the stress distribution as given by Inglis for a double-ended crack with a length of 2c in an infinite plate with a constant thickness as shown in figure 2.1, but with an infinitely sharp crack (ρ→0), Griffith derived an expression for the stored strain energy for a plate of unit width:

πσ2 c2 U = ------A - , (2.15) M E′

11 Chapter 2 where E′ is E under plane stress ("thin" plate) or E /(1+ν2) under plane strain ("thick" plate) conditions, E is the Young’s modulus and ν is the Poisson’s ratio. Furthermore, the fracture energy of unit width is set to:

UF = 2cR, (2.16) where R is the specific fracture energy, which is assumed to be a constant. In his original analysis, Griffith equalled the specific fracture energy to the specific surface energy 2γ.

The total energy given by adding equations 2.15 and 2.16 and in combination with the Griffith condition in equation 2.14 and a critical crack size cc leads to an expression for the critical stress σ c:

′ σ = ------RE . (2.17) c π cc σ σ As long as A < c, crack growth needs more energy than stored in the system and, therefore, σ σ the crack will not extent. On the other hand, as A > c, more energy is stored in the system than necessary required for crack extension resulting in crack growth. Moreover, since the first and second derivative of the total energy to the crack length are negative the crack will grow in an unstable manner.

2.3.2 Mechanical energy release rate Applying a force F on an elastic body with an edge crack as shown in figure 2.3 results in a displacement u. Furthermore, assuming traction free walls, a workless constraint at the crack tip preventing crack extension and a linear elastic response to the applied load described by Hooke’s Law

uCF= , (2.18) where C is the compliance factor (= S-1, S is the stiffness), provides an expression for the mechanical energy:

u u2 Fu F2C U ====∫Fx()dx ------. (2.19) M 2C 2 2 0 If the constraint preventing the extension of the crack is released, the compliance of the body will increase (dC > 0) with the crack extension from c to c+dc. This follows from the crack extension requirements du > 0 and dF < 0 and the differential of equation 2.18:

12 Theory

F u

cdc

Figure 2.3: Elastic body with crack as used in the mechanical energy release rate treatise.

du= CdF+ FdC. (2.20) The increase in compliance leads to a decrease in mechanical energy. It can be shown that for both constant displacement and constant force conditions the change in mechanical energy is given by:

F2dC dU = –------. (2.21) M 2 The mechanical energy release rate G is then defined as:

dU G = –------M , (2.22) dA where A is the area of newly formed surfaces. Furthermore, the mechanical release rate for a straight crack per unit width is given by:

dU G = –------M . (2.23) dc It can be proven (e.g. [2]) that under constant displacement conditions the mechanical energy release rate is a function of the stress intensity implying that the several contributions of different loading are additive:

13 Chapter 2

dU G ==–------M G ()K ++G ()K G ()K dc I I II II III III u . (2.24) K2 K2 ()1 + ν K2 = ------I + ------II- + ------III E′ E′ E

2.3.3 Intrinsic work of fracture The increase in crack length (dc > 0) results in an increase in fracture energy:

dUF = Rdc, (2.25) which introduces the intrinsic work of fracture R per unit width opposing crack extension as:

dU R = ------F- . (2.26) dc Griffith set in his analysis this work rate equal to twice the surface energy γ in a vacuum γ (Ro =2 ). However, it is known that the work rate is, among others, also dependent on the loading rate, temperature and the crack length. The latter results in the so-called R-curve behaviour. Nevertheless, the work rate remains proportional to the surface energy.

2.3.4 Failure criterion The balance between the mechanical and fracture energies can be written in a differential form as:

() dU== dUM + dUF – Gdc + Rdc =RG– dc. (2.27)

The (R-G) term defines the crack extension force. At Griffith’s equilibrium, G = GIc, this leads for mode I to:

K2 G ==------Ic R . (2.28) Ic E′ If G > R, the crack extends and if G < R, the crack "retracts". For instable crack extension also the criteria dU / dc < 0 and d2U/dc2 < 0 must be fulfilled.

The applied load/stress is only introduced in this Griffith’s energy balance via the mechanical stress rate implying a critical load/stress when a crack starts to propagate leading to (possible) failure. At this critical applied stress, the stress intensity factor is defined as the fracture toughness KIc. Theoretically, the fracture toughness is a material constant under plane strain conditions. But practically, the fracture toughness is not constant even under plane strain conditions due to its relation with the intrinsic work of fracture R as seen from equations 2.24 and 2.28.

14 Theory

The intrinsic work of fracture is dependent on the surface energy, which can be lowered by adsorbing molecules on the surface. Therefore, it is required to quantify the influence of adsorption on the intrinsic work of fracture.

2.4 Adsorption

2.4.1 Adsorption isotherm Γ Gaseous molecules X adsorb non-dissociatively on the surface on unoccupied sites * with a certain rate k1 and, simultaneously, desorb again from the surface with a rate k2. On a surface Γ Γ Γ Γ −Γ with a maximum of M active surface sites, X sites are occupied and * = M X unoccupied. In a dynamic equilibrium, the following relation holds:

Γ ()Γ Γ Γ k1pX * ==k1pX M – X k2 X , (2.29) where pX is defined as the ratio between the actual partial pressure and the equilibrium partial pressure at the operating temperature. In the case of water vapour, this equals the relative hu- θ Γ /Γ midity. Using the relative surface coverage defined as X Μ, equation 2.29 can be rewritten as:

bp θ = ------X - , (2.30) 1 + bpX where b is equal to k1 / k2.

Equation 2.30 shows the relation between the surface coverage and the applied relative partial pressure of the adsorbing molecules, which is known as the Langmuir adsorption isotherm [5- 6]. Furthermore, the constant b is also known as the Langmuir adsorption coefficient. The Lang- muir adsorption isotherm for several values of b is shown in figure 2.4.

Strictly speaking, the Langmuir adsorption isotherm only describes monolayer adsorption, but multilayer adsorption can be described by an adaption of Langmuir’s analysis by Brunauer, Em- mett and Teller known as the BET-equation [7]. Restricting to n layers, the BET-equation becomes [8]:

1 – ()n + 1 pn + npn + 1 θ = cp ------X X , (2.31) X()()() n + 1 1 – pX 1 + c – 1 pX – cpX where c is approximately given by:

15 Chapter 2

1.0

c 0.8

0.6

[-]

θ b 0.4

0.2 a

0.0 0.00.20.40.60.81.0 p [-] X Figure 2.4: Langmuir adsorption isotherms with different Langmuir adsorption coefficient b: a) 0.25, b) 1 and c) 20.

Q1 – Qv c ≈ exp ------, (2.32) RT where Q1 and Qv are the heats of condensation of the first layer and the liquid absorbing species, respectively, R is the gas constant and T is the temperature.

The restricted BET-equation and the Langmuir adsorption isotherm describe the adsorption well in the presence of opposing walls as in e.g. capillaries and cracks, since these walls phys- ically limit the number of adsorbed layers. However, in the absence of such limitations, the restricted BET-equation and the Langmuir adsorption isotherm only describe the adsorption incorrectly in the high relative pressure range.

2.4.2 Adsorption and fracture energy The intrinsic work of fracture is proportional to the surface energy in the analysis of Griffith. ∆γ The surface energy will decrease with ads in the presence of adsorbing molecules X and thus also the intrinsic work of fracture:

dU R ==R – 2------ads R – 2∆γ , (2.33) X o dA o ads where dUads is the adsorption energy. The change in surface energy can be calculated using the Gibbs-equation:

16 Theory

d ∆γ Γ() ads = kBT aX , (2.34) dlnaX Γ where aX is the chemical activity of the adsorbing molecules, is the excess concentration of the adsorbing molecules on the surface, kB is Boltzmann’s constant and T is the temperature. The change in surface energy can be written from equation 2.34 as:

pX θ ∆γ Γ ads = kBT M ∫ ------dpX , (2.35) pX 0 Γ where M is the maximum excess concentration, if the adsorption isotherm is known and also continuously increasing, since the partial pressure pX is related to the chemical activity. Apply- ing the Langmuir adsorption isotherm to equation 2.35 provides a relation between the intrinsic fracture work energy and the partial pressure given by:

Γ () RX = Ro – 2kBT M ln 1 + bpX . (2.36)

2.4.3 Adsorption and fracture toughness σ Two ways to decrease the required applied stress for fracture f are revealed by rewriting equation 2.11 to:

K σ Ic f = ------. (2.37) Ycc The applied stress can thus be decreased by both a decrease in fracture toughness and an increase in crack length.

The fracture toughness without any influence of reacting or adsorbing species is called the inert fracture toughness Ko and can be measured in vacuum. The inert fracture toughness of a body with an initial crack length co is given for a mode I loading by:

σ Ko = Y o co , (2.38) σ where o is the applied stress at fracture.

When molecules adsorb on the newly formed fracture surfaces, the surface energy is lowered, which decreases also the intrinsic work of fracture as shown in equation 2.33. This reduces the inert fracture toughness to the adsorption controlled fracture toughness Kads, which is for a body in mode I loading with an initial crack length co given by:

17 Chapter 2

σ Kads = Y ads co , (2.39) σ where ads is the applied stress at fracture.

The relation between the inert and adsorption controlled fracture toughness can be given by using the Griffith equilibrium as in equation 2.28 and the relation between the intrinsic work of fracture with and without adsorbing species:

φ () Kads = Ko 1 – ln 1 + bpX , (2.40) with the constant φ given by:

2k Γ T φ = ------B M - . (2.41) Ro The second manner for decreasing the fracture toughness is crack extension from its initial length co to cc due to the presence of gaseous molecules leading to the process of subcritical crack growth (SCG), which will be treated in the following sections. The lowering of fracture toughness due to subcritical crack growth and adsorption can be formulated as:

σ Kads = Y SCG cc , (2.42) σ σ σ where SCG is the applied stress at fracture and SCG < ads.

2.5 Subcritical crack growth

2.5.1 General The gaseous molecules in the environment do not necessarily have to be involved in adsorption processes only. Some types of molecules interact also with the bonds in the crack and, especially, with the stretched bonds in the vicinity of the crack tip. The reacting molecules facilitate crack growth below the fracture toughness due to a lowering of the activation barrier for bond breaking. Michalske and Freiman [11] have proposed a chemical model for these reactions in silica. This process is called subcritical crack growth (SCG).

Subcritical crack growth is occurring in a large number of ceramic materials (e.g. [2]) and is therefore a long studied phenomenon. For instance, Wiederhorn [12] has shown that the crack growth rate during subcritical crack growth is controlled by the rate of reaction of the active in- gredient in the environment with the chemical bonds at the crack tip. Furthermore, Freiman et al. [13] and Wiederhorn et al. [14] have shown that subcritical crack growth is not only due to water molecules, but also due to other reactive molecules as e.g. ethanol.

18 Theory

Region I II III

v vlim ln ln

vth

KI,th KI,lim KIc ln KI Figure 2.5: Typical double logarithmic plot of crack growth rate vs. stress intensity.

2.5.2 Constitutive model In many materials, a crack grows below the critical applied stress when subjected to an environment with reactive (gaseous) molecules. Under these circumstances, the rate of crack growth v (= dc / dt) as a function of the stress intensity shows typically several stages as shown in figure

2.5. Firstly, below the so-called threshold stress intensity KI,th the crack growth is absent [9]. Subsequently, the crack starts to grow slowly above this treshhold value and enters Region I. It starts to grow more rapidly with increasing crack length and thus stress intensity, where the dependence of ln v on ln KI is approximately linear. However next, at higher stress intensities, the bulk diffusion of the gaseous molecules to the crack tip is limited and not sufficient any more to increase the growth rate. The crack growth rate vlim after the limiting stress intensity KI,lim remains constant and indicates Region II. Finally, Region III is entered as the crack growth rate rapidly increases to the terminal velocity vt before reaching the fracture toughness.

2.5.3 Empirical power law Subcritical crack growth, also denoted as stress corrosion or slow crack growth, is an important parameter in life-time prediction. Subcritical crack growth in Region I is the only necessary part to take into account for the prediction of time-to-failure due to the fact that the crack will remain the most of the time-to-failure in this region. Furthermore, the crack growth rate vlim in Region

19 Chapter 2

II is too high to have any practical influence on the time-to-failure and, moreover, Regions II and III are not always present in all materials.

A linear dependence between ln v and ln KI exists in Region I for most materials. Therefore, the crack growth rate is empirically approximated by a power law expression [10]:

n KI vv= o------, (2.43) Ko where vo is a pre-exponential factor and n is the SCG-parameter equalling the slope in the ln v -lnKI curve. It is assumed that the pre-exponential and SCG-parameters are not influenced by the environment. The activity a of the reactive molecules can be introduced into the pre-exponential factor to account for its effects as:

η vo = vηa , (2.44) which can be, assuming an ideal gas, written as:

η vo = vηpX , (2.45) where pX is the partial pressure of the reactive molecules.

2.5.4 Combined kinetic and adsorption model Reactive gaseous molecules can thus interact in two ways in the process zone at the crack tip. Firstly, by forming a transition state complex with a stretched bond at the crack tip as is shown in figure 2.6. This transition state complex decreases the activation barrier for the bond breaking reaction resulting in an enhanced crack growth. Secondly, by decreasing the surface energy via adsorption as also shown in figure 2.6. This gives a lower resistance to fracture resulting also in an enhanced crack growth.

The crack growth can be seen as a molecular reaction between gaseous molecules with a certain partial pressure pX and the bonds of a solid allowing for the approximation of the crack growth rate by the rate of the rate determining reaction step. Classical theory of rate processes provides an expression for the crack growth rate:

∆ ∆ η kBT UF UB vp= Xlo------exp–------– exp–------, (2.46) h kBT kBT

20 Theory

a) b)

Figure 2.6: Schematic atomistic representation of two possible bond breaking processes: a) transition state complex formation and b) adsorption.

21 Chapter 2

η where lo is the characteristic crack growth step length, is the (pseudo-)reaction order of pX, ∆ ∆ kB is Boltzmann’s constant, T is the temperature, h is Planck’s constant and UF and UB are the activation energy barriers for the forward and backward reaction, respectively [2, 15]. How- ever, the activation energies change by the presence of an applied stress resulting in an introduction of a function of the crack extension force (RX -G) according to Griffith’s analysis:

∆ α() ∆ α() η kBT UF – GR– X UB + GR– X vp= Xlo------exp–------– exp–------, (2.47) h kBT kBT where α is the activation area. The mechanical release rate introduces the applied stress and thus the applied mechanical energy and the intrinsic work of fracture the surface energy effects.

The crack extension force can be rewritten using equations 2.24, 2.36 and 2.40 to:

K K2 GR– = ------o-----I-1– + Φln()1 + bp . (2.48) X E′ 2 X Ko Inserting this expression in equation 2.47, where the backward reaction is usually neglected, gives:

α() η GR– X vv= opX exp------kBT 2 , (2.49) η K βΦ = v p exp–β1 – -----I- ()1 + bp o X 2 X Ko with

∆ kBT UF vo = lo------exp–------(2.50) h kBT and

αR β = ------o- . (2.51) kBT When the inert fracture toughness is much larger than the applied stress intensity factors and β has a large value, equation 2.49 can be simplified using exp()–ax ≈ ()1 – x a to:

nΦ ------K n η ()2 I vv= opX 1 + bpX ------, (2.52) Ko

22 Theory where the subcritical growth factor n is given by

2αR n ==2β ------o . (2.53) kBT A further rewriting of equation 2.52 produces the familiar empirical power law formula using equation 2.40 as:

n KI vv= p------. (2.54) Kads

The contribution of the reaction kinetic vkin and adsorption vads factors in the pre-exponential factor vp are separable as:

Φ ()n vp = vkinvads 1 – ln 1 + bpX , (2.55) where

η vkin = vopX (2.56) and

Φ ------n ()2 vads = 1 + bpX . (2.57)

2.6 Experimental mechanical tests

2.6.1 Fracture toughness: SENB A method of determining the fracture toughness of materials is by fracturing a test specimen with an artificially made large notch. The key element of these methods is that the notch will be the location of fracture and it simulates a natural crack, but with a known size. Several methods with several specimen geometries are developed. One of most widely used methods is putting a single edge notched beam (SENB) specimen with a straight notch in a four-point bending set- up as shown in figure 2.7. Subsequently, a constant stress rate is applied to determine the frac- σ ture stress f, which decreases with decreasing stress rates if the material is susceptible to subcritical crack growth. Generally, the original crack length co is used to calculate the (apparent) fracture toughness Kapp omitting explicitly the occurrence of subcritical crack growth.

The apparent fracture toughness measured with SENB specimen is given by:

23 Chapter 2

a)F b) w h

li o c lo

Figure 2.7: a) Four-point bending set-up with a SENB specimen. b) Cross-section of a SENB specimen with the notch.

3F ()l – l σ f o i Kapp ==YS f co YS------co , (2.58) 2wh2 where lo and li are the outer and inner span of the four-point bending set-up, respectively, w and σ h are the width and the height of the specimen, respectively, as shown in figure 2.7, f and Ff are the applied stress and force at fracture, respectively. Furthermore, the geometry factor YS is given by the Gross-Srawley equation [16-17]:

α α2 α3 α4 YS = 1.99– 2.47 ++ 12.97 –23.17 24.8 , (2.59) α ≤α≤ where = co / h, and which is accurate within 0.2 % for the valid range of 0 0.6.

α The geometry factor YS shows a minimum around = 0.15 and, moreover, changes only slightly close to the minimum. It can be considered as a constant and, consequently, the notches in the SENB specimen are made around this minimum. Furthermore, since the crack lengths will not exceed the validity range of the Gross-Srawley equation, this equation is used.

2.6.2 Effect SCG in SENB test The omission of subcritical crack growth in SENB experiments by taking the initial notch size co leads to an underestimation of the fracture toughness of materials which are susceptible to subcritical crack growth. When the subcritical crack growth is taken into account during SENB measurements, it must be assumed that the geometry factor YS taken at the original notch length remains constant during the experiment when the notch extends from co to cc, which is only true around the minimum.

24 Theory

Substituting equation 2.58 into the derived empirical power law expression 2.54 for crack growth in the presence of reactive gaseous molecules gives:

n n n σ· Kapp vpYS s n n ⁄ 2 v ==-----dc v ------=------t c , (2.60) dt pK n ads Kads · · where the applied stress is entered as a function of time: σ = σs ⋅ t and σs = dσ / dt (stressing rate).

Separation of variables in equation 2.60 gives:

c t c n σ· n f 1 vpYS s n ∫ ------dc = ------∫t dt , (2.61) n ⁄ 2 n c Kads co 0 The right-hand side can be expressed as:

t n σ· n f n σ n + 1 v Y s n 1 v Y ------p S -∫t dt = ------p S ------f , (2.62) n n + 1 n · K K σs ads 0 ads using

σ n + 1 σ· n n + 1 f s tf = ------· . (2.63) σs The left-hand side can be expressed as:

cc 2 1 n – 2 1 n – 2 ∫ ------1 -dc = ------– ------n ⁄ 2 ()n – 2   c co cc co , (2.64) Y σ n – 2 Y σ n – 2 ------2 ------S f ------S f = ()–  n – 2 Kapp Kads using equations 2.42 and 2.58 for the adsorption/subcritical crack growth controlled fracture toughness and the apparent fracture toughness, respectively.

Combining equations 2.62 and 2.64 into equation 2.61 gives an expression for the ratio of the apparent fracture toughness and the adsorption/subcritical crack growth controlled fracture toughness:

25 Chapter 2

a)F b) w h

Figure 2.8: a) Three-point bending set-up with a strength specimen. b) Cross-section of a chamfered strength specimen.

1 2 3 ------K n – 2 v Y σ 2 – n ------app =  1 + ------p S ------f K 2()n + 1 2 σ· ads Kads s . (2.65) 1 3 ------n – 2 v K 2 – n = 1 + ------p------app 2()n + 1 c 2 · oKadsK

2.6.3 Strength: Three-point bending In practice, the large notches, as used in SENB specimens, do not occur in applied materials. Therefore, it is necessary to determine the applied stress when the fracture starts on natural de- σ fects. This applied stress is called strength f (or S), but it depends on the measuring method: e.g. tensile strength, compressive strength, shear strength or bending strength. It is important to realise the differences between these kinds of strengths, since a high compressive strength does not automatically imply a good bending strength (e.g. concrete). However, as far as only bending strength is treated, this will be shortened to strength in the remainder of this thesis.

The (bending) strength of a material can be determined using a three-point bending set-up as shown in figure 2.8. The strength of a material is given by:

F l σ 3 f o f = ------, (2.66) 2wh2 where Ff is the applied force at fracture, w is the width and h is the height of the specimen. The specimens have to be chamfered in order to avoid edge fracture from abnormal defects introduced by cutting and grinding during the specimen preparation.

26 Theory

2.6.4 Fracture toughness - strength ratio An expression for the influence of subcritical crack growth in strength specimens is given by a combination of equations 2.61 and 2.62:

1 c ------· c n + 1 σi n 1 σ = ()n + 1 ------K ∫ ------dc , (2.67) f n ads n ⁄ 2 vpYi c co σ· where Yi is the geometry factor of the initial defect and i is the applied stressing rate. Using the assumption that the adsorption/subcritical crack growth controlled strength is larger than the apparent strength, equation 2.67 becomes:

------1 - · n + 1 2()n + 1 σi n 1 n – 2 σ = ------K ------, (2.68) f ()n – 2 n ads vpYi ci where ci is the initial defect size in the strength specimen.

An expression for the apparent fracture toughness can be obtained by a similar derivation using the assumption that the adsorption/subcritical crack growth controlled fracture toughness is larger than the apparent fracture toughness:

1 ------· n + 1 ()σ 3 2------n + 1 -----s n Kapp = ()YSKads co . (2.69) n – 2 vp σ The fracture toughness-strength (Kapp- f) ratio can be calculated from equations 2.68 and 2.69:

------1 - · 3 n + 1 K σsYS c ------app = Y c ------o- . (2.70) σ i i · 3 f σ iYi ci σ If the Kapp- f ratio remains constant for all combinations of partial pressures and stressing rates, it can be concluded that the susceptibility to the effects of the gaseous molecules is a material property and does neither depend on the type of test nor on the used geometry. This allows for the conclusion that SENB experiments can be used to study subcritical crack growth behaviour since the effects of the gaseous molecules in the crack is a material property.

27 Chapter 2

2.7 Computer simulations

2.7.1 General Computer simulations are a widely used tool in engineering sciences to predict the behaviour of materials and structures. In the field of mechanical engineering finite element methods are used to model, among others, mechanical behaviour of structures. Predictions can be made from these calculations where stress concentrations occur indicating possible failure locations.

Finite element methods approach the reality on a macroscale. However, fracture mechanics are also determined by intrinsic material properties on a microscale. For instance, the fracture toughness is given by:

KIc = ER, (2.71) where E is the Young’s modulus and R is the specific fracture energy. Computer simulations can be used to calculate e.g. the elastic constants, lattice vibrations and the surface energy (γ), which appears in the expression of the specific fracture energy as:

· R = 2fT(),,ε … γ . (2.72) Generally, the calculation of these and other intrinsic material properties follows the same procedure. Atoms are set up in a starting configuration using the symmetry of the crystal lattices and using periodic boundary conditions. Subsequently, potentials describing the atomic interactions are used to move the atoms to an equilibrium configuration where the overall energy is minimized.

2.7.2 Bulk and surface properties When the equilibrium configuration is reached, several properties can be determined. Firstly, the elastic constants (Cij) can be calculated by taking the second derivative of the Helmholtz energy F under isothermal conditions, which under adiabatic conditions equals the energy U, with respect to the applied strain (using the pseudo-vector notation):

2 2 ∂ F ∂ U C ==. (2.73) ij ∂ε ∂ε ∂ε ∂ε i j T, ε' i j S, ε'

In the case of a spinel structure, the elastic constants C11, C12 and C44 can be calculated by dis- torting the lattice parameters of the equilibrium unit cell by a small amount δ but keeping the same volume. These distortions of the lattice parameters for the tetrahedral and shear distortions are given by the following matrices, respectively:

28 Theory

1 +00δ 01+0δ , and (2.74) 001()+ δ –2

1 δ 0 δ 10. (2.75) –1 001()– δ2

A second order polynomial fit of the energy-distortion curve is made from which the elastic constants can be obtained via:

2 ∂ U  = 6()C – C – 36()δC – C , and (2.76) 2 11 12 11 12 ∂δ tetra

2 ∂ U  = 4C . (2.77) 2 44 ∂δ shear Secondly, the frequencies of the lattice vibrations are derived from the dynamical matrix, which φ is on its turn constructed from the atomic force constants. The atomic force constants ij(k,l) are defined as the second order term in the expansion of the potential energy φ in small displace- ments ui of the kth atom due to a displacement uj of the lth atom:

∂2φ φ ()kl, = ------. (2.78) ij ∂ ()∂ () ui k uj l Assuming a cell with N atoms and harmonic approximation, the elements of the (3N-3)×(3N-3) dynamical matrix D are defined as:

φ (), ij kl Dij = ------, (2.79) mkml where mk and ml are the masses of the kth and lth atom, respectively. The phonon eigenfrequen- cies squared (ω2) are obtained as the solutions of the secular equation:

D-ω2I = 0 , (2.80) where I is the (3N-3)×(3N-3) unit matrix. The vibrational part of several thermophysical properties can be calculated using these phonon frequencies.

29 Chapter 2

Bulk region Near-surface region Figure 2.10: Schematic representation of the approach of Tasker. a) Structure of crystal cell for calculation of equilibrium configuration. b) Structure of the crystal cell with a near-surface and bulk region for calculation of surface structure and energy.

Finally, the surface energy can be obtained by generating a surface in the equilibrium configuration by "cleaving" the crystal and minimizing the energy of the new configuration using the same potentials as for the bulk energy calculations. The approach of Tasker [18] can be followed, where the crystal is constructed by two regions (see figure 2.10): a bulk region where the atoms are fixed at the equilibrium configuration positions and a near-surface region where the atoms are allowed to move to minimize the overall energy. The bulk region is required for the fixation of the atoms in the lower layers of the near-surface region providing a smooth transition to the bulk configuration and for the application of periodic boundaries required for the calculations. The surface energy can be calculated by dividing the difference in energy between the bulk and surface calculations by the surface area.

2.7.3 Potentials The accuracy of the computer simulations is crucially dependent on the model used of the atomic interactions. Although quantum mechanics provides the complete formalism, the implemen- tation of this formalism is rather tedious. The number of particles to be considered increases rapidly with increasing nuclei. This results in (unacceptable) time consuming calculations, restricting these quantum mechanical calculations only to systems with a relative small number of atoms. Therefore, several techniques have been developed to circumvent these problems. Techniques as pair potentials (PP, used in chapter 7) and Density Functional Theory (DFT, used in chapter 8) are widely applied.

30 Theory

2.7.3.1 Pair potentials (PP)1 The interaction between the nuclei is composed of, among others, electrostatic forces and Van der Waals forces. The electrostatic forces contain both charge and polarization interactions. No- tably, the polarization contribution of the oxygen atoms has to be taken into consideration during calculations. This is being achieved by assuming a spring system between the core and the (electron) shell of the oxygen atom with a certain spring constant (shell model). Furthermore, the Van der Waals forces are included using the Buckingham potential, since the description of the repulsive part of the Van der Waals forces (exchange forces) using an exponential term is more suited than the r-12-term in the standard Lennard-Jones 12-6 potential. The potential (V) is then given by:

1 V = ---∑V , (2.81) 2 ij ij with

q q r C V ==V ++V V ------i -j + A exp –-----ij- –------ij , (2.82) ij electro rep VdW r ij ρ 6 ij ij rij where qi and qj are the charges of ions i and j respectively, rij is the distance between the ions ρ and Aij, ij and Cij are adjustable parameters to describe the potential.

Applying the Ewald summation [19] to equation 2.81 and moving the positions of the nuclei, the total energy (U) is minimized obtaining the equilibrium configuration:

d ∑Ui = 0 . (2.83) drj In order to diminish the amount of calculations a cut-off distance is frequently used beyond which the energy contribution is neglected.

2.7.3.2 Density Functional Theory The electronic structure of solids can be quantum mechanically calculated the conventional way by solving approximately the classical Schrödinger equation (H|〉Ψ = E|〉Ψ ) via the variation principle. However, the size of the system has to be limited for computational reasons. Alterna- tively, the Density Functional Theory approach can be used to calculate the electronic structure of the solid. In the DFT-approach it is stated the complete information about the structure is

1. This chapter contains only a brief summary of the pair potential and density functional theory techniques. The reader is referred to specialized literature for an extended treatise on the underlying theo- ries and implementations.

31 Chapter 2 stored in the charge density ρ(r) implying that all properties, in particular potential and energy, are a function of the charge density (first Hohenberg-Kohn theorem). Furthermore, only the true ground-state density ρο(r) provides the true energy (second Hohenberg-Kohn theorem). Since the energy is a functional of the charge density, the Hamiltonian (H) and its components must also be:

[]ρ() []ρ() []ρ() []ρ() []ρ() H r = Hkin r +++Hne r Hee r Hxc r , (2.84) ρ( )] ρ( ) ρ( ) ρ( ) where Hkin[ r , Hne[ r ], Hee[ r ] and Hxc[ r ] are the kinetic, nucleus-electron interaction, coulombic electron-electron interaction and the exchange-correlation Hamiltonians, respectively. The first three terms can be expressed in the analytical functionals of the charge density, but not the exchange-correlation term. This problem can be circumvented by e.g. applying the Local Density Approximation (LDA), where a homogeneous electron gas system is assumed. This system is the only system where the exchange and correlation terms are known and can be expressed as a functional of the charge density.

The energy optimization in a DFT-calculation with respect to the charge density depends on the representation of the charge density. Kohn and Sham [21] introduced an approach where the density is generated from a Slater determinant with one-electron wavefunctions called "Kohn- θ Sham" orbitals i(r). The charge density is in this approach defined as:

ρ()|〉θ θ()〈|() r = ∑ i r i r . (2.85) The energies of the orbitals can be calculated using the Kohn-Sham equations:

[]θρ(), |〉ε() []θρ()|〉() FKS r i i r = i r i r , (2.86) where the Kohn-Sham operator (FKS) is related to the Hamiltonian via:

[]ρ() []ρ(), H r = ∑FKS r i . (2.87) θ By solving the Kohn-Sham equations the Kohn-Sham orbitals i(r) are obtained, which yield on their turn the charge density as shown in equation 2.85. Since the operators in the Kohn- Sham equations are defined in terms of charge density, a self-consistent iterative process has to be applied to obtain the lowest energy U. Subsequently, the structure is optimized to obtain the equilibrium configuration.

32 Theory

2.8 References [1] M.A.H. Donners, Fracture of MnZn ferrites, Ph.D. Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands (1998) [2] B.R. Lawn, Fracture of Brittle solids, 2nd Ed., Cambridge University Press, Cambridge, UK, (1993) [3] C.E. Inglis, Stresses in a plate due to the presence of cracks and sharp corners, Trans. Inst. Naval Archit., 55 (1913), 219-241 [4] A.A. Griffith, The phenomena of rupture and flow in solids, Phil. Trans. Roy. Soc. Lond., A221 (1920), 163-198 [5] I. Langmuir, Constitution and fundamental properties of solids and liquids. I. Solids, J. Am. Chem. Soc., 38 (1916), 2221-2295 [6] I. Langmuir, The adsorption of gases on plane surfaces of glass, mica and platinum, J. Am. Chem. Soc., 40 (1918), 1361-1402 [7] S. Brunauer, P.H. Emmett, E. Teller, Adsorption of gases in multimolecular layers, J. Am. Chem. Soc., 60 (1938), 309-319 [8] S. Brunauer, The adsorption of gases and vapors, Vol. 1, Princeton University Press, Princeton, USA, (1945) [9] D.K. Shetty, A.R. Rosenfield, W.H. Duckworth and G.K. Bansal, The static fatigue limit in the thermal activation approach to subcritical crack growth, J. Am. Ceram. Soc., 62 (1979), 536-537 [10] R.J. Charles, Static fatigue of glass II, J. Appl. Phys., 28 (1958), 1554-1560 [11] T.A. Michalske and S.W. Freiman, A molecular interpretation of stress corrosion in silica, Nature, 295 (1982), 511-512 [12] S.M. Wiederhorn, Influence of water vapor on crack propagation in soda-lime glass, J. Am. Ceram. Soc., 50 (1967), 407-414 [13] S.W. Freiman, G.S. White and E.R. Fuller, Environmentally enhanched crack growth in soda-lime glass, J. Am. Ceram. Soc., 68 (1985), 108-112 [14] S.M. Wiederhorn, S.W. Freiman, E.R. Fuller and C.J. Simmons, Effects of water and other dielectrics on crack growth, J. Mat. Sci., 17 (1982), 3460-3478 [15] A.S. Krausz and K. Krausz, Fracture kinetics of crack growth, 1st Ed., Kluwer Academic Publishers, Dordrecht, The Netherlands, (1988) [16] B. Gross and J.E. Srawley, Stress-intensity factors for single-edge-notch specimen in bending or combined bending and tension by boundary collocation of a stress function, Technical Note D-2603, NASA, USA, (1965) [17] W.F. Brown and J.E. Srawley, Plane strain crack toughness testing of high strength metallic materials, ASTM Spec. Tech. Publ., 410 (1967), 13-14 [18] P.W. Tasker, The surface energies, surface tensions and surface structure of the alkali halide crystals, Phil. Mag. A., 39 (1979), 119-136 [19] P. Ewald, Die Berechnung optischer und elektrostatischer Gitterpotentials, Ann. Phys., 64 (1921), 253-287

33 Chapter 2

[20] P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev., 136 (1964), B864- B871 [21] W. Kohn and L.J. Sham, Self-consistent equations including exchange and correlation effects, Phys. Rev., 140 (1965), A1133-A1138

34 - 3 -

Water vapour influence on the fracture of commercially applied oxide ceramics

3.1 Introduction The selection of materials for a certain application depends generally on a number of criteria. One of the most important criteria is the expected mechanical lifetime of the component when subjected to the practical conditions (temperature, chemical environment, etc.). One of the processes which reduces the lifetime is subcritical crack growth (SCG). Subcritical crack growth can occur when gaseous molecules from the environment enter the crack and lower the strength and/ or fracture toughness of the material. However, how the molecules from the environment interact exactly with the material is still under debate.

Two mechanisms have been proposed. Firstly, a chemical attack of the gaseous molecules on the stretched bonds at the crack tip and, secondly, adsorption of gaseous molecules on the newly formed surfaces reducing the surface energy, which opposes fracture. It is shown for hexaferrites [1] and MnZn ferrites [2] that the latter mechanism cannot be disregarded. The question is addressed whether this is a general feature of oxide ceramics. Therefore, the focus was directed to other oxide ceramics, since the complex composition and surface chemistry of MnZn ferrites and hexaferrites can play a role too.

In this chapter the results of fracture experiments at different loading rates on seven different materials are discussed. The experiments were performed on commercially applied materials

35 Chapter 3

Table 3.1: Examples of applications of the materials investigated Material Applications Alumina Bearings, artificial teeth, electronic insulating substrates Translucent alumina lamp bulbs, windows PZT Transducers, actuators, oscillators Mg-PSZ Armour, ionic conductors, structural parts

(see table 3.1): two types of polycrystalline alumina (Wesgo995 and Wesgo997), translucent alumina with and without erbium additives (DGA and DGA-Er), lead zirconate titanate (PZT), magnesium-partially stabilize zirconia (Mg-PSZ) and a traditional ceramic as applied in floor tiles. The experiments were conducted at different relative humidities in a mixture of dry nitrogen gas and (humidified) air to measure the fracture toughness of all materials using a single edge notched beam (SENB) technique and the strength of Wesgo995 with a three-point bending technique.

3.2 Experimental

3.2.1 Materials and specimens The materials were commercially obtained as blocks and cut and ground into the required dimensions of the specimens as given in table 3.2. For the fracture toughness the notch of the SENB specimens was cut up to ~15 % of the specimen’s height using a 100 µm diamond saw blade resulting in a notch width of ~125-150 µm.

Table 3.2: Set-up and specimen dimensions Material Inner span Outer span Specimen SEM specimen (w×h×l) (w×h×l) [mm] [mm] [mm] [mm] Wesgo995 15 30 3×3.75×40 2×3×50 Wesgo997 15 30 3.5×4.5×50 2×3×50 DGA 15 40 ~4.6×50a 2×3×50 DGA-Er 15 40 ~4.6×50a 2×3×50 PZT 15 40 3×4×50 3×4×50 Mg-PSZ 15 40 3×4×50 1.5×3×50 Floor tiles 15 80 ~7×~20.2×120 3×4×40 a. The DGA and DGA-Er specimen were solid round bars. The dimensions are given as diameter × length [mm]. The bars were slightly flattened off to prevent rolling. The notch was perpendicular to the flattened plane.

36 Commercially applied oxide ceramics

The specimens of Wesgo995, Wesgo997 (both from Western Gold & Platinum Company, USA), Mg-PSZ (2.5 % magnesium and 0.5 % hafnium additives [3] from ICI, UK) and PZT

(PbZr0.54Ti0.46O3 doped with 1 % Nb2O5, poled, from Morgan Electro Ceramics, The Nether- lands) were rectangular bars. The latter specimens have a composition near the morphotropic phase boundary and have an upper and bottom side covered with a 0.4 µm thick layer of a NiCrNi electrode, which is assumed not to have influence on the fracture toughness. The DGA and DGA-Er specimens (Philips Lighting, The Netherlands) were solid round bars. The specimen cut from floor tiles (Trega International, The Netherlands) were ground up to the desired height with the glaze left intact and the notch was made opposite the glaze.

3.2.2 Fracture toughness and strength A four point bending set-up was used to measure the fracture toughness of all the materials. The inner (li) and outer span (lo) of this set-up were adjusted to the dimensions of the specimens investigated and are given in table 3.2. Using this set-up, the apparent fracture toughness KIc of all the materials measured with rectangular bars is calculated by:

3Fl()– l σ o i KIc ==YS f co YS------co , (3.1) 2wh2 σ where f is the stress at fracture, co is the initial notch length, F is the force at fracture, w and h are the width and height of the specimen, respectively, and YS is the geometry factor given by the Gross-Srawley equation [4,5]:

α α2 α3 α4 YS = 1.99– 2.47 ++ 12.97 –23.17 24.8 , (3.2) α where = co / h.

The apparent fracture toughness for the round solid bars of translucent alumina is given by:

8Fl()– l K ==Yσ c Y------o i c , (3.3) Ic f o 3 o πd where d is the diameter and Y is the geometry factor given by [6]:

Y = 0.7438– 1.6478α ++ 5.9538α2 –3.7273α3 3.1061α4 , (3.4) where a = co / d. In order to prevent rolling in the set-up, the round bars were slightly double sided flattened off, but the original radius was used in the calculation of the relative notch length, since the change in moment of inertia by flattening was negligible.

37 Chapter 3

Furthermore, the fracture strength of Wesgo995 was measured using a three point bending set- up with a span of 30 mm on chamfered specimens with the same dimensions as the fracture toughness specimen.

3.2.3 Measurement The experiments were performed at room temperature on an universal Erichsen testing machine with a calibrated Test GmbH 500 N load cell except for PZT which was measured using a calibrated Erichsen 50 N load cell. Before each experiment the specimen was preloaded at 0.2 mm min-1 up to 5 or 20 N depending on the material. During the experiment, the crosshead velocity was constant and preset on the testing machine at 0.2, 2.0 and 20 mm min-1, which cor- responds to the experimentally determined average stressing rates of the materials investigated as shown in table 3.3. Furthermore, an additional experiment was performed with a lower stressing rate on special testing machine1 mounted with a calibrated Test GmbH 500 N load cell. On this testing machine, the fracture toughness of translucent alumina without erbium additives was measured with a stressing rate of 0.076 MPa s-1.

The experiments at different humidities were performed inside a perspex chamber mounted in the testing machine with a gas and hygrometer inlet. The relative humidity (RH) inside the chamber was calculated as the ratio of the partial pressure at the actual dew point inside, which was measured by a Panametrics system III hygrometer, and the equilibrium partial pressure at room temperature. The relative humidity between 2 and 80% was controlled by the inlet of dry nitrogen gas in case of humidity lowering or by the inlet of nitrogen gas led through water in case of humidity increase.

Table 3.3: Average stressing rates at different crosshead velocities Material 0.2 mm min-1 2.0 mm min-1 20 mm min-1 [MPa s-1][MPas-1][MPas-1] Wesgo995 1.37 18.6 141 Wesgo997 0.97 12.8 105 DGA 1.79 23.5 175 DGA-Er 1.79 22.7 188 Mg-PSZ 1.70 21.6 197 PZT 0.56 8.3 78

1. Testing machine constructed in the Central Workshop of the Eindhoven University of Technology; stressing rate range: ~0.025 - ~2.5 MPa s-1.

38 Commercially applied oxide ceramics

The experiments at 0% humidity were performed inside a Scanning Electron Microscope (SEM) on a special four point bending testing machine (Deben UK Ltd., UK). For these experiments special specimens, for which the dimensions are given in Table 1, are required since the maximum load of this machine is 160 N. The floor tile samples were disposed of the glaze and the DGA and DGA-Er specimens were sawn to rectangular bars. All the notches were 15% of the specimen height.

After fracture the fracture surfaces were investigated using a SEM in order to determine the fracture mode. The fracture surfaces were gold-coated by sputtering in order to prevent charging effects.

3.3 Results and discussion

3.3.1 Fractography Since the stressing rate and the relative humidity may have an influence on the fracture mode and in this way an indirect effect on the fracture toughness, the fracture surfaces were examined by SEM. Images of several typical fracture surfaces are shown in figure 3.1.

The fracture surfaces of DGA and DGA-Er show both a mixed trans- and intergranular fracture mode. Investigation of the fracture surfaces of DGA-Er using energy dispersive X-ray technique did not reveal the presence of erbium. This is probably due to the low amount of erbium additives (approximately 50 ppm [7]), which is supposed to be segregated to the grain boundaries. The surfaces of both other types of alumina do show both also a mixed inter- and transgranular fracture mode with a tendency towards transgranular. Both Wesgo995 and Wegso997 have a fracture mode ratio which is independent of the relative humidity.

The fracture surfaces of floor tiles show predominately a transgranular fracture mode. The same is also observed for Mg-PSZ, which fracture mode does not alter with increasing relative humidity. However, Leach and Lambov have found for magnesium partially stabilized zirconia an intergranular fracture behaviour below 600 °C [8]. The hafnium additives in the zirconia investigated might explain this difference. The fracture surface of PZT does show predominantly an intergranular fracture mode, which is in agreement with the results of Kim et al. [9]. The fracture mode did also not appear to be dependent on the relative humidity.

From these observations it is concluded that the fracture mode is essentially independent of the stressing rate and the relative humidity.

39 Chapter 3

a) b)

c) d)

e) f)

Figure 3.1: SEM images of typical fracture surfaces of a) Wegso995, b) Wesgo997, c) DGA, d) PZT, e) Mg-PSZ and f) floor tiles.

40 Commercially applied oxide ceramics

3.3.2 Fracture toughness The fracture toughness of the materials investigated at different relative humidities and different crosshead velocities are shown in figures 3.2-3.5. The fracture toughness of all materials decreases with increasing relative humidity. Furthermore, it is observed that the fracture toughness measured at 0.2 mm min-1 is always lower than the fracture toughness measured at 2.0 and 20 mm min-1. The fracture toughness of the latter two crosshead velocities is almost the same, indicating that fracture at these loading rates is rapidly catastrophic so that subcritical crack growth hardly occurs at these loading rates.

Taking the results of MnZn ferrites [2] into account, two types of behaviour can be distinguished. The first type, designated as Type I, is characterised by a very sharp decrease in fracture toughness between the relative humidities of 0 and 2 %. Subsequently, the fracture toughness decreases slowly or not at all with increasing relative humidity above 2 %. This behaviour is observed for all the aluminas, PZT, the floor tile ceramics and zirconia. The relative constant behaviour of the fracture toughness of Type I materials with respect to the humidity above 2 % implies that the contribution of adsorption remains constant. This suggests that either the surface is already fully covered at 2 % or that the extra water adsorption, whether chemical or physical, does not have an effect.

Adsorption experiments on the (0001) surface of a single crystal α-alumina, where the surface was heat-treated, by Liu et al. [10] showed that below 4 % humidity hydroxylation of the surface does not take place, even at longer exposure times. The small amount of hydroxyl groups detected on the surface was due to adsorption on defect sites. Also Elam et al., [11] performed adsorption experiments on the (0001) surface of single crystal α-alumina, where the surface was chemically etched, at different exposure times and vapour pressures. Their results indicated that at 2 % humidity ~48 % of the surface will be covered by hydroxyl groups in approximately 55 s. This is approximately the same time between the application of the load an the catastrophic failure in the conducted fracture experiments when subcritical crack growth occurs. The coverage increased to ~74% of a monolayer at 80 % humidity. This indicates that the largest part of the coverage is already reached in the first 2 % humidity.

The amount of defect sites can explain the discrepancy between the results of Liu and Elam due to different surface preparation procedures. Since fracture surfaces are in general not perfect,

41 Chapter 3

a) 2.0

] 1.8 1/2

1.6

1.4

1.2 Fracture toughness [MPa [MPa m toughness Fracture

1.0 0 20406080100 Relative humidity [%]

b) 5.5

] 5.0 1/2

4.5

4.0

3.5 Fracture toughness [MPa [MPa m toughness Fracture

3.0 0 20406080100 Relative humidity [%]

Figure 3.2: The fracture toughness of two type of alumina: a) Wesgo995 and b) Wesgo997 vs. the relative humidity at different loading rates (0.2 (), 2.0 (z) and 20 mm min-1 (S)). The fracture toughness at a relative humidity of 0 % is measured inside a SEM.

42 Commercially applied oxide ceramics

a) 5.5

5.0 ] 1/2

4.5

4.0

3.5

3.0 Fracture toughness [MPa [MPa m toughness Fracture

2.5 0 20406080100 Relative humidity [%]

b) 5.5

5.0 ] 1/2

4.5

4.0

3.5

3.0 Fracture toughness [MPa [MPa m toughness Fracture

2.5 0 20406080100 Relative humidity [%]

Figure 3.3: The fracture toughness of a) translucent alumina (DGA) and b) translucent alumina with erbium additives (DGA-Er) vs. the relative humidity at different loading rates (0.2 (), 2.0 (z) and 20 mm min-1 (S)). The fracture toughness at a relative humidity of 0 % is measured inside a SEM.

43 Chapter 3

1.2 ] 1/2 1.1

1.0

0.9

Fracture toughness [MPa [MPa m toughness Fracture 0.8

0 20406080100 Relative humidity [%]

b) 13.0 12.5 ]

1/2 12.0

11.5

11.0

10.5

10.0

9.5 Fracture toughness [MPa m [MPa toughness Fracture

9.0 0 20406080100 Relative humidity [%]

Figure 3.4: The fracture toughness of a) PZT and b) magnesium partially stabilised zirconia (Mg-PSZ) vs. the relative humidity at different loading rates (0.2 (), 2.0 (z) and 20 mm min-1 (S)). The fracture toughness at a relative humidity of 0 % is measured inside a SEM.

44 Commercially applied oxide ceramics

2.0

] 1.8 1/2

1.6

1.4

1.2 Fracture toughness [MPa m [MPa toughness Fracture

1.0 020406080100 Relative humidity [%] Figure 3.5: The fracture toughness of floor tile ceramics vs. the relative humidity at different loading rates (0.2 (), 2.0 (z) and 20 mm min-1 (S)). The fracture toughness at a relative humidity of 0 % is measured inside a SEM. this could imply that for the aluminas, which had both a trans- and intergranular fracture mode, below 2 % already the surface is fully covered, with at least one monolayer.

A series of experiments with translucent alumina without erbium additives at a lower stressing rate at different humidities shows also Type I behaviour as shown in figure 3.6. However, an extra experiment was performed at a low relative humidity of 0.15 % resulting in a higher fracture toughness value. This supports the idea that certainly in the first 2 % adsorption plays a role.

The second type of behaviour, designated as Type II, is characterised by a continuous decrease on the fracture toughness over the whole range of humidity. The sharp decrease below 10 % humidity is absent and from 20 % relative humidity onwards the toughness still decreases showing that the influence of adsorption is still present after 2 %. This behaviour is observed for MnZn ferrites [2]. The exact origin of the difference between Type I and Type II behaviour is still unknown.

3.3.3 Relation between fracture strength and fracture toughness for Wesgo995 The fracture strength of Wesgo995 measured at different loading rates is given in figure 3.7. It is steadily decreasing with increasing relative humidity. However, the strength at a loading rate

45 Chapter 3

5.5

5.0 ] 1/2

4.5

4.0

3.5

3.0

Fracture toughness [MPa [MPa m toughness Fracture 2.5

020406080100 Relative humidity [%] Figure 3.6: The fracture toughness of translucent alumina without erbium additives (DGA) vs. the relative humidity with a stressing rate of 0.076 () and 1.76 MPa s-1 (z, 0.2 mm min-1).

325

300

275

250

225

Fracture strength [MPa] strength Fracture 200

175 0 20406080100 Relative humidity [%] Figure 3.7: The fracture strength of Wesgo995 measured at different humidities at different loading rates (0.2 (), 2.0 (z) and 20 mm min-1 (S)).

46 Commercially applied oxide ceramics of 0.2 mm min-1 attains a kind of plateau at 40 % relative humidity indicating that the adsorption processes are not contributing any more to the decrease. A similar behaviour is found by Cho et al. [12].

σ The apparent fracture toughness vs. strength (Kapp - f) ratio is given approximately by the formula as derived by Donners et al. [2]:

------1 - σ· 3 n + 1 K SYS co ------app = Y c ------, (3.5) σ i i f σ· 3 iYi ci · · where n is the subcritical crack growth factor, σS and σi are the stressing rates of the SENB and strength specimen, respectively, YS and Yi are the geometry factors of the notch in the SENB specimen and the initial defect in the strength specimen, respectively, and co and ci are the initial length of the notch and the initial defect, respectively.

σ If the Kapp - f ratio remains constant, it can be concluded that the influence of humidity is a material property and not dependent on the used experimental set-up or specimen geometry. σ -2 -2 1/2 This Kapp - f ratio for Wesgo995 is indeed constant at a value of 1.46×10 ± 0.05×10 m , as seen in figure 3.8.

A value for the subcritical crack growth factor of n = 26 and an initial defect size of ci =62µm are obtained after fitting equation 3.5 to the data using a notch length of the SENB specimen of -1 562.5 µm, YS = 1.85 and a stressing rate of 1.37 and 2.74 MPa s for SENB and strength experiments, respectively. Furthermore, it is assumed that the initial defect in the strength specimen is a semi-circular flaw (Yi = 1.26).

The value for the subcritical crack growth factor is typical for alumina, while the initial defect size in not unreasonable. Unfortunately, fractography could not reveal the various fracture ori- gins.

3.4 Conclusions In order to investigate the effect of adsorption of water on the fracture process, the fracture toughness of seven different ceramic materials was investigated at different humidities and loading rates. It can be concluded that adsorption of water can play an important role in the fracture process of oxide ceramics next to the kinetic effect. However, the role of adsorption is dependent on the material.

47 Chapter 3

0.020

0.018 ]

1/2 0.016

ratio [m

f 0.014 σ - app K 0.012

0.010 0 20406080100 Relative humidity [%] Figure 3.8: Fracture toughness - strength ratio vs. relative humidity for Wesgo995 alumina. The dashed line is the average.

Two types of behaviour can be distinguished. For Type I materials, a sharp decrease in fracture toughness from the 0 to 2 % humidity is observed and an insignificant decrease for humidities above 2-10 %, indicating that adsorption only has an influence in the first 2-10 % humidity. The investigated aluminas, PZT, floor tile ceramics and zirconia show this behaviour.

On the other hand, Type II materials show over the whole range of humidities a steady decrease of the fracture toughness, indicating that adsorption still plays a significant role above 20%. MnZn ferrites show this behaviour.

This investigation shows that it is important to determine whether the material is a Type I or a Type II material. The reported fracture toughness values can be used for Type I materials, since they are almost independent of humidity. However, the reported fracture toughness values for Type II materials are significantly depending on the relative humidity and the proper value for the condition of use should be taken.

3.5 Acknowledgements Philips Lighting, Morgan Electro Ceramics and Trega International are acknowledged for providing the materials. Hans Onstenk (Central Workshop TU/e) is acknowledged for the specimen preparation.

48 Commercially applied oxide ceramics

3.6 References [1] G. de With, Anisotropy in mechanical properties of sintered Sr-hexaferrite, Silicates Industrielles, 295 (1984), 185-189 [2] M.A.H. Donners, L.J.M.G. Dortmans and G. de With, Adsorption and kinetic effects on MnZn ferrites, J. Mater. Res., 15 (2000), 1377-1388 [3] P.T.H. van den Berg, Zirconia ceramics and mechanical surface interaction, Ph.D. The- sis, Eindhoven University of Technology, Eindhoven, The Netherlands (1992) [4] B. Gross and J.E. Srawley, Stress-intensity factors for single-edge-notch specimen in bending or combined bending and tension by boundary collocation of a stress function, Technical Note D-2603, NASA, USA, (1965) [5] W.F. Brown and J.E. Srawley, Plane strain crack toughness testing of high strength metallic materials, ASTM Spec. Tech. Publ., 410 (1967), 12-14 [6] E. Si. Stress intensity factors for edge cracks in round bars, Eng. Frac. Mech., 37 (1990), 805-812 [7] Private communication with Dr. Th. Kappen, Philips Lighting [8] C. Leach and N. Lambov, Temperature dependence of crack propagation in magnesia partially stabilized zirconia (Mg-PSZ), Mater. Sci. Monogr., 66C (1991), 1519-1525 [9] S.-B. Kim, D.-Y. Kim, J.-J. Kim and S.-H. Cho, Effect of grain size and poling on the fracture mode of lead zirconate titanate ceramics, J. Am. Ceram. Soc., 73 (1990), 161- 163 [10] P. Liu, T. Kendelewicz, G.E. Brown, E.J. Nelson and S.A. Chambers, Reaction of water α α vapor with -Al2O3(0001) and -Fe2O3(0001) surfaces: sychnotron X-ray photoemis- sion studies and thermodynamic calculations, Surf. Sci., 417 (1998), 53-65 [11] J.W. Elam, C.E. Nelson, M.A. Cameron, M.A. Tolbert and S.M. George, Adsorption of α H2O on a single-crystal -Al2O3(0001) surface, J. Phys. Chem. B, 102 (1998), 7008- 7015 [12] S.-J. Cho, K.-J. Yoon, J.-J. Kim and K.-H. Kim, Influence of humidity on the flexural strength of alumina, J. Eur. Ceram. Soc., 20 (2000), 761-764

49 Chapter 3

50 - 4 -

Subcritical crack growth in calcium hydroxyapatite

4.1 Introduction The rigid skeletal system enables the body of humans and other vertebrates to maintain its shape, to protect vital organs, to serve as a centre of metabolic activity and to transmit forces of muscular contraction between parts of the body during movement. The bones and teeth are com- posite materials built up from oriented small needle-like mineral crystals (~4×4×50 nm3) located in a collagen fibre matrix. The mineral crystals are found with X-ray diffraction to be an analogue of the geological mineral apatite (Ca10(PO4)6(OH/F)2) [1], but the elemental compo- - - 2- 3- sition of these crystals varies (e.g. F for OH or CO3 for PO4 ). The amount and size of hydroxyapatite crystals determine the physical properties of bone and teeth. For example, the enamel, the toplayer of teeth, can withstand high compressive stresses during eating due to a high volume of large hydroxyapatite crystals, while the underlying dentin has less and smaller hydroxyapatite crystals, providing a better shock absorbance (see table 4.1). However, the physical properties can change with changing mineral content due to e.g. the unrepairable degener- ation of hydroxyapatite in enamel due to acids formed by mouth bacteria from sugars in food and (notably soft) drinks, which causes serious caries leading to cavities.

The similarity in elemental composition and structure of hydroxyapatite with the minerals in bone makes hydroxyapatite ceramics a very favourable material for implants. Dense and porous hydroxyapatite ceramics show a very good biocompatibility (no rejection by living tissue) and

51 Chapter 4 bioactivity (source of tissue generation), where the latter even allows the regeneration of new natural bone material inside the pores and channels and can later dissolve inside the body. Hy- droxyapatite can also be used as a coating on metallic implants (e.g. titanium alloy) to prevent rejection by the surrounding tissue. However, bulk hydroxyapatite ceramics show a very poor mechanical behaviour. A low fracture toughness, a pronounced sensitivity towards subcritical crack growth and a small resistance to impact restricts the use of hydroxyapatite for clinical use to coatings, composites and pastes.

Subcritical crack growth in dense hydroxyapatite has been studied by different authors under different conditions. Thomas et al. [5] found a subcritical crack growth parameter n = 80 (recal- culation of the provided data gives n = 92 [6]), implying a high resistance to fatigue in air. On the other hand, De With et al. [6,7] found n = 26 and n = 12 for dense hydroxyapatite in air and water, respectively, using dynamic fatigue testing indicating substantial subcritical crack growth. The reported results by Raynaud et al. [8] of n = 22 and n = 14 in air and water, respectively, are in agreement with the latter authors. De With et al. [6] also found a subcritical crack growth parameter of n = 11 using double torsion specimens in air.

It is thus evident that the mechanical properties like fracture toughness and strength depend on the environment. However, it is unclear whether the amount of water present (i.e. relative humidity) has an influence on these properties. It this chapter, the influence of humidity on the fracture toughness of dense hydroxyapatite ceramics is investigated. The fracture toughness was determined at different loading rates and humidities in a mixture of dry nitrogen gas and (humidified) air using a single edge notched beam (SENB) technique.

Table 4.1: Typical physical properties of bone and teetha Tissue Mineral Crystal size Elastic Tensile Compressive content modulus strength strength [wt %] width thickness [GPa] [MPa] [MPa] [nm] [nm] Enamel 96 68 26 84 10 384 Dentin 70 30 3 12 37 305 Femurb 69 4 50 18 121 167 a. Data combined from [2,3,4], exact data are dependent on sex, age and testing condition (dry/ wet) b. Longitudinal direction

52 Calcium hydroxyapatite

4.2 Experimental

4.2.1 Synthesis Hydroxyapatite powder from Merck A.G. (Darmstadt, Germany) was sieved to remove agglomerates. Next, the powder was prepressed at 5 MPa to a block in a stainless steel die with an area of 6×6cm2 using a 10 % oleic acid in petroleum ether mixture as a lubricant resulting in a block with a relative density of ρ = 25 % (theoretical density of hydroxyapatite is 3.156 g cm-3). Sub- sequently, the prepressed block was cold isostatically pressed in a rubber bag at 100 MPa, reaching a green density of ρ = 45 %. Finally, the green bodies were sintered at temperatures between 1150 °C and 1350 °C for 6 h with a heating and cooling rate of 3 and 1 °C min-1, respectively, in moist air, which was created by leading air through distilled water.

4.2.2 Characterisation The resulting hydroxyapatite blocks after firing were subjected to X-ray diffraction (XRD) on α a Rigaku Geigerflex using CuK 1 radiation in order to detect the presence of several calcium- phosphate phases, which can form during sintering. The microstructure was investigated using a scanning electron microscope (SEM, Jeol JSM 840a) on samples which were polished with diamond paste to 0.5 µm, etched for 1 min in 1 % citric acid and gold coated to prevent charging. Furthermore, the density was determined using the Archimedes’ method.

4.2.3 Fracture The fracture toughness of hydroxyapatite was determined on SENB specimens using a four point set-up with an inner (li) and outer (lo) span of 15 and 30 mm, respectively. The specimens were cut and ground from the blocks into rectangular bars of 3×4×36 mm3 (w×h×l). The notch of the specimens was cut to ~15 % of the height with a 100 µm diamond saw blade resulting in a notch width of ~150 µm.

The experiments were performed at room temperature on an universal testing machine (Erich- sen GmbH, Germany) with a calibrated 500 N load cell (Test GmbH, Germany). Prior to each experiment, the specimen was preloaded up to 5 N with a crosshead velocity of 0.2 mm min-1. During the experiment, the crosshead velocity was constant and preset to 0.2, 2.0 and 20 mm min-1.

53 Chapter 4 Intensity [-]

10 20 2030 3040 50 50 60 2θ [°] Figure 4.1: XRD spectrum of hydroxyapatite sintered at 1300 °C. at room temperature. The relative humidity between 2 and 80 % was controlled by the inlet of dry nitrogen gas in case of humidity decrease or by the inlet of nitrogen gas led through water in case of humidity increase. The experiments at 0 % humidity to obtain the inert fracture tough- × -4 ness were performed inside a SEM (pH2O <7 10 Pa) on a special four point bending testing machine (Deben UK Ltd., Debenham, UK) with an inner and outer span of 5 and 20 mm, respectively.

4.3 Results and discussion

4.3.1 Synthesis and characterisation XRD measurements on the blocks sintered at different temperatures are shown in figure 4.1 and show only the presence of hydroxyapatite (JCPDS-file 09-0432) indicating for all temperatures the prevention of formation of other stable calcium phosphate phases like tricalcium phosphate

(TCP, Ca3(PO4)2, JCPDS-file 32-0176), tetracalcium phosphate (TTCP, Ca4P2O9, JCPDS-file 11-0232) and calcium oxide (CaO, JCPDS-file 28-0775) during sintering.

The density of the blocks sintered at different temperatures is shown in figure 4.2. A relative density of ρ = 98 % is reached at 1275 °C and hardly increases with further increasing sintering temperature. The maximum relative density is in agreement with reported data in literature [6- 10].

54 Calcium hydroxyapatite

100

Relative density [%] density Relative 80

75 1100 1200 1300 1400 Sintering temperature [°C] Figure 4.2: Relative density of hydroxyapatite sintered at different temperatures.

The microstructure of polished and etched surfaces of the hydroxyapatite ceramics was inves- itigated. The porosity decreases with increasing sintering temperature as shown in figure 4.3. Simultaneously, the average grain size, estimated visually from the SEM images, increases from 1, 1, 2 to 3 µm for a sintering temperature of 1200, 1250, 1300 and 1350 °C, respectively.

4.3.2 Fractography Since the relative humidity might have an influence on the fracture mode and in this way an indirect effect on the fracture toughness, the fracture surfaces were examined by SEM, as shown in figure 4.4. The fracture is predominately transgranular as reported by De With et al. [6] and Suchanek et al. [11]. It was observed that the fracture mode is independent of relative humidity.

4.3.3 Fracture toughness The fracture toughness of hydroxyapatite, sintered at 1300 °C (ρ = 98 %), at different humidities and crosshead velocities of 0.2, 2.0 and 20 mm min-1 is shown in figure 4.5. The inert fracture toughness was measured to be 1.42 MPa m1/2 independent of the used crosshead velocities of 0.05 and 0.20 mm min-1 confirming that the inert fracture toughness does not depend on stressing rate as also shown for MnZn ferrites [12]. The fracture toughness values at ambient conditions (room temperature and RH = ~40 %) are ~1 and 1.27 MPa m1/2 for the lowest and

55 Chapter 4 a)c) b) d)

Figure 4.3: The microstructure of hydroxyapatite sintered at a) 1200, b) 1250, c) 1300 and d) 1350 °C.

56 Calcium hydroxyapatite

Figure 4.4: Fracture surfaces of hydroxyapatite ceramics sintered at 1300 °C.

1.6 ]

1/2 1.4

1.2

1.0 Fracture toughness [MPa [MPa m toughness Fracture

0.8 0 20406080100 Relative humidity [%] Figure 4.5: Fracture toughness of hydroxyapatite sintered at 1300 °C vs. relative humidity with a loading rate of 0.2 (), 2.0 (z) and 20 mm min-1 (S). The fracture toughness at 0 % humidity is measured inside a SEM.

57 Chapter 4 the two highest crosshead velocities. The fracture toughness at the lowest humidity is comparable with other values obtained by several authors [6,13], although caution should be taken for the wide variety of reported data in literature due to the high sensitivity of properties of hydroxyapatite to preparation and sintering [14], see table 4.2.

The fracture toughness is continuously decreasing with increasing humidity. The smooth decrease of fracture toughness for hydroxyapatite indicates that at all humidities the contribution of adsorption is present. This is in contrast to e.g. alumina, where a sharp decrease of the fracture toughness between 0 and 10 % relative humidity and a constant toughness onwards is observed [16]. The sharp decrease indicates that the adsorption of water contributes significantly to the decrease in fracture toughness at lower humidities, but the constant toughness onwards indicates that at higher humidities the adsorption has only a negligible contribution despite the increasing water availability. Materials showing the latter behaviour in fracture toughness (e.g. alumina) are designated as Type I materials and showing the first behaviour as Type II. Fracture toughness data reported in literature of Type II materials should be treated with extra caution, since they do depend on the humidity present during measurement. Hydroxyapatite is the second known Type II material next to MnZn ferrites.

Table 4.2: Fracture toughness data for polycrystalline hydroxyapatite ceramics a ρ Ref. KIc Method Grain Sinter- Remarks size ingb [MPa m1/2][%][µm] [6] 1.0 SN >95 ~4.0 C Sintering in moist air [7] 1.33 SN 98 1.0 C Sintering in moist air 0.93 SN 94 2.9 C Sintering in moist air [9] 0.70 CN 99.0 ~0.5 C Grain size measured with 5 % Ag particles [13] 1.0 VI 97.8 0.2 HIP 0.65 VI 98.9 0.4 HIP [15] 0.70 SN 99.9 0.64 C+HIP 0.86 SN 99.9 1.07 C This 0.99 SN 98 2 C work a. SN = straight notch, CN = cevron notch, VI = Vickers indent b. C = conventional sintering in air, HIP = hot isostatic pressing under argon

58 Calcium hydroxyapatite

4.3.4 Analysis The relation between the measured (apparent) fracture toughness (K) and the adsorption controlled fracture toughness (Kads) for Type II materials is given by [12]:

------1 - n – 2 v K3 2 – n KK= 1 + ------p------, (4.1) ads2()n + 1 c 2 · o Kads K where n is the subcritical crack growth factor, vp is the pre-exponential factor in the subcritical n crack growth power law equation v = vp(K / Ko) with Ko as the inert fracture toughness, co is the initial notch length and K· = dKt⁄ d . Using standard Matlab [17] fitting routines the parameters n, vp,RH and Kads,RH (RH = 2, 10, 20, 60, 80 %) are determined assuming that the subcritical crack growth factor n is constant for all humidities. The results of the fit procedure are shown in table 4.3.

The fit resulted in a value of 9.4 for subcritical crack growth factor n, which is low compared to data in literature (n ~26). However, the subcritical crack growth parameter in these studies is calculated without taking the contribution of adsorption into account. Moreover, the higher n- values are measured using strength specimens which have microflaws as the origin of fracture. On the other hand, a value of n = 11 has been reported using double torsion technique [6] with a macroflaw resembling the flaw in the SENB specimen used in this work.

The adsorption controlled fracture toughness data (Kads) are shown in figure 4.6. The relation between the inert fracture toughness and the adsorption controlled fracture toughness is given by

φ () Kads = Ko 1 – ln 1 + bpX , (4.2)

Table 4.3: Data from fit with n = 9.4

RH Kads vp [%] [MPa m1/2][10-4 ms-1] 2 1.44 0.1296 10 1.34 0.7140 20 1.29 3.1217 60 1.18 1.4375 80 1.20 5.6215

59 Chapter 4

1.6

1.5

1.4 ] 1/2

1.3

[MPam

ads 1.2 K

1.1

1.0 0 20406080100 Relative humidity [% ]

Figure 4.6: The fitted values of the adsorption controlled fracture toughness of hydroxyapatite vs. the relative humidity.

φ where is a constant, b is the Langmuir adsorption coefficient and pX is the relative humidity. Fitting the adsorption controlled fracture toughness data to this equation results in values of 1.48 MPa m1/2, 0.82 and 23.6 for the inert fracture toughness, φ and b, respectively. The fitted value of 1.48 MPa m1/2 for the inert fracture toughness is in good agreement with the experimental determined value of 1.42 MPa m1/2 in the SEM. The constant φ represents the ratio of energy released by adsorption to the surface energy required for generating two surfaces during fracture under inert conditions. The fitted value of 0.82 is comparable to the high φ-value of 0.94 of MnZn ferrites [12], whereas a low φ-value would indicate an insensitivity of the fracture toughness towards adsorption. The high Langmuir constant b of 23.6 indicates a rapid increase of coverage at low humidities. This implies that the monolayer coverage is rapidly reached as indicated by experimental adsorption studies of water on natural bone and synthetic hydroxyapatite [18]. These results confirm, firstly, the usefulness of the combined analysis and, secondly, the sensitivity of hydroxyapatite for subcritical crack growth.

4.4 Conclusions Dense hydroxyapatite ceramics have been made by sintering cold isostatic pressed blocks from powder at several temperatures in moist air. The fracture toughness of hydroxyapatite sintered at 1300 °C is measured using the SENB technique at different loading rates and relative humid-

60 Calcium hydroxyapatite ities. Hydroxyapatite fractures predominantly transgranular. Furthermore, it is observed that the fracture toughness decreases with increasing humidity. The smooth decrease makes hydroxyapatite a Type II material like MnZn ferrites indicating that care should be taken with the usage of reported data in literature, since the fracture toughness depends heavily on the humidity. The subcritical crack growth parameter n is determined to be 9.4 using fitting with the combined kinetic-adsorption model showing that hydroxyapatite is a very sensitive material towards subcritical crack growth. Furthermore, the inert fracture toughness obtained from fitting of 1.48 MPa m1/2 is in good agreement with the measured inert fracture toughness of 1.42 MPa m1/2.

4.5 Acknowledgements A.J.M. van Dijk (Eindhoven University of Technology) is acknowledged for his major contribution to this chapter performed within the framework of the graduation work for his Masters thesis [19].

4.6 References [1] W.F. de Jong, La substance minérale dans les os, Rec. Trav. Chim. Pays-Bas, 45 (1926), 445-448 [2] K.E. Healy, Dentin and Enamel, in: "Handbook of biomaterial properties",1st Ed., Eds: J. Black and G. Hastings, Chapman & Hall, London, UK, (1998) [3] J. Currey, Cortical bone, in: "Handbook of biomaterial properties",1st Ed., Eds: J. Black and G. Hastings, Chapman & Hall, London, UK, (1998) [4] J.B. Park and R.S. Lakes, Biomaterials: An introduction, 2nd Ed., Plenum Press, New York, USA (1992) [5] M.B. Thomas, R.H. Doremus, M. Jahro and R.L. Salsbury, Dense hydroxylapatite: fatigue and fracture strength after various treatment from diametral tests, J. Mater. Sci., 15 (1980), 891-894 [6] G. de With, H.J.A. van Dijk, N. Hattu and K. Prijs, Preparation, microstructure and mechanical properties of dense polycrystalline hydroxy apatite, J. Mater. Sci., 16 (1981), 1592-1598 [7] G. de With, H.J.A. van Dijk and N. Hattu, Mechanical behaviour of biocompatible hydroxyapatite ceramics, Proc. Brit. Ceram. Soc., 31 (1981), 181-189 [8] S. Raynaud, E. Champion, D. Bernache-Assolant and D. Tetard, Dynamic fatigue and degradation in solution of hydroxyapatite ceramics, J. Mater. Sci.: Mater. in Med., 9 (1998), 221-227 [9] X. Zhang, G.H.M. Gubbels, R.A. Terpstra and R. Metselaar, Toughening of calcium hydroxyapatite with silver particles, J. Mater. Sci., 32 (1997), 235-243

61 Chapter 4

[10] S. Raynaud, E. Champion and D. Bernache-Assollant, Calcium phosphate apatites with variable Ca/P atomic ratio II. Calcination and sintering, Biomaterials, 23 (2002), 1073- 1080 [11] W. Suchanek, M. Masamoto, M. Kakihana and M. Yoshimura, Hydroxyapatite/ hydroxyapatite-whisker composites without sintering additives: mechanical properties and microstructural evolution, J. Am. Ceram. Soc., 80 (1997), 2805-2815 [12] M.A.H. Donners, L.J.M.G. Dortmans and G. de With, Adsorption and kinetic effects on MnZn ferrites, J. Mater. Res., 15 (2000), 1377-1388 [13] S. Raynaud, E. Champion, J.P. Lafon and D. Bernache-Assollant, Calcium phosphate apatites with variable Ca/P atomic ration III. Mechanical properties and degradation in solution of hot pressed ceramics, Biomaterials, 23 (2002), 1081-1089 [14] Z.Z. Zyman, I.G. Ivanov and V.I. Glushko, Possibilities for strengthening hydroxyapatite ceramics, J. Biomed. Mater. Res., 46 (1999), 73-79 [15] T. Nonami and F. Wakai, Evaluation of crack propagation in hydroxyapatite by double- torsion method in air, water and toluene, J. Ceram. Soc. Jpn., 103 (1995), 648-652 [16] N.J. van der Laag, A.J.M van Dijk, L.J.M.G. Dortmans and G. de With, The role of water adsorption on the fracture of ceramic oxides, J. Eur. Ceram. Soc., (submitted) [17] Matlab v.5.3, The Mathworks Inc. Nathick, MA, USA [18] M.E. Dry and R.A. Beebe, Adsorption studies on bone mineral and synthetic hydroxyapatite, J. Phys. Chem., 64 (1960), 1300-1304 [19] A.J.M. van Dijk, The influence of loading rate and environment on brittle fracture of ceramics, M.Sc. Thesis, Eindhoven University of Technology, Eindhoven, The Nether- lands, (2001)

62 - 5 -

Fracture of magnesium aluminate spinel

(MgAl2O4)

5.1 Introduction Magnesium aluminate spinel (MgAl2O4) is an interesting ceramic material. It has an excellent combination of desired properties for e.g. refractory applications like a high melting temperature, high strength at elevated temperatures, good thermal shock resistance, low thermal expansion and a high inertness to chemical attack. Magnesium aluminate is in combination with its excellent optical properties suitable for making strong windows or lenses for e.g. armoured ve- hicles and cruise missiles. Furthermore, magnesium aluminate shows an unusual inertness to high-fluence neutron irradiation concerning mechanical properties. This allows the material to be used in (low) nuclear applications as a buffer material [1].

Figure 5.1 shows the phase diagram of magnesium aluminate indicating that spinel formation is possible with a large range in stoichiometry. The synthesis of magnesium aluminate powder is relative easily achieved by mixing magnesia and alumina and firing at elevated temperatures. Magnesium aluminate ceramics can be obtained from these powders using conventional ceramic processing like cold or hot pressing and conventional sintering.

One of the important questions raised during the selection process of (ceramic) materials con- cerns the life time expectancy of the material under thermomechanical loading. An important phenomena in this field is subcritical crack growth due to the presence of water. Subcritical

63 Chapter 5

2827 °C

Liquid 2600

2200 2105 °C 2054 °C

1800 Spinel

Temperature [°C] Temperature 1400

1000 0 20406080100

MgO Al2O3 Mol.% Al2O3 Figure 5.1: Phase diagram of MgO-Al2O3 [2]. crack growth in magnesium aluminate is also very intriguing due to the fact that of its constitu- ents magnesia does not show substantial subcritical crack growth behaviour, but alumina does.

Unfortunately, only one paper by Rice et al. [3] deals with subcritical crack growth in magnesium aluminate. They performed tests in liquid nitrogen (- 196 °C), where the influence of water is not present and found a 20-30 % higher value for the strength of polycrystalline magnesium aluminate than at ambient temperature and humidity (22 °C and ~30 %). Since the temperature dependence of the Young’s modulus can only explain a 3 % higher value, the remainder should be attributed to the absence of water.

In this chapter, the influence of humidity on the mechanical properties of two types of magnesium aluminate ceramics is investigated. Also, the influence of humidity on the fracture of single crystals along low index planes is determined. Furthermore, the fracture path of magnesium aluminate has been investigated using Electron Backscattering Diffraction (EBSD), in order to see whether texture and crystallite orientation are important parameters. Finally, hydrated surface investigations have been performed using Low Energy Ion Scattering (LEIS) and Diffuse Reflectance Infrared Fourier Transform Spectroscopy (DRIFTS) in order to obtain an insight in the adsorption of water on fracture surfaces.

64 Magnesium aluminate

5.2 Experimental

5.2.1 Material preparation and characterisation The mechanical properties of two types of polycrystalline magnesium aluminate ceramics are investigated. Type A is a commercially obtained ceramic from Custom Technical Ceramics (Arvada, CO, USA) in the form of 5x50x50mm3 blocks. Type B are ceramic discs (∅ 60 mm x 10 mm) made by conventional sintering (1600 °C, 4 h) in air of prepressed (40 MPa) and subsequently cold isostatically pressed (250 MPa) submicrometer magnesium aluminate powder from Baikowski (Annecy, France). Furthermore, samples cut from a Ver- neuil-grown single crystal were used for the single crystal experiments and a hot-pressed coarse grained sample for EBSD experiments.

The density of the ceramics is determined by the Archimedes’ method using distilled water. The Young’s, shear and bulk modulus and Poisson’s ratio are determined using ultrasonic measurements (Panametrics, 25DL) on plane-parallel polished samples. The average grain size was visually estimated by typically measuring some twenty grains on polished (down to 0.25 µm) and chemically etched (85 % phosphoric acid,~175 °C, 10 min) samples, which were gold sput- tered, using a Scanning Electron Microscope (SEM, Jeol 840).

5.2.2 Mechanical testing Single edge notched beam (SENB) specimens were used to measure the fracture toughness. The rectangular polycrystalline specimen were cut and ground into the dimensions shown in table 5.1 from the blocks or discs. The specimens from the single crystals were cut and ground after establishing the correct orientation using the Laue backscattering method within ~3 °. The length of the specimens was at least 5 mm longer than the outer span. The notch of the SENB specimens was cut to ~15 % of the specimen height using a 100 µm diamond blade saw resulting in a notch width of ~120 µm. The notch of the single crystal specimen were sharpened with a razor blade up to ~50 µm using a 2-6 µm diamond suspension.

The fracture toughness experiments were performed at room temperature on an universal Erich- sen testing machine with a calibrated Test GmbH 500 N load cell for the polycrystalline samples and a calibrated Erichsen 50 N load cell for the single crystals. The experiments used a four- point bending set-up with an inner (li) and outer (lo) span depending on the length of the specimen as given in table 5.1. The experiments were carried out with a constant crosshead velocity -1 -1 of 0.2 mm min for the single crystal specimens and 0.2, 2.0 and 20 mm min for the polycrys-

65 Chapter 5

Table 5.1: Set-up and specimen dimensions Sample Inner span Outer span Specimen SEM Specimen (w x h) (w x h) [mm] [mm] [mm] [mm] Type A 5 20 3x4 3x4 15 30 Type B 5 15 3 x 4 2 x 4 100 5 15 2 x 3 111 5 15 2 x 3 talline specimens. The experimentally observed stressing rates for the polycrystalline specimens are shown in table 5.2. Prior to each experiment, the specimens were preloaded upto 20 N for the polycrystalline material and 5 N for the single crystals using a crosshead velocity of 0.2 mm min-1.

The experiments at different humidities were performed inside a perspex chamber mounted in the testing machine with a gas and hygrometer inlet. The relative humidity (RH) inside the chamber was calculated as the ratio of the partial pressure (pH2O) at the actual dew point inside, which was measured by a Panametrics MM35 hygrometer, and the equilibrium partial pressure at room temperature. The relative humidity between 2 and 80 % was controlled by the inlet of dry nitrogen gas in the case of a humidity decrease or by the inlet of nitrogen gas led through water in the case of a humidity increase.

The fracture toughness experiments for the polycrystalline magnesium aluminate at RH = 0 % were performed inside the vacuum chamber of a scanning electron microscope (pH2O < 7×10-4 Pa) on a special four-point bending set-up (Deben UK Ltd., Debenham, UK). The inner and outer span were 15 and 36 mm for type A and 5 and 20 mm for type B, respectively. The crosshead velocity was preset to 0.05 mm min-1.

Table 5.2: Average stressing rates of polycrystalline magnesium aluminate ceramic Loading rate Type A Type B [mm s-1][MPas-1][MPas-1] 0.2 1.13 0.92 2 14.3 13.6 20 93 104

66 Magnesium aluminate

5.2.3 Surface investigation preparation The microtexture of a coarse grained hot-pressed magnesium aluminate sample with cracks was studied using EBSD (see also the appendix to this chapter). The measurements were performed on an electron probe microanalyser (EPMA, Superprobe Jeol JXA 8600 SX) with a standard EBSD set-up from TexSEM Laboratories (Draper, UT, USA). The magnesium aluminate sample was embedded in a conducting polymer matrix and polished down to 3 µm with diamond discs and diamond suspension. Since diamond polishing introduces mechanical damage (resid- ual stresses) preventing the generation of indexible EBSD-patterns, this damage is eliminated by polishing for 8 hours using a colloidal silica suspension. Subsequently, cracks were generated by indenting using Vickers indents.

The electrically non-conductivity of magnesium aluminate requires precautionary measures to prevent surface charging effects, without influencing the EBSD-patterns. Therefore, colloidal graphite was deposited inside the cracks and indents and the complete sample was coated with a ~20 nm thick carbon layer. Additional carbon bridges were required in some cases.

The hydroxyl groups resulting from the dissociative adsorption of water on the surface of magnesium aluminate were investigated using DRIFTS and LEIS. The magnesium aluminate powder for the DRIFTS experiment was heated to 900 °C for 6 h in air to remove the bulk water collected during storage. Subsequently, the powder put in an in-situ chamber of a SpectraTech Collector DRIFT cell in a Nicolet Protége 460 spectrometer with a MCT detector. The powder was outgassed at 550 °C for 1 h in flowing helium gas (10 ml min-1) prior to measuring the spectrum at room temperature.

The fracture surfaces of single crystals are investigated using LEIS, which provides information about the topmost atomic layer. The LEIS instrument “Calipso” at Calipso BV [5], which is described in [6], is used since this set-up is more sensitive than normal LEIS instruments. Surfac- es, (100) and (111) orientated, were created by fracturing single crystals with a relative small and high amount of water inside a glove box (pH2O < 0.7 Pa, RH = ~0.02 %) and in air, respectively. The samples fractured inside the glove box were transported to the LEIS apparatus in a special device completely isolated from the air. The (100) oriented sample that was broken in the glove-box was taken out of the apparatus and exposed to the air for 45 minutes, which was the same time that the (100) oriented sample broken in air was left in air prior to measuring. A relatively high energy 4He+ ion beam (5 and 4 keV for the (100) and (111) oriented samples)

67 Chapter 5 was used for the LEIS measurements in order to get a good distinction between the magnesium and aluminium signals. Some oxygen sputtering was applied prior to the measurements to remove possible contaminations on the surface. This process is supposed to have no influence on the surface itself.

5.3 Results and discussion of bulk properties

5.3.1 Characterisation The density of the investigated magnesium aluminate ceramics has been determined using the Archimedes’ method with water as the displaced fluid. Type A magnesium aluminate has a density of 87 % of the theoretical density and Type B 98 %, see table 5.3. The SEM images of the surfaces of the two polycrystalline ceramics are shown in figure 5.2. Type A magnesium aluminate shows a large number of pores, which are located at the grain boundaries, triple points and inside the grains indicating that enhancement in the sintering procedure is possible. On the other hand, Type B shows only small pores at the triple points. The SEM images allow for an estimation of the average grain size, which is 10 and 2 µm for Type A and Type B, respectively.

5.3.2 Mechanical characterisation The elastic properties of the two polycrystalline ceramics are shown in table 5.3. The low values for Type A can be contributed to the large fraction of pores in the material. The values for Type B are in agreement with the results from Baudín et al. [7] for a polycrystalline magnesium aluminate with similar microstructural properties.

The fracture toughness at ambient conditions (room temperature, ~40 % humidity) is measured using the SENB method with a crosshead speed of 0.2 mm min-1 for comparison with published data in literature. The toughness is 2.10 and 2.99 MPa m1/2 for Type A and B, respectively. The fracture toughness of polycrystalline stoichiometric magnesium aluminate has been reported by several authors [7-10] and is shown in table 5.4. The range of toughness data is between 1.8 and

Table 5.3: Density and elastic properties of magnesium aluminate ceramics Spinel Type A Type B ρ [g cm-1] 3.183 3.562 E [GPa] 196 259 ν [-] 0.27 0.32 G [GPa] 76.9 98.4 K [GPa] 143 238

68 Magnesium aluminate

a) b)

c) d)

Figure 5.2: SEM images of polished and chemically etched surfaces of Type A (a & b) and Type B (c & d) magnesium aluminate ceramic.

Table 5.4: Fracture toughness data for polycrystalline magnesium aluminate a ρ Reference Crosshead Method Grain size KIc speed [mm min-1][µm] [%] [MPa m1/2] [7] 0.5 SN 1.5 98 3.0 [9] 0.1 SN 35 100 1.46 CN 35 100 1.79 KI 35 100 1.81 [10] 0.005 CN 150 b 2.2 [8] 2.5 KI 5 c 1.94 12 c 1.98 25 c 1.83 38 c 1.97 This work 0.2 SN 10 87 2.08 2 98 2.99 a. SN = straight notch, CN = chevron notch, KI = Knoop microflaw induced fracture b. Minimization of impurities and porosity if emphasized c. Theoretical dense

69 Chapter 5

2.2 MPa m1/2 and is just a little higher than the fracture toughness of single crystals along the three low-index crystallographic planes: 1.18, 1.54 and 1.90 MPa m1/2 for the (100), (110) and (111) planes [11]. The small difference in toughness between the single crystals and polycrystalline samples can be explained by the fracture mode which is mixed intergranular/transgranular [8,10] or 100 % transgranular [3] implying that the fracture of a single grain can act as a starter for the overall fracture.

The value for Type A is in the range as reported in the literature. However, a 50 % higher toughness value of ~3 MPa m1/2 is found for Type B, as did Baudín et al. [7]. The grain size is the main difference between these studies and the others. A grain size effect is absent for magnesium aluminate ceramics with grains larger than 5 µm according to Stewart and Bradt [8]. How- ever, the high toughness ceramics have grain sizes of 1.5-2 µm. Bhaduri and Bhaduri [12] found a value of 3.45 MPa m1/2 for a magnesia-rich magnesium aluminate with a grain size of 0.1- 0.15 µm. The smaller grains provide an explanation for the high toughness values as they can not act as a starter for fracture on their own. The flaw starting in one grain should pass a grain boundary via e.g. subcritical crack growth and fracture a second grain in order to start the overall fracture. The grain boundary acts in this case as an extra toughening mechanism. Rice and Wu [3] have suggested this possibility of a flaw growing into a second grain, but rejected this possibility for the resulting high values of ~3.4 MPa m1/2 which they considered as unrealistic.

The fracture toughness of the magnesium aluminates as a function of relative humidity and at different crosshead velocities is shown in figure 5.3. It is observed that the fracture toughness of magnesium aluminate is decreasing substantially with increasing humidity for all crosshead velocities, indicating subcritical crack growth. This decrease was also observed by Rice and Wu [3] in bending strength experiments, but they could not detect a decrease in fracture toughness due to problems with flaw size identification. The decrease in toughness for Type B magnesium aluminate is continuous with increasing humidity indicating that adsorption of water molecules in the wake field of the crack does play an important role in the fracture process next to the chemical reaction at the crack tip. This continuous decreasing behaviour of the toughness is similar to MnZn ferrites [13] and calcium hydroxyapatite [14] making magnesium aluminate the third known material to have a Type II behaviour.

70 Magnesium aluminate

a) 3.0

2.8 ] 1/2

2.6

2.4

2.2

2.0 Fracture toughness [MPa m [MPa toughness Fracture 1.8

020406080100 Relative humidity [%]

b) 4.2

4.0 ]

1/2 3.8

3.6

3.4

3.2

3.0

2.8 Fracture toughness [MPa m [MPa toughness Fracture 2.6

020406080100 Relative humidity [%]

Figure 5.3: Fracture toughness of magnesium aluminate ceramics a) Type A and b) Type B as a function of relative humidity and at crosshead velocities of 0.2 (), 2.0 (z) and 20 mm min-1 (S). The lines provide a guide for the eye.

71 Chapter 5

However, while dense magnesium aluminate shows a Type II behaviour, porous magnesium aluminate as Type A does not. The decrease in fracture toughness with increasing humidity is characterised by a sharp drop between 0 % and 2-10 % relative humidity and a slowly decreasing toughness onwards. This is typical for a Type I behaviour as e.g. alumina or lead zirconate titanate shows [14]. This indicates that adsorption can play a role in the decrease of fracture toughness, but microstructural aspects as e.g. pores interact as well.

The exact interaction of pores on adsorption remains unrevealed. The interaction is only partly explained by the reduction of fractured surface area. Another explanation for this interaction might be that the crack tip moves too fast while travelling trough the pores impeding the catch- ing up of the (energy) contribution of the adsorption to the process zone. Furthermore, the adsorption energy from the adsorption on the surface of the pores is less than on the fractured surface as the surface is already relaxed.

The subcritical crack growth of spinel has been studied using single crystal specimens. Special- ly prepared specimen were fractured with a crosshead speed of 0.2 mm min-1 along the (100) and (111) plane at a relative humidity of 2 and 40 %. The results of three specimens at each combination of humidity and orientation are shown in table 5.5. A decrease of 17 and 18 % due to subcritical crack growth is observed for the (100) and (111) orientations, respectively, which is somewhat smaller than the decrease of 26 % obtained by Rice and Wu [3] for the (100) orientation using strength measurements. This can probably be explained by the fact that in this work the low humidity measurements were performed at 2 % humidity and by Rice in liquid nitrogen.

A single experiment was performed for comparison with the data of Stewart and Bradt [11] at a relative humidity of ~30 % and a crosshead velocity of 2.5 mm min-1 with a (111) oriented

Table 5.5: Fracture toughness and fracture surface energy of single crystal spinel 1/2 γ -2 Relative KIc [MPa m ] f [J m ] humidity [%] (100) (111) (100) (111) 2 1.30 1.97 4.33 5.21 (±0.07) (±0.12) 40 1.07 1.63 2.93 3.56 (±0.09) (±0.12)

72 Magnesium aluminate specimen. The resulting fracture toughness of 1.86 MPa m1/2 is in good agreement with the value of 1.90 MPa m1/2 of these authors.

γ The fracture surface energy ( f) can be calculated by:

K2 ()1 – ν2 γ = ------Ic , (5.1) f 2E where E and ν are the orientation dependent Young’s modulus and Poisson’s ratio, respectively, which can be calculated using the single crystal elastic parameters [15]. The decrease in fracture surface energy is ~46 %, as shown in table 5.5.

Fang et al. [16] have performed computer simulations of the reduction of surface energy of different fracture planes in spinel with increasing coverage of hydroxyl groups. They concluded using the experiments of Rice and Wu in combination with their theoretical results that the fracture planes will only be partially hydrated. The present work confirms this idea.

5.4 Results and discussion of surface properties

5.4.1 EBSD The EBSD image of a coarse grained hot-pressed magnesium aluminate ceramic with a combined crack originating from two different Vickers indents is shown in figure 5.4 together with a SEM image of the same region. The EBSD image is composed of three separate EBSD scans, which are separately edited for the confidence level, image quality and crystal growth function. The small differences between the same grain measured in two EBSD scans are due to the used interplanar angle tolerance of 5 °, sample movement during the two scans and the used growth function, which deviates near the edges of the scans. The pole figures of the image, shown in figure 5.5, indicate the absence of texture.

The Euler angles of the grains, describing the orientation of the grain with respect to the specimen axes, along the left and right side of the wide crack in figure 5.4 are shown in table 5.6. It is shown from the image and the Euler angles, that the fracture is both trans- and intergranular. The pole figures of the orientations of the grains along the crack, shown in figure 5.6, indicate that a preferenced orientation for the crack path is absent.

73 Chapter 5

a)b)

20 µm 10 µm c)

111

100 110

Figure 5.4: a) ESBD image of a magnesium aluminate ceramic with a crack showing the orientation of the grains, b) SEM image of the same region, c) transgranular fractured grains in a).

74 Magnesium aluminate

Table 5.6: Orientation of the grains from top to bottom along the crack in figure 5.4 in Euler anglesa Left side Trans Right side granular ϕ Φ ϕ ϕ Φ ϕ 1 [°] [°] 2 [°] 1 [°] [°] 2 [°] 208.4 18.7 242.1 143.5 71.0 339.2 107.8 49.8 1.4 154.8 70.6 321.0 159.7 37.4 264.0 109.6 39.3 304.9 162.4 30.3 290.5 87.8 13.0 19.3 122.2 35.3 321.1 103.2 52.3 20.6 283.7 13.3 188.7 225.0 26.1 271.1 169.1 41.4 322.1 127.1 78.3 306.7 102.8 39.9 15.2 209.6 19.3 264.3 221.4 34.0 267.3 67.8 62.1 12.1 95.0 28.7 23.1 105.9 23.9 338.7 106.1 29.0 12.2 101.4 48.5 305.0 111.5 55.1 9.9 20.2 16.6 90.5 111.2 56.6 10.2 205.8 26.1 275.9 28.0 18.6 93.4 77.9 29.6 343.0 7.2 17.0 42.6 177.4 49.2 242.9 176.8 50.2 242.4 74.1 56.0 36.3 122.3 32.5 13.5 125.9 35.6 319.5 73.8 55.7 36.8 111.8 31.9 328.9 72.7 38.0 346.1 203.1 46.3 248.1 65.8 50.9 43.3 9.5 19.3 43.2 269.5 11.5 140.0 131.2 64.6 301.2 197.0 43.0 304.8 160.1 29.2 293.9 87.7 50.4 331.3 139.8 38.4 273.0 138.5 38.3 282.6 88.4 48.9 330.3 62.5 62.6 16.3 137.5 50.2 351.5 142.7 79.8 315.7 a. The arrows indicate the grains which have been broken transgranular. These are shown in figure 5.4c.

75 Chapter 5

(100) (111)

y/TD y/TD

x/RD x/RD Figure 5.5: (100) and (111) pole figures of all the grains in figure 5.4a.

(100) (111)

y/TD y/TD

x/RD x/RD Figure 5.6: (100) and (111) pole figures of grains along the crack in figure 5.4a.

5.4.2 LEIS Since a preferred fracture plane is absent, the focus is laid on the fracture along the low-index planes. Therefore, the composition of the outmost atomic layer of the fracture surfaces of single crystals is investigated with LEIS. The LEIS spectra of the fracture surfaces of the (100) and

(111) planes broken at pH2O < 0.7 Pa and in air are shown in figure 5.7. The composition can be estimated from the peak areas after subtraction of the background and is shown in table 5.7 with the peak area of the oxygen peak set to 100. However, the peak areas are very dependent on the applied background making it easy to obtain different Al/Mg ratios between ~1.0-1.5. Therefore, it must be concluded that the composition more or less remains the same regardless of applied water vapour pressure.

76 Magnesium aluminate

a) 50 OMgAl

40 III I

II LEIS yield [counts/nC] 10

0 1400 1600 1800 2000 2200 2400 2600 2800 3000 Energy [eV]

b) 100 O Mg Al 80

60 I

LEIS yield [counts/nC] yield LEIS 20 II 0 1200 1400 1600 1800 2000 2200 2400 Energy [eV]

Figure 5.7: LEIS spectra of oxygen treated fracture surfaces of single crystals with a) (100) direction (5 keV) and b) (111) direction (4 keV). I = fractured at pH2O < 0.7 Pa, II = fractured in air and III = fractured at pH2O < 0.7 Pa and subsequently exposed to air for 45 minutes.

77 Chapter 5

Table 5.7: Relative composition of fracture surfaces of single crystal spinel Sample O Mg Al Al/Mg (100), < 0.7 Pa 100 118 149 1.26 (100), exposed to air 100 135 152 1.13 (100), air 100 117 125 1.07 (111), < 0.7 Pa 100 89 126 1.41 (111), air 100 103 100 0.97

It is noted that both magnesium and aluminium cations are present on the surface. This is remarkable, since under normal conditions magnesium cations in the tetrahedral interstices are not visible with LEIS [18]. On the other hand, computer simulations of (100) fracture surfaces indicate that a magnesium cation (in the tetrahedral interstice) terminated surface is the most stable [16]. This difference between simulation and experiment can be explained by the absence of the inversion process in the simulations. In this process, a magnesium cation in the tetrahedral interstice and an aluminium cation in the octahedral interstice swap their interstices in the lattice. In a single crystal up to 30 % of the magnesium cations can be in octahedral interstices [17], which can explain the presence of both cations. However, this cannot explain completely the Al/Mg ratio indicating that some magnesium in tetrahedral interstices must be visible on fracture surfaces for LEIS.

5.4.3 DRIFTS The stretching region of the hydroxyl groups on the surface of magnesium aluminate powder in the DRIFTS spectrum is shown in figure 5.8. The spectrum is similar to the reported spectra in literature [19-21], although it is more resolved due to outgassing under a helium flow in contrast to air used by the latter authors. Nevertheless, bulk water is still present on the sample, which is indicated by the broad peak at ~3300 cm-1 from the hydrogen bridges and the sharp small peak at ~3695 cm-1 of the hydroxyl-stretching of bulk water. The presence of bulk water can be explained by the outgassing temperature of 550 °C, since at that temperature only ~90 % of the bulk water is removed in 1 h [19].

The peak at ~3680 cm-1 is due to bridging hydroxyl groups on trivalent aluminium cations1. The sharp small peak at ~3790 cm-1 can be contributed to hydroxyl groups on an aluminium cation in a tetrahedral interstice proving the presence of inversion at the surface as indicated by the

1. The absorption region of the surface hydroxyl groups are taken from Busca et al. [21].

78 Magnesium aluminate

0.40

0.35

0.30

0.25

0.20

0.15

0.10

Absorbance [Kubelka-Munk] 0.05

0.00 4000 3800 3600 3400 3200 3000 2800 Wavenumbers [cm-1] Figure 5.8: DRIFTS spectrum of the hydroxyl-stretching region on magnesium aluminate.

LEIS experiments. The broad peak with shoulders between ~3700 and ~3780 cm-1 cannot be unambiguously assigned to aluminium cations in the tetrahedral interstice (~3800 - 3760 cm-1)/ octahedral interstice (~3750 - 3720 cm-1), to the magnesium cations in the octahedral or tetrahedral interstices (~3750 - 3700 cm-1) or to a combination of these possibilities.

5.5 Conclusions The fracture toughness of dense and porous magnesium aluminate ceramics has been investigated at ambient conditions. It is observed that the fracture toughness of dense magnesium aluminate is ~3.0 MPa m1/2, which is ~50 % higher than reported data in literature. This is due to the grain size, which is in this work 1.5-2 µm and in the literature more than 5 µm. The fracture toughness of porous magnesium aluminate is in the range of the reported data in literature.

Furthermore, the role of water on the fracture of magnesium aluminate spinel is investigated. The fracture toughness of polycrystalline magnesium aluminate, measured using the SENB technique, decreases with increasing humidity, showing the typical Type II behaviour indicating that adsorption plays an important role in the fracture process. The only other materials known to have this behaviour are hydroxyapatite and MnZn ferrite. However, porous magnesium aluminate shows a typical Type I behaviour indicating that microstructural aspects as e.g. pores have an influence too. The latter exact mechanism is still unknown.

79 Chapter 5

The subcritical crack growth along low-index planes of single crystal spinel is investigated using a SENB set-up. Comparing the decrease in fracture surface energy of ~46 % between 2 and 40 % relative humidity to the surface energy values obtained from computer simulations indicates that during fracture the surface is only partially hydrated.

The texture of the grains along a crack in magnesium aluminate is investigated using EBSD. The investigation of a non-conductive material with this technique is not straightforward (see appendix) and special precautions must be taken to use EBSD. EBSD revealed that the fracture is both trans- and intergranular. The grains along the crack do not show texture, which would have indicated a preference in fracture mode.

Although computer simulations predicted a magnesium terminated (100) surface, both magnesium and aluminium cations are found in the outermost layer of the fracture surface using LEIS. This difference can be partially attributed to the presence of inversion, which is absent in the simulations. DRIFTS measurements of hydroxyl groups on powders confirm the presence of inversion by the detection of hydroxyl groups on aluminium cations in a tetrahedral interstice. However, it is shown that the hydroxyl groups are at least present on the aluminium cations, but the presence on magnesium cations cannot be excluded.

5.6 Appendix: Fracture and EBSD An electron backscattered diffraction pattern (Kikuchi pattern) is created by focusing an electron beam on the surface of a specimen inclined with respect to the incoming beam. While strik- ing the sample, some electrons in a small volume are scattered on the atomic planes. These backscattered electrons collide with a phosphoric screen forming Kikuchi patterns, see figure 5.9. The reflection bands in these patterns correlate directly to the crystallographic planes in the diffracting volume. The images of these patterns are recorded and processed by a computer pro- gram obtaining the indexation of the reflection bands in the pattern and quality parameters like confidence index (how well could the pattern be indexed?) and image quality. Overall micro- textural information can be obtained by scanning the surface of the specimen point by point.

It is possible to study the fracture surfaces with the EBSD technique to get more information about the role of texture in the fracture process. However, there are some difficulties attached to such an investigation. Only a fraction of the grain facets are oriented adequately for obtaining

80 Magnesium aluminate

EPMA-chamber

Phosphoric screen e-

Camera Computer

specimen 70 °

Generation Detection Processing Figure 5.9: Schematic representation of an EBSD set-up. the Kikuchi diffraction patterns. This requires to reorient the specimen multiple times to get a good statistical result [22].

An easier approach is to determine the orientation of the grains along the crack normal to the crack surface and to compare these with the orientation of the other grains. Sharp cracks in conductive materials, like e.g. a titanium alloy [23], can be investigated easily in this way. Howev- er, for non conductive materials charging at the edges of the crack is throwing a spanner. This can be avoided by generating a relative wide crack, which is filled with a conductive material like e.g. colloidal graphite.

5.7 Acknowledgements B.G. Anderson (Eindhoven University of Technology) is acknowledged for the DRIFTS measurements and N. Lousberg (Eindhoven University of Technology) for the EBSD experiments. A. Knoester, A. Gildenpfenning, M. Viitanen and H.H. Brongersma (Calipso, Eindhoven Uni- versity of Technology) are acknowledged for the LEIS experiments and discussions. Further- more, H. Gorter (Institute of Applied Physics, TNO-TPD) is acknowledged for providing the Type B magnesium aluminate ceramics and A.J.M. van Dijk (Eindhoven University of Tech- nology) [24] for the Type A magnesium aluminate ceramic fracture experiments.

81 Chapter 5

5.8 References [1] T. Yano, Effects of neutron irradiation on the mechanical properties of magnesium aluminate spinel single crystals and polycrystals, J. Am. Ceram. Soc., 82 (1999), 3355-3364

[2] B. Hallstedt, Thermodynamic assessment of the system MgO-Al2O3, J. Am. Ceram. Soc. 75 (1992), 1497-1507

[3] R.W. Rice and C.CM. Wu, Slow crack growth in MgAl2O4 single- and polycrystals, J. Mater. Sci. Lett., 14 (1995), 723-727 [4] Custom Technical Ceramics Inc., Arvada, CO, USA [5] Calipso BV, Eindhoven, The Netherlands, see also www.calipso.nl [6] H.H. Brongersma, A. Gildenpfennig, A.W. Denier van der Gon, R.D. van de Grampel, W.P.A. Jansen, A. Knoester, J. Laven and M.M. Viitanen, Insight in the outside: New applications of low-energy ion scattering, Nucl. Instr. and Meth. in Phys. Rev. B, 190 (2002), 11-18 [7] C. Baudín, R. Martínez and P. Pena, High temperature mechanical behaviour of stoichiometric magnesium spinel, J. Am. Ceram. Soc., 78 (1995), 1857-1862

[8] R.L. Stewart and R.C. Bradt, Fracture of polycrystalline MgAl2O4, J. Am. Ceram. Soc., 63 (1980), 619-623 [9] A. Ghosh, K.W. White, M.G. Jenkins, A.S. Kobayashi and R.C. Bradt, Fracture resistance of a transparent magnesium aluminate spinel, J. Am. Ceram. Soc., 74 (1991), 1624- 1630 [10] K.W. White and G. P. Kelkar, Fracture mechanisms of a course-grained, transparent MgAl2O4 at elevated temperatures, J. Am. Ceram. Soc., 75 (1992), 3440-3444

[11] R.L. Stewart and R.C. Bradt, Fracture of single crystal MgAl2O4, J. Mater. Sci., 15 (1980), 67-72 [12] S. Bhaduri and S.B. Bhaduri, Microstructural and mechanical properties of nanocrystalline spinel and related properties, Ceram. Intern., 28 (2002), 153-158 [13] M.A.H. Donners, L.J.M.G. Dortmans and G. de With, Adsorption and kinetic effects on MnZn ferrites, J. Mater. Res., 15 (2000), 1377-1388 [14] N.J. van der Laag, A.J.M. van Dijk, L.J.M.G. Dortmans and G. de With, The role of water adsorption on the fracture of ceramic oxides, J. Eur. Ceram. Soc., (submitted) [15] J.F. Nye, Physical properties of crystals, 2nd Ed., Clarendon Press, Oxford, UK (1990) [16] C.M. Fang, G. de With and S.C. Parker, Computer simulation of dissociative adsorption of water on the surfaces of spinel MgAl2O4, Phys. Rev. B, (accepted) [17] K.E. Sickafus, J.M. Wills and N.W. Grimes, Structure of spinel, J. Am. Ceram. Soc., 82 (1999), 3279-3292 [18] J.-P. Jacobs, A. Maltha, J.G.H. Reintjes, J. Drimal, V. Ponec and H.H. Brongersma, The surface of catalytically active spinels, J. Catal., 147 (1994), 294-300 [19] C. Morterra, G. Ghiotti, F. Boccuzzi and S. Coluccia, An infrared spectroscopic investigation of the surface properties of magnesium aluminate spinel, J. Catal., 51 (1978), 299-313

82 Magnesium aluminate

[20] P.F. Rossi, G. Busca, V. Lorenzelli, M. Waqif, O. Saur and J.-C. Lavalley, Surface basic- ity of mixed oxides: magnesium and zinc aluminates, Langmuir, 7 (1991), 2677-2681 [21] G. Busca, V. Lorenzelli, G. Ramis and R.J. Willey, Surface sites on spinel-type and corundum-type metal oxide powders, Langmuir, 9 (1993), 1492-1499 [22] D.P. Field, Recent advances in the application of orientation imaging, Ultramicroscopy, 67 (1997), 1-9 [23] S.I. Wright and D.P. Field, Recent studies of local texture and its influence on failure, Mater. Sci. Eng., A257 (1998), 165-170 [24] A.J.M. van Dijk, The influence of loading rate and environment on brittle fracture of ceramics, M.Sc. Thesis, Eindhoven University of Technology, Eindhoven, The Nether- lands, (2001)

83 Chapter 5

84 - 6 -

Structural, mechanical, thermophysical and dielectric properties of zinc aluminate

(ZnAl2O4)

6.1 Introduction Zinc aluminate (ZnAl2O4) is a member of the spinel family, whose natural occurrence is the mineral gahnite. It was discovered near Falun (Sweden) in 1807 and was originally called au- tomolite, but later renamed after its discoverer the Swedish chemist J.G. Gahn (1745-1818) [1]. The gahnite crystals occur in an octahedral, dodecahedral or cubic shape and have a blackish, greenish appearance. In general, the crystals have magnesium, iron and - in rare cases- manga- nese impurities. Gahnite is mainly found in granitic pegmatites, but also in contact altered lime- stone and metasomatic replacement veins and ores. The major localities of gahnite are the replacement ores in Franklin, NJ, USA and Falun and in the metasomatic deposit in Western Australia [2].

At present, zinc aluminate is used as a catalyst for the dehydration of saturated alcohols to ole- fins [3], methanol and higher alcohol synthesis [4,5], preparation of polymethylbenzenes [6], synthesis of styrenes from acetophenons [7] and double bond isomerisation of alkenes [8]. Fur- thermore, zinc aluminate can also be used as a catalyst support, since it has a high thermal stability, low acidity and a hydrophobic behaviour. Moreover, it has a strong metal-support interaction preventing e.g. platinum and platinum/tin to sinter [9]. Finally, zinc aluminate can

85 Chapter 6 be used as a second phase in glaze layers of white ceramic tiles to improve wear resistance and mechanical properties and to preserve whiteness [10].

Zinc aluminate is normally synthesised via a solid-state reaction of zinc and aluminium oxides above 800 °C [11,12] or via coprecipitation [9,14], hydrothermal [13] and sol-gel methods with several organic precursors [12,14-16]. It could be used as a ceramic material like e.g. magnesium aluminate (MgAl2O4) and manganese-zinc ferrites ((Mn,Zn)Fe2O4). However, in literature only a few papers have been published about the sintering of zinc aluminate [11,17-19], implying many unknown bulk properties like e.g. Young’s modulus. In this chapter, it is attempted to manufacture dense zinc aluminate ceramics and to determine some of its bulk properties. Fur- thermore, since cation inversion in spinels can influence the properties, the effect of the preparation method of the zinc aluminate powders on this inversion is investigated.

6.2 Crystallographic aspects of zinc aluminate The crystal structure of spinel stricto sensu (MgAl2O4) has been determined independently by 7 Bragg [20] and Nishikawa [21] in 1915. It has the space group Fd3m (Oh ; number 227 in the

International Tables [22]) and has a cubic structure made of eight molecular units (AB2O4).

There are three main groups of compounds with the spinel structure and the chemical formula

AB2O4: II-III spinels, IV-II spinels and defect-spinels. In the first group the A cation is divalent and the B cations trivalent. Magnesium aluminate and zinc aluminate are examples of this group. In the second group the A cation is tetravalent en the B cations divalent. Ülvospinel

(Fe2TiO4) is a well-known member of this group. Spinels from the third group have vacancies on locations where cations should be. A well-known example is γ-alumina.

One unit cell of a compound with the spinel structure is built up from 32 oxygen atoms arranged in a fcc-lattice, giving 64 tetrahedral and 32 octahedral interstices. In a II-III spinel the divalent A cations occupy 8 tetrahedral interstices and the trivalent B cations occupy 16 octahedral interstices. This distribution of the cations is designated as normal. However, this is thermody- namically not the most stable situation, since the configurational entropy counteracts the site preference energy. Therefore, A and B cations interchange interstices via diffusion, eventually leading to the situation where all the A cations are in octahedral interstices. The latter extreme situation is designated as inverted.

All distributions between the two extremes are possible and the degree of inversion is given by the inversion parameter x, which is defined as the fraction of A cations in octahedral sites. The

86 Zinc aluminate (bulk) inversion parameter x can be obtained via careful single crystal X-ray diffraction or via nuclear magnetic resonance with magic angle spinning (MAS-NMR). In the latter case, x can be obtained for aluminates from the peak areas by:

2 x = ------, (6.1) 1Al+ ()VI ⁄ AlIV where AlVI is the peak area of aluminium cations in an octahedral surrounding (~0-10 ppm) and AlIV of the aluminium cations in a tetrahedral surrounding (~60-70 ppm).

The inversion in zinc aluminate is known, by X-ray diffraction [26-28], to be small in contrast to e.g. magnesium aluminate [29]. Moreover, in computer simulations the inversion of zinc and aluminium atoms is also found to be energetically unfavourable [30,31]. Furthermore, NMR studies to the cation inversion in zinc aluminate, which is prepared via the solid state synthesis route, confirm the virtual absence of inversion [32]. However, to some extent, inversion has been found in zinc aluminate synthesized using a sol-gel method with alkoxides depending on the temperature [16].

6.3 Instrumentation The structural properties of zinc aluminate powders were investigated by powder X-ray diffrac- α tion (XRD) on a Rigaku Geigerflex using CuK 1 radiation. Furthermore, one-dimensional solid state 27Al MAS-NMR spectra were measured with a ~6 mg sample for quantitative comparison on a Bruker DMX500 spectrometer operating at an 27Al NMR frequency of 130.32 MHz using a spinning speed of 30 kHz. Rotor-synchronised echoes were recorded using the two-pulse se- τ τ τ quence p1- -p2- -FID (Hahn spin-echo pulse [23,24]) with a dwell time ( ) of 1 µs and opti- mised pulse lengths p1 and p2 of 1.2 and 2.4 µs, respectively. A delay time of 10 s was applied between each of the 128 scans. The dielectric properties were determined on a HP 4284A LCR meter and the elastic properties with a Panametrics model 25DL, using the pulse-echo method [25]. The thermophysical properties were determined on a Compotherm XP/20X (diffusivity) and Differential Scanning Calorimeter (DSC) of Perkin Elmer model Pyris 1 (heat capacity).

87 Chapter 6

6.4 Preparation

6.4.1 Powder synthesis Zinc aluminate was made via three different routes: solid state synthesis, coprecipitation and a sol-gel method. In the solid state synthesis route equimolar amounts of zinc oxide (Merck, >99%) and γ-alumina (Degussa) with a total weight of 90 g were put into a plastic container. Distilled water was added and the mixture was mixed overnight. After evaporating most of the water, the mixture was dried at 110 °C for 2 h and, subsequently, the large agglomerates were were pulverized in an agate mortar. The mixture was divided in three batches, each put in an alumina crucible covered with an alumina lid in order to prevent contamination of the furnace with zinc. In separate experiments the batches were heated in air to 1000 ºC for 8 h, 800 ºC for 8 h and 12 h, respectively, with a heating rate of 10 ºC min-1. The batches were dried (without the lid) for 1 hour at 110 ºC prior to the heating.

In the coprecipitation method, described in [14], an equimolar amount of zinc nitrate and aluminium nitrate (both from Merck, >99 %) was dissolved in a citric acid solution of pH = 2 under stirring. A solution of 16 wt% ammonia nitrate (Merck, >99 %) was slowly added until the solution was neutral and a chelate was formed. The chelate was dried in air for 20 h and subsequently dried in a furnace at 50 ºC (68 h) and 110 ºC (20 h). The resulting powder was divided into three batches.

As in the solid state route, the batches were heated in air to 1000 ºC for 8 h, 800 ºC for 8 h and 12 h, respectively, with a heating rate of 10 ºC min-1. Although the batches were put into an alumina crucible, this time the lid was removed in order to let the nitrogenous fumes evaporate. The batches were dried at 110 ºC (1 h) and 300 °C (6 h) prior to calcining in order to remove the nitrates.

Zinc aluminate was synthesized using a sol-gel method based on the method described by Mon- rós et al. [33]. Aluminium and zinc chloride (Merck, >99 %) were dissolved in 100 ml water, resulting in an aluminium and zinc cation concentration of 0.6 M and 0.3 M, respectively. Fur- thermore, 10 g of gelatin (Merck, >99 %) is dissolved in 100 ml of water under stirring and heating to obtain a homogeneous solution. The warm gelatin solution was added to the cation solution under heating and stirring, until a homogeneous mixture was obtained, subsequently followed by cooling in a refrigerator (4 °C) for 30 min. The resulting gel was aged for 96 h at room temperature and dried at 80 °C for 40 h.

88 Zinc aluminate (bulk)

In order to burn out the gelatin, the dried gel was put in an alumina vessel and heated up to -1 400 °C at a rate of 3 °C min , and kept at that temperature for 7 h in a gas flow of 80 % N2/ -1 20 % O2. After cooling down at a rate of 3 °C min , the powder was ground and subsequently reheated at 3 °C min-1 to 800 °C. The zinc aluminate powder was kept at that temperature for

12 h in 80 % N2/20 % O2 gas mixture and, subsequently, cooled to room temperature at 3°Cmin-1.

In additional experiment this powder was reheated to 1000 °C at 10 °C min-1 and kept at that temperature for 8 h.

6.4.2 Compaction and sintering Zinc aluminate powder prepared via the solid state route at 1000 °C for 8 h was ball-milled for 24 h with a small amount of isopropanol. Subsequently, the powder was sieved in a copper sieve with a mesh of 50 µm. Finally to improve sintering, a small amount of distilled water was added to the powder, before it was stored in a closed plastic container at 70 °C for 24 h.

The powder was put in a steel die with a diameter of 12 or 15 mm, which was lubricated with a 90% petroleum ether-10% oleic acid mixture, followed by uniaxially pressing to 70 MPa or 150 MPa. The resulting tablets were put with some extra zinc aluminate powder in an alumina crucible closed with an alumina lid. The tablets were heated to different temperatures between 1200 and 1600 °C at 3 °C min-1 and kept at those temperatures for 8 h.

6.5 Results and discussion

6.5.1 Structural properties 6.5.1.1 XRD XRD measurements showed that zinc aluminate powder (JCPDS file 05-0669) has been formed (see figure 6.1). However, in all samples zinc oxide (JCPDS file 36-1451) was present, except those calcined at 1000 °C and prepared via the sol-gel method. This is in agreement with the results from Keller et al. [12], who found that at 1000 °C ~95 % zinc aluminate would be made. The presence of alumina (JCPDS file 47-1292) could not be detected in any sample, since the peak positions of alumina coincide with those of zinc aluminate.

6.5.1.2 Composition The zinc-aluminium ratio of the single phase zinc aluminate prepared via the solid state route, coprecipitation and the sol-gel method at 1000 °C is determined using atomic absorption spec-

89 Chapter 6

c I II 20 30 40 50 60 70 2θ [°] Figure 6.1: XRD spectra of zinc aluminate powders prepared via a) the sol-gel method (800 °C, 12 h), b) coprecipitation (800 °C, 12 h) and c) solid state route (1000 °C, 8 h). Fur- thermore, the JCPDS files of I) gahnite (05-0669) and II) zincite (36-1451) are shown.

troscopy (AAS). The composition is Zn0.95Al2O4, Zn0.83Al2O4 and Zn0.48Al2O4 (accuracy: ± 5%) for the solid state, coprecipitation and the sol-gel prepared zinc aluminate, respectively. The zinc-aluminium ratio of the zinc aluminate ceramics determined by energy dispersive X- ray (EDX) on polished surfaces is 0.99:2, consistent with the zinc deficient character of the starting powders.

The formation of zinc deficient zinc aluminate is due to the volatile nature of zinc oxide (zinc × -4 × -13 oxide, pZn =9 10 Pa at 1000 °C [34]; aluminium oxide, pAl =4 10 Pa [35]; magnesium × -8 oxide, pMg =4 10 Pa [35]). The large surface area and presence of a flow in the furnace, increases the deficiency for the zinc aluminate prepared via the sol-gel method as compared to the material prepared via the solid state route.

6.5.1.3 MAS-NMR The NMR spectra of the zinc aluminate powders prepared via the solid state and coprecipitation are shown in figures 6.2 and 6.3 respectively. The presence of a peak at ~60 ppm indicates that there is some inversion present in all the powders. However, the size of the peak indicates that the amount of inversion is very small. Kashii et al. [32] found the same results, but they kept zinc aluminate at 900 °C for 40 h in contrast to 800 °C for 8 h used in this investigation. Fur-

90 Zinc aluminate (bulk)

120 100 80 60 40 20 0 -20 -40 ppm Figure 6.2: 27Al MAS NMR spectra of zinc aluminate prepared following the solid state synthesis route at a) 1000 °C for 8 h, b) 800 °C for 8 h and c) 800 °C for 12 h.

120 100 80 60 40 20 0 -20 -40 ppm Figure 6.3: 27Al MAS NMR spectra of zinc aluminate prepared following the coprecipitation route at a) 1000 °C for 8 h, b) 800 °C for 8 h and c) 800 °C for 12 h.

91 Chapter 6

120 100 80 60 40 20 0 -20 -40 ppm Figure 6.4: 27Al MAS NMR spectra of zinc aluminate prepared by a) sol-gel synthesis and b) solid state synthesis for comparison. thermore, it is clear that there in no difference between the two preparation methods and that the temperature or the calcining time does not have a significant influence.

The NMR spectra of the zinc aluminate powder prepared by the sol-gel method are shown in figure 6.4 together with a spectrum of zinc aluminate prepared via the solid state synthesis route for comparison. In contrast to the other methods, the sol-gel method prepared zinc aluminate has a large degree of inversion. The normal degree of inversion, as obtained by the other methods, could not be reached after an additional reheating for 8 h at 1000 °C. This is in contrast with the results of Mathur et al. [16], who found a large degree of inversion at 600 °C, but a small one at 1000 °C.

The presence of a large inversion in the sol-gel prepared zinc aluminate could be attributed to the presence of aluminium cations with a four or five oxygen atom surroundings at the surface, since zinc aluminate prepared via the sol-gel route has a large surface area [36]. However, the NMR experiments were performed in a quantitative manner, revealing that all aluminium cations are detected and that no magnetic shielding occurred. This implies that the inversion is occurring in the bulk and is not an artefact of surface aluminium cations, since they are only a minor fraction of all the aluminium cations.

92 Zinc aluminate (bulk)

The NMR spectra have been measured on zinc deficient zinc aluminate, which implies an excess of aluminium cations forming aluminium oxide. The absence of second phase in XRD spectra suggests that a solid solution of the spinel phase of aluminium oxide (γ-alumina) and zinc aluminate has been formed.

However, in γ-alumina 25 % of the aluminium cations are located at tetrahedral interstices [37] and contribute to the peak at 60 ppm. In order to determine the inversion parameter of pure zinc aluminate from these spectra, a correction to equation 6.1 for this γ-alumina contribution must be made. The inversion parameter for zinc deficient zinc alumina (Zn1-yAl2O4) is given by:

2 1 x = ------– ---y . (6.2) 1Al+ ()VI ⁄ AlIV 2 The first term on the right hand side of equation 6.2, for all powders prepared via the solid state route and coprecipitation, equals 0.04 ±0.01 and for the sol-gel method 0.23 ±0.02. Application of the correction for the zinc aluminate prepared via the solid state route (y = 0.05) and sol-gel method (y = 0.52) gives an inversion parameter of 0.015 and -0.01, respectively. However, a systematic error is present in estimating the inversion parameter from MAS-NMR spectra due to the location of some intensity of in the spinning side bands and to baseline correction. The results obtained are within the error range concluding that MAS-NMR is not the proper method to determine the inversion parameter for zinc aluminate.

6.5.2 Mechanical properties The density of the sintered zinc aluminate tablets has been measured by the Archimedes’ method with water as the displaceable fluid for densities higher than 90 % and below 90 % by the measuring the dimensions and weight of the tablet. The density as a function of the sintering temperature is shown in figure 6.5. The density increases with sintering temperature and remains constant at 93 % from 1300 °C onwards.

Sidorov found that single phase zinc aluminate powder calcined at 1250 °C did not sinter without additives like TiO2 up to 1650 °C [17]. However, Hong et al. found that single phase powder calcined at 1300 °C did sinter, but reached at 1400 °C a maximum density of 80 % [11]. In this work, the zinc aluminate powder was calcined at 1000 °C, indicating that some zinc oxide and aluminium oxide is still present. These latter compounds react during sintering and can contribute to an extra densification. However, since the combined densities of zinc oxide

93 Chapter 6

100

Relative density [%] density Relative 60

50 1200 1300 14 00 1500 1600 Sinter temperature [°C]

Figure 6.5: Density as a function of sintering temperature of zinc aluminate uniaxially prepressed at 70 MPa.

(5.679 g cm-3) and aluminium oxide (3.990 g cm-3) is smaller that the density of zinc aluminate (4.611 g cm-3), this reaction will leave pores and therefore a 100% density cannot be reached.

The Young’s modulus has been measured using a pulse-echo method. The Young’s modulus as a function of density is shown in figure 6.6. The absence of experimental data above the density of 95 % makes it impossible to predict an accurate value for the zero porosity Young’s modulus. However, an estimation can be made by linear extrapolation, yielding 242 GPa.

6.5.3 Thermophysical properties 6.5.3.1 Heat capacity The heat capacity of zinc aluminate powder has been measured using differential scanning ca- lorimetry between 15 °C and 100 °C on zinc aluminate powder reheated up to 1400 °C for 8 h and is shown in figure 6.7.

94 Zinc aluminate (bulk)

240

230

220

210 Young's modulus [GPa] modulus Young's

200 130 120 110 70 75 90 92 94 96 Relative density [%] Figure 6.6: Young’s modulus of zinc aluminate as a function of density.

150

140 ]

-1 130 K -1

120 [J mol p C

110

100 280 300 320 340 360 380 400 Temperature [K] Figure 6.7: Heat capacity at constant pressure of zinc aluminate. The step character of the graph originates from the data acquisition during measuring.

95 Chapter 6

6.5.3.2 Thermal diffusivity The thermal diffusivity (a) of sintered zinc aluminate ceramics has been determined using the flash method on a tablet (1600 °C, 70 MPa) with a thickness of 1.37 mm and a density of 91 % using the flash method [38]. The results are given in table 6.1.

Thermal conduction in ceramics predominantly takes place via lattice vibrations (i.e. phonon conductivity). It is being determined by lattice characteristics (intrinsic properties) and defects as impurities, grain boundaries and pores (extrinsic properties).

>ΘΘ ⁄ At sufficiently high temperature, T Dr b , where Dr is the reduced Debye temperature Θ ≈ ( D) and b is a constant ( 2), the inverse of the thermal diffusivity is given by:

1 ------bA A ′ ′ --- ==Θ TB+ – --- A TB+ , (6.3) a Dr 2 where A is related to the phonon-phonon scattering processes (intrinsic lattice diffusivity), B is Θ related to the phonon-scattering processes due to impurities, grain boundaries, etc., and Dr is Θ ⁄ 3 the reduced Debye temperature defined as D n , where n is the number of atoms per primi- tive unit cell (n = 14 for zinc aluminate) [39].

In figure 6.8 the graph of the inverse thermal diffusivity vs. absolute temperature is shown for a zinc aluminate with a 91 % density. The slope A′ is 580.6 ± 18.1 s m-2 K-1 and B′ is 38.1 ± 7.36×103 sm-2 (R = 0.9981).

For the determination of the maximum thermal diffusivity (B = 0) [39], the intercept at the in- Θ verse thermal diffusivity axis (= A/2) or the intercept at the temperature axis (=Dr /2b) should ′ Θ be known in combination with the slope A (= bA/Dr ). The estimated maximum thermal diffu-

Table 6.1: Thermal diffusivity of zinc aluminate Temperature a Error [K] [10-4 m2 s-1][%] 296.15 0.04766 5 338.15 0.04220 5 378.15 0.03893 3 423.15 0.03550 3 469.15 0.03259 3 503.15 0.02992 3

96 Zinc aluminate (bulk)

360

340

320

300 ] -2 280

s m 3 260 [10 -1 a 240

220

200 250 300 350 400 450 500 550 Temperature [K]

Figure 6.8: Inverse thermal diffusivity vs. absolute temperature for zinc aluminate ceramics.

-4 2 -1 Θ sivity at 300 K is 0.08318×10 m s , using D = 896 K as calculated from the phonon spectrum [40] using n=14 and b = 2.

6.5.3.3 Thermal conductivity The thermal conductivity (κ) can be obtained by:

κ ρ = a Cp , (6.4) ρ -3 -3 where is the density [mol m ], which is for zinc aluminate 25141.8 mol m , and Cp is the heat capacity at constant pressure [J mol-1 K-1]. The thermal conductivity between 20 °C and 100 °C is shown in 6.9. The maximum achievable thermal conductivity at 300 K is about 26 W m1 K-1 as compared with the measured value of 14.7 W m-1 K-1. This large difference can be explained by the large porosity, but such a difference is not uncommon: e.g. ~28 and ~23 W m-1 K-1 for the maximum achievable and measured thermal conductivity, respectively, of fully dense MgSiN2 ceramics containing impurities [39].

6.5.4 Dielectric properties Two 0.5 mm thick parallel plate capacitors have been prepared from a tablet (1400 °C, 150 MPa) with a relative density of 90 %. On these capacitors, round silver electrode areas of 12.566 mm2 have been evaporated. Since the capacitors are not made of a fully dense polycrys-

97 Chapter 6

16 ] -1 K -1 15

13 Thermal conductivity [W m

12 280 300 320 340 360 380 400 Temperature [K]

Figure 6.9: Thermal conductivity of zinc aluminate.

′ talline material, the real part of the measured dielectric constants (k meas) is related to the real part of the single crystal dielectric constant (k′) by:

3 – ρ k' = ------k' , (6.5) 2ρ meas where ρ is the relative density [41].

The results are shown in table 6.2. An accuracy of 10 % is reached.

The polarizability (α) is the ability of ions or atoms to deform under an externally applied elec- tric field and consists of space-charge, dipole, ionic and electronic components. The dielectric

Table 6.2: Dielectric constants of zinc aluminate ′ ′ Frequency k meas k [MHz] [-] [-] ′ k 0 0.1 10.4 11.94 k′ 1 9.23 10.60 tan δ 0.01815 k′∞ 10 9.46 10.86

98 Zinc aluminate (bulk)

α polarizability ( D) is, on a microscopic level, related to the experimentally determined dielectric constant by the Clausius-Mosotti equation:

1 ()k′ – 1 α = --- V ------, (6.6) D b m()k′ + 2 π 3 ′ where b is defined as 4 /3, Vm the molar volume in Å and k is the real part of the complex dielectric constant, which is measured between 1 kHz and 10 MHz [42-44]. The dielectric polarizability is in this range only composed of the ionic and electronic components, where the electronic component itself is related to the refractive index n by the Lorenz-Lorentz equation [45,46]:

()2 α 1 n – 1 e = --- Vm------. (6.7) b ()n2 + 2

The dipole and space-charge components are not contributing to the polarizability at these frequencies. These mechanisms have a large relaxation time as compared to the frequencies used.

The dielectric polarizability of zinc aluminate calculated from k′ = 10.60 is 12.06 Å3 3 (Vm = 66.08 Å ). The electronic polarizability, calculated from refractive index n = 1.7725 [47] is 6.57 Å3, giving 5.49 Å3 for the ionic polarizability.

The concept of additivity of polarizabilities is the assumption that the polarizability of a complex compound is the sum of the polarizabilities of the simpler compounds. It has been applied to both electronic and dielectric polarizabilities (see e.g. [48]). The additivity rule for oxide compounds with the spinel structure is then given by:

α()α()α() AB2O4 = AO + B2O3 (6.8) The results for the application of the oxide additivity rule to magnesium aluminate and zinc aluminate is given in table 6.3. The difference between the measured value and the predicted value of the dielectric polarizability is 2.5 %. This is slightly larger than the typical values of 0.5- 1.0 % found for other aluminates, beryllates, borates, gallates, silicates and phosphates (references in [50]), but the large experimental error should be taken into account. However, there are several explanations for the higher estimates of the dielectric polarizability in spinels. Firstly, the discrepancy can originate from the inaccuracy in measuring or the presence of impurities with a high dielectric constant, like e.g. water molecules. Secondly, in analogy to magnesium aluminate by Shannon and Rossman [50], there exists an imbalance between the "rattling" alu-

99 Chapter 6

Table 6.3: Comparison between measured and calculated dielectric polarizabilities ′ α α ∆ Compound k Vm D(exp) D(calc) Ref. [-] [Å3][Å3][Å3][%] MgO 9.83 18.67 3.33 - - [49] ZnO 9.30 23.55 4.13 - - [49]

Al2O3 10.12 42.45 7.63 - - [49] MgAl2O4 8.18 65.93 11.10 10.96 +1.3 [48] ZnAl2O4 10.60 66.08 12.06 11.76 +2.5 This work minium cations in the octahedral interstices with a valence somewhat less than 3.0 (= 2.93 [51]) and "compressed" zinc cations in the tetrahedral interstices with a valence somewhat higher than 2.0 (= 2.11 [51]). Finally, for completeness, Grimes [52] indicates that at low frequencies, due to inversion of the cations, the anti-ferroelectric array of dipole moments is unbalanced in combination with a crystallographic reorientation from Fd3m to F43m, which affects the overall polarizability.

6.6 Conclusions The structural, mechanical, thermophysical and dielectric properties of zinc aluminate ceramics have been investigated. Zinc aluminate powders have been prepared at different temperatures, with different sintering times and via different preparation routes (solid state, coprecipitation and sol-gel). It has been revealed that zinc deficient zinc aluminate has been made, which is due to the volatile nature of zinc oxide during calcining. Furthermore, it is established that MAS- NMR is not the proper method for determining the inversion parameter in zinc aluminate.

Zinc aluminate ceramics have been made by sintering uniaxially prepressed tablets at temperatures above 1200 °C. The maximum density of 93 % is reached at 1300 °C and remains constant for higher sintering temperatures. The dielectric constant of zinc aluminate ceramic is 10.60 and its polarizability is 12.06 Å3. The prediction of this value from the polarizabilities of zinc and aluminium oxides using the oxide additivity rule gave a larger than typical difference of 2.5 %.

The thermal conductivity of zinc aluminate is determined from the measured heat capacity (124 J mol-1 K-1) and thermal diffusivity (0.04711×10-4 m2 s-1), giving a value of 14.7 W m-1 K-1. The maximum achievable thermal conductivity for pore and impurity free zinc aluminate ceramics can be estimated at 26 W m-1 K-1.

100 Zinc aluminate (bulk)

6.7 Acknowledgements M.M.R.M Hendrix (Eindhoven University of Technology, The Netherlands) is acknowledged for the XRD measurements, S. Tappe (RWTH Aachen University of Technology, Germany) for the dielectric measurements, E.M. van Oers and P.C.M.M. Magusin (Eindhoven University of Technology, The Netherlands) for the NMR experiments, B. Norder (Delft University of Tech- nology) for the heat capacity measurements. Furthermore, M.D. Snel (Eindhoven University of Technology) is acknowledged for the synthesis of the sol-gel zinc aluminate [53].

6.8 References [1] in: Efemeriden der Berg- und Hüttenkunde, Ed. C.E. Freiherr von Moll, Nürnberg, Ger- many, (1807) [2] in: Rock forming minerals, Vol 5. Non-silicates, Eds: W.A. Deer, R.A. Howie and J. Zussman, Longmans, London, UK, (1962) [3] T.K. Shioyama, Alcohol dehydration employing a zinc aluminate catalyst, U.S. Patent 4.260.845,(1981) [4] F. Le Pelier, P. Chaumette, J. Saussey, M.M. Bettahar and J.C. Lavalley, Mol. Catal. A: Chemical., 122 (1997), 131-139 [5] R. Szymansky, Ch. Travers, P. Chaumette, Ph. Courty and D. Durand, in: Studies in surface science and catalysis, vol 31, Eds: B. Delmon, P. Grange P.A. Jacobs and G. Ponce- let, Elsevier, Amsterdam, (1987), 739-748 [6] L.R. Cobb, Preparation of polymethylbenzenes, U.S. Patent 4.568.784, (1985) [7] R. Roesky, J. Weiguny, H. Bestgen and U. Dingerdissen, Appl. Catal. A: General, 176 (1999), 213-220 [8] M.B. Welch, Zinc aluminate double bond isomerization catalyst and process for its pro- duction, U.S. Patent 4.692.430, (1986) [9] G. Aguilar-Ríos, M. Valenzuela, P. Salas, H. Armendáriz, P. Bosch, G. del Toro, R. Silva, V. Bertín, S. Castillo, A. Ramírez-Solís and I. Schifter, Hydrogen interactions and catalytic properties of platinum-tin supported on zinc aluminate, Appl. Catal. A: General, 127 (1995), 65-75 [10] A. Escardino, J.L. Amorós, A. Gozalbo, M.J. Orts and A. Moreno, Gahnite devitrifica- tion in ceramic frits: mechanism and kinetics, J. Am. Ceram. Soc., 83 (2000), 2938-2944 [11] W.-S. Hong, L.C. De Jonghe, X. Yang and M.N. Rahaman, Reaction sintering of ZnO- Al2O3, J. Am. Ceram. Soc., 78 (1995), 3217-3224

[12] J.T. Keller, D.K. Agrawal and H.A. McKinstry, Quantitative XRD studies of ZnAl2O4 (Spinel) synthesized by sol-gel and powder methods, Adv. Ceram. Mater., 3 (1988), 420- 422 [13] M. Zawadzki and J. Wrzyszcz, Hydrothermal synthesis of nanoporous zinc aluminate with high surface area, Mater. Res. Bull., 35 (2000), 109-114

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[14] M.A. Valenzuela, J.P. Jacobs, P. Bosch, S. Reije, B. Zapata and H.H. Brongersma, The influence of the preparation method on the surface structure of ZnAl2O4, Appl. Catal. A: General, 148 (1997), 315-324 [15] L.K. Kurihara and S.L. Suib, Sol-gel synthesis of ternary metal oxides. 1. Synthesis and characterization of MAl2O4 (M= Mg, Ni, Co, Cu, Fe, Zn, Mn, Cd, Ca, Hg, Sr, and Ba) and Pb2Al2O5, Chem. Mater., 5 (1993), 609-613 [16] S. Mathur, M. Veith, M. Haas, H. Shen, N. Lecerf, V. Huch, S. Hüfner, R. Haberkorn, H.P. Beck and M. Jilavi, Single source sol-gel synthesis of nanocrystalline ZnAl2O4: structural and optical properties, J. Am. Ceram. Soc., 84 (2001), 1921-1928 [17] N.A. Sidorov, The sintering of gahnite (zinc aluminate), Trudy Khar’kov. Politekh. Inst., 17 (1958), 251-255 [18] I.S. Kainarskii and N.A. Sidorov, Dense porous and alkali-resistant ceramics from gahnite, Ser. Khim.-Tekhnol. 13 (1957), 165-171 [19] I.S. Kainarskii and N.A. Sidorov, Gahnite and its refractory properties, Ogneupory, 23 (1958), 19-23 [20] W.H. Bragg, The structure of the spinel group of crystals, Phil. Mag., 30 (1915), 305-315 [21] S. Nishikawa, Structure of some crystals of the spinel group, Proc. Math. Phys. Soc. Tokyo, 8 (1915), 199-209 [22] in: International tables for X-ray crystallography, vol. 1, Eds: N.F.M. Henry and K. Lonsdale, Kynoch Press, Birmingham, UK, (1952) [23] E.L. Hahn, Spin echoes, Phys. Rev., 80 (1950), 580-594 [24] H.Y. Carr and E.M. Purcell, Effects of diffusion on the free precession in nuclear magnetic resonance experiments, Phys. Rev., 94 (1954), 630-638 [25] L.C. Lynnworth, Ultrasonic measurements for process control, 1st Ed., Academic Press, London, UK, (1989)

[26] R.F. Cooley and J.S. Reed, Equilibrium cation distribution in NiAl2O4, CuAl2O4 and ZnAl2O4 spinels, J. Am. Ceram. Soc., 55 (1972), 395-398 [27] H.St.C. O’Neill and W.A. Dollase, Crystal structures and cation distributions in simple spinels from powder XRD structural refinements: MgCr2O4, ZnCr2O4, Fe3O4 and the temperature dependence of the cation distribution in ZnAl2O4, Phys. Chem. Minerals, 20 (1994), 541-555 [28] S. Lucchesi, A. Della Guista and U. Russo, Cation distribution in natural Zn-aluminate spinels, Miner. Mag., 62 (1998), 41-54 [29] R.J. Hill, J.R. Craig and G.V. Gibbs, Systematics of the spinel structure type, Phys. Chem. Miner., 4 (1979), 317-339 [30] A.N. Cormack, G.V. Lewis, S.C. Parker and C.R.A. Catlow, On the cation distribution of spinels, J. Phys. Chem. Solids, 49 (1988), 53-57 [31] R.W. Grimes, A.B. Anderson and A.H. Heuer, Predictions of cation distributions in AB2O4 spinels from normalized ion energies, J. Am. Chem. Soc., 111 (1989), 1-7

[32] N. Kashii, H. Maekawa and Y. Hinatsu, Dynamics of cation mixing of MgAl2O4 and ZnAl2O4 spinel, J. Am. Ceram. Soc., 82 (1999), 1844-1848

102 Zinc aluminate (bulk)

[33] G. Monrós, J. Carda, M.A. Tena, P.Escribando, J. Badenes and E.Cordoncillo, Spinels from gelatin-protected gels, J. Mater. Chem., 5 (1995), 85-90 [34] K. Kodera, I. Kusunoki and S. Shimizu, Dissociation pressures of various mettalic oxides, Bull. Chem. Soc. Jpn., 41 (1968), 1039-1045 [35] V.L.K. Lou, T.E. Mitchell and A.H. Heuer, Graphical displays of the thermodynamics of high-temperature gas-solid reactions and their application to oxidation of metals and evaporation of oxides, J. Am. Ceram. Soc., 68 (1985), 49-58 [36] M.A. Valenzuela, P. Bosch, G. Aguilar-Rios, A. Montoya and A. Schifter, Comparison between sol-gel, coprecipitation and wet mixing synthesis of ZnAl2O4, J. Sol-Gel Sci. Technol., 8 (1997), 107-110 [37] I. Levin and D. Brandon, Metastable alumina polymorphs: crystal structures and transition sequences, J. Am. Ceram. Soc., 81 (1998), 1995-2012 [38] W.J.Parker, R.J. Jenkins, C.P. Butler and G.L. Abbott, Flash method of determining thermal diffusivity, heat capacity and thermal conductivity, J. Appl. Phys., 32 (1961), 1679- 1684 [39] R.J. Bruls, H.T. Hintzen and R. Metselaar, A new estimation method for the intrinsic thermal diffusivity/conductivity of non-metallic compounds: a case study for MgSiN2, β AlN and -Si3N4 ceramics, J. Am. Ceram. Soc., (submitted)

[40] C.M. Fang, C.-K. Loong, G.A. de Wijs and G. de With, Phonon spectrum of ZnAl2O4 spinel from in elastic neutron scattering and first-principles calculations, Phys. Rev. B., 66 (2002), 144301 [41] M.L. Dunn, Effects of grain shape anisotropy, porosity, and microcracks on the elastic and dielectric constants of polycrystalline piezoelectric ceramics, J. Appl. Phys., 78 (1995), 1533-1541 [42] S. Roberts, Dielectric constants and polarizabilities of ions in simple crystals and barium titanate, Phys. Rev., 76 (1949), 1215-1220 [43] S. Roberts, A theory of dielectric polarization in alkali-halide crystals, Phys. Rev., 77 (1950), 258-263 [44] S. Roberts, Polarizabilities of ions in perovskite-type crystals, Phys. Rev., 81 (1951), 865-868 [45] H.A. Lorenz, Über die Beziehung zwischen der Fortpflanzungsgeschwindigkeit de Lich- tes und die Körperdichte, Ann. Phys. Chem., 9 (1880), 641-645 [46] L. Lorentz, Über die Refraktionskonstanten, Ann. Phys. Chem., 11 (1880), 70-103 [47] O. Medenbach and R.D. Shannon, Refractive indices and optical dispersion of 103 synthetic and mineral oxides and silicates measured by a small-prism technique, J. Opt. Soc. Am. B, 14 (1997), 3299-3318 [48] R.D. Shannon and M.A. Subramanian, Dielectric constants of chrysoberyl, spinel, phen- acite and forsterite and the oxide additivity rule, Phys. Chem. Minerals, 16 (1989), 747- 751 [49] R.D. Shannon, Dielectric Polarizabilities of ions in oxides and fluorides, J. Appl. Phys., 73 (1993), 348- 366

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[50] R.D. Shannon and G.R. Rossman, Dielectric constant of MgAl2O4 spinel and the oxide additivity rule, J. Phys. Chem. Solids, 52 (1991), 1055-1059 [51] I.D. Brown and D. Altermatt, Bond-valence parameters obtained from a systematic analysis of the inorganic crystal structure data base, Acta. Cryst., B41 (1985), 244-247 [52] N.W. Grimes, Dielectric constants and the oxide additivity rule - comments on a recent investigation of MgAl2O4 spinel., J. Phys.: Condens. Matter, 4 (1992), L567-L570

[53] M.D. Snel, Preparation of MnxZn1-xAl2O4 via solid-state and gelatin-protected route, Internship Report, Eindhoven University of Technology, Eindhoven, The Netherlands, (2001)

104 - 7 -

Computational investigation of bulk and surface properties of zinc aluminate

(ZnAl2O4)

7.1 Introduction Zinc aluminate (ZnAl2O4, also known as gahnite or zinc spinel) is a member of the spinel family and is being used as a catalyst and as an addition to glaze layers for floor tile and dental applications. It has the same crystallographic structure as the natural occurring mineral spinel

(MgAl2O4) having the space group Fd3m. It has a cubic structure made out of eight molecular units (ZnAl2O4), where the 32 oxygen atoms form a fcc-lattice with 8 zinc cations located in the tetrahedral holes and 16 aluminium cations located in the octahedral holes.

Zinc aluminate is from computational point of view a very interesting compound. The spinel family is a model material for the ternary oxides, which have been investigated much less than the binary oxides like MgO and Al2O3. However, the natural occurring process of inversion, where some bivalent and trivalent cations swap positions, affects the physical properties, but is generally not taken into account during computational investigations. The inversion process is almost absent in zinc aluminate due to the very large positive octahedral site preference energy of zinc cations (~+32 kJ for zinc and e.g. ~+4 kJ for magnesium) [1]. The absence of inversion makes it possible to compare computational results directly with experimental data.

105 Chapter 7

Several properties of zinc aluminate have been studied using different computational methods: cation distribution [2-4], structure [4,5], vibrational spectra [4], electronic structure [6], mechanical properties [5,7] and phonon spectra [7]. In this chapter, the surface properties of low- index planes of zinc aluminate are studied using computational methods. Furthermore, the properties of hydrated surfaces are experimentally investigated with Diffuse Reflectance Infrared Fourier Transform Spectroscopy (DRIFTS) and compared to previously reported surface investigations.

7.2 Experimental Zinc aluminate powder was made via a solid state synthesis route. Equimolar amounts of zinc oxide (from Merck) and γ-alumina (Degussa) were mixed overnight in some distilled water in a plastic container. After evaporation of the water, the mixture was dried at 110 °C for 2 h and, subsequently, pulverised in an agate mortar. The mixture was put in an alumina crucible with an alumina lid and calcined in air at 1000 °C for 8 h with a heating rate of 10 °C min-1. The resulting powder was determined to be single phase zinc aluminate using powder X-ray measure- α ments (Rigaku Geigerflux using CuK 1 radiation) and had, as determined by atomic adsorption spectroscopy (AAS), a zinc deficient composition: Zn0.95Al2O4. This zinc deficiency should be taken into account when comparing between experimental and computational results. The zinc aluminate powder for DRIFTS measurements was put in an in-situ chamber of a SpectraTech Collector DRIFT cell in a Nicolet Protége 460 spectrometer with a MCT detector. The powder was outgassed at 450 °C for 2 h in flowing helium gas (10 ml min-1) prior to measuring the spectrum.

7.3 Computational methods

7.3.1 Potentials The surfaces of zinc aluminate for all possibilities along the low-index directions are investigated using the computer code Metadise [8], which is designed for simulation of dislocations, surfaces and interfaces. Within Metadise, the zinc aluminate crystal is considered as a cell which is periodic in two dimensions and with the surface in the remaining third direction. The cell is divided in two regions according to the approach of Tasker [9]: a near-surface region with a thickness of two unit cells containing the atoms near the surface and the bulk region with a thickness of ten unit cells containing the remainder of the atoms. The atoms in the near-surface region are allowed to relax via free-energy minimisation, while the atoms in the bulk region are fixed to the equilibrium positions.

106 Zinc aluminate (surface)

Table 7.1: Potential parameters Ions Charges Core-shell interaction Core Shell [eV Å-2] Zn +2.0 Al +3.0 O +1.0 -3.0 78.30 Ion Pair Buckingham potential A [eV] ρ [eV] C [eV Å6] Al-O 1725.02 0.2897 0.0 Zn-O 529.70 0.3581 0.0 O-O 22764.3 0.1490 27.88

Full valence charges and a Buckingham short-range potential were used as the electrostatic and short-range interactions, respectively, to model the forces between the ions i and j:

q q r C V = ------i -j + A exp –-----ij- – ------ij , (7.1) ij ij ρ 6 r ij r where qi and qj are the charges of ions i and j respectively, rij is the distance between the ions ρ and Aij, ij and Cij are adjustable parameters to describe the potential. The oxygen polarizability was included via the shell model using the potential parameters derived by Lewis and Catlow [10]. The cut-off distance of the static-interaction between the ions employed in calculations is 13 Å. The parameters used in the simulations are shown in table 7.1.

7.3.2 Surface energy The surface energy (γ) is defined as the difference of the energy of the atoms in the near surface region when they are encapsulated in the bulk (Ebulk) and when they are relaxed after cleavage of the cell (Esurf):

()E – E γ = ------surf bulk , (7.2) 2A where A is the area of the surface unit cell. It is required for these calculations that the unit cell does not have a dipole perpendicular to the surface.

107 Chapter 7

Table 7.2: Experimental and calculated properties of zinc aluminate Property Experimental This work Fang [7]a Pandey [5]b Sampath [6]c Structural properties a [Å] 8.086 [11] 8.167 7.998 8.21 7.91 u [-] 0.390 [11] 0.390 0.389 0.386 0.389 Elastic and mechanical properties

C11 [GPa] 354.6 316 341.8 C12 [GPa] 202.3 169 246.8 C44 [GPa] 155.5 148 150.4 Ed [GPa] 242 [13] 303 286 255 Kd [GPa] 202 [12] 253 218 278e Gd [GPa] 117 112 94.9 νd [-] 0.29 0.28 0.34 Dielectric properties ε o [-] 11.94 [13] 8.55 11.16 ε∞ [-] 2.37 5.11 a. using first principle molecular dynamics b. using pair-potentials with oxygen polarizability c. using tight-binding muffin-thin orbital method d. Calculated from elastic constants using the Voigt-Reuss-Hill average assuming an isotropic polycrystalline material. e. Pandey et al. determined a value of 273 GPa by minimizing the energy vs. volume.

7.4 Results and discussion

7.4.1 Structural and bulk properties The lattice parameter a is 8.167 Å in the equilibrium state after relaxation and the anion parameter u is 0.390. These data are within ~1 % of the experimental determined data of 8.086 Å and

0.390 [11]. The elastic coefficients C11, C12 and C44 are 354.6, 202.3 and 155.5 GPa respectively, but, unfortunately, experimental data of single crystals is not available to verify these data. However, the Young’s, bulk and shear modulus and the Poisson’s ratio can be calculated from the elastic constants using the Voigt-Reuss-Hill averages assuming an isotropic polycrystalline material. The bulk modulus is 253 GPa and the Young’s modulus is 303 GPa, which differ from the experimental data of 202 GPa [12] and 242 GPa (estimated) [13], respectively. This large difference is quite common for calculations using pair potentials. A better agreement with experimental data is obtained by using ab-initio methods [7]. An overview of calculated data compared with experimental data is shown in table 7.2.

108 Zinc aluminate (surface)

(100)(110) (111)

↑↑↑

 O  O   O  O   O  O  O   Al  (a)  Al  (a)  O   O  O   O  O  (c)  Al  (a)  Zn  (b)

 O  O  (c)  O  (c)  Al  Zn  O  Al  O (b)  O  O  O   O  O   Zn   Zn   Al  (b)  O  O   O  O   Al   O  O   Zn 

Figure 7.1: Schematic overview of the stacking in zinc aluminate spinel.

7.4.2 Surfaces (simulation) The stacking of the atoms layers along the three low index directions in zinc aluminate is shown in figure 7.1. Just three different possible cleavage planes producing two non-polar surfaces are present along each direction and indicated by (a), (b) and (c)1. During cleavage of the a- and b- layers two equal surfaces are created, but cleavage of the c-layer gives two different, but com- γ plementary, surfaces (c1 and c2). The cleavage surface energy ( cleav) of these layers is shown γ in table 7.3 together with the surface energy after relaxation ( relax). The average of surface energy of the c1- and c2-layers should be taken for comparison with the surface energies of the other layers.

The lowest surface energy after relaxation of all possibilities is obtained by cleaving (100)b: 1.88 J m-2. The created surface is characterised by the presence of two zinc cations on top of the surface, which are somewhat nestled inside the oxygen layer (see figure 7.2a). Furthermore, this b-plane is also the preferred plane for fracture/cleavage, since it has also the lowest cleavage surface energy. The energy gain due to relaxation for this configuration is small, since the atoms do not move in the lateral directions but only in the normal direction. This results in more con-

1. The x-plane along the (uvw) direction is denoted as (uvw)x.

109 Chapter 7

Figure 7.2: Surfaces of a) (100)b, b) (110)a and c) (111)b. Red is oxygen, blue is zinc and green is aluminium.

110 Zinc aluminate (surface)

Table 7.3: Surface energy of zinc aluminate after cleaving and relaxation Plane (100) (110) (111) γ γ γ γ γ γ cleav relax cleav relax cleav relax [J m-2][Jm-2][Jm-2][Jm-2][Jm-2][Jm-2] a 13.03 3.80 9.91 2.44 19.63 3.65 b 3.57 1.88 6.70 2.66 7.35 3.06

c1 5.21 2.08 5.58 2.91 9.48 3.10 c2 5.21 2.71 5.58 2.55 11.47 3.24 tracted upper layers as indicated by the decrease of the zinc-oxygen distance from its bulk value of 1.97 Å to 1.75 Å in the outmost layer.

The large cleavage surface energy and large energy gain by relaxation of the (100)a can be explained by the presence of the four aluminium cations on top in the unrelaxed configuration. The large charge density of aluminium (σ = 3.67×1020 Cm-2) is not compensated by the absence of surrounding oxygen anions making this configuration energetically very unfavourable. A large relaxation is made by submerging the aluminium cations in the oxygen layer making this oxygen layer the outmost layer. Furthermore, the zinc cations from the subsurface layers are brought up to just below the surface or are exposed on the surface itself, which can be explained by the relative low charge density of zinc cations (σ = 0.82×1020 Cm-2) as compared to aluminium cations.

The relaxed surface energies of the planes along the (110) direction are quite similar. The (110)a (see figure 7.2b) has the lowest value, but the relaxed configuration shows again the presence of zinc cations on the surface despite the presence of aluminium cations in the outer layer. The only surface without zinc cations present on the surface is the (110)c1-layer, which is due to the absence of zinc cations in the subsurface layer resulting in the highest surface energy for this direction.

The relaxed surface energies of planes along the (111) direction resulting from the cleavage along the (111)b (see figure 7.2c) and (111)c are the lowest and quite similar. Both layers have zinc cations in their subsurface layer which emerge on the surface as stabilisers. Surfaces formed by cleaving of the a-plane do not have these zinc cations in the subsurface layer and therefore have a higher surface energy.

111 Chapter 7

Figure 7.3: Small crystals of natural gahnite with {100}, {221} and {111} facets. The width of the image is 0.5 mm. Reprinted with permission [16].

The surface energies of cleaved surfaces of magnesium aluminate have been reported in literature [7,14,15]. The (100)b plane has also the lowest surface energy: 2.27 J m-2. This value is larger than the value for zinc aluminate since the charge density of magnesium cations (σ = 1.46×1020 Cm-2) is larger than of zinc cations disallowing the same reduction as in zinc aluminate. This explains also the generally observed trend of higher surface energies of magnesium aluminate for the same surfaces. However, the (111)b direction has in magnesium aluminate a much higher surface energy than the c-planes in contrast to zinc aluminate. This is due to the different rearrangement of the aluminium cations on the surface of magnesium and zinc aluminate. The surface energy of the planes along the (110) direction is similar as is observed in zinc aluminate.

7.4.3 Surfaces (experimental) The morphology of zinc aluminate based on the surface energy calculations is expected to be {100} > {110} > {111}. However, large natural gahnite crystals occur primarily in an octahedral shape showing the opposite morphology: {111} >> {110} >> {100} [16]. At first glance this appears to be contradictive, but it should be taken into account that if the surface energy determines the morphology, kinetic mechanisms during growth must be absent. The situation where the growth is solely determined by thermodynamics is only valid for small single crystals. Small crystals of natural gahnite with large {100} and {111} facets and (probably) small {221} facets are shown in figure 7.3 indicating the transition from thermodynamical determined

112 Zinc aluminate (surface)

0.20

0.15

0.10

0.05 Absorbance [Kulbelka-Munk] Absorbance

0.00

4000 3800 3600 3400 3200 3000 2800 Wavenumber [cm-1] Figure 7.4: DRIFTS spectrum of the hydroxyl-stretching region on zinc aluminate immediately after outgassing. growth to kinetically determined growth1. Another possibility for the change in morphology is adsorption of water on the surface changing the surface energy. Computer simulations of dissociative adsorption of water on magnesium aluminate provides an indication for this morphology change [17].

The hydroxyl groups on the surface of zinc aluminate powder are investigated using DRIFTS, which is shown in figure 7.4. The spectrum, measured immediately after outgassing at 450 °C, shows a single peak at ~3660 cm-1 of the Zn-OH vibration, whereas the peak of the Al-OH vibrations (~3730-3790 cm-1) is absent [18]. The spectrum is similar to earlier reported data by Rossi et al. [19]. However, Rossi et al. had some zinc oxide contamination present in their powder in contrast to the powder used indicating that this small contamination does not play a role. A spectrum measured at room temperature immediately after cooling down showed only a single broad peak in the hydroxyl-stretching region indicating a rapid adsorption of water on zinc aluminate. Furthermore, it is remarkable that hydroxyl groups are solely located on top of zinc cations and not on aluminium cations as is on e.g. magnesium aluminate [20] indicating that zinc cations must be present at the surface.

1. It should be noted that natural crystals are almost never 100% pure compounds.

113 Chapter 7

Jacobs et al. [21] have studied the surface composition of zinc manganese aluminate

(Zn1-xMnxAl2O4; 0 < x < 1) powders synthesised by coprecipitation and a two step-calcining process (900 °C; 1150 °C) using Low Energy Ion Scattering (LEIS), which is sensitive only to the outmost atomic layer. They did not detect any zinc on the surfaces, which is not due to a low sensitivity of the apparatus for zinc, since zinc in zinc oxide could easily be detected. Valenzue- la et al. [22] confirmed this absence on their (doped) zinc aluminate powder synthesised at 800 °C via coprecipitation. They concluded from the absence of zinc cations using additional results from MnAl2O4 and CoAl2O4 that tetrahedral sites generally do not appear on the surfaces of spinel oxide powders and that the surfaces consist out of octahedral interstices.

X-ray Photoelectron Spectroscopy (XPS) by Maltha et al. on the same zinc manganese aluminate powders as measured by Jacobs et al. shows a reduction of ~25 % in zinc content in the outmost ~5 atomic layers (~15 Å), which is the penetration depth of the photoelectrons for zinc [23]. Elemental analysis of the bulk composition, which was in good agreement with the desired composition, showed that the reduction of concentration of zinc is a near-surface phenomenon, which was also shown by Inabe and Matsui using EPMA analysis of MnZn ferrite ceramics after heating [24]. The reduction indicates a depletion of zinc during calcining. This is not uncommon for reactions with zinc oxide present since zinc oxide at high temperatures has a high partial × -4 pressure (pZn =9 10 Pa at 1000 °C [25]) as compared to e.g. aluminium oxide × -13 × -8 (pAl =4 10 Pa [26]) or magnesium oxide (pMg =4 10 Pa [26]) and will vaporize [27]. Since in the outmost 15 Å of zinc aluminate 12 zinc cations per unit area are located, approximately 3 must have been evaporated, which can explain the absence of zinc cations on the surface. However, the absence of zinc cations due to depletion on its own does not provide conclusive proof for the absence of tetrahedral sites on the surface.

Jacobs et al. [21] used for their interpretation a model of the spinel surfaces based on the model from Knözinger and Ratnasamy [28]. The latter model is based on the ideal surfaces of γ-alumina, which has a defect spinel structure. However, the model used shows some imperfections. The surface structures are made by extending the bulk structure to the surface. This implies the absence of relaxation leading to possible reconstruction. Furthermore, it does not take a difference between bi- and trivalent cations on the relaxation into account, which results in a different surface geometry.

The two surfaces without tetrahedral sites according to Knözinger and Ratnasamy/Jacobs et al. correspond with the cleavage of the (111)a and (110)a planes in notation used in this work. A

114 Zinc aluminate (surface) careful examination of these surfaces after relaxation, as calculated in this chapter, shows that on both planes a hill and valley structure is present. The hills consist out of aluminium and oxygen atoms with on the (110)a plane also zinc in the "interior" of the hill. On the other hand, the valleys consist out of zinc and oxygen atoms, which is a subsurface layer that emerges on the surface. Actually, many reconstructed surfaces show zinc cations from the subsurface layer emerging on the surface.

The conclusion that the surfaces of zinc aluminate do not contain tetrahedral interstices is not unequivocally founded. The primary indication was the conclusion that zinc was not detected on the surface of zinc aluminate using LEIS. However, there are several questions to be made. Firstly, XPS measurements indicate that ~25 % of the zinc cations in the outmost 15 Å is vola- tilised during the heating stage in the synthesis process, most likely being those are the cations at the surface. Secondly, DRIFTS measurements show that hydroxyl groups are exclusively located on zinc cations. The oxygen of these hydroxyl groups can screen the underlying zinc cations from detection.

LEIS studies on surfaces made by fracturing single crystals of ZnAl2O4 in the absence of water vapour or in vacuum should indicate whether zinc is indeed absent on the {110} and {111} surfaces. An absence of zinc would confirm the conclusion that tetrahedral sites are not present on the surface. A presence of zinc would indicate that absence of zinc in the powders is due to evaporation of zinc oxide during synthesis or to screening of zinc by oxygen. Forced adsorption of water by immersion of the sample in water and subsequently drying and measuring, would be able to distinguish between the evaporation and the screening options.

7.5 Conclusions Computer simulations of the low-index planes of zinc aluminate showed that the (100)b plane has the lowest surface energy after relaxation: 1.88 J m-2, and will be the preferred plane for fracture/cleavage. The lowest surface energy for the (110) and (111) directions are 2.44 and 3.06 J m-2, respectively, for (110)a and (111)b. In general, the surface energy is low when zinc cations are on top as e.g. for (100)b, due to its much smaller charge density compared to aluminium cations. However, many cleavage planes contain aluminium cations on top. The surface energy is lowered by relaxation, where the aluminium cations are embedded in the oxygen layer and zinc cations from the subsurface layer become exposed on the surface. For instance, the (110)a and (111)b planes have a hill and valley structure, where the hills contain the aluminium cations and oxygen anions and the valleys consist out of zinc cations and oxygen anions.

115 Chapter 7

DRIFTS measurements have indicated that hydroxyl groups are exclusively located on the zinc cations indicating that zinc cations must be present at the surface. However, reported LEIS measurements indicated that zinc is not present at the surface. Therefore, using the model of Knözinger and Ratnasamy, tetrahedral interstices are not present at the surface of spinel oxides. On the other hand, reported XPS measurements show a deficiency of zinc in the outmost atomic layers, which is due to evaporation during synthesis, and might explain the absence.

The computer simulations show that zinc cations are present on the surface, but not necessary at the outmost atomic layer. This can explain the absence of zinc in the LEIS measurements. Furthermore, the oxygen anion of the hydroxyl group can screen the zinc cation during the LEIS measurements. However, both options show that the assumption of an absence of tetrahedral interstices on the surface might not be hold resulting in a possible reconsideration of the model of Knözinger and Ratnasamy.

7.6 Acknowledgements C.M. Fang (Eindhoven University of Technology) is acknowledged for the many fruitful discussions about the simulations and surfaces in general and for assistance with the usage of Metadise. B.G. Anderson (Eindhoven University of Technology) is acknowledged for the DRIFTS measurements.

7.7 References [1] A. Navrotsky and O.J. Kleppa, The thermodynamics of cation distributions in simple spinels, J. Inorg. Nucl. Chem., 29 (1967), 2701-2714 [2] A.N. Cormack, G.V. Lewis, S.C. Parker and C.R.A. Catlow, On the cation distribution of spinels, J. Phys. Chem. Solids, 49 (1988), 53-57 [3] R.W. Grimes, A.B. Anderson and A.H. Heuer, Predictions of cation distributions in AB2O4 spinels from normalized ion energies, J. Am. Chem. Soc., 111 (1989), 1-7

[4] L.J. Alvarez, P. Bosch and M.A. Valenzuela, Molecular dynamics studies of ZnAl2O4 spinel, Catal. Lett., 22 (1993), 361-372 [5] R. Pandey, J.D. Gale, S. K. Sampath and J.M. Recio, Atomistic simulation study of spinel oxides: zinc aluminate zinc gallate, J. Am. Ceram. Soc., 82 (1999), 3337-3341 [6] S.K. Sampath, D.G. Kanhere and R. Pandey, Electronic structure of spinel oxides: zinc aluminate and zinc gallate, J. Phys.: Condens. Matter, 11 (1999), 3635-3644

[7] C.M. Fang, C-K-Loong, G.A. de Wijs and G. de With, Phonon spectrum of ZnAl2O4 spinel from inelastic neutron scattering and first-principles calculations, Phys. Rev. B, 66 (2002), 144301

116 Zinc aluminate (surface)

[8] G.W. Watson, E.T. Kelsey, N.H. de Leeuw, D.J. Harris and S.C. Parker, Atomistic simulation of dislocations, surfaces and interfaces in MgO, J. Chem. Soc. Faraday Trans., 92 (1996), 433-438 [9] P.W. Tasker, The surface energies, surface tensions and surface structure of the alkali halide crystals, Phil. Mag. A, 39 (1979), 119-136 [10] G.V. Lewis and C.R.A. Catlow, Potential models for ionic oxides, J. Phys. C: Solid State Phys., 18 (1985), 1149-1161 [11] R.J. Hill, J.R. Craig and G.V. Gibbs, Systematics of the spinel structure type, Phys. Chem. Miner., 4 (1979), 317-339 [12] S. Lucchesi, A. Della Guista and U. Russo, Cation distribution in natural Zn-aluminate spinels, Miner. Mag., 62 (1998, 41-45 [13] N.J. van der Laag, this thesis, chapter 6 [14] M.J. Davies, S.C. Parker and G.W. Watson, Atomistic simulations of the surface structure of spinel, J. Mater. Chem., 4 (1994), 813-816 [15] C.M. Fang, S.C. Parker and G. de With, Atomistic simulation of surface energy of spinel MgAl2O4, J. Am. Ceram. Soc., 83 (2000), 2082-2084 [16] P.J. Dunn, Franklin and Sterling Hill, New Jersey: the world’s most magnificent mineral deposits; also (partly) available at: www.simplethinking.com/dunn/ [17] C.M. Fang, G. de With and S.C. Parker, Computer simulation of dissociative adsorption of water on the surfaces of spinel MgAl2O4, J. Am. Ceram. Soc., 84 (2001), 1533-1558 [18] G. Busca, V. Lorenzelli, G. Ramis and R.J. Willey, Surface sites on spinel-type and corundum-type metal oxide powders, Langmuir, 9 (1993), 1492-1499 [19] P.F. Rossi, G. Busca, V. Lorenzelli, M. Waqif, O. Saur and J.-C. Lavalley, Surface basic- ity of mixed oxides: magnesium and zinc aluminates, Langmuir, 7 (1991), 2677-2681 [20] N.J. van der Laag, this thesis, chapter 5 [21] J.P. Jacobs, A. Maltha, J.H.G. Reintjes, J. Drimal, V. Ponec and H.H. Brongersma, The surface of catalytically active spinels, J. Catal., 147 (1994), 294-300 [22] M.A. Valezuela, J.P. Jacobs, P. Bosch, S. Reijne, B. Zapata and H.H. Brongersma, The influence of the preparation method on the surface structure of ZnAl2O4, Appl. Catal. A: General, 148 (1997), 315-324 [23] A. Maltha, H.F. Kist, B. Brunet, J. Ziolkowski, H. Onishio, Y. Iwasawa and V. Ponec, The active sites of manganese- and cobalt-containing catalysts in the selective gas phase reduction of nitrobenzene, J. Catal., 149 (1994), 356-363 [24] H. Inaba and T. Matsui, Vaporization and diffusion of manganese-zinc ferrite, J. Sol. State Chem., 121 (1996), 143-146 [25] K. Kodera, I. Kusunoki and S. Shimizu, Dissociation pressures of various mettalic oxides, Bull. Chem. Soc. Jpn., 41 (1968), 1039-1045 [26] V.L.K. Lou, T.E. Mitchell and A.H. Heuer, Graphical displays of the thermodynamics of high-temperature gas-solid reactions and their application to oxidation of metals and evaporation of oxides, J. Am. Ceram. Soc., 68 (1985), 49-58 [27] J.A. Hedvall and R. Jagitsch, Hochtemperatursflüchtigkeit und Festdiffusion bei Pulver- reaktionen, Z. Anorg. Allg. Chem. 262 (1950), 49-53

117 Chapter 7

[28] H. Knözinger and P. Ratnasamy, Catalytic aluminas: surface models and characterization of surface sites, Catal. Rev.-Sci. Eng., 17 (1978), 31-69

118 - 8 -

Predictive calculation of intrinsic thermophysical properties from ab-initio phonon density of states: a case study for oxide ceramics

8.1 Introduction The exact mathematical formalism of a number of thermophysical phenomena in solids is known since the early years of the twentieth century. However, this mathematical formalism was too complex to deal with in those times. Therefore, approximations have been made. An example of such a relative simple approximation is the Debye model, which describes a number of thermophysical properties with only long waves. The Debye model performs (surprisingly) rather well for different solids and is widely used for many years.

However, the improvements in memory and speed of processors of computers in the recent dec- ades in combination with improvements of computational methods for quantum mechanics have it made possible to tackle the exact mathematical functions, without far-reaching simpli- fications. The question addressed here is: are these improvements to obtain a more accurate physical description of solids, really necessary for engineering purposes or can we still use the simple approximation?

The thermophysical properties determined from the Debye model and the phonon density of states calculated with a recent ab-initio computational method are compared with experimental

119 Chapter 8

abc →ω →ω →ω

ω D

Γ Γ → ω → ω KXXWLL F( ) FD( ) Figure 8.1: a) Ab-initio calculated dispersion curves of magnesium aluminate along different symmetry directions. b) Phonon density of states of magnesium aluminate determined from the dispersion curves. c) Debye spectrum of magnesium aluminate with the Debye frequency. data in order to investigate the necessity for practical applications of the phonon density of states.

8.2 Theory

8.2.1 Phonons In a crystal lattice, the atoms vibrate around their equilibrium position like a harmonic oscillator due to thermal excitation. These lattice vibrations can be described by elastic waves with wave vector k. Although the wave vector can have all values, those between -π/a and π/a (First Brillouin Zone) have physical significance, where a is the lattice constant of cubic structures. Furthermore, the atoms can vibrate in three independent directions called modes (λ), which for high symmetry directions in a crystal structure can be separated in one longitudinal and two transversal modes.

A phonon is the quantum of the (quantizised) energy of the lattice vibration, in analogy with the photon of the electromagnetic energy. Each phonon state has a phonon frequency unambiguously coupled to the wave vector and mode: ω(k,λ). By determining the number of phonon states in the interval [ω,ω+∆ω] the phonon density of states (F(ω)) is obtained, see figure 8.1.

By defining the normalisation of F(ω) as

120 Thermophysical properties of oxide ceramics

ω max ∫ F()ωω d = 3 , (8.1) 0 the phonon density of states can be obtained from the dispersion relations ω(k,λ) as:

V 3 F()ω = ------∑∫δ()ω()ωk, λ – d k , (8.2) ()π 3 n 2 λ where V is the crystal volume, n is the number of atoms in the unit cell and δ(x) is the Dirac delta-function. For low frequencies the density of states can be described by:

ω []ω2 ω4 … F()= a1 ++a2 . (8.3) The phonon density of states can be obtained from experiment or from ab-initio computer simulations. From now on the labels k and λ are omitted.

8.2.2 Thermophysical properties The total value of a thermophysical property (X(T)) is the integral over all the states of the contribution of each state (X(ω,T))1:

ω max ()ω, ()ωω Xhar()T = nX∫ T F d . (8.4) 0 As a result, in thermal equilibrium, the energy associated with a phonon state (X=E) is:

1 "ω E()ω, T = ---"ω + ------, (8.5) ()ω ⁄ 2 exp" kBT – 1 where the first term on the right hand side is the zero-point vibration at T = 0 K.

The expressions for the Helmholtz energy and the entropy are, respectively, given by:

()ω, ()ω, ()ω, ()()ω ⁄ A T ==E T – TS T kBTln 2sinh " 2kBT , and (8.6) ω ω ω ()ω, " " " S T = kB ------coth------– ln2sinh------. (8.7) 2kBT 2kBT 2kBT

Since only strictly harmonic vibrations are used, the heat capacities at constant pressure (Cp) and at constant volume (Cv) are equal (= C). The contribution of a single phonon state to the heat capacity is:

1. The temperature dependence is solely incorporated in the description of the thermophysical properties, although in principle the phonon density of states are also temperature dependent.

121 Chapter 8

"ω 2 exp()"ω ⁄ k T ω, ------B - C()T = kB ()ω ⁄ . (8.8) kBT exp" kBT – 1

8.2.3 Debye model The frequency of a phonon state in the Debye model can be rewritten to ω()k, λ = C()k, λ k . In the simplest Debye-model [1] the frequency is assumed to be linear in k for all wave vectors (), λ k and a constant value for C k =CD . The maximum frequency in this model is called the ω Debye frequency (D ), see figure 8.1c. Applying these approximations to equation 8.2, the De- bye density of states from 0 to the Debye frequency is given by:

9 2 F ()ω = ------ω . (8.9) D ω 3 D Θ In thermophysical applications it is common practice to use the Debye temperature ( D) as a measure of the maximum Debye frequency, see figure 8.1c. They are related to each other via:

ω Θ " D = kB D . (8.10) Applying equation 8.9 to equation 8.4 with equations 8.5, 8.7 and 8.8, the Debye model equiv- alents for energy, entropy and heat capacity, respectively, are obtained:

Θ ⁄ D T 3 x ()Θ , --9- Θ ------T------ED D T = nkB D + 9nkBTΘ ∫ () dx , (8.11) 8 D exp x – 1 0

Θ ⁄ D T T 3 x –Θ Θ , ------------D- SD()3D T = nkB 4Θ ∫ () dx – ln1 – exp, (8.12) D exp x – 1 T 0

Θ ⁄ D T 3 4 () Θ , ------T------x exp x - CD()9D T = nkBΘ ∫ () dx . (8.13) D exp x – 1 0 Θ C Θ S The "heat capacity Debye temperature" ( D ), "entropy Debye temperature" ( D ) and "zero- Θ H point energy Debye temperature" ( D ) can be obtained from:

() ()Θ C, Char T = CD D T , (8.14)

() ()Θ S, Shar T = SD D T and (8.15)

() ()Θ H, Ehar T = ED D T . (8.16) Θ E The "elastic constants Debye temperature" ( D ) can be determined by:

122 Thermophysical properties of oxide ceramics

Table 8.1: Vibrational properties described by a single parameter Physical property Debye temperature Θ C()Θ→ () Heat capacity; low T D T 0 = D –3 Θ S()Θ→ () Entropy; low T D T 0 = D –3 Θ E() Θ() Elastic constants; any T D T = D –3 Θ H()Θ() Zero-point energy (T=0) D T = 0 = D 1 Θ S()Θ> Θ () Entropy; high T D T D = D 0 Θ C()Θ> Θ () Heat capacity; high T D T D = D 2

13⁄ 6π2nN ρ Θ E " A D = ------CD , (8.17) kB M ρ where n is the number of atoms in the unit cell, NA is Avogadro’s number, is the mass density,

M is the weight of a mole of the material and CD is the Debye sound velocity. The Debye sound velocity is given for isotropic (polycrystalline) materials by:

1 12– ν 32⁄ –13⁄ C = C --2- + ------, (8.18) D T3 3 21()– ν ν where CT is the experimental transversal sound velocity and is the Poisson’s ratio. The trans- ⁄ ρ versal sound velocity is related to the shear modulus G by CT = G .

8.2.4 Moment representation of phonon density of states In the limit of low or high temperatures, a thermophysical property can be calculated from a certain average of F()ω , a frequency moment, or its related Debye temperature moment (see table 8.1). The n-th moment of F()ω is defined as:

ω ω max max 〈〉ωn ==∫ ωnF()ωω d ⁄ω∫ F()ωω d n()n 0 0 . (8.19) and for n=0 as:

ω ω max max ln[]ω()0 = ∫ ln()ω F()ωω d ⁄ ∫ F()ωω d . (8.20) 0 0 The Debye temperature is related to the frequency moment in case of a strict Debye density of states as:

123 Chapter 8

3 1 ⁄ n k 3 1 ⁄ n ω()n ==------ω ----B------Θ ()n . (8.21) n + 3 D " n + 3 D

For n = 0 and n = -3 the relation is defined as:

k ω()0 = ----B-exp()Θ–31 ⁄ () 0 (8.22) " D and

a 13⁄ ω() 1 → Θ () " 9 –3 ==----- D –3 ---------- . (8.23) 3 kB a1 Θ In a strict Debye model D(n) are equal for all n. A large deviation in Debye temperatures for different moments would indicate that the Debye model is not applicable to the compound investigated. For a more elaborate treatment of this type of theory, see the book by Grimvall [2].

8.3 Criteria for comparison The entropy and the heat capacity can easily been obtained from the density of states for every temperature. The values at 298.15 K and 1 atm (standard conditions) are compared to the experimental data in literature. For the calculation of the entropy and the heat capacity at standard conditions from the Debye-model, also the Debye-temperature should be known. The Debye temperature derived from the elastic constants at room temperature is chosen, since this temperature is generally used due to the large availability of experimental data. However, it should be noted that in general it is wrongly assumed that the Debye temperature from elastic constant is temperature independent, despite the knowledge of the temperature dependency of the elastic constants. The corresponding Debye temperature is shown in table 8.2.

Another way to validate the phonon density of states is to determine the Debye-temperature at different temperatures from the calculated thermophysical properties and compare their values with those obtained from the moments of the density of states, as shown in table 8.1.

Table 8.2: Debye temperature of elastic bulk properties νa a Θ E Material G CD D [-] [GPa] [m s-1][K] MgO 0.175 128 6580 981

Al2O3 0.233 163 7082 1089 MgAl2O4 0.268 107 6078 903 a. from ref. [3]

124 Thermophysical properties of oxide ceramics

8.4 Computer simulation The phonon spectra of MgO [4], Al2O3 [5], MgAl2O4 [6] and ZnAl2O4 [7] have been reported in literature. These phonon spectra were calculated using Density Functional Theory within the Local Density Approximation with the computer code VASP (Vienna ab-initio simulation pro- gram) [8-11]. Subsequently, the interatomic forces were calculated using the Hellmann-Feyn- man theorem by slightly displacing (0.01-0.05 Å) some selected atoms from their equilibrium positions. Next, the phonon spectra, including phonon frequencies and eigenvectors along some high-symmetry points in the First Brillouin Zone (see figure 8.1a), were obtained by straightforward diagonalisation of the dynamical matrix which was acquired from the force constants of the interatomic forces. Finally, the phonon density of states was obtained by linear tetrahe- dron integration of the phonon frequencies.

The calculated infrared- and Raman frequencies at the Γ-point in these phonon spectra are in excellent agreement with experimental data in literature [4-7]. Recently, Fang et al. also performed neutron inelastic scattering experiments for ZnAl2O4, where the measured generalised phonon density of states exhibits a gapless spectrum agreeing with the theoretical calculations [7].

8.5 Results and discussion

8.5.1 Density of states The phonon density of states of the investigated materials are shown in figure 8.2. The density of states of all the materials does not show any band gap. Furthermore, the width of the density of states is approximately the same for aluminium oxide, magnesium aluminate and zinc aluminate. The width for magnesium oxide is smaller than of the others.

The contribution of the magnesium and zinc atoms is primarily in the low frequency range. The contribution of the aluminium atoms is influenced by the presence these atoms, since in the aluminates the centre of gravity of the aluminium contribution is shifted towards higher frequencies. The contribution of the oxygen atoms is in a first approximation the same in alumina and the aluminates. In this first approximation it can be concluded the contribution of the oxygen atoms does not depend on whether they form a bcc- or fcc-sublattice.

125 Chapter 8

abcd

Mg/

Intensity [arb. units] [arb. Intensity O

Total

0 200 0 200 0 200 0 200 Frequency [Trad s-1]

Figure 8.2: Phonon density of states of a) MgO; b) Al2O3; c) MgAl2O4 and d) ZnAl2O4.

8.5.2 Heat capacity The heat capacity of the investigated materials is shown in figure 8.3. All curves of the heat ca- Θ pacity approach in the high-temperature limit (T > D) the Dulong-Petit limit of 3nR, where R is the gas constant.

The harmonic heat capacity can be obtained from the experimentally accessible anharmonic heat capacity at constant pressure via:

()αγ Cp = 13+ T C , (8.24) where αγ is the linear thermal expansion coefficient, is the thermodynamical Grüneisen parameter and T is the temperature. The experimental heat capacities of the investigated materials are listed in table 8.3 and are compared with the calculated values from the phonon density of states and from the Debye model in table 8.4. The difference for the heat capacity from the density of states is within ~2 %. However, for the simple oxides large relative differences are found for the Debye model.

126 Thermophysical properties of oxide ceramics

a)60 b) 140

120 50 ] ] -1 -1 100

K 40 K -1 -1

80 30

20 40

Heat capacity mol [J 10 Heat capacity mol [J 20

0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Temperature [K] Temperature [K]

c)200 d) 200

160 160 ] ] -1 -1 K K -1 -1 120 120

80 80

40 40 Heat capacity mol [J Heat capacity mol [J

0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Temperature [K] Temperature [K] Figure 8.3: Heat capacity calculated from phonon density of states of a) MgO, b) Al2O3, c) MgAl2O4 and d) ZnAl2O4.

Table 8.3: Heat capacity at constant volume at T = 298.15 K a αb γc Material Cp(exp) C(exp) [J mol-1 K-1][10-6 K-1][K T-1] [Jmol-1 K-1] MgO 37.23 11.88 1.53 36.63

Al2O3 79.76 5.60 1.27 79.26 MgAl2O4 115.88 7.02 1.28 114.96 a. from ref. [12] b. from ref. [13] c. from ref. [14]

Table 8.4: Heat capacity comparison at T = 298.15 K Material C (exp) C (DOS) Difference C (Debye) Difference [J mol-1 K-1][Jmol-1 K-1] [%] [J mol-1 K-1][%] MgO 36.63 36.06 -1.55 30.65 -16.3

Al2O3 79.26 78.95 -0.39 69.35 -12.5 MgAl2O4 114.96 112.49 -2.15 114.82 -0.12

127 Chapter 8

Table 8.5: Entropy at T = 298.15 K Material S (exp)a S (DOS) Difference S (Debye) Difference [J mol-1 K-1][Jmol-1 K-1] [%] [J mol-1 K-1][%] MgO 26.94 25.98 -3.5 18.15 -32.6

Al2O3 50.89 51.46 +1.1 38.51 -24.3 MgAl2O4 80.63 76.72 -4.8 73.79 -6.84 (no inversion)

MgAl2O4 80.63 82.72 +2.6 79.79 -1.09 (with inversion) a. from ref. 15

8.5.3 Entropy The experimentally determined entropy values are shown in table 8.5 together with the calculated values from the ab-initio phonon density of states. The difference in entropy values is somewhat higher than the difference in heat capacity. Furthermore, a large difference is observed with the calculated values from the Debye model, as in the case for the heat capacity.

The absence of configurational entropy in the simulations can explain the relative large difference for magnesium aluminate since aluminates show inversion, where divalent cations have migrated from tetrahedral interstices to octahedral interstices and trivalent cations visa versa. The entropy term related to this configurational contribution is given by:

x  S = –Rxln() x + ()1 – x ln()1 – x + xln --- + ()2 – x ln 1 – --x- , (8.25) conf 2 2 where R is the gas constant and x is the fraction of bivalent cations in octahedral interstices (inversion parameter). The configurational entropy is +6.00 J mol-1 K-1 for magnesium aluminate which has an inversion parameter of ~0.10 [16]. Adding the configurational entropy to the vibrational entropy from the ab-initio density of states gives 82.72 J mol-1 K-1, which decreases the error to +2.6 %, and to the entropy from the Debye model gives 79.79 J mol-1 K-1 resulting in an error of -1.04 %.

8.5.4 Debye temperature Θ C The Debye temperature ( D ) of the materials investigated is shown in figure 8.4. The Debye temperature of all the materials shows a substantial drop in the low temperature range (0 - 100 K), but the Debye temperature of magnesium oxide remains relative constant from the minimum onwards, see table 8.6. Furthermore, the increasing molar weight of the cations caus-

128 Thermophysical properties of oxide ceramics

a)1200 b) 1200

1000 1000

800 800 Debye temperature [K] Debye temperature [K]

600 600 0 500 1000 1500 2000 0 500 1000 1500 2000 Temperature [K] Temperature [K]

c) 1200 d) 1200

1000 1000

800 800 Debye temperature [K] Debye temperature [K]

600 600 0 500 1000 1500 2000 0 500 1000 1500 2000 Temperature [K] Temperature [K] Θ C Figure 8.4: Debye temperature ( D ) dependence on temperature for a) MgO, b) Al2O3, c) MgAl2O4 and d) ZnAl2O4.

Table 8.6: Debye temperature Θ C Θ C()ΘC() Material Tmin Dmin D 298.15 D 1000 [K] [K] [K] [K] MgO 114 775 786 790

Al2O3 88 883 947 970 MgAl2O4 58 816 927 951 ZnAl2O4 40 616 896 925 ing an increase of density of states at lower frequencies manifests itself in the decrease of the temperature (Tmin) where the minimum Debye temperature is located.

The Debye temperature depends on the summation over the contributing density of states. Since at high temperatures this integral approaches the Dulong-Petit limit for all materials, the width of the density of states is a measure for the maximum Debye frequency and thus the Debye temperature. The similar width of aluminium oxides and both the aluminates reflects in the similar Debye temperature and the small width for magnesium oxide reflects in the relative low Debye temperature.

129 Chapter 8

8.5.5 Moments of density of states Several thermophysical properties in the low or high temperature limit are related to the frequency moments of the density of states. The Debye temperature in these limits calculated from the entropy, heat capacity and zero-point energy are compared to the Debye temperature equiv- alent of the frequency moments for these properties (via equation 8.10) and are shown in table 8.7

All calculated Debye temperatures from the thermophysical properties are almost equal to the Debye temperature from the moment of the density of states. The only exceptions are the Debye temperatures in the low temperature limit, which are due to the numerical instabilities at low temperatures of the calculation of the Debye temperature of the entropy and heat capacity.

8.5.6 Debye temperature from elastic constants The Debye temperature as derived from the elastic constants is for every temperature equal to the Debye temperature of the -3rd frequency moment of the phonon density of states and can be calculated by equations 8.17 and 8.18. The results are shown in table 8.8.

The Debye temperature of magnesium and aluminium oxide are approximately equal to the value as obtained from the -3rd moment Debye temperature. However, the -3rd moment Debye tem-

Table 8.7: Comparison Debye temperatures from thermophysical properties and frequency moments Θ D MgO Al2O3 MgAl2O4 ZnAl2O4 Θ C a D [K] 1060 1112 1120 855 Θ S a D [K] 1032 1068 1081 817 Θ () D –3 [K] 984 1058 1170 885 Θ S D [K] 785 935 899 831 Θ () D 0 [K] 784 932 902 833 Θ C D [K] 789 970 952 927 Θ () D 2 [K] 792 973 955 929 b -1 Evib [kJ mol ] 14.70 44.40 60.83 58.22 Θ H D [K] 786 950 930 890 Θ () D 1 [K] 786 952 929 889 a. determined at 12 K b. at 0 K

130 Thermophysical properties of oxide ceramics

Table 8.8: Debye temperature of elastic constants Θ E Θ () Material D D –3 [K] [K] MgO 981 984

Al2O3 1089 1058 MgAl2O4 903 1170 ZnAl2O4 760 885 perature of the aluminates is different from the experimental observed value. This possibly can be explained by the absence of inversion in the simulations, giving an incorrect acoustic part of the phonon density of states.

8.6 Conclusions The experimental values of several thermophysical properties of several oxide ceramics are compared with the results of two theoretical models: an ab-initio phonon density of states from density functional theory calculations using local density approximation and the Debye model. However, the calculations do not include (structural) disorder, like e.g. the inversion in aluminates, which possibly results in an improper description of the acoustic part of the density of states leading to an improper prediction of the Debye temperature from elastic constants. On the other hand, the absence of inversion in the calculation of entropy can easily be solved by adding a configurational term.

The ab-initio phonon density of states gives much more accurate results for heat capacity and entropy than the Debye model using the experimentally observed Debye temperature from elastic constants. The typical error from phonon density of states is ~1-4 % whilst the typical error from the Debye model is larger than 10 % for simple oxides. Moreover, the phonon density of states predicts also correctly the behaviour of the thermophysical properties in the low and high temperature limit. Therefore, the usage of the ab-initio phonon density of states should be preferred over the usage of the Debye model.

8.7 Acknowledgements C.M. Fang (Eindhoven University of Technology) is acknowledged for the many fruitful discussions about phonon spectra and the underlying theory.

131 Chapter 8

8.8 References [1] P. Debye, Theory of specific heats, Ann. Physik, 39 (1913), 789-839 [2] G. Grimvall, Thermophysical properties of materials, 1st Ed, North Holland, Amster- dam, The Netherlands, (1986) [3] D.J. Green, An introduction to the mechanical properties of ceramics, 1st Ed., Cam- bridge University Press, Cambridge, UK, (1998), 25 [4] G.A. de Wijs and C.M. Fang, unpublished results [5] R. Heid, D. Strauch and K.-P. Bohnen, Ab-initio lattice dynamics of sapphire, Phys. Rev. B, 63 (2000), 8625-8627 [6] G.A. de Wijs, C.M. Fang, G. Kresse and G. de With, First-principles calculation of the phonon spectrum of MgAl2O4 spinel, Phys. Rev. B, 65 (2002), 094305/1-5

[7] C.M. Fang, C-K. Loong, G.A. de Wijs and G. de With, Phonon spectrum of ZnAl2O4 spinel from inelastic neutron scattering and first-principles calculations, Phys. Rev. B, 66 (2002), 144301 [8] G. Kresse and J. Hafner, Ab-initio molecular dynamics of liquids, Phys. Rev. B, 47 (1993), 558-561 [9] G.Kresse and J. Hafner, Ab-initio molecular-dynamics simulation of the liquid-metal- amorphous-semiconductor transition in germanium, Phys. Rev. B, 49 (1994), 14251- 14269 [10] G. Kresse and J. Furthmueller, Efficiency of ab-initio total energy calculations of metals and semiconductors using a plane-wave basis set, Comput. Mater. Sci., 6 (1996), 15-50 [11] G. Kresse and J. Furthmueller, Efficient iterative schemes for ab-initio total energy calculations using a plane-wave basis set, Phys. Rev. B, 54 (1996), 11169-11186 [12] Y.S. Touloukain and E.H. Bayco, in: Thermophysical properties of matter, Volume 5: Specific heat of nonmetallic solids, New York, USA, (1977) [13] Y.S. Touloukain, R.K. Kirby, R.E. Taylor and T.Y.R. Lee, in: Thermophysical properties of matter, Volume 13: Thermal expansion of nonmetallic solids, New York, USA, (1977) [14] O.L. Kuskov and V.A. Kronrod, Core sizes and internal structure of Earth’s and Jupiter’s satellites, Icarus, 151 (2001), 204-227 [15] CRC Handbook of Chemistry and Physics: 1’st student edition, R.C. West (Ed), CRC Press, Boca Raton, Fl, USA, (1988) [16] R.J. Hill, J.R. Craig and G.V. Gibbs, Systematics of the spinel structure type, Phys. Chem. Minerals, 4 (1979), 317-339

132 - 9 -

Epilogue: overview, conclusions & future work

9.1 Introduction The primary aim at the start of this Ph.D. research was to understand the influence of water adsorption on the fracture process and further assess the theory derived by Donners et al. [1] for MnZn ferrites. Therefore, a three step approach was pursued. Firstly, investigative experiments were performed on several oxide ceramics to determine whether the influence of adsorption was not limited to hexaferrite and MnZn ferrites only. Secondly, magnesium aluminate (spinel) was chosen as a model material to investigate in more detail the influence of adsorption using surface analysis techniques, since the cation stoichiometry in MnZn ferrites is rather complex 2+ 2+ 2+ 3+ 3+ (Mn Fe Zn , Mn Fe O4) and both polycrystalline ceramics and single crystals are available. However, inversion -the process where magnesium and aluminium cations swap tetrahedral and octahedral interstices- is still present in magnesium aluminate, possibly disrupting a clear interpretation. Therefore, zinc aluminate ceramics were manufactured, since zinc aluminate does not show inversion.

Finally, computer simulations of fracture surfaces were performed for magnesium aluminate and zinc aluminate. The results were compared to the experimental data. Furthermore, within the framework of expanding the process zone to meso-scale level, computer simulations using density functional theory methods were used to obtain the phonon spectra. The thermophysical

133 Chapter 9 properties were calculated from these phonon spectra and compared to those obtained via the Debye model and experimental data.

9.2 Overview and conclusions

9.2.1 Fracture and adsorption The (apparent) fracture toughness as a function of the stressing rate and humidity was measured for several materials revealing two groups. Type I materials (alumina, translucent alumina, PZT, zirconia and floor tiles) are characterised by a sharp decrease in fracture toughness between 0 and ~10 % relative humidity and constant fracture toughness from 20 % onwards. Type II materials (MnZn ferrites, calcium hydroxyapatite and magnesium aluminate) are characterised by the absence of a sharp initial decrease and have a more smooth decrease of fracture toughness over the complete range of relative humidity. Accordingly, Type II materials have to be treated with care, since documented fracture toughness values in literature are dependent on the applied humidity during measuring, which is not the case for Type I materials.

The exact reason for this distinction is still unclear, but some conclusions can be drawn from comparisons of structural, microstructural and fracture aspects.

Structural aspects:

• The crystal structure is not an explanation since both calcium hydroxyapatite and alumina have a hexagonal structure and PZT and MnZn ferrites a cubic structure. 2+ 3+ • The presence of cations in different oxidation states as e.g. Mn and Mn in MnZn ferrites can also be excluded as an explanation, since for example, magnesium and aluminium in magnesium aluminate do not have several oxidation states. • Charge displacement might play a role as it is occurring in all Type II materials as e.g. inversion in spinels, but not in Type I materials. Therefore, the adsorbated hydroxyl groups can relax the structure of Type II materials more than of Type I materials. A further investigation into this matter is required, where also the nature of the bonds (ionic-covalent character ratio) should be taken into account. Microstructural and fracture aspects:

• Pores have an influence on the adsorption during fracture as porous magnesium aluminate ceramics shows Type I behaviour and dense Type II behaviour. This can partly be explained

134 Epilogue

by the reduction of fracture surface. Additionally, the instantaneous progression of the crack front might deny the adsorption to contribute. • The difference in behaviour between dense and porous magnesium aluminate might also be partly explained by the different grain sizes requiring further research. • The fracture mode seems not to play an important role. Both PZT and Mg-PSZ show a Type I behaviour, but have an intergranular and transgranular fracture mode, respectively.

Most ceramic materials investigated were (commercially) obtained except for calcium hydroxyapatite and zinc aluminate ceramics. Unfortunately, fully dense zinc aluminate ceramic could not be manufactured due to processing imperfections, which could be solved in future work. Although several (thermophysical) properties of zinc aluminate ceramics could be measured, measuring the influence of adsorption on the fracture toughness was not possible due to the large porosity.

9.2.2 Surface and computer modelling Single crystals of magnesium aluminate have been broken at low humidity and in air along the (100) and (111) planes. The fracture toughness of these single crystals were smaller than the polycrystalline specimens and those in air were lower than those at low humidity. The fracture surfaces of these single crystals were investigated using Low Energy Ion Scattering (LEIS) in order to determine the composition.

Different surface compositions for (100) fracture surfaces were predicted by computer modelling. The most likely fracture surface composition would contain magnesium cations, while another fracture surface composition would contain aluminium cations. LEIS measurements indicated a surface composition with both cations present with an Al/Mg ratio of ~1.0-1.5. This could partially be explained by the inversion process, which is not taken into account in simulations. Furthermore, the composition did not alter much when fractured in air or at very low humidities.

Diffuse Reflectance Infrared Fourier Transform Spectroscopy (DRIFTS) measurements in the hydroxyl stretching region on magnesium aluminate powder could not distinguish whether hydroxyl groups are solely located on aluminium, on magnesium or on both making an atomistic description of adsorption during fracture more difficult. However, DRIFTS measurements on zinc aluminate revealed that hydroxyl groups are solely located on zinc cations and on cobalt aluminate solely on the alumina cations [2]. Computer modelling should point out whether this

135 Chapter 9 preference will have also an effect on the preferred fracture surface, which could then be more easily detected by LEIS studies on fracture surfaces of these single crystals. Although, a computational study to the fracture surfaces of zinc aluminate was conducted, hydration studies of these surfaces were not possible due to lack of good potentials (garbage in = garbage out). Fi- nally, the computer modelling of fracture surfaces should be extended to other than low-index planes, since these do occur often.

As fracture is often treated on either an atomistic level or on a continuum level, a number of phenomena, as e.g. microstructure, are generally not taken into account. Therefore, the process- zone must be scaled up from an atomistic level to a meso-level requiring, among others, thermophysical properties to be taken into account. As a first attempt, a comparison was made between thermophysical properties determined from phonon spectra calculated using density functional theory methods, from the Debye model and experimental data. It is found that the phonon spectra gave much better predictions (~1-4 % deviation) than the Debye model (~10 % deviation).

9.3 Thoughts for future work Two questions considering the role of adsorption on the fracture process have not been addressed in this thesis, but are to be answered to fully understand the effects of adsorption. What is the influence of adsorption on the fracture toughness relative to other toughening mechanisms and how is adsorption involved in the process zone mechanisms, since adsorption operates in the wake of the advancing crack?

9.3.1 Toughening It is assumed that the intrinsic work of fracture R is solely dependent on the surface energy, when applied under same experimental conditions (i.e. same temperature, same loading rate, etc.) However, many materials show so-called R-curve behaviour, where the fracture toughness, and thus the intrinsic work of fracture, is increasing with the advancing crack. Several explanations for this effect have been given:

• Bridging effects (occurring in e.g. alumina [3]) caused by crack-border interaction in the wake of the advancing crack. These bridging effects can be subdivided in intact bridges (e.g. unbroken fibres or whiskers) and interlocking bridges (e.g. broken grains, but remain wedged), which restrain crack opening, and frictionally sliding bridges, which transfer fric- tion stresses.

136 Epilogue

• Stress-induced phase transformation, as in e.g. zirconia [4]. • Domain switching in piezoelectric materials as e.g. PZT [5]. • Microcracking or crack branching generating more surfaces with increasing crack length increasing the resistance to fracture. A more elaborate overview is given in e.g. [6].

It is surprising to see that the Type I materials investigated show an R-curve behaviour. An explanation for this coincidence might be that several processes prevent adsorption to happen sufficient near the crack tip. For example, interlocking grains can block the transport towards the crack tip. The remaining question whether the Type II materials do not show an R-curve behaviour needs to be answered. Therefore, further (theoretical) research is required into this subject. Especially, since it is shown that bridging stresses influence the subcritical crack growth in alumina and can even arrest advancing cracks [7].

9.3.2 Wake field Adsorption is a process which takes place in the wake of the advancing crack implying the ex- istence of some kind of connection between the process zone around the crack tip and the adsorption site. However, the nature of this connection is unknown. Therefore, some fundamental theoretical research must be performed to understand the energetic balancing inside the process zone in relation to the released energy due to adsorption near the crack tip. This includes also an investigation to the size of the process zone and to distance of the adsorption site to the crack tip in order to determine when adsorption will contribute to the decrease of the fracture toughness.

Not only the process zone processes are of importance, but also the transport of gaseous molecules to the adsorption sites, since this will influence the adsorption. Although it is assumed that adsorption will take place in the same way as on powders, this is not the case inside a crack due to the presence of an opposite surface in very close proximity. The effect of these two near surfaces on the transport of gaseous molecules towards the crack tip/process zone is up for a fundamental (computational) research. However, not only perfect surfaces should then be considered, but also rough surfaces and grain boundaries.

Future work should point out the relevance of both the energetic balancing and the transport of gaseous molecules and their respective contributions to the process zone.

137 Chapter 9

9.4 References [1] M.A.H. Donners, L.J.M.G. Dortmans and G. de With, Adsorption and kinetic effects on MnZn ferrites, J. Mater. Res., 15 (2000), 1377-1388 [2] G. Busca, V. Lorenzelli and V. Bolis, Preparation, bulk characterisation and surface chemistry of high-surface-area cobalt aluminate, Mater. Chem. Phys., 31 (1992), 221- 228 [3] L. Llorca and R.W. Steinbrech, Fracture of alumina: an experimental and numerical study, J. Mater. Sci., 26 (1991), 6383-6390

[4] A.H. Heuer, Transformation toughening in ZrO2-containing ceramics, J. Am. Ceram. Soc., 70 (1987), 689-698 [5] S. Baik and S.M. Lee, R-curve behaviour of PZT ceramics near the morphotropic phase boundary, J. Mater. Sci., 29 (1994), 6115-6122 [6] D. Munz and T. Fett, Ceramics, Mechanical properties, Failure behaviour, materials selection, Springer-Verlag, Berlin, Germany, (1999), chapter 4 [7] T. Fett and D. Munz, Influence of bridging interactions on the lifetime behaviour of coarse-grained Al2O3, J. Eur. Ceram. Soc., 12 (1993), 131-138

138 Summary

The chemical environment plays an important role in the fracture process of many materials and in particular of inorganic materials. Water molecules from the environment can enter a crack through diffusion and, when a load is applied, can react with the stretched bonds near the crack tip. Subsequently, a transition complex is formed lowering the activation energy for the breaking of the bond. This kinetic mechanism results in so-called subcritical crack growth limiting the time-to-failure of a ceramic component. However, since new surfaces are created during fracture, the surface energy - an opposing force to fracture - increases. Adsorption of water molecules on the newly created surfaces lowers the surface energy, rendering crack growth again depending on relative humidity. The question is raised whether the adsorption mechanism has a significant contribution to subcritical crack growth like the kinetic mechanism.

The first approach to answer this question was to obtain an overview of the fracture behaviour of several oxide ceramics. Commercially applied oxide ceramics such as e.g. (translucent) alumina, PZT, zirconia and spinel, were subjected to Single Edge Notched Beam (SENB) measurements at different loading rates and different relative humidities. Two distinctive types of behaviour were found: Type I materials showing a rapid decrease in fracture toughness with increasing humidity between 0 and 10 %, but remaining constant at higher humidities than 10 % (aluminas, PZT, zirconia) and Type II materials showing a continuous decrease of the fracture toughness over the whole range of humidities (spinel, calcium hydroxyapatite). The materials showing the latter property have to be treated with care, since the fracture toughness data reported in literature depend on the relative humidity.

The second approach was to understand the atomistic origin of the distinction between Type I and Type II materials by studying a model material. The ternary oxide spinel (MgAl2O4, mag-

139 Summary nesium aluminate) was chosen, since its constituent magnesia does not show subcritical crack growth, but alumina does and both polycrystalline ceramics and single crystals were available. Firstly, the emphasis was laid on the fracture process itself. Fracture experiments on polycrystalline materials with different densities revealed Type II behaviour. Electron Backscattering Diffraction (EBSD) measurements on a polycrystalline sample with a crack induced by inden- tations did not reveal a preferred plane for fracture. Therefore, single crystal samples were fractured at different orientations and humidities showing subcritical crack growth. This implies that the effect of water on the fracture could be studied on the fracture surfaces of these crystals.

Subsequently, the focus was directed to the atomistic aspects of fracture. The composition of (100) fracture surfaces was studied using Low Energy Ion Scattering (LEIS) revealing both magnesium and aluminium cations present on the surface. This is rather surprising, since computer simulations have predicted that a surface with only magnesium cations on top is the most stable surface. However, the presence of aluminium cations on top can partially be explained by the occurrence of the inversion process, where divalent (magnesium) cations in tetrahedral interstices swap lattice positions with trivalent (aluminium) cations in octahedral interstices, which could not be taken into account during simulations. Diffuse Reflectance Infrared Fourier Transform Spectroscopy (DRIFTS) measurements of hydroxyl groups on the surface of powders indeed confirmed the presence of aluminium cations on the tetrahedral interstices. Howev- er, the DRIFTS measurements could not provide information whether the adsorption took place solely on aluminium cations, solely on magnesium cations or on both. The inversion prevented a clear atomistic interpretation.

In order to avoid inversion complicating a clear interpretation, focus was laid on zinc aluminate spinel (ZnAl2O4) ceramics. First, the possible influence of the synthesis route (solid state, coprecipitation and sol-gel) of the zinc aluminate powders on the inversion was investigated using 27Al MAS-NMR. Only a small amount of inversion was present, in comparison with magnesium aluminate, with the solid state and coprecipitation route and a considerable amount with the sol-gel route. The latter could be explained by a lack of zinc, as determined by Atomic Absorp- tion Spectroscopy (AAS), due to the evaporation of zinc oxide during firing. Second, zinc aluminate ceramics were made using powder from the solid state route. Unfortunately, processing problems prevented the synthesis of fully dense ceramics required for fracture experiments. However, several (thermo)physical/mechanical properties were determined, which were compared to results of computational simulations and those reported in literature.

140 Summary

Although fracture experiments could not be performed, computational simulations of fracture surfaces indicated that a (100) fracture surface with zinc cations on top would be the most stable. Furthermore, DRIFTS measurements on zinc aluminate powder showed that hydroxyl groups are only present on zinc cations. This all indicates that zinc aluminate would be an ideal material to investigate in future studies.

Finally, since adsorption takes place in the wake of the advancing crack, a mechanism of thermophysical interactions must be present inside the process zone. In a first attempt, two theoretical models describing these interactions are compared. It is found that thermophysical properties calculated from phonon spectra from Density Functional Theory (DFT) simulations are more accurate than those calculated from the Debye model using experimental (mechanical) data when compared to experimental thermophysical data. The main discrepancy of the Debye model is mainly due to the incorrect description of the distribution of the lattice vibrations resulting clearly in a wrong description.

The adsorption mechanism has a large influence on the fracture properties of some materials, but the exact origin of this influence is still unrevealed. However, several causes such as e.g. fracture mode, crystal structure and oxidation state of the cations can be ruled out as possible explanations. The results from the experiments indicate that local charge disturbances in the lattice, e.g. due to inversion, might play an important role, which could be further investigated with zinc aluminate. Furthermore, the knowledge of the exact (thermophysical) interactions between adsorption site and the process zone should be extended.

141 Summary

142 Samenvatting

De chemische omgeving speelt een belangrijke rol in het breukproces van materialen en in het bijzonder van anorganische keramische materialen. Het is bekend dat watermoleculen uit de omgeving in de scheur naar de scheurtip kunnen diffunderen. Als het voorwerp waarin de scheur zit ook nog eens belast wordt, dan kunnen deze watermoleculen reageren met de opge- rekte bindingen aan de scheurtip. Zij vormen daarmee een transitie-complex, dat de activering- senergie voor het breken van de verbinding verlaagt. Dit kinetische proces noemt men langzame of subkritische scheurgroei en resulteert in een verkorte levensduur van de keramische component.

Er speelt echter ook nog een ander mechanisme een rol. Bij het breken ontstaan nieuwe opper- vlakken, waardoor de oppervlakte-energie zal toenemen. Deze toename kan verminderd worden door adsorptie van watermoleculen, waardoor het oppervlak relaxeert. Aangezien een hoge oppervlakte-energie het breukproces tegenwerkt, resulteert een verlaging van deze energie door adsorptie in een versnelling van het breukproces door subkritische scheurgroei, welke afhankelijk is van de vochtigheid. De vraag is nu of het adsorptiemechanisme, naast het kinetische mechanisme, een significante bijdrage levert aan de totale subkritische scheurgroeisnelheid.

In eerste instantie is gekeken naar het breukgedrag van verscheidene oxidische keramische materialen. Deze commercieel toegepaste materialen, zoals alumina, PZT, zirconia en spinel, werden onderworpen aan SENB experimenten bij verschillende belastingssnelheden en vochtigheden. Twee verschillende typen breukgedrag werden gevonden: Type I materialen, waarbij de breuktaaiheid tussen 0 en 10 % relatieve vochtigheid drastisch afnam, maar bij hogere vochtigheden constant bleef (alumina’s, PZT, zirconia) en Type II materialen, waarbij de breuktaaiheid over de gehele vochtigheidsrange bleef dalen (spinel, calcium hydroxyapatiet).

143 Samenvatting

Bij Type I materialen is de rol van adsorptie net name aanwezig voor relatieve vochtigheden lager dan 10 %, maar bij Type II materialen speelt adsorptie een duidelijke rol bij alle vochtigheden. Deze laatste materialen moeten dan ook met een zekere voorzichtigheid worden behandeld, omdat de breuktaaiheidwaarden in de literatuur afhankelijk zijn van de relatieve vochtigheid tijdens de meting.

In tweede instantie is op atomaire schaal gekeken naar de oorsprong van het verschil tussen Type I en Type II materialen d.m.v. de bestudering van een modelmateriaal. Hiervoor is het ter- nair oxidische spinel (MgAl2O4, magnesium aluminaat) gekozen, vanwege het feit dat een van zijn bestandsdelen (magnesia) geen subkritische scheurgroei vertoont, maar het andere (alumina) wel en dat zowel polykristallijn keramiek als éénkristallen verkrijgbaar zijn. Als eerste werd gekeken naar het breukproces zelf. Breukexperimenten op twee soorten materiaal met verschillende dichtheden toonde Type II gedrag aan. Electron Backscattering Diffraction (EBSD) metingen op een polykristallijn proefstukje waarin een scheur gemaakt was m.b.v. indentaties toonden aan dat er geen voorkeursorientatie was voor het breukvlak. Daarom werden éénkris- tallen gebroken langs verscheidene kristallografische vlakken bij verschillende vochtigheden. Hierbij vond subkritische scheurgroei plaats, wat inhield dat de rol van adsorptie bestudeerd kon worden met de breukvlakken van deze éénkristallen.

Vervolgens werd gekeken naar de atomaire aspecten van het breukproces. De compositie van (100)-breukvlakken werd bestudeerd d.m.v. Low Energy Ion Scattering (LEIS). Dit breukoppervlak bestond naast zuurstof zowel uit magnesium- als aluminiumkationen. Dit was enigzins verrassend, omdat computersimulaties voorspelden dat een breukoppervlak met alleen magnesiumkationen het meest stabiele breukoppervlak zou zijn. Echter, de aanwezigheid van aluminiumkationen kan gedeeltelijk verklaard worden door het in materialen met een spinelstructuur vaak voorkomende proces van inversie, welke niet in computersimulaties meegenomen wordt. Bij inversie verwisselen de bivalente (magnesium) en trivalente (aluminium) kationen onder- ling hun respectievelijke tetraedrische en octaedrische omgeving. Diffuse Reflectance Infrared Fourier Transform Spectroscopy (DRIFTS) metingen aan hydroxylgroepen op het oppervlak van spinelpoeder bevestigden de aanwezigheid van aluminiumkationen in een tetraedrische omgeving. De DRIFTS metingen konden echter geen uitsluitsel verschaffen, waar de adsorptie exact plaatsvindt: alleen op de aluminiumkationen, alleen op de magnesiumkationen of op alle- bei. De inversie voorkomt een eenduidige atomaire interpretatie.

144 Samenvatting

Derhalve is de aandacht gericht op zink aluminaat spinel (ZnAl2O4), waarbij geen inversie werd verwacht. Als eerste is de mogelijke invloed van het poedersyntheseproces (vastestofreactie, coprecipitatie en sol-gel) op de inversie onderzocht m.b.v. 27Al MAS-NMR. Slechts een kleine mate van inversie (in vergelijking met magnesium aluminaat) was aanwezig in de poeders gesynthetiseerd via de vastestofreactie en de coprecipitatie route, maar een grote mate van inversie in het sol-gel gesynthetiseerde poeder. Dit laatste is echter veroorzaakt door een tekort aan zink, zoals vastgesteld met Atomic Adsorption Spectroscopy (AAS) t.g.v. verdamping van zinkoxide tijdens het verhitten. Vervolgens werd polykristallijn zink aluminaat keramiek gemaakt van het poeder gesynthetiseerd d.m.v. de vastestofreactie. Helaas, zorgden processingproblemen ervoor dat een zeer dicht materiaal niet maakbaar was, wat wel noodzakelijk is voor breukexperimenten. Met het gemaakte materiaal zijn enkele (thermo)fysische eigenschappen bepaald, welke vergeleken zijn met voorspellingen van computersimulaties.

Hoewel er geen breukexperimenten gedaan konden worden, zijn de breukoppervlakken van zink aluminaat gesimuleerd. Deze simulaties toonden aan dat het (100)-breukvlak met een ter- minatie van zinkkationen het meest stabiele is. Verder toonden DRIFTS metingen op zink aluminaat poeder aan dat er inderdaad zinkkationen aan het oppervlak zitten en bovendien de hydroxylgroepen uitsluitend hier op aanwezig zijn. Dit alles toont aan dat zink aluminaat een ideale kandidaat is voor verdere bestudering.

Tenslotte, omdat adsorptie plaatsvindt achter de scheurtip moet er een mechanisme zijn van thermofysische interacties in de proceszone. Als een eerste poging zijn twee theoretische mo- dellen vergeleken die deze interacties beschrijven. Het blijkt dat de thermofysische eigenschappen berekend uit phonon spectra verkregen uit Density Functional Theory (DFT)-simulaties nauwkeuriger zijn dan de eigenschappen berekend via het Debye model vergeleken met experimentele thermofysische data.

Het adsorptiemechanisme heeft voor enkele materialen een grote invloed op de breukeigen- schappen, maar de exacte reden is (nog steeds) onbekend. Enkele oorzaken, zoals trans- of in- tergranulaire breukwijze, kristalstructuur en oxidatietoestand van de kationen, kunnen worden uitgesloten. De experimentele resultaten doen vermoeden dat lokale ladingsverstoringen in het rooster een grote rol spelen. Dit dient verder onderzocht te worden, zoals met zink aluminaat. Verder dient ook de kennis over de exacte (thermofysische) interacties tussen adsorptie en mi- crostructurele aspecten van de proceszone nader bestudeerd te worden.

145 Samenvatting

146 Samenvatting voor niet-wetenschappers

Woord vooraf “Denkend aan een wetenschapper zie ik een suf persoon in een labjas die onbegrijpelijke wartaal uitslaat”. Dit zou een beginregel kunnen zijn van een gedicht, maar het is een (overdreven) voorbeeld van een beschrijving van het nerd-karakter van (natuur)wetenschappers1. Iedereen weet wel dat het niet waar is, maar er zit wel een kern van waarheid in. Niet het suffe, maar wel de onbegrijpelijke wartaal. Dit moet men hen echter niet zo aanrekenen, want dit is hun vakjargon, zoals elk beroep dat heeft. Voeg bij dit onbegrijpelijke vakjargon ook nog het feit toe dat vakken als natuur-, schei- en wiskunde op de middelbare school als moeilijk te boek staan, dan heb je alle ingrediënten te pakken voor een negatief beeld van de natuurwetenschappen en de techniek.

Dit negatieve beeld leidt tot gevaarlijke ontwikkelingen, zoals de jarenlange daling van het aantal eerstejaars studenten in de technische en natuurwetenschappelijke studies. Deze studenten zijn echter wel de uitvinders van de toekomst. De (Nederlandse) economie bestaat niet alleen uit diensten & services, maar voor een groot deel ook uit het ontwikkelen en maken van “echte” producten en daarbij zijn ze hard nodig. Deze daling van het aantal studenten zal op termijn ook een grote impact hebben op de economie.

Een tweede gevaarlijke (en misschien wel de gevaarlijkste) ontwikkeling is dat met de afkeer van de natuurwetenschappen men deze kennis ook niet tot zich neemt. Men kan nog wel lachen als men antwoordt dat de Zon om de Aarde draait of dat er seizoenen zijn, omdat de Aarde soms dichter bij de Zon staat en soms verder weg2. Het wordt ernstiger als men pseudo-wetenschap-

1. Chemici, wiskundigen, fysici.

147 Samenvatting voor niet-wetenschappers pelijke stellingen niet meer kan doorzien en er dus maar in gaat geloven. Men moet hierbij den- ken aan stellingen als "Er is een chemische stof ontdekt die dodelijk is bij inname en zij verspreidt zich ook door de lucht. Dit chemische product is diwaterstofoxide. Uw regering doet niets! Steun onze strijd financieel! Maak uw bijdrage over op rekening....3" of kretologie als "Significant beter" of "Megaperls met microwaskracht".

Het is duidelijk dat beide groepen elkaar moeten naderen. De niet-natuurwetenschappers moeten gaan begrijpen dat niet alles wat natuurwetenschappers doen per definitie gevaarlijk is en verboden moet worden. Aan de andere kant moeten natuurwetenschappers ook hun best doen om hetgeen wat ze doen voor iedereen begrijpbaar uit te leggen. In dit kader is hieronder een samenvatting geschreven van dit proefschrift, waarbij de nadruk ligt op het hoe en waarom van het onderzoek met aandacht voor de achtergrond.

Inleiding Dit proefschrift gaat over keramische materialen4. Keramische materialen worden in het dage- lijks leven zeer veel gebruikt, maar zijn niet altijd zichtbaar. Zo bestaat bijvoorbeeld een mo- biele telefoon van Nokia voor 16 % uit glas/keramiek. Keramische materialen hebben naast veel interessante eigenschappen ook enkele zwakke. Een van deze zwakke eigenschappen heeft te maken met de mechanische eigenschappen. Zo treedt er bij veel keramische materialen het proces van subkritische scheurgroei op onder invloed van moleculen uit de omgeving. Dit proces verzwakt in de loop der tijd het keramische materiaal, waardoor het op termijn zal breken. Voor toepassing van keramische materialen is het van het allergrootste belang om te weten wanneer het zal breken en dus dient subkritische scheurgroei zeer goed bestudeerd te worden.

Subkritische scheurgroei is reeds door veel wetenschappers onderzocht. De meest gegeven uitleg van subkritische scheurgroei is aan de hand van het kinetische mechanisme. Maar onderzoek aan het keramische materiaal mangaanzinkferriet toonde aan dat er nog een tweede mechanisme een rol speelt: het adsorptiemechanisme. De eerste vraag is nu of dit adsorptiemechanisme ook bij andere keramische materialen een rol speelt. De tweede vraag is hoe dat adsorptiemechanisme zich op atomair niveau manifesteert. Deze twee vragen staan centraal in het proefschrift.

2. Bijna alle afgestudeerden van twee Amerikaanse top-universiteiten gaven dit antwoord op de vraag waarom er seizoenen zijn. Dit antwoord is fout en derhalve aan de lezer de taak om wel het goede antwoord te vinden. 3. Flauw chemisch grapje, want het diwaterstofoxide is gewoon water. 4. Een eenduidige definitie van keramiek is zelfs voor wetenschappers niet makkelijk te maken. De beste (wetenschappelijke) definitie van keramiek is een anorganisch niet-metallisch materiaal gevormd door toepassing van hoge temperaturen. Een nadere uitleg volgt verderop in deze samenvatting.

148 Samenvatting voor niet-wetenschappers

In deze samenvatting zal op deze twee vragen worden ingegaan. Daarvoor zal eerst een beknop- te uitleg gegeven worden over keramiek en subkritische scheurgroei dat als basis dient. Vervol- gens wordt in drie stappen het onderzoek en de conclusies, die in dit proefschrift beschreven staan, behandeld. Hierbij wordt getracht om vakjargon te vermijden.

Keramische materialen Vaste materialen kunnen grofweg worden ingedeeld in drie klassen: metalen, polymeren en keramiek, zie tabel 1. Deze verdeling is gebaseerd op de chemische samenstelling en de aard van de onderlinge bindende krachten. De meeste materialen behoren tot een van deze klassen, maar een tussenvorm is ook mogelijk.

Bij keramiek denkt men direct aan producten zoals porselein, kopjes, bordjes, baksteen en dakpannen. Deze producten zijn al duizenden jaren bekend bij de mens en daarom wordt het ook wel traditionele keramiek genoemd. Dit keramiek wordt volgens een vaststaand procédé gemaakt. Allereerst wordt de grondstof, klei in het geval van het traditionele keramiek, in een mal geperst om het een vorm te geven. Het object dat zo ontstaat heet groene keramiek. Vervolgens gaat dit groene keramiek de oven in en wordt het gebakken, oftewel in vaktermen gesinterd, waarbij de losse kleikorrels door de hoge temperatuur aan elkaar "gelijmd" worden, zoals weer- gegeven in figuur 1. Hierbij kunnen nog loze ruimtes tussen de korrels overblijven, die poriën genoemd worden. Tenslotte wordt het eindproduct soms nog wat nabewerkt, bijvoorbeeld door slijpen of polijsten.

Het uiteindelijke product hangt af van vele variabelen in het productieproces. Zo hebben bijvoorbeeld de korrelgrootte van de grondstoffen, de druk tijdens het persen en de oventemperatuur een grote invloed op het uiteindelijke resultaat. Een voorbeeld hiervan is het verschil in kleur tussen rode en zwarte dakpannen dat ontstaat omdat in de laatste fase van het bakken er wel of geen lucht in de oven wordt geblazen.

Tabel 1: Overzicht van materialen Klasse Bindingstype Voorbeelden (toepassingen) metalen metallisch goud (ringen), aluminium (blikje), staal (fietsframe) polymeren covalent polyetheen (boterhamzakje), nylon (kousen), cellu- lose (hout) keramiek ionisch/cova- aluminiumsilikaten (porselein, mokken, baksteen), lent aluminiumoxide (horlogeglas)

149 Samenvatting voor niet-wetenschappers

Figuur 1: Schematische weergave van het sinteren van de losse korrels tot één vast keramisch materiaal. De overblijvende ruimtes tussen de korrels zijn poriën.

Keramische materialen hebben veel interessante eigenschappen. Zo kunnen sommige materialen bijvoorbeeld heel goed tegen grote temperatuursverschillen die optreden in een razend korte tijd, waardoor ze kunnen worden gebruikt als hitteschild op de Space Shuttle. Verder kunnen ze goede (electrische) isolatoren zijn en zijn ze in het algemeen stijf en hard, waardoor ze respectievelijk minder doorbuigen en minder snel beschadigen. Helaas hebben ze ook een aantal minder goede mechanische eigenschappen. Allereerst breken ze op een brosse wijze5, waardoor er geen waarschuwing vooraf is in de vorm van een verbuiging. Verder is er een relatief grote spreiding in de belasting waaronder keramiek breekt. Tenslotte is er ook nog het subkritische scheurgroeiproces, waarbij moleculen uit de omgeving ervoor zorgen dat een scheur kan groeien, terwijl de kritische belasting waarbij het pas zou mogen gebeuren, nog niet bereikt is. De combinatie van goede en zwakke eigenschappen is van wezenlijk belang voor het antwoord op de vraag of men deze materialen daadwerkelijk kan toepassen.

In de afgelopen dertig jaar hebben wetenschappers in universitaire en industriële laboratoria onderzoek gedaan naar nieuwe methoden om zowel de goede als de zwakke eigenschappen van keramiek te verbeteren. Dit werd bereikt door in plaats van klei zuivere poeders te gebruiken en de verschillende variabelen in het productieproces, zoals de oventemperatuur, te optimaliseren. Dit resulteerde in het zogenaamde geavanceerde of technische keramiek met sterk verbeterde eigenschappen. Technisch keramiek is bijvoorbeeld veel sterker of is compleet doorzichtig. Voorbeelden van toepassingen van dit technisch keramiek zijn de "natriumlampen" van de straatverlichting (doorzichtig aluminiumoxide), de sensor die de airbag activeert (loodzircon- aattitanaat) en de kernen van transformatoren (mangaanzinkferriet), zie figuur 2. Uit deze voor-

5. Het onderscheid tussen een brosse en een ductiele breuk kan getoond worden door het breken van een potlood (brosse wijze) en een kauwgumpje (ductiele wijze). Een potlood zal in één klap breken, terwijl het zijn oorspronkelijke vorm terug krijgt als men voor het breekt, stopt met belasten en een gekauwd kauwgumpje zal eerst heel erg uit elkaar worden getrokken en vervormen voordat het breekt, maar zal de oorspronkelijke vorm niet terugkrijgen als je in de tussentijd stopt met trekken.

150 Samenvatting voor niet-wetenschappers

Figuur 2: Enkele voorbeelden van toepassingen van traditioneel en technisch keramiek: een kern van een transformator van mangaanzinkferriet (rechtsboven), een lagerring van silicium- carbide (rechtsonder), een halogeen autolampje van siliciumdioxide en een vloertegel (onder- op). beelden blijkt wel dat het in de moderne samenleving belangrijke materialen zijn. Ze zijn echter wel onttrokken aan het oog van de mensen, waardoor ze minder bekend zijn.

Subkritische scheurgroei in keramiek Als een keramisch onderdeel breekt, dan gebeurt dat altijd op een bestaande onregelmatigheid, zoals een scheurtje. Dit altijd aanwezige scheurtje, in vaktermen een defect, kan door verschillende oorzaken ontstaan. Als men zo’n scheurtje gaat belasten, d.w.z. een kracht uitoefenen op het onderdeel waar het in zit, dan kan zo’n scheurtje uitgroeien tot een grote scheur, wat uiteindelijk leidt tot de breuk van het onderdeel. De belasting waaronder het onderdeel breekt, wordt sterkte genoemd. Hoe groter de scheur, hoe lager de sterkte.

De sterkte is echter wel afhankelijk van de wijze waarop belast wordt. Trekken, buigen en op druk belasten leveren respectievelijk de trek-, buigen compressiesterktes op. De term "een sterk materiaal" moet dan ook met de nodige voorzichtigheid worden gebezigd, omdat meestal slechts één enkele sterkte van de verschillende sterktes hoog is. Zo heeft cement een uitstekende compressiesterkte, maar een uitermate geringe buigsterkte.

Bij subkritische scheurgroei wordt de scheur al groter voordat de kritische belasting bereikt is. Dit kan beschreven worden door het kinetische mechanisme, waar watermoleculen uit de lucht de scheur in zweven en reageren met de chemische bindingen, die nog net niet zijn gebroken. Tijdens deze reactie breken de bindingen wel met als gevolg dat de scheur groter wordt en de sterkte dus afneemt. De snelheid waarmee de scheur groeit, is verschillend voor verschillende

151 Samenvatting voor niet-wetenschappers materialen, waardoor het uiteindelijke breken bij sommige materialen slechts een paar dagen duurt, maar bij andere tientallen jaren. Kortom, als men wil voorspellen of een bepaald onderdeel vandaag, morgen, over een maand, over een jaar of over tien jaar zal gaan breken, is het dus van belang om enig inzicht te hebben in subkritische scheurgroei.

Naast het beschreven kinetische mechanisme is er echter ook nog een tweede mechanisme, waarmee watermoleculen uit de omgeving de scheur kunnen laten groeien: het adsorptiemechanisme. De watermoleculen kunnen adsorberen op het nieuwe oppervlak, wat ontstaat bij het breken, en kunnen daarmee de oppervlakte-energie verlagen. Aangezien een hoge oppervlakte- energie de scheurgroei tegenhoudt, zal een verlaging door wateradsorptie scheurgroei bevor- derd worden. Dit resulteert uiteindelijk in een lagere sterkte.

Onderzoek I: een overzicht Het kinetische mechanisme komt veelvuldig voor en is door de wetenschappers geaccepteerd. Het adsorptiemechanisme is daarentegen minder bekend en is tot voorafgaand aan dit promotieonderzoek slechts alleen aangetoond voor het mangaanzinkferriet, wat een technisch keramisch materiaal is behorende tot het oxidische keramiek. Dit leverde de twee centrale vragen op: Komt het adsorptiemechanisme bij elk oxidisch keramisch materiaal voor en wat is de exacte atomaire oorzaak van dit mechanisme?

De eerste vraag kan worden beantwoord door de breuktaaiheid6, de weerstand tot scheuren, van een aantal verschillende oxidische technisch keramische materialen te meten bij verschillende vochtigheden. De aanwezigheid van het adsorptiemechanisme kan namelijk worden aangetoond door een verlaging van de breuktaaiheid met toenemende vochtigheid. In tabel 2 staan de materialen vermeld die onderzocht zijn. Uit deze materialen, die commercieel verkrijgbaar zijn of in het geval van calcium hydroxyapatiet zelf gemaakt zijn, zijn proefstukjes van ongeveer 3 mm breed, 4 mm hoog en 40 mm lang gemaakt. In deze zogenaamde SENB-proefstukjes is kunstmatig een veel grotere scheur aangebracht dan dat er van nature aanwezig is om er zeker van te zijn dat het proefstukje daar zal breken en om de zogenaamde breuktaaiheid in plaats van de sterkte te meten. Vervolgens wordt het proefstukje in een zogenaamde vierpuntsbuigproef- opstelling gelegd tussen vier gehard stalen rolletjes, zie figuur 3. Deze hele opstelling wordt

6. De reden om de breuktaaiheid te meten i.p.v. de sterkte is een technische en een economische. Omdat de spreiding in de sterktemetingen veel groter is dan in de breuktaaiheidmetingen, moeten er voor een betrouwbare sterktemeting minimaal vijfentwintig proefstukjes gebroken worden en voor de breuktaaiheid ongeveer zes tot acht. Overigens zijn de sterkte en breuktaaiheid aan elkaar gerelateerd.

152 Samenvatting voor niet-wetenschappers

Tabel 2: Overzicht van onderzochte materialen Materiaal Formule Toepassing

alumina Al2O3 lagers, kunstmatige tanden, electronisch isolerende ondergrond

doorzichtig alu- Al2O3 lampjes, ruiten mina

PZT Pb(Zr,Ti)O3 actuatoren, oscilatoren, sensoren zirconia ZrO2 pantser, zwaar belastbare componenten calcium Ca10(PO4)6(OH)2 biomateriaal, implantaat hydroxyapatiet

spinel MgAl2O4 pantserglas, buffer materiaal in nucleaire installaties vloertegels AlxSiyOz spreekt voor zich weer in een tweede opstelling geplaatst, waarin de vochtigheid geregeld kan worden tijdens het breken. Tenslotte wordt met een bepaalde snelheid de belasting (F) op de twee bovenste rollers opgevoerd totdat het proefstukje breekt. De breuktaaiheid wordt berekend aan de hand van de belasting bij breuk.

Alle materialen in ogenschouw genomen, zijn er twee typen van reacties te herkennen, zie figuur 4. Type I materialen worden gekenmerkt door een scherpe daling in de breuktaaiheid bij lage vochtigheden en Type II materialen worden gekenmerkt door een continu dalende breuktaaiheid bij alle vochtigheden. Dit betekent dat voor Type I materialen de rol van het adsorptiemechanisme alleen bij de lage vochtigheden een rol speelt, maar bij de hogere vochtigheden niet meer, en voor Type II materialen speelt het adsorptiemechanisme wel degelijk een belangrijke rol bij alle vochtigheden. Dit houdt in dat men moet oppassen met het toepassen van deze Type II materialen, omdat de breuktaaiheid in de tropen bijvoorbeeld anders is dan in de woestijn. Ge-

Figuur 3: Zijaanzicht van een vierpuntsbuigopstelling met een SENB-proefstukjes tussen de rollers.

153 Samenvatting voor niet-wetenschappers

J J Breuktaaiheid Breuktaaiheid

0% Vochtigheid J 80% 0%Vochtigheid J 80%

Figuur 4: Typische breuktaaiheidafname bij verschillende vochtigheden voor Type I materialen (links) en Type II materialen (rechts). De grafieken zijn niet op schaal. lukkig zijn de meeste materialen van Type I, maar calcium hydroxyapatiet en spinel zijn, net zoals mangaanzinkferriet, Type II materialen.

Onderzoek II: een modelmateriaal De tweede vraag gaat over de oorzaak waarom bepaalde materialen wel of geen Type II gedrag vertonen. Het antwoord op deze vraag is niet erg eenvoudig. Derhalve is het onderzoek gecon- centreerd op één bepaald materiaal, dat als modelmateriaal dient om op atomair niveau nader onderzocht te worden. Om verscheidene wetenschappelijke redenen is daarvoor het spinel

(MgAl2O4) gekozen.

Als eerste zijn natuurlijk weer de breuktaaiheidsmetingen gedaan op twee soorten spinelkeramiek, die een verschillende dichtheid hebben. Deze experimenten toonden aan dat het dichte spinelkeramiek Type II gedrag vertoonde en het poreuzere Type I gedrag. Bij het breken van het spinelkeramiek ontstaat een breukvlak met vele korreltjes, waaruit het keramiek is opge- bouwd. Er is gekeken of er een voorkeursoriëntatie van die korrels op het breukvlak aanwezig is met behulp van een techniek genaamd Electron Backscattering Diffraction. Dit bleek niet het geval te zijn, waardoor er voor is gekozen om twee oriëntaties, het (100)- en het (111)-breukvlak, nader te bestuderen op éénkristallen. Éénkristallen zijn als het ware zeer grote, georiën- teerde korrels met een diameter van enkele centimeters, terwijl een korrel in een keramisch materiaal typische kleiner is dan ongeveer 20 micrometer. Uit breukexperimenten op deze één- kristallen bleek dat het (100)-breukvlak het makkelijkst te maken is en ook nog eens het gevoe- ligst is voor vochtigheid.

154 Samenvatting voor niet-wetenschappers

De vraag is nu of er aan de atomaire samenstelling van het breukvlak gezien kan worden wat het Type II gedrag veroorzaakt. Misschien bestaat er een voorkeur voor een bepaalde plaats, dan wel atoom, om op te adsorberen. Dit is onderzocht voor het (100)-breukvlak m.b.v. de Low Energy Ion Scattering (LEIS) techniek7, waarmee de samenstelling van het breukvlak bepaald kan worden. Hieruit bleek dat er naast zuurstof (O) ook veel magnesium (Mg) en aluminium (Al) aanwezig was. Dit kwam niet geheel overeen met de voorspellingen die verkregen waren uit computersimulaties. Zijn de computersimulaties fout of is het experiment niet goed uitgevoerd?

De waarheid is wat gecompliceerder, want zowel de simulaties als het experiment zijn goed uitgevoerd. Er is bij materialen met de spinelstructuur bekend dat er een proces voorkomt met de naam inversie. Bij inversie verwisselen de magnesium- en aluminiumatomen hun positie in het kristalrooster. De computersimulaties voorspellen een (100)-breukvlak bestaande uit alleen zuurstof- en magnesiumatomen, omdat in deze berekeningen inversie niet is meegenomen. Maar in werkelijkheid kunnen een aantal van die magnesiumatomen door inversie vervangen zijn door aluminiumatomen, wat inderdaad ook gemeten is.

De vraag is nu of het inversieproces problemen oplevert met de bepaling waar het water adsorbeert op het oppervlak en dus een eventuele eenduidige interpretatie in de weg staat. Daarvoor is er met een infrarood techniek (DRIFTS)8 gekeken naar de lokatie van de hydroxylgroepen aan het oppervlak. Deze hydroxylgroepen zijn de overblijfselen van het geadsorbeerde water en geven dus aan waar ze geadsorbeerd zijn. Het enige wat met zekerheid gezegd kan worden is dat het water in ieder geval adsorbeert op aluminiumatomen, maar het is onduidelijk of ze ook op magnesiumatomen adsorberen. De inversie staat dus een duidelijke interpretatie in de weg.

Onderzoek III: een nieuw modelmateriaal? Omdat het inversieproces een duidelijke interpretatie bemoeilijkt, is er gezocht naar een materiaal met dezelfde structuur als spinel, maar dat geen inversie vertoont. Dit materiaal is zink aluminaat (ZnAl2O4). Aangezien zink aluminaat niet een standaard keramisch materiaal is, is er

7. Met de LEIS techniek wordt het breukvlak als het ware bestookt met speciale atomen. Als deze atomen een atoom op het breukvlak raken, verliezen ze iets van hun snelheid. Dit verlies kan gemeten worden en is alleen afhankelijk van de massa van het atoom waarop het gebotst is. Aangezien de massa van het atoom karakteristiek is voor dat atoom, kun je de samenstelling van het breukvlak bepalen. 8. Elke chemische binding kan trillen met een bepaalde karakteristieke frequentie. Met DRIFTS wordt bepaald welke frequenties aanwezig zijn in het zogenaamde infrarode frequentiegebied. Aan de hand van de gevonden frequenties kan bepaald worden welke chemische groepen, zoals hydroxylgroepen, aan het oppervlak zitten.

155 Samenvatting voor niet-wetenschappers eerst onderzoek gedaan naar de invloed van verschillende syntheseroutes van het startmateriaal op de inversie. Het zink aluminaat keramiek is uiteindelijk gemaakt volgens de standaardme- thode van persen en sinteren met het startmateriaal wat gemaakt is met de vastestofreactie. He- laas is de kwaliteit van het zink aluminaat keramiek nog niet goed genoeg om er breukexperimenten mee te doen. Dit kan in de toekomst wel gebeuren als het productieproces geoptimaliseerd wordt.

Ondanks het feit dat er geen breukexperimenten gedaan konden worden met zink aluminaat, zijn er wel een aantal onbekende (thermo-)fysische eigenschappen bepaald. Verder is er ook nog met de DRIFTS-techniek gekeken naar de lokatie van de hydroxylgroepen aan het oppervlak. Deze hydroxylgroepen zijn alleen te vinden op zinkatomen en niet op aluminiumatomen, in tegenstelling tot het spinel. Dit betekent dat het nu bekend is dat het water alleen adsorbeert op de zinkatomen. Bovendien hebben computersimulaties aangetoond dat een (100)-breukvlak met alleen zuurstof- en zinkatomen de voorkeur geniet. Dit alles maakt een toekomstige interpretatie gemakkelijker.

Conclusies Het adsorptiemechanisme speelt een rol tijdens het breukproces van het oxidische keramiek. Er zijn echter twee soorten gedrag te onderscheiden. Bij Type I materialen is de rol van het adsorptiemechanisme al heel snel bij lage vochtigheden uitgespeeld, terwijl het bij Type II materialen bij alle vochtigheden een rol speelt. Om inzicht te krijgen in de exacte wisselwerking van het adsorptiemechanisme en het breukproces is spinel als een modelmateriaal intensief onderzocht. Hieruit bleek dat het inversieproces een eenduidige interpretatie van het adsorptiemechanisme in de weg staat. Derhalve is aan zink aluminaat keramiek, dat geen inversie vertoont, gewerkt. Hiermee zijn echter wegens productieproblemen nog geen breukexperimenten uitgevoerd, maar additionele (computer) experimenten toonden wel aan dat het een potentieel zeer interessant modelmateriaal is voor verder onderzoek.

Op dit moment kan nog geen eenduidig antwoord gegeven worden op de vraag wat de exacte interactie tussen de adsorptie en het breukproces is. Er kunnen echter wel een aantal redenen uitgesloten worden danwel een aantal redenen aangewezen worden als mogelijke oorzaak. Ver- der onderzoek is en blijft dus noodzakelijk. Dit toont aan dat wetenschappelijk onderzoek een proces is van kleine stapjes vooruit.

156 Dankwoord

Een collega van mij heeft het reeds gemeld. Met het afronden van een promotieonderzoek komt er een einde aan je formele opleiding van bijna vijfentwintig jaar. Als je dan terug kijkt, dan zijn er veel mensen geweest die daar in min of meerdere mate aan hebben bijgedragen. Het gaat te ver om ze allemaal te bedanken, waar ze ook mogen zijn, maar ik zou graag van de gelegenheid gebruik willen maken om enkelen van hen met naam te bedanken die hebben bijgedragen in de fase van het promotieonderzoek.

Als eerste wil ik graag professor Bert de With bedanken voor de mogelijkheid tot het verrichten van mijn promotieonderzoek in het laboratorium voor Vaste Stof en Materiaalchemie, waarbij ik ook de mogelijkheid heb gekregen om mijn eigen chemische interesses in een ten dele inge- nieurswerk in te passen. Verder wil ik je bedanken voor de vele discussies die we hebben gehad; niet alleen over het onderzoek en de wetenschap, maar ook over onderwijs, popularisering van de wetenschap en mineralogie.

Graag wil ik ook mijn begeleider Ardi Dortmans bedanken voor de vele discussies die we hebben gehad over de meetresultaten en vooral hun implicaties. Furthermore, I would like to thank ChangMing Fang for his assistance with and discussions about computer simulations and their comparison with experiments. Tenslotte wil ik graag de leden van mijn leescommissie de pro- fessoren R.C. Bradt, H. Brongersma en L. Van Poucke bedanken voor hun nuttige commentaar.

Met de teruglopende studentenaantallen is het niet meer vanzelfsprekend dat elke promovendus de mogelijkheid krijgt om studenten te begeleiden tijdens hun stages. Ik mag me daarom gelukkig prijzen dat ik tot twee keer toe die mogelijkheid wel heb gehad en bovendien ook nog met

157 Dankwoord twee uitstekende studenten: Joep van Dijk en Marcel Snel. Bedankt voor jullie bijdrage, dat zeker terecht is gekomen in dit proefschift, en voor jullie aanwezigheid.

Een proefschrift maak je niet alleen is een veel gehoorde kreet. En dat is zo! Daarom wil ik graag alle medewerkers van SVM (inclusief SVM-Coatings), aan wie ik stuk voor stuk persoon- lijk een mooie herinnering bewaar, bedanken voor het "iets" maken, het "iets" typen, het "iets" meten, het "iets" bepraten en domweg voor de gezelligheid binnen en buiten het werk. Met name wil ik daarin ook graag mijn AiO-generatiegenoten noemen, waarmee ik door Europa ben getrokken: Maru, Mark, Pascal en Dennis. Naturalemente, quiero agradecer también mi com- pañero de oficina Emilio, por todas las discussiones, ayuda y "gezelligheid" en y fuera de STO 2.24 durante estos cuatro años.

Een proefschrift met experimenteel werk kan nooit tot stand komen zonder hulp van bereidwil- lige en meedenkende technici. De TU/e mag zich dan ook gelukkig prijzen met de vakbekwame technici zoals de medewerkers van Fijnmechanica o.l.v. Theo Maas en Jos de Laat en de Glas- blazerij/Zagerij o.l.v. Frans Kuijpers, die ik hierbij ook bedank. Tenslotte, wil ik graag speciaal Hans Onstenk bedanken voor het maken van de vele, soms lastige, proefstukjes.

Een leven op de universiteit kan niet zonder te ontspannen buiten de universiteit. Mijn twee grote hobby’s stelden me gelukkig in staat om me af te leiden van het promotiewerk: volleybal (als speler en scheidsrechter) en stijldansen. Allereerst wil ik alle Hajranen met wie ik (al dan niet) heb samengewerkt, en, in het bijzonder, mijn teamgenoten van Hajraa H2/H3/H4/H3 hartelijk bedanken voor de zeer leuke tijd met vele hoogtepunten (o.a. twee kampioenschappen). Verder wil ik in het bijzonder ook mijn danspartner Yvet Beckers bedanken voor het zwevend over de dansvloer doen vergeten van de problematiek gekoppeld aan werk.

Tenslotte wil ik Bart, Stieneke en Sigrid bedanken voor hun steun en toewijding tijdens mijn opleiding in al die jaren.

158 Curriculum Vitae

Niels van der Laag is op 24 juni 1974 geboren te Boxmeer en getogen in Cuijk. Na het behalen van het Atheneum diploma aan het Merletcollege te Cuijk in 1992 startte hij met de studie Scheikunde aan de Katholieke Universiteit Nijmegen. In de doctoraalfase specialiseerde hij zich in de afstudeerrichting Informatische Chemie, waarvoor hij in het kader van het hoofdvakstage onderzoek deed bij de vakgroep Vaste Stof NMR o.l.v. prof. dr. B.H. Meier naar de performance van het NMR-simulatieprogramma GAMMA. Verder deed hij tijdens een extra bijvakstage bij de vakgroep Vaste Stof Chemie o.l.v. dr. F.K. de Theije en dr. W.J.P. van Enckevort onderzoek naar het etsen van diamantoppervlakken met zuurstof. Dit alles resulteerde in 1998 in het behalen van het doctoraalexamen (met genoegen).

Aansluitend is hij begonnen met promotieonderzoek bij de vak-/capaciteitsgroep Vaste Stof en Materiaalchemie aan de Technische Universiteit Eindhoven. Onder leiding van prof. dr. G. de With deed hij onderzoek naar de invloed van water op het breukgedrag van het oxidische keramiek. De resultaten van dit promotieonderzoek zijn weergeven in dit proefschrift.

Niels van der Laag is born on 24 june 1974 in Boxmeer and bred in Cuijk (The Netherlands). After obtaining his Atheneum diploma (Dutch pre-university education) at the Merletcollege, Cuijk in 1992, he commenced studying chemistry at the Catholic University of Nijmegen. Dur- ing the final phase of his study, he specialised himself in computing chemistry and performed an internship at the department of Solid State NMR under the guidance of prof. dr. B.H. Meier on the performance of the NMR simulation package GAMMA. Furthermore, he did an extra internship at the department of Solid State Chemistry under the guidance of dr. F.K. de Theije and

159 Curriculum vitae dr. W.J.P. van Enckevort on the etching mechanism of diamond surfaces by oxygen. He obtained his Masters degree in 1998.

Subsequently, he started his Ph.D.-research project at the Laboratory of Solid State and Mate- rials Chemistry at the Eindhoven University of Technology. Under the guidance of prof. dr. G. de With, he investigated the influence of humidity on the fracture of oxide ceramics. The results are published in this Ph.D. thesis.

160

UITNODIGING

U bent van harte welkom bij de openbare

De promotieplechtigheid zal plaatshebben op

woensdag 18 december 2002 om 16.00 uur

in het Auditorium van de Technische Universiteit Eindhoven.

Aansluitend aan deze plechtigheid zal een receptie plaatsvinden waarvoor U ook van harte bent uitgenodigd.

Niels van der Laag der van Niels Niels van der Laag Niels van der Laag F.D. Rooseveltlaan 239 5625 AZ Eindhoven 040-2486540 [email protected]

± 170 pag. = 11mm rug f.c. glanslaminaat